Copyright by William Harold Asquith 2003 The Dissertation Committee for William Harold Asquith certifies that this is the approved version of the following dissertation: MODELING OF RUNOFF-PRODUCING RAINFALL HYETOGRAPHS IN TEXAS USING L-MOMENT STATISTICS Committee: __________________________________ John M. Sharp, Supervisor __________________________________ Clark R. Wilson __________________________________ Zong-Liang Yang __________________________________ David R. Maidment __________________________________ David B. Thompson MODELING OF RUNOFF-PRODUCING RAINFALL HYETOGRAPHS IN TEXAS USING L-MOMENT STATISTICS by William Harold Asquith, B.S., M.S. Dissertation Presented to the Faculty of the Graduate School of the University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin May 2003 Dedication This dissertation is dedicated to my parents, George and Anne, who have supported me throughout my life, encouraged me to become an author, and to become a scientist by an education through engineering and then the geosciences. This dissertation is also dedicated to my wife, D’Anne, for many wonderful years of boundless encouragement and support during times when my attention was directed on school, and for giving me Nathan and Caroline. These individuals have continuously inspired me to appreciate the complexity of the world around us all. Acknowledgements I would like to thank the following people/institutions for their assistance, encouragement, and friendship. I thank the dissertation committee members: Drs. John Sharp, Clark Wilson, Liang Yang, David Maidment, and David Thompson. I especially thank John Sharp for years of friendship and for assuming the role of supervisor. I thank Dr. James Famiglietti for encouraging me to return to school, sponsoring my admission to the Department of Geosciences, and for several extremely rewarding years. I thank the Geology Foundation for financial support. I thank the United States Geological Survey (USGS) for providing an academically favorable environment, for providing unlimited opportunities for research, and for periodic tuition support. I thank the Texas Department of Transportation Research Management Committee No. 3 for over ten years of research sponsorship including large portions of the effort behind this dissertation. I thank Raymond Slade, Jr. for hiring me at the USGS, for guiding me through my early career, and for being a friend. I thank the USGS technical writing and editorial staff, Peter Bush, Gail Sladek, and Judy Voigt, for greatly enhancing my technical writing skills and encouraging my pursuit of excellence in graphical and tabular presentation. I thank David Stolpa, Rudy Herrmann, and Amy Ronnfeldt and Drs. Ted Cleveland and Xing Fang for their assistance. I thank Drs. Jonathan Hosking, James Wallis, Robert Serfling, and Warren Gilchrist for reading and commenting on drafts of this dissertation. I also thank Jonathan Hosking and countless others for the beautiful world of L-moment statistics. I thank Larry Wall and the rest of the Perl community for developing a wonderfully expressive open-source programming language upon which nearly all algorithms used in this dissertation are based. And I acknowledge Tkg2, a Perl-based data-plotting software package that generated all data graphics in this dissertation and was my programming passion between 1999 and 2001. May 2003 v MODELING OF RUNOFF-PRODUCING RAINFALL HYETOGRAPHS IN TEXAS USING L-MOMENT STATISTICS Publication No. _________ William Harold Asquith, Ph.D. The University of Texas at Austin, 2003 Supervisor: John M. Sharp Temporal distributions of storm rainfall are known as hyetographs. Design hyetographs are important for cost-effective risk-mitigated rainfall-runoff modeling. The hyetographs considered are known to produce or generate runoff on small watersheds (typically about 50 square kilometers) in Texas. L-moment statistics and the nonparametric median are used to summarize the dimensionless representations of over 1,600 observed hyetograph distributions. A focus is made on storm depths in excess of about 25 mm and durations of 0–12, 12–24, and 24 hours and greater. Statistical distributions are fit to the L-moments of the dimensionless hyetographs including the newly described L-gamma. L-gamma hyetograph models are anticipated to be reliable predictors of expected hyetographs. Finally, a separate permeability- related L-moment application to the popular Carman-Kozeny equation is described. vi TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Lower Case Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Upper Case Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hypotheses and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Expected Hyetograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Model Verification and Suitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Chapter 2. Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Intensity-Duration Frequency (IDF) based Hyetograph Methods . . . . . . . . . . . . 34 Actual Rainfall Record based Hyetograph Methods . . . . . . . . . . . . . . . . . . . . . . 36 Hyetograph Research by Huff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Hyetograph Research by Pani and Haragan . . . . . . . . . . . . . . . . . . . . . . . . . 41 Hyetograph Research by Schaefer and Parrett . . . . . . . . . . . . . . . . . . . . . . . 46 Hyetograph Research by Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Example of the Balanced Storm Hyetograph Method for Austin, Texas . . . . . . . 52 Chapter 3. Triangular Model of Dimensionless Rainfall Hyetographs Known to Produce Runoff in Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 vii Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Triangular Dimensionless Hyetograph Definition . . . . . . . . . . . . . . . . . . . . . . . . 73 Estimation of Triangular Hyetograph Parameters . . . . . . . . . . . . . . . . . . . . . . . . 76 Example Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Chapter 4. Sample L-moment Estimation using prior-Probability Weighted Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Moments of a Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 L-moments and Probability Weighted Moments of a Distribution . . . . . . . . . . . 91 Sample L-moments and Probability Weighted Moments . . . . . . . . . . . . . . . . . . 97 Unbiased Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Plotting-Position Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Prior-Probability Weighted Moments (p-PWMs) . . . . . . . . . . . . . . . . . . . . . . . 101 p-PWM Suitable Data and Real-World Examples . . . . . . . . . . . . . . . . . . . . . . . 109 p-PWM Sampling Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Kappa Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Kappa based Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Uniform Distribution of Nonexceedance Probability . . . . . . . . . . . . . . 118 Nonuniform Distribution of Nonexceedance Probability . . . . . . . . . . . 123 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Chapter 5. L-moments of Dimensionless Rainfall Hyetographs Known to Produce Runoff in Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Relations between Storm Depth and Hyetograph Statistics . . . . . . . . . . . . . . . . 131 Storm Durations of 0 to 12 Hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Storm Durations of 12 to 24 Hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Storm Durations of 24 Hours and Greater . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Section Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Monthly and Seasonal Influences on Hyetograph Statistics . . . . . . . . . . . . . . . 158 Geographic Influences and Data Base Differences on Hyetograph Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Double One-Percent Trimming of Hyetograph Tails . . . . . . . . . . . . . . . . . . 163 viii No Trimming of Hyetograph Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Modal Analysis of Hyetograph Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Chapter 6. L-gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Parameter Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 L-moments of the Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Parameter Estimation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Verification of Theoretical L-moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Chapter 7. L-gamma Model of Dimensionless Rainfall Hyetographs Known to Produce Runoff in Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 L-gamma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Beta Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Model Suitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Empirical Hyetograph Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Additional Empirical Hyetographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Model Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Example Application of L-gamma Hyetograph . . . . . . . . . . . . . . . . . . . . . . . . . 243 Chapter 8 Modification of the Carman-Kozeny Equation for Application of L-moment Statistics for Estimation of the Intrinsic Permeability of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Linear Model of the Particle-Size Distribution . . . . . . . . . . . . . . . . . . . . . . 253 Two-Parameter Power Model of the Particle-Size Distribution . . . . . . . . . 255 Three-Parameter Power Model of the Particle-Size Distribution . . . . . . . . 257 Four-Parameter Kappa Model of the Particle-Size Distribution . . . . . . . . . 258 Four-Parameter Generalized Lambda Model of the Particle-Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Additional Remarks on Models of the Particle-Size Distribution . . . . . . . . 262 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 ix Chapter 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Appendices A. Reference list of U.S. Geological Survey reports used for development of the hyetograph data base . . . . . . . . . . . . . . . . . . . . . . . . 274 B. Background on L-moment Statistical Theory . . . . . . . . . . . . . . . . . . . . . . . . 294 C. Supplemental non-uniform simulations showing biases in the unbiased, plotting-position, and prior-Probability Weighted Moment L-moment estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 D. Parameter Space Maps of L-gamma Distribution . . . . . . . . . . . . . . . . . . . . . 318 E. Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 F. Supplemental Tables for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 x LIST OF TABLES 1. U.S. Geological Survey streamflow–gaging stations within the c68c88c86c87c76c81 data base module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. U.S. Geological Survey streamflow–gaging stations within the c71c68c79c79c68c86 data base module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. U.S. Geological Survey streamflow–gaging stations within the c73c82c85c87c90c82c85c87c75 data base module . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. U.S. Geological Survey streamflow–gaging stations within the c86c68c81c68c81c87c82c81c76c82 data base module . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. U.S. Geological Survey streamflow–gaging stations within the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 data base module . . . . . . . . . . . . . . . . . . . . 12 6. Median, 10-, and 90-percentile dimensionless hyetograph coordinates for the southern High Plains of Texas derived from Pani and Haragan (1981) . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7. Depth-duration frequency of precipitation for Austin, Texas . . . . . . . . . . . . . 53 8. Intensity-duration frequency of precipitation for Austin, Texas . . . . . . . . . . . 54 9. Hyetograph for 3-hr 25-year rainfall for Austin, Texas . . . . . . . . . . . . . . . . . . 55 10. Median, 10-, and 90-percentile dimensionless cumulative hyetograph coordinates for the southern High Plains of Texas derived from Pani and Haragan (1981) . . . . . . . . . . . . . . . . . . . . . . 71 11. Statistical summary of dimensionless hyetograph averages for 0–12 hr and 12–24 hr storm durations . . . . . . . . . . . . . . . . . . . . . . . . . 77 12. Statistical summary of dimensionless hyetograph averages for 24 hr and greater storm duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 13. Dimensionless runoff-producing cumulative hyetographs for 0–24 hr and 24 hr and greater storm durations computed by triangular hyetograph model . . . . . . . . . . . . . . . . . . . . . . . . . 80 14. Probability Weighted Moment and L-moment weight factors by estimator class for first short example . . . . . . . . . . . . . . . . . . . . . . . . . 106 15. Probability Weighted Moment and L-moment weight factors by estimator class for second short example . . . . . . . . . . . . . . . . . . . . . . 108 16. Probability Weighted Moments and L-moments by estimator class for second short example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 17. Application of prior-Probability Weighted Moments on an observed hyetograph expressed in percent duration and percent depth for May 23, 1981 storm for watershed of station 08156800 Shoal Creek at 12th Street, Austin, Texas . . . . . . . . 112 18. Comparison of sample biases for a simulated Kappa distribution using a uniform distribution of probability . . . . . . . . . . . . . . 120 xi 19. Comparison of sample standard deviations for a simulated Kappa distribution using a uniform distribution of probability . . . . . . . . 121 20. Comparison of relative efficiencies for a simulated Kappa distribution using a uniform distribution of probability . . . . . . . . 122 21. Comparison of sample biases for a simulated Kappa distribution using a non-uniform distribution of probability by redrawing F is initial F was greater than 0.5 . . . . . . . . 125 22. Comparison of sample standard deviations for a simulated Kappa distribution using a non-uniform distribution of probability by redrawing F is initial F was greater than 0.5 . . . . . . . . . . 126 23. Comparison of relative efficiencies for a simulated Kappa distribution using a non-uniform distribution of probability by redrawing F is initial F was greater than 0.5 . . . . . . . . . . 127 24. Summary of observations made and weighted average values for statistics from the box plot graphs on figures 26–46 . . . . . . . . 157 25. Summary mean and median statistics for the mean, median, and L-scale values of dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and both tail trimming methods for one inch or greater storm depths . . . . . . 164 26. Comparison of theoretical L-moment and sample L-moment statistics for L-gamma distribution chosen for the verification example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 27. L-gamma distribution parameter estimates for modeling dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths . . . . 211 28. Beta distribution parameter estimates for modeling dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths . . . . 216 29. Example computation of streamflow hydrograph convolution of a balanced storm hyetograph and a unit hydrograph . . . . . . . . . . . . . . . . . 245 30. Example computation of streamflow hydrograph convolution of 0–12 hr L-gamma hyetograph and a unit hydrograph . . . . . . . . . . . . . 246 31. Examples of the particle-diameter multiplier on the Carman-Kozeny equation based on quantile function models of the particle-size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 32. Complex examples of the particle-diameter multiplier on the Carman-Kozeny equation based on four-parameter quantile function models of the particle-size distribution. . . . . . . . . . . . . . . . . . . . 265 xii LIST OF FIGURES 1. Example of a cumulative rainfall hyetograph . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Map showing locations of USGS streamflow-gaging stations represented in the hyetograph data base . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. Example data file c68c88c86c87c76c81c18c54c75c82c68c79c38c85c72c72c78c18c86c87c68c19c27c20c24c25c27c19c19c66c71c18c75c92c72c87c82c86c18 c85c68c76c81c66c86c87c68c19c27c20c24c25c27c19c19c66c20c28c27c20c66c19c24c21c22c17c71c68c87 from the c68c88c86c87c76c81 data base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Hyetographs for c68c88c86c87c76c81 data base module— 401 storm events represented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5. Hyetographs for c71c68c79c79c68c86 data base module— 240 storm events represented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6. Hyetographs for c73c82c85c87c90c82c85c87c75 data base module— 194 storm events represented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7. Hyetographs for c86c68c81c68c81c87c82c81c76c82 data base module— 215 storm events represented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8. Hyetographs for c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 data base module— 609 storm events represented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9. Dimensionless or percentile representation of the 194 hyetographs from c73c82c85c87c90c82c85c87c75 data base and three hypothetical expected hyetographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 10. Median dimensionless hyetographs at a point for first, second, third, and fourth quartile heavy rainfall storms derived from Huff (1990, table 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 11. Median dimensionless hyetograph on areas of 10 to 50 square miles for first, second, third, and fourth quartile heavy rainfall storms derived from Huff (1990, table 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 40 12. Median dimensionless hyetograph for second and third quartile storms for the southern High Plains of Texas derived from Pani and Haragan (1981, figs. 3 and 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 13. Median, 10-, and 90-percentile dimensionless hyetographs for the southern High Plains of Texas derived from Pani and Haragan (1981, fig. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 14. Definition of a triangular instantaneous hyetograph model after Yen and Chow (1980) and Chow and others (1988) . . . . . . . . . . . . . 50 15. General instantaneous hyetograph patterns observed by Chukwama and Schwab (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 16. Dimensionless balanced storm and SCS Type II hyetographs for 3-hr 25-year rainfall for Austin, Texas derived from Asquith (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 xiii 17. Map showing locations of USGS streamflow-gaging stations represented in the hyetograph data base . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 18. Median dimensionless hyetograph for second and third quartile storms for the southern High Plains of Texas derived from Pani and Haragan (1981, figs. 3 and 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 19. Median, 10-, and 90-percentile dimensionless hyetographs for the southern High Plains of Texas derived from Pani and Haragan (1981, fig. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 20. Definition of a triangular instantaneous hyetograph model after Yen and Chow (1980) and Chow and others (1988) . . . . . . . . . . . . . . . . . . 72 21. Definition of a triangular instantaneous hyetograph model in terms of quantile density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 22. Dimensionless runoff-producing cumulative rainfall hyetograph for 0–24 hr and 24 hr and greater storm durations computed by triangular hyetograph models for Texas and composite hyetograph by Pani and Haragan (1981) . . . . . . . . . . . . . . . . . 81 23. Two dimensionless hyetograph representations of May 23, 1981 storm for watershed of station 08156800 Shoal Creek at 12th Street, Austin, Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 24. Dimensionless streamflow hydrograph for May 11, 1965 storm for station 08187000 Escondido Creek subwatershed #1 near Kenedy, Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 25. Explanation of box plots and ancillary glyphs shown in figures 26–46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 26. Box plots showing distribution of hyetograph mean for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 135 27. Box plots showing distribution of hyetograph median for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 136 28. Box plots showing distribution of hyetograph L-scale for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 137 29. Box plots showing distribution of hyetograph coefficient of L-variation for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 30. Box plots showing distribution of hyetograph L-skew for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 139 31. Box plots showing distribution of hyetograph L-kurtosis for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 140 32. Box plots showing distribution of hyetograph Tau5 for 0–12 hr storm durations for integer storm depth categories . . . . . . . . . . . 141 xiv 33. Box plots showing distribution of hyetograph mean for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 142 34. Box plots showing distribution of hyetograph median for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 143 35. Box plots showing distribution of hyetograph L-scale for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 144 36. Box plots showing distribution of hyetograph coefficient of L-variation for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 37. Box plots showing distribution of hyetograph L-skew for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 146 38. Box plots showing distribution of hyetograph L-kurtosis for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 147 39. Box plots showing distribution of hyetograph Tau5 for 12–24 hr storm durations for integer storm depth categories . . . . . . . . . . 148 40. Box plots showing distribution of hyetograph mean for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 41. Box plots showing distribution of hyetograph median for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 42. Box plots showing distribution of hyetograph L-scale for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 43. Box plots showing distribution of hyetograph coefficient of L-variation for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 44. Box plots showing distribution of hyetograph L-skew for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 45. Box plots showing distribution of hyetograph L-kurtosis for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 46. Box plots showing distribution of hyetograph Tau5 for 24 hr and greater storm durations for integer storm depth categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 47. Comparison of monthly mean values for the mean, median, L-scale, and L-CV of dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations for storms having greater than one inch of precipitation . . . . . . . . . . . . . . . . 161 xv 48. Relation between hyetograph mean or median and L-scale for 0–12 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 49. Relation between hyetograph mean or median and L-scale for 12–24 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 50. Relation between hyetograph mean or median and L-scale for 24 hr and greater storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 51. Relation between hyetograph mean or median and L-scale for 0–12 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed . . . . . . . . . . . . . 171 52. Relation between hyetograph mean or median and L-scale for 12–24 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed . . . . . . . . . . . . . 172 53. Relation between hyetograph mean or median and L-scale for 24 hr and greater storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed . . . . . . 173 54. Illustration of geometric reasoning behind the limiting of L-scale values with the hyetograph mean as seen in graphs A of figures 48–53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 55. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 0–12 hr and depths equal to or greater than one inch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 56. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 12–24 hr and depths equal to or greater than one inch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 xvi 57. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 24 hr and greater and depths equal to or greater than one inch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 58. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 0–12 hr and depths equal to or greater than one inch . . . . . 179 59. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 12–24 hr and depths equal to or greater than one inch . . . . 180 60. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 24 hr and greater and depths equal to or greater than one inch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 61. Example shapes of L-gamma distribution with selected pairs of (b, c) parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 62. Comparison between the L-gamma distribution fit to a median of 0.54 and L-scale of 0.24 and Govindarajulu and Power distributions fit to the same median and the Beta distribution fit to L-scale and the mean of the L-gamma distribution . . . 201 63. Gamma and Incomplete Gamma function computation results for theoretical L-moments of L-gamma distribution verification example—G(a) is the Gamma Function and P(a,x) is the Incomplete Gamma function . . . . . . . . . . . . . . . . . . . . . . . . 205 64. Comparison between statistic estimation method for L-gamma distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 65. Comparison between L-gamma distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths for each statistic estimation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 66. Comparison between statistic estimation method for Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths . . . . . . . . . . . . . . 217 67. Comparison between Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths for each statistic estimation method . . . . . 218 68. Comparison of L-gamma distribution and triangular hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 xvii 69. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the mean statistics for parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 70. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the median statistics for parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 71. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the graphical mode statistics for parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 72. Empirical hyetograph analysis for 0–12 hr storm duration and one inch and greater storm depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 73. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 0–12 hr storm duration and one inch and greater storm depths . . . . . . . . . . . . . . . . . . . . 232 74. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 12–24 hr storm duration and one inch and greater storm depths . . . . . . . . . . . . . . . . . . . . 234 75. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 24 hr and greater storm duration and one inch and greater storm depths . . . . . . . . 236 76. Median empirical hyetographs for storm depths of one inch and greater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 77. Median empirical hyetographs for storm depths of three inches and greater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 78. Hyetographs for storms having depths between 2.5 and 3.5 inches and 0–12 hr duration . . . . . . . . . . . . . . . . . . . . . . . . . 242 79. Comparison of streamflow hydrographs derived from L-gamma and balanced storm hyetographs shown in tables 29 and 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 80. Comparison of correction factors for the Carman-Kozeny equation as a function of the coefficient of L-variation (L-CV) of the particle-size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 xviii NOMENCLATURE Conventions The editorial and format style of this dissertation follows that of the U.S. Geological Survey (Hansen, 1991). One topic requiring clarification is when specific references to content in chapters, figures, tables, or pages within the works of others are made in this dissertation, the content is indicated after the year of publication. For example, “Pani and Haragan (1981, fig. 5)” refers to figure 5 of Pani and Haragan (1981) and not figure 5 of this dissertation, or “Pilgrim and Cordery (1975, p. 81)” refers to page 81 of Pilgrim and Cordery (1975) and not page 81 of this dissertation. Conversion Factors The inch-pound unit system (English units) used in this dissertation can be converted to International System of Units (SI) by the following conversion factors: Within the body of the text both units systems are reflected for the convenience of international readers. For example in the text, 1 in. is written as 1 in. (25.4 mm). Both unit systems are shown on figures when context permits, and the head notes of each table contain unit conversion details as well. In some places in the dissertation, hyetographs are classified into categories that are label by depth, since these are proper names, a conversion to SI is not universally shown. Multiply By To obtain inch (in.) 25.40 millimeter (mm) foot (ft) 0.3048 meter (m) mile (mi) 1.609 kilometer (km) square mile (mi 2 ) 2.590 square kilometers (km 2 ) cubic foot per second (ft 3 /s) 0.02832 cubic meter per second (m 3 /s) xix Lower Case Symbols and Acronyms Parameter of the triangular and Beta distributions Ratio of internal surface to volume of solids for porous media Mean Parameter of the triangular, L-gamma, and Beta distributions Parameter of the L-gamma distribution Second shape parameter of the Generalized Lambda and Kappa distributions Intrinsic permeability Mean intrinsic permeability Sample mean The first sample L-moment computed from unbiased estimators. Sample L-scale or sample L-variation The second sample L-moment computed from unbiased estimators. Sample third L-moment A measure of sample skew computed from unbiased estimators, see L-skew. Sample fourth L-moment A measure of sample kurtosis computed from unbiased estimators, see L-kurtosis. Sample size Plotting position The plotting position for the th ascending order observation of a random sample of size . A common formula is for . psd Particle-size distribution Sample coefficient of L-variation The first sample L-moment ratio, , computed from unbiased estimators. a a v a v a v b c h k k l 1 l 2 l 3 l 4 n p j:n j np j:n j δ+()n ε+()⁄= δε 1–>> t l 2 l 1 ⁄ xx Sample L-skew The second sample L-moment ratio, , computed from unbiased estimators. Sample L-kurtosis The third sample L-moment ratio, , computed from unbiased estimators. Unbiased weight factor The unbiased weight factor on a specific for computation of sample L-moments by unbiased estimators. Plotting-position weight factor The plotting-position weight factor on a specific for computation of sample L-moments by plotting-position estimators. Prior-probability weight factor The prior-probability weight factor on a specific for computation of sample L-moments by p-PWMs. Sample order statistic Sample order statistics of an sized sample are . Upper Case Symbols and Acronyms Coefficient of variation CDA Cumulative distribution analysis CDF Cumulative distribution function CK Carman-Kozeny equation Grain size diameter Diameter of mean grain size DDF Depth-duration frequency of precipitation Second noncentral product moment t 3 1– t 3 1≤≤ t 4 1 4 --- 5t 3 2 1–()t 4 ≤ 1< w jr, x j:n w ˜ jr, x j:n w ˆ jr, x j:n x m:n nx 1:n x 2:n … x n:n ≤≤≤ C D D E 2 xxi Third noncentral product moment Nonexceedance probability, Percent of storm duration or in percent , the cumulative probability of an observation, the compliment of the exceedance probability, recurrence interval in years equals . Gini’s mean difference statistic Inter-cile range The intercile range of order . IDF Intensity-duration frequency of precipitation Inter-quartile range One quarter of the data is within the inter-quartile range. Inter-tercile range One third of the data is within the inter-tercile range, . Lower tercile Two-thirds of the data is greater than the lower tercile. Median Fifty percent of the data values are greater than or less than the median. The median is also the quantile for a nonexceedance probability of 0.50. M-MLM Modified Method of L-moments The median and inter-tercile range of the distribution are estimated using the mean and L-scale values. Then the parameters of the L-gamma distribution are solved to match the median and inter-tercile range. This method differs from the common Method of L-moments in which the parameters are estimated directly from the L-moments. The probability weighted moments, PWMs . The PWMs are useful for defined measure of distribution expressible in inverse or quantile form. MVUE Minimum Variance Unbiased Estimator NRCS Natural Resources Conservation Service PDF Probability Density Function E 3 F 0 F 1≤≤ 0 F 100≤≤ 11F–()⁄ G IcR r r IQR ITR ITR UT LT–= LT M M prs,, M prs,, ExF() p F r 1 F–() s []= xxii p-PWMs Prior-Probability Weighted Moments PWMs Probability Weighted Moments Rainfall in percent RE Relative Efficiency Variance of an unbiased L-moment estimator divided by either plotting-position estimator or the p-PWM estimator variance. SCS Soil Conservation Service Uniformly Minimum Variance Unbiased Estimate of the Standard Deviation TxDOT Texas Department of Transportation USGS U.S. Geological Survey Upper tercile Two-thirds of the data is less than the upper tercile. Random variable Functions Quantile function of Beta function Grain size diameter quantile function Intrinsic permeability quantile function Natural, Naperian, or hyperbolic exponent of . Probability density function Natural, Naperian, or hyperbolic logarithm of The logarithm have a base of as opposed to the Briggsian or base-10 logarithms. Quantile function Quantile function definition canonical with L-moment literature. R s UMVU UT X a v F() a v Bab,() DF() kF() e x x e 2.71828…= f x() x()log x e xF() xxiii Combinatorial function The formula for the combinations of distinct items taken at a time is and by definition . The expectation operator In terms of the probability density function is , and in terms of the quantile function is . The first expectation of a random variable is the mean. Quantile function The quantile for nonexceedance probability ; also is used to denote the quantile function for random variable . The derivative of the quantile function, is the quantile density function. Gamma function Quantile function of the L-gamma distribution Parameters are not defined but implied. Quantile function of the L-gamma distribution Parameters are specifically defined. Legendre polynomial The shifted Legendre polynomial. The polynomial is used in the derivation and definition of the L-moments. Incomplete gamma function Greek Symbols Scale parameter of the Kappa and Linear distribution The alpha-series of probability weighted moments , see probability weighted moments. a b   nr a b  C ab a! b! ab–()! ------------------------ == a 0  1= E [] f x() EX r [] x r fx()xd ∞– ∞ ∫ = xF() EX r [] xF() r Fd 0 1 ∫ = QF() FxF() X Fd d QF() Γ x() Λ F() Λ bcF,,() P r 1– * F() r 1– Pmnx,() α α r α r M 10r,, = xxiv The beta-series of probability weighted moments . The beta-series is preferred for L-moment derivation, see probability weighted moments. Shape parameter for the Power and Govindarajulu distributions Product moment skew Location parameter of two-parameter Power distribution Location parameter of the Generalized Lambda and Kappa distributions Integral term for the Carman-Kozeny equation First shape parameter of the Generalized Lambda and Kappa distributions Mean The first L-moment. L-scale or L-variation The second L-moment. Third L-moment A measure of skew, see L-skew. Fourth L-moment A measure of kurtosis, see L-kurtosis. Standard deviation Product moment variance Coefficient of L-variation The first L-moment ratio, . L-skew The second L-moment ratio, . L-kurtosis The third L-moment ratio, . β r β r M 1 r 0,, = β γ ε ξ ζ κ λ 1 λ 2 λ 3 λ 4 σ σ 2 τ λ 2 λ 1 ⁄ τ 3 1– τ 3 1≤≤ τ 4 14⁄()5τ 3 2 1–()τ 4 ≤ 1< xxv Sample mean The first sample L-moment computed from plotting-position estimators. Sample L-scale or sample L-variation The second sample L-moment computed from plotting-position estimators. Sample third L-moment A measure of sample skew computed from plotting-position estimators, see L-skew. Sample fourth L-moment A measure of sample kurtosis computed from plotting-position estimators, see L-kurtosis. Sample coefficient of L-variation The first sample L-moment ratio, , computed from plotting-position estimators. Sample L-skew The second sample L-moment ratio, , computed from plotting- position estimators. Sample L-kurtosis The third sample L-moment ratio, , computed from plotting- position estimators. Sample mean The first sample L-moment computed from prior-PWM estimators. Sample L-scale or sample L-variation The second sample L-moment computed from prior-PWM estimators. Sample third L-moment A measure of sample skew computed from prior-PWM estimators, see L-skew. Sample fourth L-moment A measure of sample kurtosis computed from prior-PWM estimators, see L-kurtosis. λ ˜ 1 λ ˜ 2 λ ˜ 3 λ ˜ 4 τ ˜ l 2 l 1 ⁄ τ ˜ 3 1– t 3 1≤≤ τ ˜ 4 1 4 --- 5t 3 2 1–()t 4 ≤ 1< λ ˆ 1 λ ˆ 2 λ ˆ 3 λ ˆ 4 xxvi Sample coefficient of L-variation The first sample L-moment ratio computed from prior-PWM estimators. Sample L-skew The second sample L-moment ratio computed from prior-PWM estimators. Sample L-kurtosis The third sample L-moment ratio computed from prior-PWM estimators. Tortuosity τ ˆ τ ˆ 3 τ ˆ 4 ϒ xxvii 1 CHAPTER 1 INTRODUCTION The time history of rainfall depth on the ground for a specific location or over a specific area is described by a hyetograph (Schaefer, 1993, p. 18). For this dissertation, however, the term “hyetograph” specifically refers to the accumulation of rainfall depth with the duration of the storm—a cumulative hyetograph. The instantaneous time history of rainfall rate, also known as an intensity in a length scale per time, is referred to as an instantaneous hyetograph. The separate distinction for the instantaneous hyetograph is made so that the one-word term hyetograph is reserved for the cumulative hyetograph. A dimensionless hyetograph has the units of time (storm duration) and cumulative rainfall depth (storm depth) expressed in percentages of the respective totals; dimension can be removed by other techniques. The dimensionless hyetograph is convenient in many applications and comprises a substantial concept for the research presented here. An instantaneous dimensionless hyetograph also can be constructed from the first derivative of the dimensionless hyetograph. Hyetographs are useful for computer based rainfall-runoff modeling and other applications. Because the shape and timing of the runoff hydrograph is primarily driven by the magnitude and temporal distribution of rainfall, the hyetograph is an important component of the modeling. The modeling is important for cost-effective and risk-mitigated hydrologic design of hydraulic (open-channel) structures. Discussion of hyetograph basics and relation of hyetographs to hydraulic design is found in numerous hydrologic engineering textbooks (for example, Chow and others, 2 1988, p. 75, 136; Haan and others, 1994, pp. 44–52). An example event-specific hyetograph that is known to produced runoff is provided on figure 1. Figure 1. Example of a cumulative rainfall hyetograph It is obvious on figure 1 that two short duration bursts or rainfall sub-events (less than a few hours each) occurred on or about May 24, 1981. There is one burst starting just before about 00:00 hours (hr) and the other burst starting at about 18:00 hr. Each burst could be considered a distinct hyetograph, but their proximity in time suggests that one storm event, likely generated by similar meteorological mechanisms, is represented. If the duration of the storm event is considered as lasting about one day, then the event is characterized as having multiple bursts—two in this case. A long unchanging leading tail of the hyetograph is present because of a zero depth data value DEP T H O F RAI N F A LL, I N I NCHES 23 0H 6H 12H 18H 24 0H 6H 12H 18H 0H May, 1981 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 Accumulated weighted rainfall for watershed of U.S. Geological Survey streamflow-gaging station 08156800 Shoal Creek at 12th Street, Austin, Texas EXPLANATION DEP T H OF RAI N F A L L , I N MI LLI METERS burst no. 2 Representative flat leading tail burst no. 1 inter-burst total duration duration 3 at 00:00 hr on May 23rd. The tail is an artifact of the method by which the original data was recorded and preserved (see Data Sources section). The tail leaves the numerical impression that the storm duration was just over two days long. Preprocessing of the underlying data to remove unchanging tails before subsequent analysis would be required to more accurately represent the duration shown in the figure by reducing the numerical duration of the data. This preprocessing is referred to as “tail trimming” and is described and used in several chapters of this dissertation. Problem A recurring problem in the study of hyetographs is the reproduction of the observed rainfall hyetograph with a small number of values. These values could include entries in a look-up table, parameters of empirical models, or the statistical moments of the hyetograph distribution. There is considerable interest by designers of hydraulic structures (such as culverts, inlets, roadways, water-quality “ponds”) in estimating an average or expected hyetograph for a location based on observed data. Further, often predictions of the expected hyetograph are required at ungaged or un- monitored locations. The predictions typically are made through statistical regionalization. Statistical regionalization is an analysis technique that abounds in the hydrologic sciences (for example, Kite, 1988, chapter 13; Stedinger and others, 1992, pp. 18.33– 37; Asquith, 1998; Asquith and Slade, 1997; Hosking and Wallis, 1997). For hyetograph analysis, regionalization requires statistical transference of specific 4 characteristics of the hyetograph between gaged and ungaged locations by quantifiable relations between potentially influential factors such as storm duration, season of occurrence, total rainfall depth, geographic position, topography, and regional vegetation and soil types. The use of the term location reflects geographic position as well as other “coordinates” of the multidimensional parameter space represented the identified factors. Hypotheses and Objectives The hypotheses of this research are that expected hyetographs in Texas are definable using L-moment statistics (Hosking, 1990) and quantile distribution functions (Gilchrist, 2000). The expected hyetograph is defined in the next section. A secondary hypothesis is that reliable parameter estimates for candidate hyetograph quantile function models are possible. The term reliable in this context reflects that the model, including the parameter estimates, is statistically defendable even if a rigorous assessment of model validity is not possible. In particular, it is hypothesized that the sample L-moments and the nonparametric sample median of hyetograph distributions will provide a basis for statistical analysis by summarizing actual storm events. The hyetographs for individual storm events are collectively referred to as observed hyetographs. The L-moments are regionalized by investigation of relations between the L-moments and several of the influential factors already identified. Not all factors can be investigated within the scope of this dissertation. The regionalized L-moments could provide a basis for interpretation of 5 the processes that generate rainfall time distributions, such as synoptic scale and meso scale weather systems. L-moments are useful for parameter estimation for candidate models, such as: where is rainfall, is cumulative time, and the parameters and require estimation. Although the theory of L-moments and its application in hydrologic magnitude and frequency regional analyses became popular in the last decade or so (Asquith, 1998; Asquith, 2002; and references therein; and many other publications), L-moment application to the study of hyetographs appears unprecedented. The lack of previous work is attributable in part to the comparatively limited adoption of L-moments compared to the well-known classical moments for statistical analysis in general and the limited number of hyetograph investigators in particular. The analytical scope of this dissertation generally is limited to L-moment statistics. Expected Hyetograph The expected hyetograph is a synthetic storm that is intended to represent the typical characteristics of a storm when the analyst is given values of the potentially influential factors. Examples of these factors have already been identified in this dissertation. The expected hyetograph represents a hydrostatologic (study of water statistics) model based on the statistical characteristics of observed hyetographs from a data base instead of a meteorological model in which a direct coupling of fundamental components of the atmosphere and principles of physics such as RF b e c 1 F–() = RF bc 6 humidity, temperature, pressure, and conservation of energy or momentum are represented. As defined here, the expected hyetograph also does not represent the average of all storms that could be aggregated for either a given location or larger geographical area, but rather represents the average for storms that are known to produce runoff—an important condition in rainfall-runoff analysis. The fact that only runoff-producing storms are considered distinguishes the research here when compared to most of the works described in chapter 2. Although the expected hyetograph as developed here is not meteorologically based, it should still be possible to make inferences—that is, statistically-based inferences—of the meteorological and physical processes that generate and influence the temporal distribution of storm events. To that end, statistical characteristics of observed hyetographs are used to measure fundamental properties of the rainfall time distribution. The nonparametric median and L-moment statistics of observed hyetographs distributions are used. These statistics measure fundamental properties of a distribution such as location on the real-number line (mean), scale or spread on the real-number line (variance), and various “so-called” higher measures of distribution shape (skew and kurtosis). Data Sources A data base of rainfall and concomitant runoff values for small watersheds in Texas is available from ongoing (as of 2002) collaborative rainfall-runoff characterization research projects (project nos. 0–4193 and 0–4194) funded by the 7 Texas Department of Transportation (TxDOT) and the U.S. Geological Survey (USGS), and performed by researchers at Texas Tech University (Lubbock), the University of Houston, Lamar University (Beaumont), and the USGS (Austin). The data base includes 1,659 storms for 91 small rural and urban watersheds of USGS streamflow-gaging stations in Texas. The locations of the stations are shown on figure 2. Each storm within the data base is represented by a single rainfall data file. The data was manually entered into an open-source computer text file versioning software system (http://www.cvshome.org) on a RedHat Linux server (http:// www.redhat.com) over the course of two years by project staff scattered between the four research entities. The data was derived from over 220 historical USGS data reports that occupy approximately eight linear feet of book shelf space. A comprehensive citation list for the reports is provided in Appendix A. 8 Figure 2. Map showing locations of USGS streamflow-gaging stations represented in the hyetograph data base The composite data base is divided into five data base “modules”. These modules are broken into four urban centers and are called c68c88c86c87c76c81, c71c68c79c79c68c86, c73c82c85c87c90c82c85c87c75, and c86c68c81c68c81c87c82c81c76c82, for the Austin, Dallas, Fort Worth, and San Antonio areas of Texas, respectively. The fifth module is referred to as the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 data base and contains clusters of intensively monitored small rural watershed study units within the Brazos River, Colorado River, San Antonio River, and Trinity River basins of Texas. The stations for which incremental values of concomitant streamflow and rainfall for storm events are available in the data base and are listed in tables 1–5. C o l o r a d o R i v e r B a si n B ra z os R iv e rB a si n T r i n i t y R i v e r B a s i n San Antonio River Basin Dallas Fort Worth Austin San Antonio Waco 01020Miles Dallas Fort Worth Austin San Antonio Big Spring Waco Map Base Information Albers Projection Basins from Texas Water Development Board Cities from Texas Natural Resources Information System EXPLANATION Location of streamflow- gaging station 9 Table 1. U.S. Geological Survey streamflow–gaging stations within the c68c88c86c87c76c81 data base module [Approx., approximate; mi 2 , square miles. SH, State highway; FM, Farm to market road; IH, Interstate highway. One mi 2 equals 2.59 square kilometers (km 2 ).] Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 08154700 Bull Creek at Loop 360, Austin, Texas 30 o 22’19” 97 o 47’04” 22.3 1978–1986 14 08155200 Barton Creek at SH 71, Oak Hill, Texas 30 o 17’46” 97 o 55’31” 89.7 1978–1982 6 08155300 Barton Creek at Loop 360, Austin, Texas 30 o 14’40” 97 o 48’07” 116.0 1979–1986 8 08155550 West Bouldin Creek at Riverside Drive, Austin, Texas 30 o 15’49” 97 o 45’17” 3.12 1977–1985 10 08156650 Shoal Creek at Steck Avenue, Austin, Texas 30 o 21’55” 97 o 44’11” 2.79 1975–1982 13 08156700 Shoal Creek at Northwest Park, Austin, Texas 30 o 20’50” 97 o 44’41” 7.03 1976–1983 17 08156750 Shoal Creek at White Rock Drive, Austin, Texas 30 o 20’21” 97 o 44’50” 7.56 1976–1980 14 08156800 Shoal Creek at 12th Street, Austin, Texas 30 o 16’35” 97 o 45’00” 12.3 1975–1986 24 08157000 Waller Creek at 38th Street, Austin, Texas 30 o 17’49” 97 o 43’36” 2.31 1967–1980 41 08157500 Waller Creek at 23rd Street, Austin, Texas 30 o 17’08” 97 o 44’01” 4.13 1967–1980 40 08158050 Boggy Creek at U.S. 183, Austin, Texas 30 o 15’47” 97 o 40’20” 13.1 1976–1985 10 08158100 Walnut Creek at FM 1325, Austin, Texas 30 o 24’35” 97 o 42’41” 12.6 1976–1986 15 08158200 Walnut Creek at Dessau Road, Austin, Texas 30 o 22’30” 97 o 39’37” 26.2 1976–1986 17 08158380 Little Walnut Creek at Georgian Drive Austin, Texas 30 o 21’15” 97 o 41’52” 5.22 1985–1986 2 08158400 Little Walnut Creek at IH 35, Austin, Texas 30 o 20’57” 97 o 41’34” 5.57 1976–1981 10 08158500 Little Walnut Creek at Manor Road, Austin, Texas 30 o 18’34” 97 o 40’04” 12.1 1976–1981 15 08158600 Walnut Creek at Webberville Road, Austin, Texas 30 o 16’59” 97 o 39’17” 51.3 1974–1986 21 08158700 Onion Creek near Driftwood, Texas 30 o 04’59” 98 o 00’29” 124 1980–1987 6 08158800 Onion Creek at Buda, Texas 30 o 05’09” 97 o 50’52” 166 1980–1983 2 08158810 Bear Creek below FM 1826, Driftwood, Texas 30 o 09’19” 97 o 56’23” 12.2 1980–1986 8 08158820 Bear Creek at FM 1626, Manchaca, Texas 30 o 08’25” 97 o 50’50” 24.0 1980–1983 2 08158825 Little Bear Creek at FM 1626, Manchaca, Texas 30 o 07’31” 97 o 51’43” 21.0 1980–1983 2 10 Table 2. U.S. Geological Survey streamflow–gaging stations within the c71c68c79c79c68c86 data base module [Approx., approximate; mi 2 , square miles; IH, Interstate highway. One mi 2 equals 2.59 square kilometers (km 2 ).] 08158840 Slaughter Creek at FM 1826, Austin, Texas 30 o 12’32” 97 o 54’11” 8.24 1979–1986 11 08158860 Slaughter Creek at FM 2304, Austin, Texas 30 o 09’43” 97 o 49’55” 23.1 1981–1982 2 08158880 Boggy Creek (south) at Circle “S” Road, Austin, Texas 30 o 10’50” 97 o 46’55” 3.58 1976–1986 14 08158920 Williamson Creek at Oak Hill, Texas 30 o 14’06” 97 o 51’36” 6.3 1979–1984 14 08158930 Williamson Creek at Manchaca Road, Austin, Texas 30 o 13’16” 97 o 47’36” 19.0 1976–1984 18 08158970 Williamson Creek at Jimmy Clay Road, Austin, Texas 30 o 11’21” 97 o 43’56” 27.6 1976–1986 16 08159150 Wilbarger Creek near Pflugerville, Texas 30 o 27’16” 97 o 36’02” 4.61 1967–1977 29 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 08055580 Joes Creek at Royal Lane, Dallas, Texas 32 o 53’43” 96 o 41’36” 1.94 1974–1979 7 08055600 Joes Creek, Dallas, Texas 32 o 51’41” 96 o 52’27” 7.51 1973–1979 10 08055700 Bachman Branch, Dallas, Texas 32 o 51’37” 96 o 50’13” 10.0 1964–1979 41 08056500 Turtle Creek, Dallas, Texas 32 o 48’26” 96 o 48’08” 7.98 1964–1979 42 08057020 Coombs Creek at Sylvan Avenue, Dallas, Texas 32 o 46’01” 96 o 50’07” 4.75 1965–1979 7 08057050 Cedar Creek at Bonnie View Road, Dallas, Texas 32 o 44’50” 96 o 47’44” 9.42 1974–1979 3 08057120 Spanky Branch at McCallum Lane, Dallas, Texas 32 o 57’58” 96 o 48’11” 6.77 1973–1978 5 08057130 Rush Branch at Arapaho Road, Dallas, Texas 32 o 57’45” 96 o 47’44” 1.22 1973–1979 7 08057140 Cottonwood Creek at Forest Lane, Dallas, Texas 32 o 54’33” 96 o 45’54” 8.5 1973–1978 6 08057160 Floyd Branch at Forest Lane, Dallas, Texas 32 o 54’33” 96 o 45’34” 4.17 1974–1979 8 08057320 Ash Creek at Highland Road, Dallas, Texas 32 o 48’18” 96 o 43’04” 6.92 1973–1978 5 08057415 Elam Creek at Seco Boulevard, Dallas, Texas 32 o 44’14” 96 o 41’36” 1.25 1973–1979 8 08057418 Fivemile Creek at Kiest Boulevard, Dallas, Texas 32 o 42’19” 96 o 51’32” 7.65 1976–1979 7 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 11 Table 3. U.S. Geological Survey streamflow–gaging stations within the c73c82c85c87c90c82c85c87c75 data base module [Approx., approximate; mi 2 , square miles. IH, Interstate highway; W, West. One mi 2 equals 2.59 square kilometers (km 2 ). --, not available] 08057420 Fivemile Creek at U.S. 77, Dallas, Texas 32 o 41’15” 96 o 49’22” 13.2 1973–1979 10 08057425 Woody Branch at U.S. 77, Dallas, Texas 32 o 40’58” 96 o 49’22” 11.5 1973–1979 10 08057435 Newton Creek at IH 635, Dallas, Texas 32 o 39’19” 96 o 44’41” 5.91 1976–1979 4 08057440 Whites Branch at IH 635, Dallas Texas 32 o 39’26” 96 o 44’25” 2.53 1976–1979 4 08057445 Prairie Creek at U.S. 175, Dallas, Texas 32 o 42’17” 96 o 40’11” 9.03 1976–1979 8 08061620 Duck Creek at Buckingham Road, Garland, Texas 32 o 55’53” 96 o 39’55” 8.05 1973–1979 8 08061920 South Mesquite Creek at SH 352, Mesquite, Texas 32 o 46’09” 96 o 37’18” 13.4 1973–1979 9 08061950 South Mesquite Creek at Mercury Road, Mesquite, Texas 32 o 43’32” 96 o 34’12” 23.0 1969–1979 31 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events –– Seminary South Shopping Center and associated drainage area, Fort Worth, Texas –– –– 0.38 1970–1976 21 08048520 Sycamore Creek at IH 35–W, Fort Worth, Texas 32 o 39’55” 97 o 19’16” 17.7 1970–1977 24 08048530 Sycamore Creek tributary above Seminary South Shopping Center, Fort Worth, Texas 32 o 41’08” 97 o 19’44” 0.97 1970–1977 28 08048540 Sycamore Creek tributary at IH 35– W, Fort Worth, Texas 32 o 41’18” 97 o 19’11” 1.35 1970–1976 24 08048550 Dry Branch at Blandin Street, Fort Worth, Texas 32 o 47’19” 97 o 18’22” 1.08 1969–1976 25 08048600 Dry Branch at Fain Street, Fort Worth, Texas 32 o 46’34” 97 o 17’18” 2.15 1969–1977 27 08048820 Little Fossil Creek at IH 820, Fort Worth, Texas 32 o 50’22” 97 o 19’22” 5.64 1969–1977 20 08048850 Little Fossil Creek at Mesquite Street, Fort Worth, Texas 32 o 48’33” 97 o 17’28” 12.3 1969–1977 25 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 12 Table 4. U.S. Geological Survey streamflow–gaging stations within the c86c68c81c68c81c87c82c81c76c82 data base module [Approx., approximate; mi 2 , square miles. FM, Farm to market road. One mi 2 equals 2.59 square kilometers (km 2 ).] Table 5. U.S. Geological Survey streamflow-gaging stations within the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 data base module [Approx., approximate; mi 2 , square miles; Sub., Subwatershed; The * notes that two different drainage areas have been published for this station. One mi 2 equals 2.59 square kilometers (km 2 ).] Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 08177600 Olmos Creek tributary at FM 1535, Shavano Park, Texas 29 o 34’35” 98 o 32’45” 0.33 1970–1981 14 08177700 Olmos Creek at Dresden Drive, San Antonio, Texas 29 o 29’56” 98 o 30’36” 21.2 1969–1978 23 08178300 Alazan Creek at St. Cloud Street, San Antonio, Texas 29 o 27’29” 98 o 32’59” 3.26 1969–1979 30 08178555 Harlendale Creek at West Harding Street, San Antonio, Texas 29 o 21’05” 98 o 29’32” 2.43 1977–1980 10 08178600 Panther Springs Creek at FM 2696 near San Antonio, Texas 29 o 37’31” 98 o 31’06” 9.54 1969–1975 13 08178620 Lorence Creek at Thousand Oaks Boulevard, San Antonio, Texas 29 o 35’24” 98 o 27’47” 4.05 1981–1981 3 08178640 West Elm Creek at San Antonio, Texas 29 o 37’23” 98 o 26’29” 2.45 1976–1979 8 08178645 East Elm Creek at San Antonio, Texas 29 o 37’04” 98 o 25’41” 2.33 1976–1979 6 08178690 Salado Creek tributary at Bitters Road, San Antonio, Texas 29 o 31’36” 98 o 26’25” 0.26 1969–1981 41 08178736 Salado Creek tributary at Bee Street, San Antonio, Texas 29 o 26’38” 98 o 27’13” 0.45 1972–1976 12 08181000 Leon Creek tributary at FM 1604, San Antonio, Texas 29 o 35’14” 98 o 37’40” 5.57 1970–1979 10 08181400 Helotes Creek at Helotes, Texas 29 o 34’42” 98 o 41’29” 15.0 1969–1981 15 08181450 Leon Creek tributary at Kelly Air Force Base, Texas 29 o 23’12” 98 o 36’00” 1.19 1969–1979 30 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 08042650 North Creek Sub. 28A near Jermyn, Texas 33°14’52" 98°19’19" 6.82 1973–1979 14 08042700 North Creek near Jacksboro, Texas 33°16’57" 98°17’53" 21.6 1959–1979 58 08050200 Elm Fork Sub. 6 near Muenster, Texas 33°37’13" 97°24’15" 0.77 1961–1970 34 13 An example hyetograph data file from the data base is listed on figure 3. This data was used to produce the observed hyetograph on figure 1. From the figure, the three main components of the data file are visible. The header of the file is identified by the lines specified by the leading c6 sign. A single field line containing the label or column titles follows the header. This line is delimited by one or more spaces. Finally, the data records, which also are space delimited, make up the remainder of the file. Two 08052630 Little Elm Creek Sub. 10 near Gunter, Texas 33°24’33" 96°48’41" 2.10 1966–1976 29 08052700 Little Elm Creek near Aubrey, Texas 33°17’00" 96°53’33" 75.5 1959–1976 58 08057500 Honey Creek Sub. 11 near McKinney, Texas 33°18’12" 96°41’22” 2.14 1960–1970 32 08058000 Honey Creek Sub.12 near McKinney, Texas 33°18’20" 96°40’12" 1.26 1959–1970 29 08063200 Pin Oak Creek near Hubbard, Texas 31°48’01" 96°43’02" 17.6 1959–1971 33 08094000 Green Creek Sub. 1 near Dublin, Texas 32°10’00" 98°20’30" *3.18/ 4.19 1959–1971 29 08096800 Cow Bayou Sub. 4 near Bruceville, Texas 31°19’59" 97°16’02” *5.25/ 5.04 1959–1975 51 08098300 Little Pond Creek near Burlington, Texas 31°01’35" 96°59’17" 22.2 1964–1972 19 08108200 North Elm Creek near Cameron, Texas 30°55’52" 97°01’13" 48.6 1964–1972 21 08136900 Mukewater Creek Sub. 10A near Trickham, Texas 31°39’01" 99°13’30" 21.8 1966–1973 22 08137000 Mukewater Creek Sub. 9 near Trickham, Texas 31°41’40" 99°12’18" 4.02 1961–1973 38 08137500 Mukewater Creek at Trickham, Texas 31°35’24" 99°13’36" 70.4 1959–1961 5 08139000 Deep Creek Sub. 3 near Placid,Texas 31°17’25" 99°09’22" 3.42 1960–1971 29 08140000 Deep Creek Sub. 8 near Mercury, Texas 31°24’08" 99°07’17" *4.32/ 5.41 1960–1971 30 08182400 Calaveras Creek Sub. 6 near Elmendorf, Texas 29°22’49" 98°17’33" 7.01 1961–1971 25 08187000 Escondido Creek Sub. 1 near Kenedy, Texas 28°46’41" 97°53’41" 3.29 1959–1971 32 08187900 Escondido Creek Sub. 11 near Kenedy, Texas 28°51’39" 97°50’39" 8.43 1962–1970 21 Station no. Station name Latitude Longitude Drainage area (mi 2 ) Approx. period of record No. of storm events 14 portions of the data lines have been removed for the figure because of limited space. As described for figure 1, the long flat leading tail of the hyetograph is an artifact of the data recording method. The 23.5 hr jump between zero precipitation and the first 0.02 inch (in.) (0.508 mm) is unacceptably too long for the purposes of hyetograph analysis. Algorithms are required to trim each tail of the observed hyetographs prior to subsequent analysis. Not all tails will require trimming, however. c6c3c43c60c40c55c50c42c53c36c51c43c3c41c44c47c40 c6c3c86c76c87c72c32c19c27c20c24c25c27c19c19c3c54c75c82c68c79c3c38c85c72c72c78c3c68c87c3c20c21c87c75c3c54c87c85c72c72c87c15c3c36c88c86c87c76c81c15c3c55c72c91c68c86c17 c6c3c79c68c87c76c87c88c71c72c32c22c19c11c12c20c25c10c22c24c5 c6c3c79c82c81c74c76c87c88c71c72c32c28c26c11c12c23c24c10c19c19c5 c6c3c71c85c68c76c81c68c74c72c68c85c72c68c3c11c80c76c21c12c32c20c21c17c22 c6c3c39c36c55c40c66c55c44c48c40c32c71c68c87c72c3c68c81c71c3c87c76c80c72c3c76c81c3c48c48c18c39c39c18c60c60c60c35c43c43c29c48c48 c6c3c51c53c40c38c44c51c20c32c20c54c43c47c3c85c68c90c3c85c72c70c82c85c71c72c71c3c71c68c87c68 c6c3c51c53c40c38c44c51c21c32c21c54c43c47c3c85c68c90c3c85c72c70c82c85c71c72c71c3c71c68c87c68 c6c3c36c38c38c56c48c66c58c55c39c66c51c53c40c38c44c51c32c68c70c70c88c80c88c79c68c87c72c71c3c90c72c76c74c75c87c72c71c3c82c85c3c86c76c81c74c79c72c3c74c68c74c72 c6c3c83c85c72c70c76c83c76c87c68c87c76c82c81c3c76c81c3c76c81c70c75c72c86 c39c36c55c40c66c55c44c48c40c3c3c3c3c43c50c56c53c54c66c51c36c54c54c40c39c3c3c3c3c51c53c40c38c44c51c20c3c3c3c3c51c53c40c38c44c51c21c3c3c3c3c36c38c38c56c48c66c58c55c39c66c51c53c40c38c44c51 c19c24c18c21c22c18c20c28c27c20c35c19c19c29c19c19c29c19c19c3 c19c17c19c19c19c19c3c3c3c3c19c17c19c19c19c19c3c3c3c3c19c17c19c19c19c19c3c3c3c3c19c17c19c19c19c19 c19c24c18c21c22c18c20c28c27c20c35c21c22c29c22c19c29c19c19c3c3c3c3c21c22c17c24c19c19c19c3c3c3c3c19c17c19c26c19c19c3c3c3c3c19c17c19c19c19c19c3c3c3c3c19c17c19c21c19c19 c19c24c18c21c22c18c20c28c27c20c35c21c22c29c23c24c29c19c19c3c3c3c3c21c22c17c26c24c19c19c3c3c3c3c19c17c24c26c19c19c3c3c3c3c19c17c20c28c19c19c3c3c3c3c19c17c21c27c19c19 c19c24c18c21c23c18c20c28c27c20c35c19c19c29c19c19c29c19c19c3c3c3c3c21c23c17c19c19c19c19c3c3c3c3c19c17c26c23c19c19c3c3c3c3c19c17c24c26c19c19c3c3c3c3c19c17c25c20c19c19 c19c24c18c21c23c18c20c28c27c20c35c19c19c29c22c19c29c19c19c3c3c3c3c21c23c17c24c19c19c19c3c3c3c3c20c17c19c23c19c19c3c3c3c3c19c17c28c21c19c19c3c3c3c3c19c17c28c24c19c19 c19c24c18c21c23c18c20c28c27c20c35c19c20c29c19c19c29c19c19c3c3c3c3c21c24c17c19c19c19c19c3c3c3c3c20c17c20c22c19c19c3c3c3c3c20c17c19c26c19c19c3c3c3c3c20c17c19c27c19c19 c17c17c17c3c83c82c85c87c76c82c81c86c3c70c88c87c3c73c82c85c3c69c85c72c89c76c87c92c3c17c17c17 c19c24c18c21c24c18c20c28c27c20c35c19c19c29c20c24c29c19c19c3c3c3c3c23c27c17c21c24c19c19c3c3c3c3c27c17c24c20c19c19c3c3c3c3c26c17c27c27c19c19c3c3c3c3c27c17c19c22c19c19 c19c24c18c21c24c18c20c28c27c20c35c19c19c29c22c19c29c19c19c3c3c3c3c23c27c17c24c19c19c19c3c3c3c3c27c17c24c26c19c19c3c3c3c3c26c17c28c27c19c19c3c3c3c3c27c17c20c21c19c19 c19c24c18c21c24c18c20c28c27c20c35c19c19c29c23c24c29c19c19c3c3c3c3c23c27c17c26c24c19c19c3c3c3c3c27c17c25c24c19c19c3c3c3c3c27c17c19c21c19c19c3c3c3c3c27c17c20c26c19c19 c19c24c18c21c24c18c20c28c27c20c35c19c20c29c19c19c29c19c19c3c3c3c3c23c28c17c19c19c19c19c3c3c3c3c27c17c26c22c19c19c3c3c3c3c27c17c19c26c19c19c3c3c3c3c27c17c21c22c19c19 c19c24c18c21c24c18c20c28c27c20c35c19c20c29c22c19c29c19c19c3c3c3c3c23c28c17c24c19c19c19c3c3c3c3c27c17c26c24c19c19c3c3c3c3c27c17c19c27c19c19c3c3c3c3c27c17c21c23c19c19 c19c24c18c21c24c18c20c28c27c20c35c19c21c29c19c19c29c19c19c3c3c3c3c24c19c17c19c19c19c19c3c3c3c3c27c17c26c28c19c19c3c3c3c3c27c17c20c23c19c19c3c3c3c3c27c17c22c19c19c19 Figure 3. Example data file c68c88c86c87c76c81c18c54c75c82c68c79c38c85c72c72c78c18c86c87c68c19c27c20c24c25c27c19c19c66c71c18c75c92c72c87c82c86c18 c85c68c76c81c66c86c87c68c19c27c20c24c25c27c19c19c66c20c28c27c20c66c19c24c21c22c17c71c68c87 from the c68c88c86c87c76c81 data base Some other aspects of the data file on figure 3 that need description are the values for c51c53c40c38c44c51c20, c51c53c40c38c44c51c21, and c36c38c38c56c48c66c58c55c39c66c51c53c40c38c44c51. The c36c38c38c56c48c66c58c55c39c66c51c53c40c38c44c51 (accumulated weighted precipitation) is the best available estimate for precipitation on the entire watershed and was derived from the c51c53c40c38c44c51c20 and c51c53c40c38c44c51c21 values. Two rain gages, labeled c51c53c40c38c44c51c20 and c51c53c40c38c44c51c21, were operating during the May 1981 storm 15 event in the Shoal Creek watershed. If additional precipitation gages were available during the event, then the labeling follows the c51c53c40c38c44c51c6c3convention. The raw data is shown in the c51c53c40c38c44c51c6 fields. Sometimes the rainfall data collected for a watershed might include one or more weighing or volumetric rain gages, which presumably provided (presumed by the previous USGS analysts in developing the data reports) more accurate data than the recording charts or tipping bucket type rain gages. Volumetric rain gage data is not reported in the file and hence is not represented in the data base. The c36c38c38c56c48c66c58c55c39c66c51c53c40c38c44c51 values were computed by the previous analysts by amalgamated weighting of partial areas on the watershed and corrections to the volumetric values when present. The 1,659 hyetographs for the five data base modules are shown on figures 4–8. In each of the five figures, the values composing each hyetograph are plotted against the concomitant time values in the top graphs (graphs A). The period of record available in each module is seen in the graphs A. The separate watersheds within each data base module are not differentiated. It is evident in graphs A that the number of storms represented per year is relatively uniform for the four urban data base modules (figs. 4–7). Some minor clustering of events on a yearly basis is exhibited; this reflects concepts of “wet” and “dry” years. The c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 module (fig. 8) exhibits a marked reduction in the number of storm events per year starting about 1972 as the study units were decommissioned. The last storms available in the data base occur in 1987 for the c68c88c86c87c76c81 module. Although the USGS and collaborating agencies continue 16 to intensively collect data in rural and urban areas, compilation and publication of rainfall and runoff for individual storms is no longer occurring. It is important to note that the total number of visually distinct hyetographs in graphs A does not equal the number of events seen in the other graphs (B and C) or indicated in the figure title. This is because numerous events typically occur simultaneously because a single storm often effects many stations in the area surrounding the storm and the path along the storm track. This has the effect of causing many hyetographs to become masked in graphs A. Because many of the hyetographs occur on the same day or other intervals of time, the observed hyetographs are not each independent. Therefore, meteorological mechanisms generating each hyetograph are not random or statistically independent—the data base does not contain as much independent information as suggested by the total number of hyetographs. In order to visualize seasonal differences in the magnitude and the number of events, the year was converted to a common base (on a leap year) for the middle graphs (graphs B) on figures 4–8. A strong seasonal clustering of events is seen for the c68c88c86c87c76c81 module (fig. 4). One cluster occurs between the middle of April and the middle of June, and a secondary cluster occurs in October. The c71c68c79c79c68c86 module (fig. 5) also exhibits a primary event cluster between the middle of March and the middle of June. Distinct clusters are harder to visualize in the c73c82c85c87c90c82c85c87c75 module (fig. 6); although a weak cluster near the end of May might exist. The c86c68c81c68c81c87c82c81c76c82 module 17 (fig. 7) exhibits a strong primary cluster between the middle of April and the middle of June. This is consistent with the Austin data base module. A secondary cluster for the c86c68c81c68c81c87c82c81c76c82 module is less distinct and much longer lasting than that for the c68c88c86c87c76c81 module. The cluster begins at the beginning of August and gradually tapers off towards the end of the year. The c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 module, which has the greatest range in geographic location of the five modules and includes areas proximate to other four modules, exhibits two or three event clusters. One cluster occurs between the middle of April and the middle of June; a possible cluster near the end of July; and a final cluster between the end of August to the middle of October. From the graphs B on figures 4–8, distinct meteorological mechanisms might be evident. The strong tendency for double clustering of events (spring and fall) in the Dallas area and in the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 module (which has many watersheds in the upper half of Texas) might be attributed to the more northerly watershed locations. The northern watersheds around Dallas and Fort Worth areas are more frequently influenced by cold fronts over a longer time interval or fraction of the calendar than those for Austin. It is possible that Austin experiences more rainfall and runoff producing cold fronts in the late spring than in the fall. Although San Antonio is considered geographically proximate to Austin, a considerably longer fall event cluster is evident; this might be attributed to more proximity of the San Antonio area to the Gulf of Mexico and hence a moisture source than the Austin area. 18 In the bottom graphs (graphs C) on figures 4–8, each hyetograph is plotted against the c43c50c56c53c54c66c51c36c54c54c40c39 field. From this figure, it is obvious that a substantial variation in the temporal pattern of the storms exists. As a result, prediction of the expected hyetograph is not a problem with a straightforward or unique solution. Also from the figure, the presence of unacceptably long leading and trailing tails such as the leading tail exhibited on figures 1 and 3 are visible. The hyetographs in the graphs C of figures 4–8 can be re-expressed in terms of cumulative percentage on both rainfall depth and time axes. The re-expression into percentages enables individual storms to be compared and simplifies the analysis and graphical presentation of the data. Other researchers have used this technique (for example, Huff, 1967, fig. 2). 19 Figure 4. Hyetographs for c68c88c86c87c76c81 data base module—401 storm events represented 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 CALENDAR YEAR 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 Jan 14 28 Feb 14 28 Mar 14 28 Apr 14 28 May 14 28 Jun 14 28 Jul 14 28 Aug 14 28 Sep 14 28 Oct 14 28 Nov 14 28 Dec 14 28 COMMON YEAR BASE 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 HOURS PASSED SINCE BEGINNING OF DATA RECORD ACCUMULA T ED PRECI P I T A T I O N O VER EACH W A T E R S H E D I N DA T A BAS E, I N I NCHES ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N MI LLI METERS 0 102030405060 72 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 A. B. C. 20 Figure 5. Hyetographs for c71c68c79c79c68c86 data base module—240 storm events represented 1964 1966 1968 1970 1972 1974 1976 1978 CALENDAR YEAR 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 Jan 14 28 Feb 14 28 Mar 14 28 Apr 14 28 May 14 28 Jun 14 28 Jul 14 28 Aug 14 28 Sep 14 28 Oct 14 28 Nov 14 28 Dec 14 28 COMMON YEAR BASE 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 HOURS PASSED SINCE BEGINNING OF DATA RECORD ACCUMULA T ED P R ECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N I NCHES ACCUMULA T ED PRECI P I T A T I O N O VER EACH W A T E R S H E D I N DA T A BAS E, I N MI LL I M ETERS 0 102030405060 72 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 A. B. C. 21 Figure 6. Hyetographs for c73c82c85c87c90c82c85c87c75 data base module—194 storm events represented 1969 1970 1971 1972 1973 1974 1975 1976 1977 CALENDAR YEAR 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 Jan 14 28 Feb 14 28 Mar 14 28 Apr 14 28 May 14 28 Jun 14 28 Jul 14 28 Aug 14 28 Sep 14 28 Oct 14 28 Nov 14 28 Dec 14 28 COMMON YEAR BASE 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 HOURS PASSED SINCE BEGINNING OF DATA RECORD ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N I NCHES ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N MI LLI METERS 0 102030405060 72 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 A. B. C. 22 Figure 7. Hyetographs for c86c68c81c68c81c87c82c81c76c82 data base module—215 storm events represented 1970 1972 1974 1976 1978 1980 CALENDAR YEAR 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 Jan 14 28 Feb 14 28 Mar 14 28 Apr 14 28 May 14 28 Jun 14 28 Jul 14 28 Aug 14 28 Sep 14 28 Oct 14 28 Nov 14 28 Dec 14 28 COMMON YEAR BASE 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 HOURS PASSED SINCE BEGINNING OF DATA RECORD ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N I NCHES ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N MI LLI METERS 0 102030405060 72 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 A. B. C. 23 Figure 8. Hyetographs for c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 data base module—609 storm events represented 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 CALENDAR YEAR 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 Jan 14 28 Feb 14 28 Mar 14 28 Apr 14 28 May 14 28 Jun 14 28 Jul 14 28 Aug 14 28 Sep 14 28 Oct 14 28 Nov 14 28 Dec 14 28 COMMON YEAR BASE 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 HOURS PASSED BEGINNING OF DATA RECORD ACCU MULA TED PRECI P I T A T I O N OVER EACH W A TERSHED I N D A T A BASE, I N I NCHES ACCUMULA T ED PRECI P I T A T I O N OVER EACH W A TERSHED I N DA T A BASE, I N M I LLI METERS 0 102030405060 72 0 0 1 25.4 2 50.8 3 76.2 4 101.6 5 127 6 152.4 7 177.8 8 203.2 9 228.6 10 254 11 279.4 12 304.8 A. B. C. 24 Dimensionless or percentile hyetographs of those on figure 6 are shown on figure 9. The hyetographs for the c73c82c85c87c90c82c85c87c75 module were chosen for figure 9 because the data base had the smallest number of events, and hence, it is most attractive for display. A substantial variation in hyetograph shape is evident in the figure. From the figure, it is clear that a large number of storms have lengthy leading and trailing tail lengths compared to the general time length of the precipitation bursts. It has been observed in general that both leading and trailing tails are artificially lengthened by the data recording method. Further, the figure indicates a large number of multiple burst events; whereas for other events, the line steadily increases with duration. It is important to consider that the rainfall bursts are the predominate flood runoff producing fractions of individual hyetographs for small watersheds. Three hypothetical expected hyetographs have been superimposed on figure 9 to illustrate the concept of the expected hyetograph. The anticipated values for the mean, median, and variability or scale statistics are indicated for each of the three expected hyetographs drawn on the figure; a complimentary figure illustrating anticipated values for the statistics is provided in figure 54 (referenced out of sequence). The primary objective of this research is to define the expected hyetograph. The expected hyetographs do not represent a fit or a conclusion of research but rather are for illustration only. It is important to note that there is a wide range in durations (few hours to days) and a wide range in precipitation magnitude (fractional inches to many inches) represented in the figure. An important component of the analysis of rainfall 25 hyetographs is separating the potential influences on the duration or magnitude of the event on the hyetograph. These factors are subjects of subsequent portions of this dissertation. Figure 9. Dimensionless or percentile representation of the 194 hyetographs from c73c82c85c87c90c82c85c87c75 data base and three hypothetical expected hyetographs The large number of events within the entire data base provides a basis for regional analysis of hyetographs. The temporal variation of recorded storms is so great that a large data base is necessary to estimate the expected hyetograph. It is hypothesized that rainfall hyetographs in the data base can be statistically regionalized by investigation of the regional characteristics of the L-moments of hyetograph PERCENT OF ELAPSED TIME PERCENT OF TOT A L RAI N F A LL DEPTH 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Dark lines show hypothetical expected hyetographs Expected hyetograph with small mean or median and moderate variance (steepness). Expected hyetograph with large mean or median and small variance (steepness). Expected hyetograph with moderate mean or median and large variance (steepness). 26 cumulants (ordinates of the cumulative hyetograph). An extremely important consideration of the data base is that often the recorded values for each storm are not suitable for sample L-moment estimation techniques in current practice because there is a prior expectation that the data points within the data base are not random or evenly spaced observations of the hyetograph distribution representing individual events. Because of this expectation, a new technique for L-moment computation of a sample is required. This technique is described in chapter 4. Model Verification and Suitability Verification of regional statistical hydrologic models, such as hyetograph models, is an extremely difficult and perhaps an impossible task because true controlled experiments are difficult in the natural world. Heuristic arguments outlined below can be used to evaluate the suitability of the model. Thus, assessing the suitability of a hyetograph model is a more tractable goal than explicit model verification. Because expected hyetograph models are statistically based and not founded on physics, analytical thought coupled with controlled experiments can not provide a basis for verification. Proper experimental methods applicable to hyetograph analysis and prediction are difficult to envision. Often the suitability of a hydrologic model is conducted through contrived regional statistical simulation methods on computer. Such simulations have a long and valuable history in regional hydrologic frequency analyses; numerous papers are available in journals such as Journal of Hydrology (such as Haktanir and Bozduman, 27 1995) and Water Resources Research (such as Fill and Stedinger, 1995). Hosking and Wallis (1993a) provide an excellent example of such simulations applicable to L-moments and frequency analysis, and the references therein are important. These schemes do not constitute a model verification technique. At the present time, it is unclear how to construct a viable hyetograph simulation framework or whether numerical weather and climate models could assist in that effort. The suitability of a hydrologic model can be assessed graphically or by statistical errors based on the fit of a model to the observed data. Although verification in a physical sense of a statistically-based hyetograph model is difficult, the suitability of the model can be assessed by comparing features of the model to aspects of observed storms. For example, storms can be classified into three general categories of front, center, and back loaded, where the time in which the bulk of the precipitation occurs is reflected in the name of the classification. A front-loaded storm might have 70 percent of cumulative depth in the first half of time. A suitable model that is adequately fitted to a front-loaded storm should have an upper right hand tail that gradually or even asymptotically approaches the total storm depth. This implies that the second derivative is negative and decreases with increasing duration near the end of the storm, which means that rainfall rates are diminishing. Other arguments for assessing model suitability are anecdotal accounts. Common experience and review of observed short duration (few hours or so) hyetographs suggests that runoff producing storms on small watersheds appear to be front-loaded. 28 Periodic observation of storms in the Austin area by public weather radar by the author indicates that many runoff producing storms on small watersheds hit hard with great intensity and then gradually diminish as time progresses; whereas occasionally other storms start gradually and then diminish. The skewness of an observed hyetograph is a measure that reflects the loadedness of the storm: front loaded equates to positive skew, center loaded equates to near zero skew, and back loaded equates to negative skew. Very short durations (minutes to multiple hours) storms might be expected to be less front loaded than long duration (days) storms because of a limited time interval to spread the rainfall depth. Hence, skewness should be a function of duration. This is a tertiary hypothesis that is considered in chapter 5. Dissertation Organization The dissertation is organized so that each chapter subsequent to the Introduction chapter is either a stand-alone paper or nearly stand alone discussion of a specific topic pertinent to the research. Moderate overlap between chapters naturally is present, and occasionally there is a need to directly recall material from previous chapters or foreshadow material in later chapters. Some of the topics are methodological as opposed to topics provided that hyetograph specific research results. This is because some of the methods used here are new and substantial research into methodology was required. 29 Previous Studies. This chapter provides a comprehensive review of rainfall hyetograph studies and illustrates a classic method for estimating a hyetograph based entirely on depth-duration frequency values of annual precipitation maxima. Other chapters provide similar or supplemental information about the prior work of other researchers as context dictates. Triangular Model of Dimensionless Rainfall Hyetographs Known to Produce Runoff in Texas. This chapter is organized as a stand-alone paper that has been submitted to the Journal of the American Water Resources Association. The chapter describes a triangular model for dimensionless hyetograph estimation, and the model is a simple example of a regional analysis. Such models have precedence for hyetograph approximation. The motivation for this chapter is that since the triangular model is straightforward to understand, the apply, and the fact that L-moments are not required, the expected hyetographs defined by the triangular model should provide a comparative basis for other analyses in the dissertation. This chapter precedes the much more complicated research and analysis chapters. This was done to provide the reader with a greater understanding of the remainder of the dissertation. In the paper, the triangular model is fit to the observed hyetographs of the data base after dimension is removed. The fitted model produces an expected hyetograph. A separate model is fit to hyetographs from storms having 0–24 hr and 24 hr and greater durations. An important note about nomenclature is needed; this dissertation uses “24 hr and greater” to refer to storm durations between 24 hr up to about 3 days. 30 The hyetograph from each model is compared to a hyetograph from an earlier study of hyetographs conducted for Texas rainfall data. Some material from chapters 1 and 2 is repeated, and moderate amounts of salient previous work is freshly described. The repetition is required in order to make a distinct paper out of the chapter. Sample L-moment Estimation using prior-Probability Weighted Moments. This chapter provides the description of an alternative method from established procedures for L-moment estimation. This method requires description because of the nature of the way that the observed hyetographs were recorded and preserved for the data base. The issue is that the data values defining the hyetograph are not either uniformly or truly randomly distributed (constant interval digitized) and adjustment to sample L-moment estimation techniques is needed. The new method requires explanation and demonstration by example by limited statistical simulation experiments. The experiments also show that the author’s custom software is properly functioning. L-moments of Runoff-Producing Dimensionless Rainfall Hyetographs in Texas. This chapter provides an analysis of the L-moment (and median) values of the hyetograph distributions. Because the L-moments of hyetograph distributions quantify various measures of the distributional shape, it is important that an understanding of potential influences on the numeric values of the L-moments is made. An investigation into the dependency of L-moment values on storm depth for three duration ranges is conducted. Another investigation into the seasonal and monthly behavior of the L-moments also is conducted. And finally, geographic influences on the L-moments 31 are investigated along with consideration of compatibility between the five hyetograph data modules. A sequence of important conclusions completes the chapter. L-gamma Distribution. This chapter introduces a new statistical quantile function for modeling the quantile distribution of a random variable that is bounded by 0 and 1. The distribution is compatible with the theory of L-moments. This distribution is appealing for application to hyetographs because dimensionless hyetographs by their very definition when expressed in fractional percentages are also bounded by 0 and 1. Further, the L-gamma has an explicit quantile function form and simple first and second derivatives. Each of these facts contributes to the utility of the distribution. L-gamma Model of Dimensionless Rainfall Hyetographs Known to Produce Runoff in Texas. This chapter describes an application of the L-gamma distribution for modeling the shape of the expected hyetograph for several durations. The models are based on statistics presented in chapter 5. The L-gamma hyetograph models are compared to both the triangular model and the well-known Beta distribution. The L-gamma and triangular models exhibit distinct differences that are discussed. The L-gamma and Beta distributions exhibit relatively minor differences. The suitability of the L-gamma model is assessed through an alternative hyetograph analysis technique. Finally, criticisms of the hyetograph models are made, and an application example of hyetographs for generation of streamflow hydrographs is presented. Modification of the Carman-Kozeny Equation for Application of L-moment Statistics for Estimation of the Intrinsic Permeability of Porous Media. This chapter is 32 unrelated to the hyetograph theme of all other chapters in this dissertation. The chapter is included because an promising application of L-moments to the estimation of permeability is envisioned following the lead of product moment based analysis of previous researchers. The application is based on the L-moments of grain-size distributions and quantile functions modeling the distribution. The prior-Probability Weighted Moments described in chapter 4 have a natural application to sample L-moment computation for grain (particle) size data. Conclusions. This chapter enumerates and describes the major conclusions and commentary of the research hypotheses. No new material is presented in this chapter. Appendices. Several appendices are provided. Appendix A provides a comprehensive citation list of the USGS data reports that provided the rainfall hyetograph data. Appendix B provides a comprehensive background of L-moment statistical theory and is included for readers unfamiliar with L-moments. A manual example computation of the unbiased L-moments for a sample is shown in Appendix B to demonstrate how the L-moments are computed without the aid of computer software. Appendix C provides the results of supplemental simulations used to explore the suitability of prior-Probability Weighted Moments for sample L-moment estimation described in a separate chapter. Appendix D provides extensive tables mapping the solution space of the L-gamma distribution. Appendix E provides certain critical computer programs written as part of the research that require documentation. Appendix F provides supplemental tables for chapter 7. 33 CHAPTER 2 PREVIOUS STUDIES Comparatively few papers treat expected hyetographs in general as opposed to other aspects of hydrologic design such as unit hydrographs (Thompson, personal commun., 2002). Two excellent starting points in the literature and references therein are Pilgrim and Cordery (1975) and Veneziano and Villani (1999). Pilgrim and Cordery (1975) and Veneziano and Villani (1999) categorize approaches to developing expected hyetographs. Each reference provides slightly different hyetograph categories, but despite differences these can be generalized as: 1) simple geometric shapes anchored to a single point on the rainfall intensity-duration frequency (IDF) curve, 2) use of the entire IDF curve, 3) standardized or statistical hyetograph profiles developed from rainfall records, 4) simulation using stochastic rainfall models, and 5) arbitrary temporal patterns. Another category is suggested by Haan and others (1994, p. 44); this is the adoption of an actual storm from the historical record in the vicinity of interest that has occurred and that is known to cause substantial flooding and damage. This category might be lumped into category 5. Category 4 is not desirable from a basic hydrologic design requirement of repeatability. The stochastic simulation requires sophisticated computational resources, considerable evaluation of model suitability, and high overhead for general use. The arbitrary temporal patterns (category 5) likely are not based on actual rainfall data. The historical rainfall approach is advantageous as being conceptually simple and easy to use. Haan and others (1994) report a distinct disadvantage of the historical 34 rainfall approach; the approach produces (uses) a storm with an unknown frequency of occurrence. Intensity-Duration Frequency (IDF) based Hyetograph Methods Use of the IDF curve as a foundation for hyetograph generation (categories 1 and 2) is very common. Most hydrologic textbooks (for example, Chow and others, 1988; Haan and others, 1994) dealing with rainfall-runoff design problems contain hyetograph procedures based in some fashion on the IDF curve. Keifer and Chu (1957) estimated design hyetographs based somewhat on the IDF curves, and their method is sometimes known as the Chicago Method. This method is more formulaic than the widely used more ad hoc IDF methods detailed in this section. The Soil Conservation Service (SCS, 1973), now the National Resources Conservation Service (NRCS), developed a hyetograph modelling approach very similar to the IDF curve method. The method provides dimensionless hyetographs classified into four types dependent upon specific regions of the United States. The Type II and III hyetographs are represented in Texas and elsewhere in the United States. The Type II hyetograph is applicable for most of Texas, is the most intense, and is in common use (Herrmann, written commun., 2002 and Stolpa, written commun., 2002). An example of the Type II hyetograph is shown on figure 16 (referenced out of sequence), and the Type III hyetograph is very similar in shape. The curves are essentially generalizations (by the author’s reading of SCS (1973) of the balanced storm techniques based on the now outdated rainfall depth-duration frequency (DDF) 35 values, equivalently IDF values, of Hershfield (1962). Frederick and others (1977) provide DDF values for very short storm durations (5–60 minutes) and compliment the 30 minutes to 24 hr storm durations of Hershfield (1962). An updated report of DDF for Texas is provided by Asquith (1998). The durations considered in the Texas report include those of Hershfield (1962) and Frederick and others (1977). There are several reasons for the widespread use of the IDF approach. First, it is assumed by many practitioners that frequency levels (recurrence intervals or return periods) can be assigned to synthetic hyetographs derived from IDF curves (IDF hyetographs). Second, these hyetographs are repeatable—they provide consistent (not necessarily accurate) results. Third, IDF hyetographs are known to represent reasonable (or at least so broadly accepted and used by the hydrologic and engineering community that “reasonable” is seldom questioned and the potential problems with the hyetograph are mitigated by other aspects of the design process) temporal storm patterns for a given frequency. Fourth, it is common to develop the IDF hyetograph in such a fashion so that the hyetograph produces a rainfall depth or intensity whose frequency is independent of the storm duration (Haan and others, 1994, p. 45). Because of the duration independence, it is presumed that these hyetographs are applicable across an entire range of watershed scales. A limitation of the IDF based approaches is that a physical storm must have an incredibly small chance of occurring exactly as the IDF hyetograph suggests. It is important to note that the IDF curve is constructed by the most intense bursts of 36 rainfall, which are preserved in the historical record as annual maximum intensities for a given duration, that occur within large storms. The IDF hyetograph is not based in any fashion on the temporal nature of real events. The IDF curve based hyetograph is hence a worst case scenario and does not represent an “expectation” in a statistical sense. Pilgrim and Cordery (1975, p. 81) observe that “design rainfall derived from frequency-duration data does not generally represent the rainfall in complete storms.” Further, it seems to the author and observed by Pilgrim and Cordery that an implicit assumption in IDF hyetograph justification and usage is that a hyetograph of a given frequency level produces a flood peak and volume of the same frequency level. Thus, a 3-hr 25-year hyetograph produces the 25-year flood on a watershed characterized by a 3-hr time scale. Testing of this assumption is outside the scope of the dissertation. Actual Rainfall Record based Hyetograph Methods The standardized or statistical hyetograph developed from actual rainfall records (category 3) is the last general technique of hyetograph specification. A broad range of analytical and statistical methods can be used. This category is attractive because the hyetographs become expectations of real data and entire storm durations, and the hyetographs are not derived from abstractions of the non-whole storm based IDF curve. Some important papers involving the analysis of actual rainfall records include: Huff (1967, 1990), Yen and Chow (1980), Chukwuma and Schwab (1983), Bonta and 37 Rao (1988a,b; 1989), Schaefer (1989), Parrett (1998), and references therein. The research by Huff is perhaps the most widely known and cited. Bonta and Rao (1989) and Huff (1990) cite Pani and Haragan (1981)—no other citations to Pani and Haragan are known. The Pani and Haragan paper provides dimensionless hyetographs for a small rain gage network in the southern High Plains of Texas. Because these hyetographs are based on Texas data, the hyetographs are directly comparable to the resulting hyetographs in this dissertation. Hyetograph Research by Huff Huff (1990), based largely on Huff (1967), presented dimensionless rainfall hyetographs as families of curves derived from storms that Huff classified as first-, second-, third-, and fourth-quartile. Huff defines a “storm” as a “rain period separated from preceding and succeeding rainfall by six hours or more” (Huff, 1967, p. 1007). Huff does not provide sensitivity analysis of the presumably arbitrarily chosen 6 hr definition. The quartile designation depends on whether the greatest percentage to total rainfall occurs in the first, second, third, or fourth quarter of the storm duration. Huff’s data base includes a total of 261 storms on a 400 mi 2 (1,036 km 2 ) network of 49 recording rain gages in east-central Illinois sampled between 1955 and 1966. Each storm had an areal mean rainfall of at least 0.5 in. (12.7 mm), a duration between 3 and 48 hr, and at least one rain gage within the area had to record at least 1 in. (25.4 mm) of precipitation. An interesting inconsistency is that the 1955–1966 period is said by 38 Huff of contain both 11 years of data (Huff, 1967, p. 1007) and 12 years of data (Huff, 1990, p. 3). The exact algorithm used by Huff for the storm classification is not provided by Huff, although a digital computer was used in the analysis. Huff (1967, p. 1008) does say that the classification depended “on whether the heaviest rainfall occurred in the first, second, third, or fourth quarter of the storm period.” Huff (1967, table 1) determined that the relative frequencies of the quartiles were 30, 36, 19, and 15 percent for the first, second, third, and fourth quartile, respectively. These relative frequencies change somewhat in Huff (1990) by considering differences between precipitation at a point and over an area. Huff concludes that short duration storms (less than 6 hr) were often associated with first-quartile storms, moderate duration storms (6–12 hr) were often associated with second-quartile storms. Third-quartile storms often had durations of 12–24 hr; whereas, fourth-quartile storms had durations greater than 24 hr. The median (50th percentile) dimensionless hyetograph for each quartile classification based on point rainfall values—that is rainfall data specifically for the recording device—is presented on figure 10. These curves were generated for this dissertation from a summary data table provided by Huff (1990, table 3). Huff also provides curves representing other percentiles ranging from 10 to 90 percent which envelop the median curve. From the figure, it is clear that each storm classification has a considerably different shape than its neighbors. This variation is inherently related to 39 factors (Huff, 1967, p. 1009) such as developmental stage of the storm, size and complexity of the storm system, rainfall type, synoptic storm type, location of the sampling points with respect to the storm center, and the movement of the storm system across the sampling region. Figure 10. Median dimensionless hyetographs at a point for first, second, third, and fourth quartile heavy rainfall storms derived from Huff (1990, table 3) The factors identified and described by Huff result in subtle differences when the dimensionless hyetographs are generated over finite areas instead of at points. Using summary data from Huff (1990, table 4), figure 11 shows the four median dimensionless hyetographs for areas ranging from 10 to 50 mi 2 (25.9 to 129 km 2 ). Comparison of corresponding curves on figures 10 and 11 will show that the difference between the curves generally is small. Huff concludes that the point hyetographs also are valid for areas but that the validity of the curves diminishes as area increases. PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 F O U R T H - Q U A R T I L E T H I R D - Q U A R T I L E S E C O N D - Q U A R T I L E F I R S T - Q U A R T I L E 40 Figure 11. Median dimensionless hyetograph on areas of 10 to 50 square miles for first, second, third, and fourth quartile heavy rainfall storms derived from Huff (1990, table 4) Huff (1967) provides considerable discussion of analysis of rainfall bursts (see figure 1 of this dissertation for a graphical description of rainfall bursts). Huff (1967) investigates the relations between rain type, storm type, and storm shape and orientation on the temporal distribution of rainfall. Huff (1990) concludes for hydrologic design applications that first-quartile hyetographs should be used for design time scales of about 6 hr or less and second- quartile hyetographs should be used for time scales of about 6–12 hr. No direct recommendation for third- or fourth-quartile hyetographs for design applications appears to have been made. PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 F O U R T H - Q U A R T I L E T H I R D - Q U A R T I L E S E C O N D - Q U A R T I L E F I R S T - Q U A R T I L E 41 Hyetograph Research by Pani and Haragan The hyetograph research by Pani and Haragan (1981), although not widely known, appears to be the only hyetograph analysis done explicitly on Texas rainfall data. Their analysis followed Huff (1967), and they compare their results favorably to those of Huff. Details of the Huff approach including clarification of terminology used here are provided in the previous section. Pani and Haragan classified storms into four quartile categories depending upon which quarter of dimensionless storm time had the greatest change in storm depth. They analyzed 117 storms that occurred between May 15 through July 31 for a 3-year period (1978–80) over the High Plains Cooperative Program (Texas HIPLEX) rain gage network (Texas Department of Water Resources, 1980). The network was located in southern High Plains of Texas near the town of Big Spring, Texas and covered about 2,600 mi 2 (6,744 km 2 , reported by Pani and Haragan) The data consisted of 15-minute values. The mid May to July period was selected because the authors were interested in the hyetographs of convective-generated precipitation. The authors defined a storm as a “rain period of at least 0.75 hour duration separated by at least 1 hour” (Pani and Haragan, 1981, p. 77). Interestingly, Pani and Haragan appear to not have identified a minimum storm depth considered for analysis, but Bonta and Rao (1989, table 1) report that Pani and Haragan (1981) used about a 0.75 in. (a value of 19.3 mm was reported by Bonta and Rao) minimum depth. 42 Pani and Haragan (1981, table 1) determined that the relative frequencies of the quartiles were 13, 41, 32, and 14 percent for the first, second, third, and fourth quartile, respectively. These frequencies indicate that most events (73 percent) are characterized as second and third quartile. This observation differs from Huff (1967, 1990) who found that most events (66 percent) are characterized as first and second quartile. The authors used the χ 2 -test (Davis, 1986, pp. 80–86) to determine whether their hyetographs are similar to Huff’s. The test showed significant differences in the relative frequencies (Pani and Haragan, 1981, p. 78). Pani and Haragan recognize the differences in relative frequencies and elaborate that the differences can be attributed to advecting storm systems moving across a network that was six times larger in areal extent than the Illinois network used by Huff. The authors conclude that “if a storm’s areal extent was larger than the network, which is often the case in Illinois, the resultant temporal distribution would show more rain falling duration the early portion of the network lifetime and produce a classification in the first quartile.” Hence as a network area increases, storms will increasingly be characterized as central peaking. Huff (1967) reached a similar conclusion. The proceeding conclusion is critical because of the relevance to the research presented here. The watershed rainfall depths (the c36c38c38c56c48c66c58c40c44c42c43c55c40c39c66c51c53c40c38c44c51 field) for each storm contained in the data base, which were derived from one to as many as six rain gages, is assumed representative of watershed drainage areas ranging from 0.26 to 43 166 mi 2 (0.673 to 430 km 2 ) (see tables 1–5). Most watersheds are less than about 20 mi 2 (51.8 km 2 ), and only two or three rain gages were in operation. Hence, the “network scale” of the data base is very small relative to the scales of the network used by Pani and Haragan and the Illinois network used by Huff. Following this discussion, it is expected then that the analysis presented later in this dissertation will favor first and second quartile storm types. Pani and Haragan (1981, figs. 3 and 4) provide median dimensionless hyetographs for only second and third quartile storms. First and fourth quartile hyetographs are not provided because of limited sample sizes (Pani and Haragan, 1981, p. 78). For this dissertation the median hyetographs were graphically extracted from the figures of Pani and Haragan and are reproduced on figure 12. Unlike Huff (1990), Pani and Haragan do not provide tables of the coordinate values for their figures. Pani and Haragan also provide the 10 and 90 percentile hyetographs to illustrate uncertainty. The authors used the χ 2 -test at the 0.1 percent (0.001) significance level to test whether their hyetographs are similar to Huff’s. The test showed no significant differences (Pani and Haragan, 1981, p. 79). Visually the third quartile storms have the most potential differences—compare the third quartile hyetographs on figures 11 and 12. Therefore, although the relative frequencies of quartile storm types between Illinois and Texas are statistically different, the resultant hyetographs from each apparently are not. 44 Figure 12. Median dimensionless hyetograph for second and third quartile storms for the southern High Plains of Texas derived from Pani and Haragan (1981, figs. 3 and 4) Because the third quartile storm exhibits its greatest rate near 55 percent of the duration (see figure 12), Pani and Haragan decided that the third quartile storm was close enough to a second-quartile classification that for application purposes all second and third quartile events should be combined (Pani and Haragan, 1981, fig. 5). The median composite hyetograph along with the 10 and 90 percentile curves are shown on figure 13. The hyetograph coordinates again were graphically extracted the Pani and Haragan figure. PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 T H I R D - Q U A R T I L E S E C O N D - Q U A R T I L E 45 Figure 13. Median, 10-, and 90-percentile dimensionless hyetographs for the southern High Plains of Texas derived from Pani and Haragan (1981, fig. 5) The coordinates used to generate figures 12 and 13 are shown in table 6. These are provided in facilitate the work of other researchers. The coordinates are rounded to the nearest quarter interval. As a final remark about the Pani and Haragan study is that the authors did not consider the influences of storm duration on the hyetograph. This is probably not an oversight but partly attributable limited sample sizes. PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 50 p er c e n t ile 10 p e rc e n t ile 90 p e rc e n t ile 46 Table 6. Median, 10-, and 90-percentile dimensionless hyetograph coordinates for the southern High Plains of Texas derived from Pani and Haragan (1981) [Note: Table entries manually extracted from figure by Pani and Haragan (1981, fig. 5) and rounded to the nearest quarter interval.] Hyetograph Research by Schaefer and Parrett Schaefer (1989) and Parrett (1998) used similar approaches for the regional analysis of hyetographs for Washington State and Montana, respectively. Parrett (1998) largely was based on the same methodology as Schaefer. The work of Parrett is described here. Parrett analyzed 188 large storms from 87 National Weather Service (NWS) recording rain gages in and proximate to Montana. A storm was considered for Storm duration 10th percentile dimensionless hyetograph Median (50th percentile) dimensionless hyetograph 90th percentile dimensionless hyetograph (percent) (percent) (percent) (percent) 0000 5 0 1.25 3.5 10 0 2.75 6.75 15 .75 5.5 12.75 20 1.5 9.25 19.5 25 3 14.5 28.75 30 5 21.5 40 35 7.75 30 52.75 40 11.25 38.5 63.25 45 15.75 47 74.5 50 22.5 56 82.5 55 29.5 65 88 60 39 74 91.5 65 50 81.5 94.5 70 64.5 87 96.75 75 74.5 92 97.75 80 82 95 98.5 85 88 97.5 99.25 90 92.25 99 99.75 95 96.25 99.5 100 100 100 100 100 47 analysis if the annual exceedance probability of the total storm depth was about 0.10 or less (recurrence intervals greater than 10). The duration categories used were 2, 6, and 24 hr; the core NWS data types for each category were 5 minute, 15 minute, and 1 hr, respectively. Too few 5-minute NWS rain gages existed for analysis. A multiplier of three was used on the duration for purposes of cataloging by duration (Parrett, 1998, p. 7). For example, storms in the 2, 6, and 24 hr category could have durations as long as 6, 18, and 72 hr. The storms were then cataloged into three-geographic generally-homogeneous regions of Montana (Regions 1, 2, and 3). Region 1 represents western mountains, region 2 represents the front range of the Rocky Mountains, and region 3 includes the eastern plains of the state. Seasonal analysis of the relative frequencies of storms on a monthly basis was performed. The “time-to-peak” intensity of each storm was measured. The graphic definition of the time-to-peak is indicated by variable “a” on figure 20 (referenced out of sequence) of this dissertation. The time-to-peak rainfall intensity influences peak discharge of the runoff process. Further, Parrett determined sequencing patterns of the three adjacent largest intensity incremental storm depths. This was an important step because the sequencing of the high-intensity portions of the event also are believed to influence peak discharge. Following the lead of Schaefer (1989), Parrett (1998) divided each storm into equal thirds that were termed “trisectors” so that general temporal patterns of rainfall could be classified into “macro-patterns” (Parrett, 1998, fig. 3). The macro-patterns 48 reflect the relative total storm depth in each trisector and whether or not rainfall was continuous throughout the total storm duration. Twelve macro-patterns were considered and each of the 188 storms was assigned a pattern type. Macro-pattern 1 (a single front-loaded burst type event) was the most common pattern exhibited by the storms in all regions for the 2- and 6-hr durations for each region. Macro-pattern 3 (a multiple decreasing magnitude bursting event) was the most common for the western Mountains of Montana. The 2-hr duration storms were much more likely to have the macro-pattern 1 than either the 6- or 24-hr duration storms (Parrett, 1998, fig. 5). Each of the 188 storms considered by Parrett (1998) was converted to dimensionless depths by division of each incremental depth by the maximum depth for the duration of interest within the event. For example, each hourly depth for a storm lasting 72 hr (the 24-hr duration category) was divided by the maximum 24-hr storm depth of that storm. Parrett does not remove dimension from the time or horizontal axis of the hyetograph; “decimal hours passed” are favored. Finally, the incremental (now dimensionless) values are cumulated into a “depth-duration curve” (Parrett’s terminology). The depth-duration curve is analogous to the “dimensionless cumulative hyetograph” considered in the dissertation. The dimension removal method by Schaefer (1989) and Parrett (1998) defers considerably from the percentile conversion favored by Huff (1967, 1990) and Pani and Haragan (1981) and used in this dissertation. 49 Parrett subsequently fitted the four parameter Beta distribution (Benjamin and Cornell, 1970) to the dimensionless depth data each duration within the total storm duration while maintaining the regional and storm duration categories (Regions 1, 2, and 3; and Durations 2, 6, and 24 hr). The four-parameter Beta distribution—a less well known variant of the two-parameter Beta distribution that is familiar to many statistical analysts (Ross, 1994, pp. 235; Press and others, 1992, pp. 219–221; Wilks, 1995, pp. 95–97; Evans and others, 2000, pp. 34–42; and Karian and Dudewicz, 2000, pp. 79–81)—is a flexible distribution that can take on a wide variety of shapes and is useful for describing random variables having fixed lower and upper bounds. The Method of Moments was used for parameter estimation. The Beta distribution is only expressible as a probability density function and has no explicit cumulative distribution form or quantile function form. Numerical methods are required to use the Beta distribution. To formalize a method for hyetograph estimation after the Beta distribution was fitted, Parrett continues with complex and lengthy correlation and regression analysis so that smoothed depth-duration curves can be computed for a given region, duration, and probability level. Parrett concludes with three application examples to assist the reader in estimating the instantaneous synthetic (expected) hyetograph. Hyetograph Research by Others Yen and Chow (1980), with a focus on small drainage system design, presented a simple triangular instantaneous hyetograph shape for runoff computations with 50 drainage areas less than about 10 mi 2 (25.9 km 2 ). They used hourly precipitation data from the National Weather Service from rain gages in Boston, MA, Urbana, IL, and Asheville, N.C. The Method of Moments was used to determine their hyetograph parameters. Their triangular hyetograph model is shown on figure 14. Figure 14. Definition of a triangular instantaneous hyetograph model after Yen and Chow (1980) and Chow and others (1988) French (1983) following Yen and Chow (1980) also uses a statistical analyses using the Method of Moments to fit triangular hyetographs to the mean of the hyetograph distribution. French reports (1983, p. 6) that more complicated geometric shapes used to approximate the instantaneous hyetograph require more data and computational effort than the triangular hyetograph. French concludes that “the use of the first moment of the precipitation distribution and the assumption of a triangular shape is a reasonable compromise between accuracy and practicality.” Chukwuma and Schwab (1983) presented results from analysis of 43 years of rainfall records from the North Appalachian Experimental Watershed, Coshocton, R A IN F A L L I N T E N S IT Y , i TIME, t ab h D P = total precipitation P = 0.5 × D × h (area of triangle) D = duration r = advancement coefficient r = a / D h = 2 × P / D h = peak intensity a = time to peak = r × D b = recession time = D - a 51 Ohio (NAEW). Their records comprised 454 storms of duration from 11 to nearly 24 hr that produced more than 0.5 in. (12.7 mm) of rainfall, and a break-in-time of at least 1 hr between storm periods. Chukwuma and Schwab (1983) categorized storm events according to which third of the storm duration contained the most intense rainfall. Their general classification scheme is shown on figure 15. They subsequently performed statistical analysis on the storms within each classification to develop six type curves (not reproduced here). Figure 15. General instantaneous hyetograph patterns observed by Chukwama and Schwab (1983) I NCREASI NG RELA TI VE I N TENSI T Y OF RAI N F A L L INCREASING TIME Advanced Patterns Intermediate Patterns Delayed Patterns Type C 52 An important contribution of Chukwama and Schwab is that they examined the relation between sample size and the variance of the rainfall percentage within specific storm periods. They determined that a sample size of 50 storms should be sufficient to produce reasonable estimates of expected rainfall hyetographs. The authors also compared the SCS Type II hyetograph to their most comparable type curve (Type C). They report that the SCS Type II hyetograph has considerably more rainfall on a proportional basis in the central portion of the hyetograph and maximum 30-minute rates of rainfall for any of their six types would only amount to 40 percent or less than the maximum 30-minute rainfall rate of the SCS Type II hyetograph. This is an important observation and is consistent with the results presented in this dissertation. The peak intensities of observed hyetographs are smaller than those implied by the SCS Type II hyetograph. Example of the Balanced Storm Hyetograph Method for Austin, Texas The intensity-duration frequency (IDF) method for hyetograph construction, also known as the balanced storm method (Haan and others, 1994, pp. 45–46) or the alternating block method (Chow and others, 1988, p. 466), is based on the IDF curve. The method considers the IDF curve for a location such as might be derived from depth-duration frequency (DDF) values from Asquith (1998). A listing of DDF for Austin, Texas is listed in table 7. The values in the table can be converted to IDF values by dividing each depth by its corresponding duration. The IDF values are listed 53 in table 8. The 3-hr 25-year hyetograph can be computed from the shaded values in table 8. Table 7. Depth-duration frequency of precipitation for Austin, Texas [Note: Values in table derived from Asquith (1998). The location used to define the parameters of the precipitation distribution was Tom Miller Dam on the Colorado River near the center of the Austin area located at latitude 30°17’39” and longitude 97°47’12”. min., minutes; in., inches. One in. equals 25.4 millimeters (mm).] To construct the 3-hr 25-year hyetograph in table 9, a quarter hour time interval was chosen and the intensity values were linearly interpolated to durations not available in Asquith (1998). The linearly interpolated values are represented by bracketing parentheses. Subsequently, the ranked intensity values for each duration are converted to an accumulated depth [column 1 (duration) × column 3 (intensity)]. The accumulated depth is then converted to an incremental depth by differencing column 4. The incremental depths finally were rearranged in an alternating block pattern centered on the mid point of the duration to derive the hyetograph. Dimension on the Annual non- exceed- ance prob- ability Recur- rence inter- val Precipitation depth for indicated duration 15 min. 30 min. 1 hr 2 hr 3 hr 6 hr 12 hr 1 day 2 days 3 days (percent) (years) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) 0.500 2 0.98 1.32 1.72 2.16 2.32 2.67 3.06 3.44 3.81 4.04 .60 2.5 1.051.421.862.352.532.913.333.84 4.28 4.51 .700 3.33 1.14 1.54 2.04 2.58 2.79 3.19 3.64 4.33 4.84 5.08 .80 5 1.261.712.282.893.133.564.074.9 5.60 5.85 .900 10 1.47 1.98 2.68 3.42 3.71 4.21 4.81 6.10 6.88 7.14 .960 25 1.76 2.36 3.28 4.20 4.55 5.14 5.90 7.64 8.63 8.91 .980 50 2.01 2.68 3.79 4.88 5.28 5.94 6.86 8.87 10.0 10.3 .90 10 2.93.044.375.666.1 6.857.9610.2 1.5 1.8 .96 250 2.733.575.266.867.388.249.6712.0 13.613.9 .98 50 3.1 4.026.067.948.519.471.213.5 15.215.6 54 hyetograph was removed through division by the total storm depth. The computational steps are shown in table 9. Truncation of the dimensionless values for the 2.50 and 2.75 hr durations to 1.00 was made. Table 8. Intensity-duration frequency of precipitation for Austin, Texas [Note: Values in table derived from Asquith (1998). The location used to define the parameters of the precipitation distribution was Tom Miller Dam on the Colorado River near the center of the Austin area located at latitude 30°17’39” and longitude 97°47’12”. min., minutes; hr., hours; in./hr., inches per hour. One in. equals 25.4 millimeters (mm).] Annual non- exceed- ance prob- ability Recur- rence inter- val Precipitation intensity for indicated duration 15 min. 30 min. 1 hr 2 hr 3 hr 6 hr 12 hr 1 day 2 days 3 days (percent) (years) (in./hr.) (in./hr.) (in./hr.) (in./hr.) (in./hr.) (in./ h.r) (in./hr.) (in./hr.) (in./hr.) (in./hr.) 0.500 2 3.92 2.64 1.72 1.08 0.773 0.445 0.255 0.143 0.0794 0.0561 .600 2.5 4.20 2.84 1.86 1.18 .843 .485 .278 .160 .0892 .0626 .700 3.33 4.56 3.08 2.04 1.29 .930 .532 .303 .180 .101 .0706 .80 5 5.043.422.281.451.04 .593.39.208.17 .0812 .900 10 5.88 3.96 2.68 1.71 1.24 .702 .401 .254 .143 .0992 .960 25 7.04 4.72 3.28 2.10 1.52 .857 .492 .318 .180 .124 .980 50 8.04 5.36 3.79 2.44 1.76 .990 .572 .370 .209 .143 .90 10 9.166.084.372.832.041.14 .63.424.239 .164 .996 250 10.9 7.14 5.26 3.43 2.46 1.37 .806 .501 .283 .193 .998 500 12.4 8.04 6.06 3.97 2.84 1.58 .934 .564 .318 .216 55 Table 9. Hyetograph for 3-hr 25-year rainfall for Austin, Texas [Note: Rainfall intensity values in parentheses were estimated by linear interpolation. min., minutes; hr., hours, in., inches; -- dimensionless or not applicable. One in. equals 25.4 millimeters (mm).] The dimensionless hyetograph is shown on figure 16. The dimensionless hyetograph acquires the traditional nearly symmetrical S-shape of all IDF based hyetographs. The hyetograph rises slow to moderately from the origin, becomes quite steep at the midpoint of the duration, and then the ordinates flatten as the storm approaches completion. The peak intensity of the IDF hyetograph is arbitrarily shifted to the left of the storm midpoint by convention. A major facet of this research is to determine whether actual storms that are known to produce runoff would take on a shape such as this. The SCS Type II hyetograph (SCS, 1973) also was computed for Austin and is shown as the dashed line on figure 16 and the last two (right) columns of Duration Percent of duration Rainfall intensity Accum- ulated depth Incre- mental depth Alter- nating depth Hyeto- graph Dimen- sionless hyeto- graph Hyeto- graph based on SCS Type II Dimen- sionless hyeto- graph based on SCS Type II (hr) (in./hr.) (in.) (in.) (in.) (in.) (--) (in.) (--) 0.00 -- -- 0.00 0.00 0.00 0.00 0.000 0.000 0.000 0.25 0.0833 7.04 1.76 1.76 .06 .06 .0132 .115 .025 .50 .167 4.72 2.36 .60 .16 .22 .0482 .122 .052 .75 .250 (4.00) 3.00 .64 .28 .50 .110 .168 .089 1.00 .333 3.28 3.28 .28 .46 .96 .211 .199 .133 1.25 .417 (2.99) 3.74 .46 .64 1.60 .351 .795 .308 1.50 .50 (2.69) 4.04 .30 1.76 3.36 .737 2.109 .771 1.75 .583 (2.40) 4.20 .16 .60 3.96 .868 .374 .854 2.00 .667 2.10 4.20 .00 .30 4.26 .934 .176 .892 2.25 .750 (1.96) 4.41 .21 .21 4.47 .980 .176 .931 2.50 .833 (1.81) 4.53 .12 .12 4.59 1.00 .107 .955 2.75 .917 (1.67) 4.59 .06 .00 4.59 1.00 .107 .978 3.00 1.00 1.52 4.56 -.03 -.03 4.56 1.00 .099 1.00 Total 4.56 4.56 4.55 56 table 9. The two curves reasonably approximate each other, but such agreement does not constitute a validation because each technique is based on the same fundamental methodology. Figure 16. Dimensionless balanced storm and SCS Type II hyetographs for 3-hr 25-year rainfall for Austin, Texas derived from Asquith (1998) PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Balanced storm hyetograph from table 9 SCS Type II hyetograph 57 CHAPTER 3 TRIANGULAR MODEL OF DIMENSIONLESS RAINFALL HYETOGRAPHS KNOWN TO PRODUCE RUNOFF IN TEXAS Abstract An instantaneous hyetograph (hyetograph), is the temporal distribution of rainfall occurring over a point or area during a storm. Synthetic hyetographs are estimates of the expected time distribution for a design storm and principally are used in small- watershed hydraulic-structure design. Combining a hyetograph with a unit hydrograph provides the designer with a synthetic streamflow hydrograph. A data base of more than 1,600 observed cumulative hyetographs that produced runoff from selected small watersheds in parts of Texas provided estimates of parameters of a simple triangular- shaped hyetograph model. The model provides an estimate of the average or expected hyetograph in dimensionless form for storm durations of 0–24 hr and 24 hr and greater (up to about 3 days). The modeled hyetographs are formulated, graphed, and tabulated to facilitate use in design applications. In this study, the expected dimensionless hyetographs of 0–12 hr and 12–24 hr durations were similar and were combined with minimal information loss. Also, dimensionless hyetographs are independent of the frequency level or recurrence interval of total storm depth. The frequency independence should enhance the suitability of dimensionless hyetographs for design applications. Introduction An instantaneous hyetograph, or simply a hyetograph, is the temporal distribution of rainfall occurring over a point or area within a storm. Other forms of the hyetograph 58 are common; a hyetograph integrated with time produces a cumulative hyetograph. The hyetograph in its various forms has several applications. For example, in order to time distribute the depth of a design storm, an expected or synthetic (design) hyetograph for a particular location is often used by engineers and hydrologists during the design process of hydraulic structures such as culverts or runoff detention basins in small watersheds. A design storm is characterized by the depth of rainfall having a specified duration and recurrence interval (exceedance probability) predicated by the design criteria; for example, the 12-hr 100-year storm. When a hyetograph is convoluted with a unit hydrograph, a synthetic streamflow hydrograph is produced (Chow and others, 1988, chap. 7). A convolution example is provided in chapter 7. Purpose and Scope The purpose of this chapter is to estimate synthetic dimensionless cumulative hyetographs for storms known to produce significant runoff in small watersheds in Texas. The synthetic hyetographs are estimated using a simple triangular model of the instantaneous hyetograph. A dimensionless cumulative hyetograph has units of percent storm duration on the horizontal axis and percent storm depth on the vertical axis. Dimension is easily restored to the hyetograph through multiplication of the storm duration and depth with the percentages of the horizontal and vertical axes respectively. The triangular model is appealing over more complicated functions or geometrical shapes because of its simplicity and ease of application. 59 A specialized hyetograph data base described in the next section provides the basis for the analysis. Three ranges of storm duration considered for this study were 0–12 hr, 12–24 hr, and 24 hr and greater (up to about three days). The majority of hydrologic design applications that require synthetic hyetographs for small watersheds have time scales on the order of 24 hr or less, but some might require longer durations. Hyetographs defined for these duration ranges are useful to practitioners of small watershed hydraulic design. Data Sources A data base of cumulative hyetographs for storm events known to produce runoff from small watersheds was compiled during the execution of multiple on-going (as of 2002) research projects that are sponsored by the Texas Department of Transportation (project nos. 0–4193 and 0–4194) and the U.S. Geological Survey (USGS). The projects are being performed by investigators at Texas Tech University, Lamar University, University of Houston, and USGS. The fact that the hyetograph data represents storms known to produce runoff is important for applications involving rainfall-runoff relations and distinguishes the analysis presented here. At this time over 1,600 events for 91 USGS streamflow-gaging stations are available. The locations of the stations are shown on figure 17; the figure is identical to figure 2. 60 Figure 17. Map showing locations of USGS streamflow-gaging stations represented in the hyetograph data base Each event has two data files; one file contains a rainfall hyetograph and another file contains a streamflow hydrograph. The data are recorded as either variable time spaced (break-point data) or constant time-interval. In either case, the time increments range from five minutes to several hours. The stations are now mostly discontinued and no longer in operation. The stations were located in areas around the Texas cities of Austin, Dallas, Fort Worth, and San Antonio, and special small watershed study areas in rural regions of the central to north-central portions of Texas. Because the majority of the stations were not located in the eastern and western portions of Texas, C o l o r a d o R i v e r B a si n B ra z os R iv e rB a si n T r i n i t y R i v e r B a s i n San Antonio River Basin Dallas Fort Worth Austin San Antonio Waco 01020Miles Dallas Fort Worth Austin San Antonio Big Spring Waco Map Base Information Albers Projection Basins from Texas Water Development Board Cities from Texas Natural Resources Information System EXPLANATION Location of streamflow- gaging station 61 precipitation processes in east and west Texas are likely not well represented by the data base. The storm durations range from a few hours up to about five days and the total storm depths range from fractional inches to over 10 in. (254 mm). The rainfall data for each storm was published between the early 1960s and the middle 1980s in about 220 USGS reports. Several runoff-producing storms per year per station are documented in the reports. The reports list the incremental values (unit values) of concomitant cumulative rainfall depth and time. A citation list of the reports is listed in Appendix A. The rainfall depths contained in the data base are assumed representative across the drainage area of the streamflow-gaging station. The drainage areas ranged from 0.26 to 166 mi 2 (0.67 to 430 km 2 ), and most of the watersheds had drainage areas less than about 20 mi 2 (52 km 2 ). The “watershed rainfall depths” were derived from as little as one to as many as six rain gages simultaneously operated in or proximal to the watershed. For events in which a single rain gage was in operation (about 25 percent of data base), the watershed depth is a point value. For about 65 percent of the events, two or three rain gages were in operation for most watersheds so the majority of hyetographs are areal values. The spatial extent is limited and the number of gages small, so it is assumed that the areal depths represent an approximate point process. Thus for the analysis here, no distinction between the areal or point values is made. The spatial scale of the data base is limited and, importantly, on the same order of areal 62 scale in which rainfall-runoff design using hyetographs is performed. The scale similarity is important because the hyetographs are representative of watershed sizes for which engineers and hydrologists often perform design. Each cumulative hyetograph in the data base was expressed in a double- dimensionless fashion, which facilitates comparative analysis. The time increments were converted to percent of storm duration, and the rainfall depths were converted to percent of storm depth. The storm duration and depth generally were defined by the extent of the available data provided by the interpretation of previous analysts when the USGS reports were prepared. The converted hyetographs are dimensionless cumulative hyetographs. Previous Studies Pilgrim and Cordery (1975) and Veneziano and Villani (1999) categorize approaches to developing expected hyetographs. Each reference provides slightly different hyetograph categories. Despite the differences, the categories can be generalized as: 1) simple geometric shapes anchored to a single point on the rainfall intensity-duration frequency (IDF) curve, 2) use of the entire IDF curve, 3) standardized or statistical hyetograph profiles developed from rainfall records, 4) simulation using stochastic rainfall models, and 5) arbitrary temporal patterns including specific storms from the historical record (Haan and others, 1994, p. 44). Category 3 includes equation-based models (such as Yen and Chow, 1980; and French, 1983) and purely empirical definitions (such as Huff, 1967). Bonta and Rao (1988a) 63 compare hyetographs derived from some of the categories and are favorable to hyetographs defined by category 3 as exemplified by Huff (1967). Aron and Adl (1992) investigate the influences of hyetograph shape on runoff hydrographs; the hyetograph shapes considered were from categories 2 and 3. A rather unique hyetograph analysis modeling approach based on a so-called nonstationary Gauss- Markov model is provided by Cheng and others (2001); Cheng and others restricted their analysis to the rainfall associated with annual maximum events. An appealing aspect of the Gauss-Markov model is summarized by the last numerated benefit described by Cheng and others—the model produces irregular hyetograph shapes that resemble those of real rainfall hyetographs in contrast to regular functional shapes (such as the triangular model utilized here). Use of intensity-duration analysis, including the IDF curve, as a foundation for synthetic hyetograph generation is common. Most hydrologic textbooks (for example, Chow and others, 1988; Haan and others, 1994) that present rainfall-runoff design problems contain hyetograph estimation procedures based in some fashion on the IDF curve. Occasionally, text books also acknowledge other methods. Preul and Papadakis (1973) define a synthetic hyetograph for an urban watershed in Ohio using intensity- duration analysis on three rain gages; their approach followed one described by Keifer and Chu (1957). The Soil Conservation Service (SCS, 1973), now the National Resources Conservation Service, developed a hyetograph modelling approach very similar to the 64 IDF curve method. The method provides four types of dimensionless cumulative hyetographs. Each type is assigned to specific regions of the United States. The Type II and III hyetographs are represented in Texas. The Type II hyetograph is the most intense—predicts the greatest instantaneous rainfall rates—of the four types and is well known and in common use by engineers and hydrologists in Texas. The Type II is symmetric about the middle of the storm. The curves are generalizations of the balanced storm techniques based on the venerable rainfall depth-duration frequency (DDF) values, equivalently IDF values, of Hershfield (1962). An updated report of DDF for Texas is provided by Asquith (1998). There are several reasons for the widespread use of the IDF approach. First, it is believed that frequency levels (recurrence intervals or return periods) can be assigned to synthetic hyetographs derived from IDF curves (IDF hyetographs). Second, IDF hyetographs are repeatable—they provide consistency. Third, IDF hyetographs are known to represent reasonable (or at least are so broadly accepted and in common use that “reasonable” is seldom questioned and the potential problems with the IDF hyetograph are mitigated by other aspects of the design process) temporal storm patterns for a given frequency. Fourth, it is common to construct the IDF hyetograph so that the hyetograph produces a rainfall depth or intensity whose frequency is independent of the storm duration (Haan and others, 1994, p. 45). Because of the duration independence it is presumed that these hyetographs are applicable across the range of watershed scales represented by the data base. 65 It is important to note that the IDF curve is constructed from data derived from the most intense bursts of rainfall that occur within annual maximum storms; the bursts are preserved in the historical record as annual maximum intensities for a given duration. The IDF hyetograph is not based on the temporal nature of real events. The IDF hyetograph is hence a worst case scenario and does not represent an “expectation” in a statistical sense. Pilgrim and Cordery (1975, p. 81) observe that “design rainfall derived from frequency-duration data does not generally represent the rainfall in complete storms.” Pilgrim and Cordery remark that a hyetograph of a given frequency level produces a flood peak and volume of the same frequency level—this is an implicit assumption in IDF hyetograph justification and usage. The standardized or statistical hyetograph developed from actual rainfall records (category 3) is the last general category of hyetograph specification. A broad range of analytical and statistical methods can be used. For example, various equations or functions that mimic the hyetograph shape are available to the investigator, and several parameter estimation schemes, such as least squares or the Method of Moments, can be used. This category is attractive because the hyetographs become expectations of real data and entire storm durations are considered. The hyetographs are not derived from abstractions of the non-whole storm based IDF curve. It is a critical assumption in the IDF-based approaches that the storm structure can be modeled using fractions of storms that represent the DDF curves. 66 Some important papers involving the analysis of actual rainfall records include: Huff (1967, 1990), Drufuca and Rogers (1978), Yen and Chow (1980), Chukwuma and Schwab (1983), Bonta and Rao (1988a, 1988b), Schaefer (1989, 1993), Parrett (1998), and references therein. The research by Huff for the analysis of Illinois data is perhaps the most widely known and cited. In Bonta and Rao (1989) and Huff (1990) a reference to Pani and Haragan (1981) is made—no other citations are known to the authors. The Pani and Haragan paper provides dimensionless hyetographs for a specialized rain gage network in the southern High Plains of Texas. The research by Pani and Haragan (1981) appears to be the only hyetograph analysis done explicitly on Texas rainfall data. Their analysis followed the empirical approach used by Huff (1967), and they compared their results to those of Huff. Pani and Haragan classified storm hyetographs into four quartile categories depending on which quarter of dimensionless storm time had the greatest change in storm depth. They analyzed 117 storms that occurred between May 15 and July 31 for a 3-year period (1978–80) over the High Plains Cooperative Program (Texas HIPLEX) rain gage network (Texas Department of Water Resources, 1980). The network was located in the southern High Plains of Texas near the town of Big Spring, Texas and covered about 2,600 mi 2 (6,744 km 2 , reported by the authors). The data consisted of 15-minute values. The convective-precipitation dominated period of mid May to July was selected because Pani and Haragan were interested in the hyetographs of convective-generated precipitation. Pani and Haragan (1981, p. 77) defined a storm as 67 a “rain period of at least 0.75 hour duration separated by at least 1 hour.” Pani and Haragan do not appear to have identified a minimum storm depth considered for analysis, but Bonta and Rao (1989, table 1) report that Pani and Haragan (1981) used about 0.75 in. (a value of 19.3 mm was reported by Bonta and Rao) minimum depth. Pani and Haragan (1981) determined that the relative frequencies of the quartiles were 13, 41, 32, and 14 percent for the first, second, third, and fourth quartile, respectively. These frequencies indicate that most events (73 percent) are characterized as second and third quartile types. This observation differs from Huff (1967, 1990) who found that most events (66 percent) are characterized as first and second quartile. Pani and Haragan used the χ 2 -test to determine whether their hyetographs are similar to those of Huff (1967). The test showed significant differences in the relative frequencies between Pani and Haragan (1981, p. 78) and those of Huff (1967). Pani and Haragan (1981, p. 78) recognize the differences in relative frequencies and elaborate that the differences can be attributed to advecting storm systems of a network that was six times larger in areal extent than the Illinois network used by Huff. Pani and Haragan conclude that “if a storm’s areal extent was larger than the network, which [was] often the case in Illinois, the resultant temporal distribution would show more rain falling during the early portion of the network lifetime and produce a classification in the first quartile.” Hence storms will increasingly be characterized as central peaking—those having the largest portions of total depth near 68 the duration midpoint—as a network area increases relative to the typical areal extent of storms. Huff (1967) reached a similar conclusion. The preceeding discussion is important because of the relevance to the research presented here. As reported in the description of the data used, the spatial or “network scale” of the data base is very small relative to the scales of the network used by Pani and Haragan and the Illinois network used by Huff. Following this discussion, this analysis is expected to favor first to second quartile storm types because of the limited areal extents of typical small-watershed runoff-producing storms in Texas. Asquith (1999, figs. 17 and 18), in the analysis of areal-reduction factors for the one-day design storm in Texas, shows that the areal extent of storms having large depths is limited to a spatial scale of a few hundred square miles for areas around Austin, Dallas, and Houston, Texas. Pani and Haragan (1981, figs. 3 and 4), through empirical methods, provide median dimensionless hyetographs for only second and third quartile storms. First and fourth quartile hyetographs are not provided because of limited sample sizes (Pani and Haragan, 1981, p. 78). The ordinates of the hyetographs were graphically extracted from the figures of Pani and Haragan and are shown on figure 18. The quartiles of storm duration are emphasized by the vertical grid lines. Pani and Haragan also provide the 10th and 90th percentile hyetographs to illustrate uncertainty. Pani and Haragan used the χ 2 -test to determine whether their hyetographs were similar to those of Huff. The test showed no significant differences (Pani and Haragan, 1981, p. 79). 69 Therefore, although the relative frequencies of quartile storm types between Illinois and Texas are statistically different, the resultant dimensionless cumulative hyetographs from each apparently are not. Figure 18. Median dimensionless hyetograph for second and third quartile storms for the southern High Plains of Texas derived from Pani and Haragan (1981, figs. 3 and 4) The second quartile storm on figure 18 produces the greatest portion of depth near 40 percent; whereas the third quartile storm produces the greatest portion near 55 percent of the duration. Pani and Haragan decided that the ordinates of a third quartile storm were sufficiently close to a second-quartile classification that, for application purposes, all second and third quartile events should be “composited” (Pani and Haragan, 1981, fig. 5). The median composite hyetograph along with the 10th and 90th percentile curves are shown on figure 19. The hyetograph ordinates again were graphically extracted from the Pani and Haragan figure. PERCENT OF STORM DURATION PERCENT OF ST O R M DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 T H I R D - Q U A R T I L E S E C O N D - Q U A R T I L E 70 Figure 19. Median, 10-, and 90-percentile dimensionless hyetographs for the southern High Plains of Texas derived from Pani and Haragan (1981, fig. 5) The coordinates used to generate figures 18 and 19 are shown in table 10. These are provided to facilitate later comparisons to the results of this study. The coordinates are rounded to the nearest quarter interval. A final remark about the Pani and Haragan study is that the influences of storm duration on the hyetograph were not considered. The oversight might be attributed to limited sample sizes and computational resources at the time of the Pani and Haragan study. PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 50 t h p er c e n t ile 10 t h pe r c e n t ile 90 t h p e rc e n t ile 71 Table 10. Median, 10-, and 90-percentile dimensionless cumulative hyetograph coordinates for the southern High Plains of Texas derived from Pani and Haragan (1981) [Note: Table entries manually extracted from figure by Pani and Haragan (1981, fig. 5) and rounded to the nearest quarter interval.] Yen and Chow (1980), with a focus on small drainage system design, presented a simple triangular instantaneous hyetograph shape for runoff computations with drainage areas less than 10 mi 2 (25.9 km 2 ). They used hourly precipitation data from the National Weather Service from rain gages in Boston, MA, Urbana, IL, and Asheville, NC. The Method of Moments was used to determine their hyetograph parameters. Their triangular hyetograph model is shown on figure 20. Storm duration 10th percentile dimensionless hyetograph Median (50th percentile) dimensionless hyetograph 90th percentile dimensionless hyetograph (percent) (percent) (percent) (percent) 00 0 0 5 0 1.25 3.5 10 0 2.75 6.75 15 .75 5.5 12.75 20 1.5 9.25 19.5 25 3 14.5 28.75 30 5 21.5 40 35 7.75 30 52.75 40 11.25 38.5 63.25 45 15.75 47 74.5 50 22.5 56 82.5 55 29.5 65 88 60 39 74 91.5 65 50 81.5 94.5 70 64.5 87 96.75 75 74.5 92 97.75 80 82 95 98.5 85 88 97.5 99.25 90 92.25 99 99.75 95 96.25 99.5 100 100 100 100 100 72 Figure 20. Definition of a triangular instantaneous hyetograph model after Yen and Chow (1980) and Chow and others (1988) French (1983) also used a triangular based hyetograph model. Following Yen and Chow (1980), French used the Method of Moments to fit triangular hyetographs to the mean of the hyetograph distribution. In the Method of Moments, the theoretical moments of a distribution (such as the mean) are set equal to the moments of the data (such as the sample mean), and the parameters are solved to maintain the equality. The triangular hyetograph is a one parameter model and only the mean statistic is used for parameter estimation. French (1983, p. 6) reports that more complicated geometric shapes used to approximate the instantaneous hyetograph require more data and computational effort than the triangular hyetograph. French concludes (p. 6) that “the use of the first moment of the precipitation distribution and the assumption of a triangular shape is a reasonable compromise between accuracy and practicality.” R A IN F A L L IN T E N S I T Y , i TIME, t ab h D P = total precipitation P = 0.5 × D × h (area of triangle) D = duration r = advancement coefficient r = a / D h = 2 × P / D h = peak intensity a = time to peak = r × D b = recession time = D - a 73 Triangular Dimensionless Hyetograph Definition Similar to the geometry of the preceding triangular model, a triangular model based on the rainfall intensity (a quantile density) and fractional percent of elapsed time is shown on figure 21, and the Method of Moments can be used for parameter estimation. The fractional percent time can be interpreted as a nonexceedance probability for purposes of model derivation and subsequent parameter estimation. The model on figure 21 clearly is related to the model on figure 20. Figure 21. Definition of a triangular instantaneous hyetograph model in terms of quantile density The following definitions for the triangular quantile density on figure 21 apply. The distances and provide , thus (1) .(2 QU A N T I LE DENSI T Y , dQ FRACTIONAL PERCENT OF ELAPSED TIME, F ab h 10 0 T r i a n g l e 1 Triangle 2 Ω 1 Ω 2 ab ab+1= b 1 a–= 74 The area of the entire triangle is the total precipitation, which is unity for a dimensionless cumulative hyetograph, and is computed as the sum of the areas of two right triangles ( and ). .(3) It follows from eq. 3 that . The quantile function and fractional duration for the triangular instantaneous hyetograph can be computed by integration from left to right of the two right triangles. The first triangle provides a quantile function .(4) The second triangle represents a more complex geometry of the quantile function and an integration limit shift is required. The triangle provides a quantile function of ,(5) where and is a triangular integration. It follows that , so (6) .(7 Ω 1 Ω 2 Ω 1 Ω 2 + 1 2 --- ha 1 2 --- hb+1== h 2= QF() F Q 1 0 Fa≤≤() h a --- FFd 0 F ∫ 1 2 --- h a --- F 2 == Q 2 aF1≤<()Q 1 a() Q ˜ 2 0 F ˜ b≤<()+= F ˜ Fa–= Q ˜ 2 Q 2 aF1≤<()Q 1 a() h h b --- F ˜ –   F ˜ d 0 F ˜ ∫ += Q 2 aF1≤<() 1 2 --- ha hF ˜ 1 2 --- h b --- F ˜ 2 –+= 75 The mean of a quantile function is defined as ,(8) which for the triangular quantile function becomes ,(9) , and (10) .(1) Substitution of into eq. 11 yields . (12) Then substitution of into eq. 12 yields , (13) , and (14) . (15) Using eqs. 14 and 15, the parameters of a triangular model can be estimated when a value of the mean dimensionless hyetograph is available. (16) µ µ QF()Fd 0 1 ∫ = µ Q 1 F()Fd 0 a ∫ Q 2 F ˜ ()F ˜ d 0 b ∫ += µ 1 2 --- h a --- F 2 Fd 0 a ∫ 1 2 --- ha F ˜ d 0 b ∫ hF ˜ F ˜ d 0 b ∫ 1 2 --- h b --- F ˜ 2 F ˜ d 0 b ∫ –++= µ 1 2 --- h a --- 1 3 --- a 3 1 2 --- hab 1 2 --- hb 2 1 2 --- h b --- 1 3 --- b 3 –++= h 2= µ a 2 3 ----- ab b 2 b 2 3 -----–++= b 1 a–= µ 2 a– 3 ------------= a 23µ–= b 3µ 1–= 76 Estimation of Triangular Hyetograph Parameters Estimates of the triangular hyetograph parameters and are provided in this section. Analysis of the mean ordinate of the dimensionless hyetographs is needed. The analysis considers whether the mean ordinate is a function of the storm duration and the total storm depth for a given duration range. In other words, the question is whether a dependency between the temporal characteristics of the storm and the total storm depth exists for small watersheds. To examine this question, three duration ranges of 0–12 hr, 12–24 hr, and 24 hr and greater are considered and within each duration range storms having a sequence of depth intervals are analyzed. For example, storms having a depth range of 1.5–2.5 in. (38.1–63.5 mm), those with 2.5–3.5 in. (63.5–88.9 mm), and so on. For each depth interval, the average of the mean of the dimensionless cumulative hyetograph for each storm therefore is computed. For example, 290 storms had durations less than 12 hr and a depth range of 0.5–1.5 in. (12.7–38.1 mm), 290 values for the mean of the dimensionless cumulative hyetograph are computed, and the average of the 290 mean values is 0.579. For purposes of tabulation, the 0.5–1.5 in. range is cataloged as 1 in. (25.4 mm), 1.5–2.5 in. range is cataloged as 2 in. (50.8 mm), and so on. The results of the analysis are provided in table 11. The coefficients of variation and sample sizes (no. of storms) also are listed in the table. ab 77 Table 11. Statistical summary of dimensionless hyetograph averages for 0–12 hr and 12–24 hr storm durations [hr, hour; CV, coefficient of variation: standard deviation divided by mean; in., inches; na, not available; --, empty cell. One in. equals 25.4 millimeters (mm).] From the results listed in table 11 for the 0–12 hr duration storm, it is clear that no functional relation between storm depth and the average of the dimensionless cumulative hyetograph exists. Therefore, a bulk or weighted average of 59 percent is considered the most representative for the 0–12 hr duration. For the 12–24 hr storm duration, it is possible that there is a weak relation between storm depth and the average of the mean values. The averages monotonically increase from 56.4 for a 1 in. (25.4 mm) depth to 61.2 for a 4 in. (101.6 mm) depth; however, given the large coefficients of variation and fact that the average for greater than 4 in. Storm depth category Range of depth represented by category 0 to 12 hr durations 12 to 24 hr durations Average of the mean for the dimen- sionless hyetograph CV of the mean for the dimen- sionless hyetograph No. of storms (sample size) Average of the mean for the dimen- sionless hyetograph CV of the mean for the dimen- sionless hyetograph No. of storms (sample size) (in.) (in.) (percent) ( ) ( ) (percent) ( ) ( ) 1 0.5–1.5 57.9 0.340 290 56.4 0.444 68 2 1.5–2.5 61.0 .302 253 57.8 .386 180 3 2.5–3.5 58.5 .309 108 58.0 .316 119 4 3.5–4.5 58.8 .279 41 61.2 .292 80 5 4.5–5.5 58.4 .248 11 59.3 .302 30 6 5.5–6.5 49.6 .411 3 70.7 .158 10 7 6.5–7.5 74.0 na 1 57.0 .362 2 8 7.5–8.5 79.8 na 1 38.2 na 1 9 8.5–9.5 22.9 na 1 48.3 na 1 10 9.5–10.5 na na 0 na na na Rounded weighted average and sample total 59 .317 709 59 .352 491 Parameter a of hyetograph .23 -- -- .23 -- -- Parameter b of hyetograph .77 -- -- .77 -- -- 78 (101.6 mm) depth decreases and then increases, it is concluded that a weighted average of 59 percent also is most representative for 12–24 hr storm durations. Because the two weighted averages, when rounded to the nearest two-significant digits, are the same, the averages for these two duration ranges can be combined into a single 0–24 hr value. The approximate equality in the weighted averages was unexpected. The weighted average of 59 percent produces and parameters of 0.23 and 0.77 respectively. The parameters also are listed in table 11. A similar analysis was performed for storm durations of 24 hr and greater. The results are listed in table 12. Based on the average of the mean (column 3 on table 3), the mean of the hyetograph distribution diminishes with increasing storm depth. This trend might be expected because the duration range is unbounded (duration can increase indefinitely). It is possible that longer duration storms (multiple days) are more likely to produce larger depths and have multiple bursts, which would increase the potential for significant rainfall to occur in later portions of the duration. The variation is large and the sample size is relatively small for the 0.5–1.5 in. and greater than 5 in. ranges. Therefore, it is concluded that, for purposes of design applications, a single weighted average should be used to represent the 24 hr or greater dimensionless cumulative hyetograph. The weighted average of 55 is preferred and produces and parameters of 0.35 and 0.65 respectively. The parameters also are listed in the table. ab a b 79 Table 12. Statistical summary of dimensionless hyetograph averages for 24 hr and greater storm duration [hr, hour; CV, coefficient of variation: standard deviation divided by mean; in., inches (1 in. = 25.4 mm); --, empty cell. One in. equals 25.4 millimeters (mm).] The ordinates of both the 0–24 hr and the 24 hr and greater duration dimensionless cumulative hyetographs are listed in table 13. The hyetographs in the table were computed by the triangular hyetograph models. This table facilitates application of the hyetographs. The equation pairs used to calculate each hyetograph are listed below. The pairs are derived by parameter substitution into eqs. 4 and 7. Storm durations between 0 and 24 hr , and (17) (18) Storm depth category Range of depth represented by category 24 hr and greater durations Average of the mean for the dimen- sionless hyetograph CV of the mean for the dimen- sionless hyetograph No. of storms (sample size) (in.) (in.) (percent) ( ) ( ) 1 0.5–1.5 65.0 0.369 23 2 1.5–2.5 56.8 .364 123 3 2.5–3.5 55.7 .380 131 4 3.5–4.5 55.8 .296 72 5 4.5–5.5 53.7 .355 43 6 5.5–6.5 50.6 .317 41 7 6.5–7.5 51.6 .362 12 8 7.5–8.5 50.2 .422 11 9 8.5–9.5 42.2 .527 4 10 9.5–10.5 57.0 .0368 2 Rounded average and total sample size 55 .355 462 Parameter a of hyetograph .35 -- -- Parameter b of hyetograph .65 -- -- Q 1 0 F 0.23≤≤()4.35F 2 = Q 2 0.23 F 1≤<()1.30F 2 –2.60F 0.299–+= 80 Storm durations of 24 hr and greater , and (19) (20) Table 13. Dimensionless runoff-producing cumulative hyetographs for 0–24 hr and 24 hr and greater storm durations computed by triangular hyetograph model [hr, hour] The dimensionless cumulative hyetographs for the 0–24 hr (heavy line) and 24 hr and greater (thin line) storm durations are illustrated on figure 22. The composite hyetograph by Pani and Haragan (1981) (dashed line) is shown for comparison. By inspection of figure 22, it is clear that the triangular model produces dimensionless cumulative hyetographs that are consistent in shape and satisfactorily close to the Storm duration Dimensionless hyetograph for 0–24 hr storm duration Dimensionless hyetograph for 24 hr and greater storm duration (percent) (percent) (percent) 0 0.00 0.00 51.9 .71 10 4.35 2.86 15 9.78 6.43 20 17.4 11.4 25 27.0 17.9 30 36.4 25.7 35 45.1 35.0 40 53.3 44.6 45 60.7 53.5 50 67.5 61.5 55 73.7 68.9 60 79.2 75.4 65 84.1 81.2 70 88.3 86.2 75 91.9 90.4 80 94.8 93.9 85 97.1 96.5 90 98.7 98.5 95 99.7 99.6 100 100 100 Q 1 0 F 0.35≤≤()2.86F 2 = Q 2 0.35 F 1≤<()1.54F 2 –3.08F 0.538–+= 81 empirically determined ordinate of the Pani and Haragan hyetograph. However, the 0– 24 hr and 24 hr and greater hyetographs peak earlier and are more front-loaded than the Pani and Haragan hyetograph, and each is a second-quartile storm. Figure 22. Dimensionless runoff-producing cumulative rainfall hyetograph for 0–24 hr and 24 hr and greater storm durations computed by triangular hyetograph models for Texas and composite hyetograph by Pani and Haragan (1981) Example Application The following is a brief example of the application of the 0–24 hr dimensionless cumulative hyetograph, which is shown on figure 22. Asquith (1998) provides depth- duration frequency values of precipitation in Texas. For a location coincident with Tom Miller Dam in Austin, Texas (latitude 30°17’39” and longitude 97°47’12”), the 50-year 12-hr design storm using procedures outlined by Asquith (1998) has a depth of about 6.9 in. (175.3 mm). Using the suggested 0–24 hr hyetograph, at 6 hr into the Dimensionless hyetograph for 0–24 hr storm durations Dimensionless hyetograph for 24 hr and greater storm durations PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Pani and Haragan recommended dimensionless hyetograph EXPLANATION 82 design event (50 percent of storm duration), the dimensionless hyetograph has an ordinate of about 67.5 percent, which provides a cumulative rainfall depth of (6.9×0.675)=4.66 in. (118.4 mm). The derivative of the quantile function at is the rainfall rate. The derivative at the point is located along the slope of the second right triangle on figure 16, so equals 2-(2/0.77)(0.5-0.23) = 1.30 percent of storm depth per storm duration percent. So the rainfall rate is 0.748 in./hr (1.30 × 6.9 in. / 12 hr) or 19.0 mm/hr at 50 percent of the storm duration. Chapter Conclusions Triangular hyetograph models were fit to the mean values for the dimensionless cumulative hyetographs from over 1,600 storms known to produce runoff on small watersheds in Texas. Although three storm durations of 0–12 hr, 12–24 hr, and 24 hr and greater originally were considered, the two sub-daily durations were combined into a single 0–24 hr duration classification. It is suggested that the two dimensionless cumulative hyetographs be considered in hydraulic design applications for small watersheds in Texas having drainage areas less than about 20 mi 2 (about 52 km 2 ). For each duration classification, the mean values for each of several categories of total depth of the event are analyzed (tables 11 and 12). It is concluded from available data for each duration range that there is no substantial relation between the mean and total depth. Hence, a single average of the dimensionless hyetograph distribution for F 0.5= dQ F 0.5= dQ h h b⁄()F ˜ – hhb⁄()Fa–()–== 83 the two duration ranges is suggested. Because no relation is observed between the mean of the hyetograph distribution and the storm magnitude as measured by the total storm depth and since storm depth is a function of the frequency level (recurrence interval or return period), a logical conclusion is that the hyetographs suggested here are independent of the frequency level. The frequency independence should enhance the suitability of the dimensionless hyetographs for design applications. Parameters of a triangular hyetograph model are estimated from the two averages and the resulting dimensionless hyetographs are provided (eqs. 17–20, fig. 22, table 13). The triangular model is mathematically straightforward and is believed to provide a practical means for estimating the design hyetograph. A comparison to a Texas-based dimensionless cumulative hyetograph from Pani and Haragan (1981) is shown on figure 22. The triangular model produces dimensionless cumulative hyetographs that are consistent in general shape to the Pani and Haragan hyetograph. However, the 0–24 and 24 hr and greater dimensionless hyetographs peak earlier and are more front-loaded than the Pani and Haragan hyetograph. The inconsistencies could be attributed to a large difference in the spatial scale of the rainfall data network, to meteorological differences between study areas, to limitations of the one-parameter triangular model, and other unidentified factors. 84 CHAPTER 4 SAMPLE L-MOMENT ESTIMATION USING PRIOR-PROBABILITY WEIGHTED MOMENTS Preface L-moment statistics (Hosking, 1990) are universally used, unless otherwise noted, in this dissertation in lieu of the well-known product or central moment statistics. In that light, several sections of this chapter provide the analytical and historical context of L-moments. The purpose of this chapter is to describe what modifications to the computational method for the L-moments of a sample are needed so that the L-moments of the observed hyetographs or dimensionless hyetographs are more accurately computed. This chapter also serves as verification that the custom computer program, c79c80c82c80c72c81c87c86c17c83c79 (Appendix E, referenced out of sequence), functioned properly. Introduction An alternative type of sample L-moment estimator based on what the author calls prior-Probability Weighted Moments (p-PWMs) is described and evaluated in this chapter. This estimator is relevant to the hyetograph research in the dissertation as it provides a method to estimate the L-moments. The p-PWMs are similar to the usual PWMs (Greenwood and others, 1979) except that the p-PWMs are capable of utilizing additional information provided by estimates of the cumulative probability of a given observation. The p-PWMs are only applicable to data satisfying two conditions. 85 The first condition is that, in addition to the observations themselves, an estimate of the cumulative probability (nonexceedance probability) or equivalently the percentile of each observation is available as part of the data collection or processing procedure. Computation of empirical cumulative probabilities after the data were collected, for example by plotting-position formula, is explicitly excluded in the first condition. The second and more restrictive condition is that the distribution of the nonexceedance probabilities also are known to be non-randomly distributed. The second condition is the principal motivation for the p-PWMs. Examples of data sets that can satisfy both conditions are data such as grain-size distributions (Mahler, personal commun., 2002), rainfall hyetographs in the TxDOT research data base (Thompson, personal commun., 2001), and streamflow hydrographs also in the TxDOT research data base (Thompson, personal commun., 2001). For comparison, the sample L-moments are computed in this chapter for some example data by unbiased, plotting-position, and p-PWM estimators. Limited simulation experiments presented in this chapter and in Appendix C (referenced out of sequence) suggest that the p-PWMs are unbiased and have smaller sampling variance than unbiased and plotting-position estimators for moderate to large samples when the simulation uses a uniform distribution of cumulative probability. When the simulation used a nonuniform distribution of cumulative probability, the p-PWMs substantially outperform the usual unbiased L-moment and plotting-position L-moment estimators 86 for most sample sizes. The simulations suggest that the p-PWMs might not perform as well for very small samples. Regardless, p-PWMs are an attractive means to extend L-moment application to previously incompatible data sets such as the rainfall hyetographs considered here. The simulations also show that custom computational software written by the author performs properly. Background Cumulative distribution analysis (CDA) is an important tool for an extensive range of scientific endeavors because it provides a framework to investigate distributional characteristics from finite samples. CDA commonly establishes a basis for subsequent exploratory data analysis (EDA). In some contexts CDA and EDA are synonymous terms. CDA generally is initiated by the statistical characterization of sample data sets. The characterization might include statistics such as the mean, the median, and others. Some of the other statistics, such as the inter-quartile range, are commonly used and others are not. Additional components of CDA might include tasks such as evaluating and selecting suitable parametric probability distributions, including the well-known Normal or Log-normal distributions, for distribution modeling, estimating distribution parameters, and statistical regionalization. Regionalization is a broad term but typical usage is summarized as the process of statistically transferring information about distributions of random variables from locations of data collection to locations lacking data. The ability to transfer the information is based on systematic distributional changes because of influences from location specific variables. 87 Within the past two decades two new branches of statistics for CDA have been developed as alternatives to the well known central or product moments and the widely used Method of Moments. These two new branches are the theories of Probability Weighted Moments (PWMs) and linear moments or L-moments. The theories principally were developed and utilized by the hydrologic research community specializing in extreme value or magnitude and frequency analysis (mainly flood and precipitation frequency applications). PWMs subsequently were followed by and provided a foundation for the theory of L-moments. L-moment statistics have remarkably changed, invigorated, and spawned much basic and applied research and study of probability distributions, extreme value analysis, and regional analysis of data from the environmental, physical, hydrologic, and other sciences. The theory of PWMs is comprehensively described by Hosking (1986); although PWMs were already being utilized (Greenwood and others, 1979; Landwehr and others, 1979a,b; Hosking and others, 1985; among others). The theory of L-moments is described in the unifying work of Hosking (1990) and followed in book form that focused on regional extreme value analysis by Hosking and Wallis (1997). Other books or chapters therein that contain L-moment and PWM description are Stedinger and others (1992), Hosking (1995), Gilchrist (2000), and Dingman (2002) L-moment statistics remain popular today among an established and growing cadre of investigators. PWM statistics generally are used as a means to facilitate 88 application of the L-moments. However, research into the PWM continues. Hosking (1995), Wang (1996a), and Zafirakou-Koulouris and others (1998) describe the so called “partial PWMs” for extension of L-moment theory to left and right censored data. Appendix B provides a substantial introduction to L-moment theory for readers who are unfamiliar with L-moments. In typical circumstances of CDA, data samples comprise ensembles of observations generated from the distribution of a real-valued random variable having cumulative probability . The cumulative probability is the random component, ranges from 0 to 1 by definition, and is uniformly distributed. The uniform distribution implies that each sampling of is equally likely as any other. This has considerable ramifications on the execution of CDA. A familiar case is the computation of the arithmetic mean, otherwise known as the sample expectation of , in which a weight factor on each observation is used. Hence, without additional information than the data values themselves, the incremental probability between the observations are assumed identical, and each observation is given a uniform weight equal of . The constant incremental probability assumption and usage in computation of the expectation conflicts with the second condition provided of the hyetograph data are considered in this dissertation. Finite sample estimators for L-moments and PWMs are well established but require an uniform distribution of assumption for their computation. These xF() X F F X n 1– n 1– F 89 estimators might not be applicable for data satisfying both of the conditions described. Fortunately, with minor modifications, the theory of PWMs is compatible with prior knowledge of and a non-random distribution of in a sample. A candidate PWM sample estimator is introduced here that will use the prior knowledge of . The new estimator is referred to as the prior-Probability-Weighted Moment (p-PWM) estimator. A new class of L-moment sample estimator also is presented based on the p-PWMs. Moments of a Distribution Distribution description is an important component of CDA and is conducted by the statistical summarization of sample observations of a random variable . Traditionally, the data are statistically summarized by the product moments of the data (arithmetic mean, standard deviation or variance, skew, and kurtosis). The mathematical definitions and the sampling properties of the product moments are widely known. The reader is referred to section “Moments of a Distribution” in Appendix B for a more detailed description of the moments, specifically, the product moments. The definitions provided are for reference in subsequent discussions in this chapter. The theoretical product moments of random variable are , (21.1) , (21.2) , and (21.3) FF F X X µ EX[]= σ 2 EXµ–() 2 []= µ 3 EXµ–() 3 []= 90 (21.4) where , , , and denote the theoretical mean, variance, third moment, and fourth moment, respectively. In general, the higher moments are . (22) The is the expectation operator and in terms of the probability density function is , (23) and in terms of the quantile function is . (24) Note that the term in eq. 24 is analogous to the incremental probability previously described for finite samples. A special treatment of the term plays a central role in the p-PWMs. The product moments also are defined for finite samples. The first sample product moment is the mean and is . (25) µ 4 EXµ–() 4 []= µσ 2 µ 3 µ 4 r 2≥ µ r EXµ–() r []= E .[] f x() EX r [] x r fx()xd ∞– ∞ ∫ = xF() EX r [] xF() r Fd 0 1 ∫ = dF dF m 1 n --- x j j 1= n ∑ = 91 The quantity is a minimum variance unbiased estimator (MVUE) of the theoretical mean . The higher product moments are . (26) are not unbiased because for less independent information (information content proportional to sample size) is available to compute a given statistic because the mean also required estimation from the sample. The bias is inversely proportional to sample size. Unbiased product moment estimators are presented in Appendix B. The important concept here is that each data value is given a weight of . This weight is another representation of the term in eq. 24. L-moments and Probability Weighted Moments of a Distribution The product moments are not satisfactory for many types of data sets, particularly those with large ranges, non-normal distributions, and a tendency to contain outliers (unexpectedly large or small values) because the biases and sampling variances are often so large as to render the statistics unattractive (Kirby, 1974; Wallis and others, 1974). Many data sets in the hydrologic sciences including floods, droughts, and extreme precipitation exhibit the above characteristics. Barnett and Lewis (1995) provide an excellent presentation and review of outliers in statistical data. The product moments are incompatible with distributions that can only be expressed in quantile m µ r 2≥ m r 1 n --- xm–() r j 1= n ∑ = m r r 2≥ m n 1– dF 92 form. Most of the common distributions (Normal, Beta) are expressible in at least a PDF form and sometimes a CDF form, and frequently no explicit quantile form exists. The quantiles of distributions that do not have explicit quantile expressions have to be solved by numerical integration. The L-moments provide an attractive theoretical framework because of some statistical considerations and their applicability to quantile functions. The L-moments have many well documented statistical advantages over the product moments. Specifically, L-moments are less sensitive to the presence of outliers in the data, exhibit less bias, are more accurate in small samples, and do not require logarithmic or other power transformations of the data. Transformations are traditionally used to reduce the skewness of the data and compensate for the shortcomings of the product moments by reducing the influence of large data values. Transformations, particularly logarithmic, are not always possible with data possessing zero or negative values. Furthermore, logarithmic transformations generally inflate the influence of small values, especially positive values considerably less than one. The L-moments also provide more secure inferences of distributional form than do the product moments. The L-moments specify a distribution even if some of the product moments do not exist (Hosking, 1990, p. 108), and Hosking also reports that an L-moment specification of a distribution is unique, which is not true of a product moment specification. 93 Some of the more important references (some repeated from earlier citation for list completeness) on L-moment theory and related statistics include: Greenwood and others (1979), David (1981), Hosking (1986, 1990, 1992, 1996), Hosking and Wallis (1993a,b), Hosking and Wallis (1995), Landwehr and others (1979a,b), Vogel and Fennessey (1993), Wang (1996a,b), and Zafirakou-Koulouris and others (1998). Only a limited number of books or chapters therein are available with descriptions of L-moments: Stedinger and others (1992), Hosking (1995), Hosking and Wallis (1997), Gilchrist (2000), and Dingman (2002). A comprehensive review of L-moments is provided in Appendix B. Some portions of Appendix B are repeated in this chapter in order to properly set the context of the p-PWMs. The primary concept is that the L-moments are exact analogs to the product moments in that features of a distribution such as the mean, variance, skew, kurtosis, and higher measures. The L-moment analogs have similar interpretations but do not acquire similar numerical values as the product moments. As in the product moment case, consider a real-valued random variable with a cumulative distribution function and a quantile function . As before, is a cumulative probability or nonexceedance probability and . If a random sample of size is drawn from the distribution of , and the sample is arranged in ascending order, the values become the order statistics of . X Fx() xF() F 0 F 1≤≤ nX X 1:n X 2:n … X n:n ≤≤≤ X 94 The expectation of an order statistic can be expressed as . (27) The order statistics for are theoretical random observations—no ties occur. In practice, real samples can contain ties. Hosking (written commun., 2002) reports that the presence of ties should not influence the accuracy of the order statistics and hence the L-moments when defined in terms of order statistic expectations. When the integral definitions of the L-moments of quantile functions (continuous functions, no ties are possible) are considered the situation changes in the presence of ties, but the topic has not been researched. The L-moments are the expectations of specific linear combinations of the order statistic expectations. In general, the L-moments are defined by . (28) The first four L-moments derived from eq. 28 are , (29.1) , (29.2) , and (29.3) EX j:r [] r! j 1–()! rj–()! ---------------------------------- xF()F j 1– 1 F–() rj– Fd 0 1 ∫ = X λ r 1 r --- 1– k r 1– k  EX rk:r– [] k 0= r 1– ∑ ≡ λ 1 EX 1:1 []= λ 2 1 2 --- EX 2:2 X 1:2 –[]= λ 3 1 3 --- EX 3:3 2X 2:3 – X 1:3 +[]= 95 . (29.4) The first L-moment is the mean ( ). The mean is the expected value of a single observation of . The second L-moment is a measure of the dispersion or spread of much like the usual standard deviation. The value is read as one half the expected difference between the two order statistics of a sample of size , and is referred to as L-scale or L-variation. The value is known as the coefficient of L-variation. The values of , , and when divided by become the L-moment ratios and measure skew ( or L-skew), kurtosis ( or L-kurtosis), and higher measures of shape ( or Tau5). Before further development, a segue into PWMs will be useful. The PWMs defined by Greenwood and others (1979) for quantile function and cumulative probability are . (30) The PWM is the mean. The higher PWMs and are not easily interpreted. However, the L-moments and PWMs can be expressed as linear combinations of each other. Because of linearity, procedures based on L-moments or the PWMs are equivalent. The PWMs actually predate the L-moments, but the L-moments usually are far more convenient and directly interpretable as measures of λ 4 1 4 --- EX 4:4 3X 3:4 –3X 2:4 X 1:4 –+[]= λ 1 µ X λ 2 X λ 2 n 2= τλ 2 λ 1 ⁄= λ 3 λ 4 λ 5 λ 2 τ 3 τ 4 τ 5 xF() F M prs,, ExF() p F r 1 F–() s []= M 100,, ExF()[]= r 0> s 0> 96 distributions. The PWMs thus are generally considered in recent times as a means to compute the L-moments. Particularly useful PWMs for L-moment theory are and . The author, following convention by other researchers, focuses here on , and remarks that and can be shown as linear combinations of each other and (31.1) . (31.2) Contrast with the usual product moment definition (eq. 24), which is repeated below . (32) The L-moments defined in linear terms of the PWMs are the quantities for , (33) where . (34) α r M 10r,, = β r M 1 r 0,, = β r α r β r α r xF()1 F–() r Fd 0 1 ∫ = β r xF()F r Fd 0 1 ∫ = β r EX r [] xF() r Fd 0 1 ∫ = λ r 1+ p rk, * β k k 0= r ∑ = r 01… n 1–,, ,= p r,k * 1– rk– r k  rk+ k  1– rk– rk+()! k!() 2 rk–()! ---------------------------------== 97 The first four L-moments are , (35.1) , (35.2) , and (35.3) , (35.4) or equivalently in terms of the quantile function as , (36.1) , (36.2) , and (36.3) . (36.4) Sample L-moments and Probability Weighted Moments L-moments and PWMs are defined for the quantile distribution function , but in general are estimated for finite samples of size by arranging the sample in ascending order to acquire the sample order statistics of random variable . Presently there are two classes of L-moment estimators for finite samples, the unbiased estimators and the plotting-position estimators. An additional class of estimator based on p-PWMs is proposed in a later section of this chapter. λ 1 β 0 = λ 2 2β 1 β 0 –= λ 3 6β 2 6β 1 – β 0 += λ 4 20β 3 30β 2 –12β 1 β 0 –+= λ 1 xF()Fd 0 1 ∫ = λ 2 xF()2F 1–()Fd 0 1 ∫ = λ 3 xF()6F 2 6F–1+ Fd 0 1 ∫ = λ 4 xF()20F 3 30F 2 –12F 1–+ Fd 0 1 ∫ = xF() n x 1:n x 2:n … x n:n ≤≤≤ X 98 Unbiased Estimators Unbiased estimates of , hence , can be made by for . (37) The “unbiased” weight factor on a specific is and is given by . (38) The multiplication to the left of the summation has been included in the unbiased weight factor to facilitate later analysis. The first four unbiased PWM estimators are , (39.1) , (39.2) , and (39.3) . (39.4) β r λ r b r 1 n --- n 1– r   1– j 1– r   x j:n jr1+= n ∑ = r 01… n 1–,, ,= x j:n w jr, w jr, j 1– r  n n 1– r   -------------------- = b 0 1 n --- x j:n j 1= n ∑ = b 1 1 n --- j 1– n 1– ------------ x j:n j 1= n ∑ = b 2 1 n --- j 1– n 1– -----------   j 2– n 2– ------------   x j:n j 1= n ∑ = b 3 1 n --- j 1– n 1– -----------   j 2– n 2– ------------   j 3– n 2– -----------   x j:n j 1= n ∑ = 99 Unbiased estimates of the first four L-moments in terms of the unbiased PWM estimates are , (40.1) , (40.2) , and (40.3) . (40.4) Hence, in general the unbiased L-moment and L-moment ratios are estimated as for , (41) , and (41.1) for . (41.2) A manual example computation of the unbiased L-moments for a sample is shown in Appendix B to demonstrate how the L-moments are computed without the aid of computer software. Plotting-Position Estimators Estimates of and also can be made with a second class of estimator called plotting-position estimators. A plotting position is a distribution free or nonparametric estimator of cumulative probability . Historically, plotting positions commonly have been used for graphical display of random samples of (Stedinger and others, l 1 b 0 = l 2 2b 1 b 0 –= l 3 6b 2 6b 1 – b 0 += l 4 20b 3 30b 2 –12b 1 b 0 –+= l r 1+ p rk, * b k k 0= r ∑ = r 01… n 1–,, ,= tl 2 l 1 ⁄= t r l r l 2 ⁄= r 3≥ β r λ r Fx j:n () X 100 1992, pp. 18.23–18.27), but they also can be used for parameter estimation (Gilchrist, 2000, chapter 9; Karian and Dudewicz, 2000, chapter 4). Often recognized reasonable choices include for , where is the plotting position for the th ascending order observation of a random sample of size . Hence, the PWMs are estimated as . (42) The plotting-position weight factor on a specific is and is given by . (43) The constant is included in the plotting-position weight factor to facilitate later comparisons. The plotting-position L-moments and L-moment ratios are estimated as , (44.1) , and (44.2) for . (44.3) p j:n j δ+()n ε+()⁄= δε 1–>> p j:n j n β ˜ r 1 n --- p j:n () r x j:n i 1= n ∑ = x j:n w ˜ jr, w ˜ jr, p j:n () r n ---------------= n 1– λ ˜ r 1+ p rk, * β r ˜ k 0= n ∑ = τ ˜ λ ˜ 2 λ ˜ 1 ⁄= τ ˜ r λ ˜ r λ ˜ 2 ⁄= r 3≥ 101 The unbiased L-moments generally are preferred over plotting-position estimators (Hosking and Wallis, 1997, pp. 31–34). In rare cases, plotting-position estimators can produce L-moments with theoretically impossible values such as or ; such occurrences are not possible with unbiased estimators. This is a very important consideration for algorithm development involving plotting-position PWM estimation and has ramifications of the p-PWM method described next. Confirmation that the bounds of the L-moments are satisfied is absolutely required. Prior-Probability Weighted Moments (p-PWMs) Thus far the PWMs and hence the L-moments for finite samples are computed only through weight factors ( or ) applied to an ordered sample. The ordered sample is derived from a random sample of . Because the sample is random, the weight factors are chosen because no prior or no additional information about specific observations of is known. This ignores the observation by the author that selection of a correct plotting-position formula arguably implies some sort of prior knowledge of the distribution generating the data. An interesting question to ask is how might the sample estimators of PWMs be effected by a priori knowledge of the cumulative probability of each ordered observation of in a sample? A follow up question is how are sample PWMs computed in situations in which this additional information is available? These questions are explored and a suitable method is developed in this section. The term “prior-Probability Weighted Moments” (p-PWMs) is used to λ 2 0<τ 3 1> w jr, w ˜ jr, X X X 102 describe how the method stresses that the cumulative probability of the sample observations is utilized in the computational process. It is useful to repeat the definition of the theoretical PWMs (eq. 31.2). . (45) How could be estimated for a sample in the situation in which the cumulative probabilities are available? If the values for were available, it is possible to evaluate with a finite difference approximation. Extension of the above integral to a sample results in (46) where is a finite difference estimator of the incremental cumulative probability for the th ascending ordered observation. The operator has a natural estimator of (47) or F β r xF()F r Fd 0 1 ∫ = β r FF dF β ˆ r x j:n F j () r dF j * j 1= n ∑ = dF j * j dF j * dF j * F j 1+ F j – 2 ---------------------- F j 0 j–+1= F j 1+ F j – 2 ---------------------- F j F j 1– – 2 ----------------------- 1 jn<<+ 1 F j – F j F j 1– – 2 ---------------------- j+ n=          = 103 . (48) Finally, the estimator must satisfy the condition of . (49) Equations 46 and 48 were first considered by the author in October 2001 stemming from communication with Thompson (personal commun., 2001). It should be pointed out that eq. 48 is similar to equations in Shen and Julien (1993, eqs. 12.1.3, 12.1.4) and an equation in Haan and others (1994, eq. 7.24). Although, Shen and Julien and Haan and others only describe a procedure for estimating the mean grain size of a sediment sample and not the higher product moments and not the L-moments. c36c3c79c68c87c72c3c81c82c87c72c3c11c39c72c70c17c3c21c19c19c21c12c3c85c72c74c68c85c71c76c81c74c3c87c75c72c3c83c16c51c58c48c3c80c82c80c72c81c87c3c72c86c87c76c80c68c87c82c85c3 c86c75c82c90c81c3c76c81c3c72c84c17c3c23c27c3c76c86c3c81c72c72c71c72c71c17c3c36c70c70c82c85c71c76c81c74c3c87c82c3c70c82c80c80c88c81c76c70c68c87c76c82c81c3c90c76c87c75c3 c43c82c86c78c76c81c74c3c11c21c19c19c21c12c3c68c81c3c68c79c87c72c85c81c68c87c76c89c72c3c72c86c87c76c80c68c87c82c85c3c70c68c81c3c69c72c3c71c72c89c72c79c82c83c72c71c17c3c55c75c76c86c3 c72c86c87c76c80c68c87c82c85c3c68c71c77c88c86c87c86c3c87c75c72c3c71c68c87c68c3c68c81c71c3c81c82c87c3c87c75c72c3c83c85c82c69c68c69c76c79c76c87c92c3c82c73c3c87c75c72c3c71c68c87c68c17c3 c43c82c86c78c76c81c74c3c86c88c74c74c72c86c87c86 c73c82c85c3 c3c68c81c71c3 c15c3c68c81c71c3c87c75c72c81c3c71c72c73c76c81c72c3c87c75c72c3c86c68c80c83c79c72c3 c72c86c87c76c80c68c87c82c85 c17 dF j * F j 1+ F j + 2 ----------------------- j 1= F j 1+ F j 1– – 2 ------------------------------ 1 jn<< 1 F j F j 1– + 2 ----------------------- j– n=          = dF j * dF j * j 1= n ∑ 1= x ˜ F() x j FF j – F j 1+ F j – ---------------------- x j 1+ x j –()+= F j FF j 1+ << j 0 … n,,= B ˜ r x ˜ F()F r Fd 0 1 ∫ = 104 c43c82c86c78c76c81c74c3c68c79c86c82c3c86c87c68c87c72c86c3c180c47c16c80c82c80c72c81c87c86c3c82c69c87c68c76c81c72c71c3c69c92c3c87c75c76c86c3c83c85c82c70c72c71c88c85c72c3c68c85c72c3 c87c75c72c3c83c82c83c88c79c68c87c76c82c81c3c47c16c80c82c80c72c81c87c86c3c82c73c3c68c81c3c68c70c87c88c68c79c3c83c85c82c69c68c69c76c79c76c87c92c3c71c76c86c87c85c76c69c88c87c76c82c81c3 c11c87c75c72c3c71c76c86c87c85c76c69c88c87c76c82c81c3c90c75c82c86c72c3c84c88c68c81c87c76c79c72c3c73c88c81c70c87c76c82c81c3c76c86c3 c12c15c3 c182c87c75c72c82c85c72c87c76c70c68c79c79c92c3c76c80c83c82c86c86c76c69c79c72c183c3c89c68c79c88c72c86c3c82c73c3c87c75c72c3c47c16c80c82c80c72c81c87c86c3c70c68c81c3c81c82c87c3 c82c70c70c88c85c17c181c3c41c88c85c87c75c72c85c15c3c43c82c86c78c76c81c74c3c85c72c83c82c85c87c86c3c87c75c68c87c3c87c75c72c3c76c71c72c68c3c82c73c3c72c86c87c76c80c68c87c76c81c74c3 c51c58c48c86c3c69c92c3c70c82c81c86c87c85c88c70c87c76c81c74c3c68c81c3c72c80c83c76c85c76c70c68c79c3c71c76c86c87c85c76c69c88c87c76c82c81c3c76c86c3c71c88c72c3c87c82c3c45c82c81c72c86c3 c11c20c28c27c24c12c3c76c81c3c68c81c3c76c81c87c72c85c81c68c79c3c85c72c83c82c85c87c17c3c41c76c81c68c79c79c92c15c3c87c75c72c3c68c79c87c72c85c81c68c87c76c89c72c3 c72c86c87c76c80c68c87c82c85c3c90c68c86c3c81c82c87c3c88c87c76c79c76c93c72c71c3c73c82c85c3c87c75c72c3c85c72c86c72c68c85c70c75c3c83c85c72c86c72c81c87c72c71c3c76c81c3c87c75c76c86c3 c71c76c86c86c72c85c87c68c87c76c82c81c17c3c54c76c81c70c72c3c87c75c72c82c85c72c87c76c70c68c79c79c92c3c76c80c83c82c86c86c76c69c79c72c3c47c16c80c82c80c72c81c87c86c3c90c72c85c72c3c81c82c87c3 c88c86c72c71c3c68c81c71c3c87c75c72c3c86c76c80c88c79c68c87c76c82c81c86c3c76c81c3c87c75c76c86c3c70c75c68c83c87c72c85c3c76c81c71c76c70c68c87c72c3c87c75c68c87c3c87c75c72c3 c68c88c87c75c82c85c183c86c3c76c80c83c79c72c80c72c81c87c68c87c76c82c81c3c82c73c3c87c75c72c3c72c84c17c3c23c27c3c72c86c87c76c80c68c87c82c85c15c3c87c75c72c3c85c72c86c72c68c85c70c75c3 c85c72c86c88c79c87c86c3c68c85c72c3c86c87c68c87c76c86c87c76c70c68c79c79c92c3c71c72c73c72c81c86c76c69c79c72c17 The “prior probability” weight factor on a specific is and is given by . (50) Contrast this weight factor with the plotting-position weight factor. The term in the plotting-position weight factor is a constant approximation to the term of the PWM integral (eq. 45) that is based solely on the sample size. Each data point in the plotting-position weight factor represents a uniform fraction of the probability range. Whereas is not uniform in magnitude across the interval. It is difficult to compare the constant term of the unbiased weight factor (eq. 38) to the term of the PWM integral. The prior probability L-moments and L-moment ratios therefore are estimated as , (51.1) x ˜ F() x j:n w ˆ jr, w ˆ jr, F j () r dF j * = n 1– dF F 01,[] dF j * 01,[] n n 1– r   1– dF λ ˆ r 1+ p rk, * β ˆ r k 0= n ∑ = 105 , and (51.2) for . (51.3) An intuitive argument of the suitability of as an estimator of can be made. Because an arbitrary sample possessing pair-wise and values can have assuming arbitrary probability, so must the term. Furthermore, the plotting- position formula is an arbitrary estimator of , so the term is derivable from plotting-positions as well. Hence, the most surely measure the same characteristics of the distribution as and should take on an approximately equivalent value for sufficiently large . It follows by and equivalence that , , and should all be approximately equal. To test without loss of generality the argument that , , and are approximately equal by computing each for order and for a sample of size and computed by the plotting position formula . This plotting-position formula was selected because it has precedence for some studies utilizing L-moments. To illustrate, the plotting position probability is assumed equal to the prior probability and is used in p-PWM computation. In practice, this is not the intent of p-PWM usage. A short example of PWM and L-moment computation by τ ˆ λ ˆ 2 λ ˆ 1 ⁄= τ ˆ r λ ˆ r λ ˆ 2 ⁄= r 3≥ β ˆ r β r xF F dF FdF β ˆ r β ˜ r n β ˜ r b r b r β ˜ r β ˆ r b r β ˜ r β ˆ r r 0=1 n 3= Fp j:n j 0.35–()n⁄= 106 estimator class for a sample of size is listed in table 14. The weight factors of each estimator class for each order on each observation are provided in the table. Table 14. Probability Weighted Moment and L-moment weight factors by estimator class for first short example The PWMs for each estimator class for are computed as follows , , and . The PWMs for are computed as follows , , and . These PWMs produce the following first L-moments j 1 0.217 1 0.333 0.333 0.384 -- 0.0723 0.0832 2 .550 3 .333 .333 .333 0.167 .183 .183 3 .883 4 .333 .333 .284 .333 .294 .250 n 3= Fp j:n = x j:n w 0 w ˜ 0 w ˆ 0 w 1 w ˜ 1 w ˆ 1 r 0= b 0 1 3 --- 134++()2.67== β ˜ 0 1 3 --- 134++()2.67== β ˆ 0 0.384 1 1⋅⋅()0.333 3 1⋅⋅()0.284 4 1⋅⋅()+ + 2.52 r 1= b 1 0.167 3 0.333 4⋅+⋅()1.83== β ˜ 1 1 3 --- 1 0.217⋅ 3 0.550⋅ 4 0.883⋅++ 1.80 β ˆ 1 1 0.217 0.384⋅⋅()3 0.550 0.333⋅⋅()4 0.883 0.284⋅⋅()+ + 1.64== 107 , , and ; and the following second L-moments , , and . Thus for the example in table 14, the first two PWMs and L-moments are similar to each though they are not necessarily equal. Some readers will note with possible concern that there are two different estimates of , which is just the mean. First, note that the unbiased property for the estimate still remains intact as the sum of the weight factors is equal to unity along with the sums of the weight factors and . Second, note that the weights give more leverage to the first value and less to the third. The “prior probability mean” has a value of 2.52 which accordingly is less than the usual mean (2.67). The , , relations are expected if and only if the were already known. By l 1 b 0 2.67== λ ˜ 1 β ˜ 0 2.67== λ ˆ 1 β ˆ 0 2.52== l 2 2b 1 b 0 –1== λ ˜ 2 2β ˜ 1 β ˜ 0 –0.93== λ ˆ 2 2β ˆ 1 β ˆ 0 –0.76== β 0 β ˆ 0 w ˆ 0 w 0 w ˜ 0 w ˜ 0 λ ˆ 1 l 1 < λ ˆ 1 λ ˜ 1 <λ ˜ 1 l 1 = F 108 intuition, the prior-probability mean should be less than arithmetic mean because the first data point is less far into the left tail of the distribution than the third data point is from the right tail ( ). Necessarily a different choice in the plotting position estimator would yield different differences in and as would a different estimator of the term. To illustrate p-PWMs further, consider the following data set in table 15 containing the median, 10th percentile, and 90th percentile observations. The unbiased and plotting position estimators do not utilize this (median, 10th percentile, and 90th percentile). The weight factors have been computed. The PWMs and L-moments of the distribution that generated this data are summarized in table 16. The third PWM and the third L-moment ratio (L-skew) have been included. Table 15. Probability Weighted Moment and L-moment weight factors by estimator class for second short example From the PWM and L-moment estimates in table 16 it is clear that the p-PWMs produce comparable mean and L-scale values. The L-skew values do not appear comparable although the PWMs appear comparable. The unbiased L-skew is negative, which indicates that the smallest observation is further away from the second observation than the second is from the third observation. j 1 0.100 1 0.333 0.333 0.3 -- 0.0333 0.03 -- 0.00333 0.003 2 .500 3 .333 .333 .4 0.167 .167 .2 -- .0833 .100 3 .900 4 .333 .333 .3 .333 .30 .27 0.333 .27 .243 0.217 0– 1 0.883–> β ˜ r β ˆ r dF r 012,,= Fp j:n = x j:n w 0 w ˜ 0 w ˆ 0 w 1 w ˜ 1 w ˆ 1 w 2 w ˜ 2 w ˆ 2 r 2= 109 Table 16. Probability Weighted Moments and L-moments by estimator class for second short example p-PWM Suitable Data and Real-World Examples A basic question is what types of data sets are characterized by the availability of cumulative probabilities? Some data sets would be those in which the percentiles of are known through the sampling procedure. If a sampling procedure reports percentiles of each data point, these can be considered as estimates of cumulative probabilities. A grain-size distribution in which the percentages of total sample mass retained by sieves of a specific diameter or the cumulative depth of storm runoff from a watershed as a percentage of elapsed time could be considered as data having prior probability. For the grain size data, the probabilities are really the random variable as the sieve size is almost always established by the sampling procedure. For the storm rainfall (runoff) data, the rainfall (runoff) depth and percentage of elapsed time are Unbiased estimates Plotting- position estimates Prior probability estimates Probability Weighted Moments 2.67 2.67 2.70 1.83 1.73 1.71 1.33 1.33 1.28 L-moments 2.67 2.67 2.70 1.00 .80 .72 -.33 .27 .12 .37 .30 .27 -.33 .34 .125 β 0 β 1 β 2 λ 1 λ 2 λ 3 τ τ 3 X 110 determined by their concomitant association. Regardless, in either example, neither plotting-position or unbiased L-moment or PWM estimators appear applicable. A limitation of the p-PWMs estimators is that, depending upon the distribution of the prior probabilities, theoretically impossible L-moments values can be computed. Impossible L-moment values usually occur for small samples or wildly non-uniform probabilities. Theoretically impossible L-moment values are also possible from the plotting-position L-moment estimators. The weight factors for the two estimators (eq. 42 and 46) are similar. However, theoretically impossible L-moments are more likely with p-PWMs than with plotting positions because the probabilities used in the weight factors on the data values can attain an unrestricted range of values; whereas, the plotting position weight factors are constrained by the nature of the plotting-position formula. Comparison between sample moment computation type is made by considering the rainfall for the May 23, 1981 storm for 08156800 Shoal Creek at 12th Street, Austin, Texas. Both unbiased and p-PWM computations for the trimmed and untrimmed data are listed in table 17. This storm was considered in the first chapter. The dimensionless hyetograph for the storm is shown on figure 23. From the figure, the recorded points defining the hyetograph typically are not well distributed. If they were, there would be the expectation that the points would appear uniformly distributed along the horizontal axis. This is clearly not the case. This situation exists because the points were chosen by previous analysts in such a fashion 111 as to accurately represent the steeper portions of the hyetograph. Although not applicable to the hyetograph on figure 23, throughout the data base, steeper portions of the hyetograph commonly occur in the first half of the time period, and are typically defined by more points than straighter or more gradually changing portions. The straighter portions typically are on the trailing or recession end of the hydrograph. Further complicating matters is that this particular hyetograph has two distinct bursts. A prudent double-tail one-percent trimming of the hyetograph prior to expression in a percentage basis was performed to reduce the length of the leading and trailing tail. Specifically, the event is considered to start once one percent of more of the total depth occurred and to stop once 99 percent of the depth occurred. After the trimming was performed, the percentages of duration and total depth were recomputed. Figure 23. Two dimensionless hyetograph representations of May 23, 1981 storm for watershed of station 08156800 Shoal Creek at 12th Street, Austin, Texas PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 20 40 60 80 100 Accumulated weighted rainfall Accumulated weighted rainfall with a double-tail one-percent trimming Note the clustering of values near steeper portions of the curve. Note the big change in the tail because of the trimming. 112 For purposes of clarity, only comparisons between the moment estimation types for the trimmed data (triangles in fig. 23) will be made with an observation that the original (untrimmed) data (circles in fig. 23) yields similar conclusions. The symmetrical one-percent trimming of the data shortens the leading tail for this particular hyetograph of the hyetograph considerably (see arrow in fig. 23), and therefore, the trimmed data is more representative of the distribution of the hyetograph. Furthermore, the comparison will focus only on the L-moments as the linearity between PWMs and L-moment makes additional comparison of the PWMs unnecessary. Table 17. Application of prior-Probability Weighted Moments on an observed hyetograph expressed in percent duration and percent depth for May 23, 1981 storm for watershed of station 08156800 Shoal Creek at 12th Street, Austin, Texas [The moment type symbology is defined in the text. The trimmed data represents a double one-percent tail trimming of the values from the hyetograph prior to expression as a cumulative percent.] Moment type Untrimmed data Trimmed data Unbiased estimators Prior probability estimators Unbiased estimators Prior probability estimators 46.40 13.91 43.94 20.35 33.29 11.71 31.92 14.35 26.02 10.20 25.23 11.98 21.27 9.122 20.78 10.59 17.90 8.308 17.58 9.61 46.50 13.91 43.94 20.35 20.09 9.513 19.90 8.350 .4320 .6840 .4529 .4103 .1439 .5119 .1906 .7349 -.1146 .3081 -.1101 .5061 -.1057 .2715 -.1041 .2069 β 0 β 1 β 2 β 3 β 4 λ 1 λ 2 τ τ 3 τ 4 τ 5 113 The mean for the unbiased estimator having a value of 43.94 is considerably greater than the mean for the p-PWM estimator of 20.35. This is because only the p-PWMs “see” the long flat inter-burst portion of the curve. This flat portion represents approximately 70 percent of the nonexceedance probability range. The ordinate of the flat portion is approximately 12 percent. Since the data continues to increase from 12 percent to 100 percent of storm depth between the 80 and 100 percent of the storm duration, the fact that the mean is somewhat greater than 12 is acceptable. The unbiased mean estimate is entirely out of line with careful inspection of figure 23. The inherent variation in the data as represented by the values appears similar between the estimation methods. It is hard to visualize in the figure. Distribution skewness is easier to visualize in the figure. The skewness of the data as represented by is quite different between the estimation methods (0.1906 compared to 0.7349). The L-skew for the p-PWMs is much larger which indicates that the hyetograph as measured by p-PWMs is far more left tailed. This occurs because the second burst is so much bigger than the first burst and occurs relatively late in the event. The higher measures of distribution shape (L-kurtosis) and actually have opposite signs. It is very difficult to visualize these higher measures on figure 23. τ τ 3 τ 4 τ 5 114 p-PWM Sampling Properties By intuition and by example, the p-PWM estimators of the sample L-moments produce reasonable values. Intuitive reasoning and examples are necessary but not sufficient to justify p-PWM usage on qualifying data sets. Investigation of the sampling properties of p-PWM estimators such as bias and their relative efficiency compared to the unbiased estimators is required. An exhaustive investigation is difficult to perform. The statistical experiments described here were performed in Fall of 2001 on a streamflow hydrograph that was arbitrarily selected. The dimensionless hydrograph is for the May 11, 1965 storm on 08178000 Escondido Creek subwatershed #1 near Kenedy, Texas. This hydrograph is shown on figure 24. It likely is somewhat confusing to many readers why a hydrograph and not a hyetograph is considered and why the simulations reported on in this section used data for a hydrograph instead of data for a hyetograph. On an historical note, when application for the p-PWMs was first considered for this dissertation, application to streamflow hydrographs was chosen. The change in application was made because of a change in research interests by the author. Although the focus of the simulations is on runoff data and not specifically rainfall data, the generalization to hyetographs is should be self evident. 115 Figure 24. Dimensionless streamflow hydrograph for May 11, 1965 storm for station 08187000 Escondido Creek subwatershed #1 near Kenedy, Texas In order to assess the sampling properties of a statistical estimator, such as PWM or L-moment estimators, statistical experimentation or simulation can be used. Statistical simulation (also known as the Monte-Carlo method) is based on random number generators. Because of the desire to compute L-moment estimates, only results for the L-moment estimators previously described and not for the PWMs are reported. In simulation a parent distribution and parameters are chosen and various sample sizes are drawn from the distribution. The L-moments are computed for each simulated sample of a specific sample size. The differences between the simulated L-moments PERCENT TIME PASSED R UNOFF I N I NCHES 0 0.2 0.4 0.6 0.8 1 0.00 0.05 0.10 0.15 0.20 Accumulated runoff hydrograph Runoff hydrograph EXPLANATION Total duration is about three hours. 116 and the specified or true L-moments of the distribution also are computed. When a sufficient number of simulation runs are performed, the mean difference is determined. This mean is referred to as bias and is defined here as the simulated mean statistic minus the true value. An unbiased estimator will have a mean difference of zero. The variance of the differences commonly also is computed. This variance is known as the sampling variance. Relative efficiency (RE) between estimators is defined as the ratio of the sampling variances. RE is computed here as the variance of the unbiased estimator divided by either the plotting-position estimator or the p-PWM estimator. Values of RE greater than one indicate that the candidate estimator outperforms the unbiased L-moment estimator. The Kappa distribution (Hosking, 1994) was selected for simulation using the inverse transformation method (Ross, 1994, p. 445). This method is convenient with the Kappa distribution because the distribution is expressed as a quantile function. The Kappa distribution is described in the next section. Kappa Distribution The Kappa distribution is a four parameter distribution that is commonly used in L-moment studies requiring simulated or artificial data to assess the accuracy of statistical methods. Because it has four parameters, the distribution is capable of assuming a wide range of scales and shape combinations not possible with more common two or three parameter distributions such as the Normal or the Generalized 117 Extreme Value (GEV). The Kappa distribution quantile function has the following form . (52) The four parameters are (location), (scale), (shape 1), and (shape 2). There is no simple expression for the cumulative distribution function (CDF) or the probability density function (PDF) for the distribution. The quantile is subject to the following restrictions. For the upper bound of the restriction is and (53.1) . (53.2) For the lower bound of the restriction is , (54.1) , and (54.2) . (54.3) The L-moments of the Kappa distribution are defined if and , or if and and are , (55.1) , (55.2) , and (55.3) xF() ξ α κ --- 1 1 F h – h ---------------   κ –    += ξακ h Fx() f x() x x ξακ⁄ if κ 0>+≤ x ∞ if κ 0≤= x x ξα1 h κ– –()κ⁄ if h 0>+≥ x ξακ⁄ if h 0 and κ 0<≤+≥ x ∞– if h 0≤ and κ 0≥≥ h 0≥κ1–> h 0< 1 κ 1 h⁄–<<– λ 1 ξα1 g 1 –()κ⁄+= λ 2 α g 1 g 2 –()κ⁄= τ 3 g 1 3g 2 2g 3 –+–()g 1 g 2 –()⁄= 118 (55.4) where (56) and (57) is the Gamma function. The Kappa parameters in terms of the L-moments have no simple expressions. Equations 55.3 and 55.4 can be solved for and by Newton-Raphson iteration. Hosking (1996) describes a suitable algorithm. Kappa based Simulation Results Uniform Distribution of Nonexceedance Probability A logical starting point in the experimentation is to assess the p-PWM estimator using a uniform distribution of . A uniform distribution of is normally done in statistical simulation. The p-PWM L-moment estimates of 0.114, 0.333, -0.148, and 0.0476 for , , , and , respectively were used for experimentation. These τ 4 g– 1 6g 2 10g 3 –5g 4 ++()g 1 g 2 –()⁄= g r rΓ 1 κ+()Γrh⁄() h 1 κ+ Γ 1 κ rh⁄++ ------------------------------------------------- for h 0> rΓ 1 κ+()Γκ– rh⁄–() h–() 1 κ+ Γ 1 rh⁄–() ------------------------------------------------------- for h 0<        = Γ k() u k 1– e u– ud 0 ∞ ∫ = κ h FF λ 1 ττ 3 τ 4 119 L-moment values were derived from a representative cumulative streamflow hydrograph (fig. 24) for Escondido Creek. The L-moments above correspond to Kappa parameters of , , , and . Eight sample sizes, which ranged from 5 to 1,000, were chosen for experimentation. The number of simulations per estimator was 50,000. Hence, the number of simulation runs for was 10,000 and the number of simulation runs for was 50. The biases for the mean, L-scale, L-CV, L-skew, and L-kurtosis are listed in table 18. The bias in the mean is relatively small for all three estimators, but the p-PWM estimator has substantially larger bias for small sample sizes ( ). As sample size increases, the p-PWM estimator shows the least bias. A similar observation is made about the bias for L-scale. The p-PWM estimator again shows the largest bias for small samples sizes. For larger sample sizes, the biases of the three estimators are approximately equal, but the p-PWM estimator consistently has the least bias for . For all three estimators, the bias of L-CV decreases in absolute magnitude as sample size increases. The p-PWM estimator consistently has the least bias for . For L-skew, it appears that the unbiased and p-PWM estimators have similar bias with the edge going to the unbiased estimator. The plotting-position estimators possess considerably larger bias for , and exhibit very large bias for small ξ 0.0699= α 0.1439= κ 1.022= h 0.5045= n 5= n 1,000= n 20≤ n 200≥ n 100≥ n 500< 120 sample sizes. Finally, for L-kurtosis, it is apparent that the p-PWM estimator is very biased for , but the bias diminishes rapidly as sample size increases. Table 18. Comparison of sample biases for a simulated Kappa distribution using a uniform distribution of probability [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] From the simulation experiments, it is apparent that all three L-moment estimators are unbiased at sufficiently large sample sizes. Most importantly for the objectives of Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias n=5 UB -0.0002 -0.0001 0.0451 0.0127 0.0209 PP .0001 -.0008 .0524 .1769 .0424 XF .0021 -.0032 -.0359 -.0023 .1947 n=10 UB .0001 -.0002 .0179 .0052 .0100 PP -.0001 -.0004 .0165 .1161 .0350 XF .0011 -.0029 -.0278 -.0524 .0585 n=20 UB .0002 .0001 .0101 -.0021 .0035 PP .0002 -.0001 .0072 .0610 .0213 XF .0005 -.0012 -.0112 -.0287 -.0075 n=50 UB .0002 .0 .0031 -.0030 .0012 PP .0003 -.0001 .0017 .0227 .0100 XF .0002 -.0003 -.0030 -.0019 -.0162 n=100 UB -.0006 .0003 .0063 -.0002 .0001 PP -.0002 .0001 .0036 .0109 .0044 XF .0001 -.0001 -.0010 -.0005 -.0051 n=200 UB .0003 -.0001 -.0008 -.0012 .0022 PP -.0001 .0 .0010 .0084 .0028 XF .0 .0 -.0003 -.002 -.0022 n=500 UB .0003 -.0002 -.0019 .0003 -.0001 PP -.0004 .0001 .0029 .0038 .0020 XF .0 .0 -.0001 .0001 -.0001 n=1,000 UB .0002 -.0001 -.0013 -.0016 .0024 PP -.0004 .0002 .0030 -.0001 .0012 XF .0 .0 .0 .0 .0 n 5= 121 this dissertation, p-PWM estimators appear to be competitive against the unbiased estimators and the plotting-position estimators for sample sizes greater than 10. For very large sample sizes it appears as though the p-PWM estimators have the least bias. Although an estimator might be unbiased, unbiasedness alone should not justify its use, as the sampling variability of the estimator could be so large as to render it unusable against other more biased estimators possessing minimal variance. Often it is possible to remove the bias from an estimator if sufficient understanding of its sampling distribution is known. Hence, the sampling variability requires analysis. The sampling standard deviations for the simulation runs in table 18 are listed in table 19. It is cumbersome to directly compare standard deviations, so corresponding relative efficiencies to the unbiased estimator are listed in table 20. Table 19. Comparison of sample standard deviations for a simulated Kappa distribution using a uniform distribution of probability [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. The sample standard deviation (SD) is defined as the square root of the sampling variance of the estimator. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] Sample size Estimator type SD for Mean SD for L-scale SD for L-CV SD for L-skew SD for L-kurtosis n=5 UB 0.0299 0.0126 0.2640 0.2934 0.3681 PP .0303 .0093 3.036 .1477 .1305 XF .0135 .0134 .1185 .3977 .2843 n=10 UB .0210 .0082 .1276 .1590 .1408 PP .0215 .0070 .1185 .1165 .0905 XF .0057 .0068 .0627 .2497 .1897 n=20 UB .0151 .0056 .0851 .0994 .0760 PP .0146 .0051 .0786 .0836 .0649 XF .0022 .0029 .0274 .1370 .0979 122 Table 20. Comparison of relative efficiencies for a simulated Kappa distribution using a uniform distribution of probability [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Relative efficiency (RE) is defined as the variance of the unbiased estimator to the variance of the plotting position or prior-probability weighted moment estimator. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] n=50 UB .0095 .0034 .0521 .0554 .0418 PP .0093 .0032 .0485 .0549 .0469 XF .0006 .0007 .0075 .0330 .0438 n=100 UB .0065 .0024 .0359 .0358 .0275 PP .0069 .0024 .0370 .0383 .0399 XF .0002 .0002 .0026 .0087 .0234 n=200 UB .0049 .0015 .0248 .0305 .0195 PP .0049 .0016 .0254 .0286 .0316 XF .0001 .0001 .0009 .0035 .0114 n=500 UB .0030 .0011 .0161 .0195 .0148 PP .0029 .0010 .0154 .0164 .0322 XF <.0001 <.0001 .0003 .0005 .0006 n=1,000 UB .0021 .0008 .0118 .0122 .0086 PP .0018 .0006 .0087 .0129 .0340 XF <.0001 <.0001 <.0001 <.0001 <.0001 Sample size Estimator type RE for Mean RE for L-scale RE for L-CV RE for L-skew RE for L-kurtosis n=5 UB11111 PP .974 1.84 .0076 3.95 7.96 XF 4.91 .884 4.96 .544 1.68 n=10 UB11111 PP .954 1.37 1.16 3.47 2.42 XF 13.6 1.45 4.14 .405 .551 n=20 UB11111 PP 1.07 1.21 1.17 1.41 1.37 XF 47.1 3.73 9.65 .526 .603 n=50 UB11111 PP 1.04 1.13 1.15 1.02 .794 XF 250 23.6 48.3 2.82 .911 n=100 UB11111 PP .89 1 .94 1.01 .475 XF 1060 144 191 19.6 1.38 Sample size Estimator type SD for Mean SD for L-scale SD for L-CV SD for L-skew SD for L-kurtosis 123 The p-PWM estimators of the mean, L-scale, and L-CV are more efficient than the unbiased estimators and the plotting-position estimators by the fact that the REs are often very much greater than one. However, for small sample sizes, plotting-position estimators have greater efficiency for L-skew and L-kurtosis. By or so, the p-PWM estimators are more efficient for these two statistics as well. In general the p-PWM estimator appears to be more efficient than the other two estimator types. The greater efficiency for the p-PWM estimator is attributed to its incorporation of more information of the simulated variable by using the cumulative probability. Obviously, the uniform simulations reported here represent a single distribution form and parametric specification. So it is uncertain how the bias and RE of the p-PWM L-moment estimator would change in different experiments. Nonuniform Distribution of Nonexceedance Probability The p-PWMs are suggested for application on qualifying data sets. As a reminder, a qualifying data set has both the cumulative probabilities of the observations known and knowledge or reasonable expectation that the distribution of the cumulative n=200 UB11111 PP 1 .879 .953 114 .381 XF 2400 225 759 7590 2.93 n=500 UB11111 PP 1.07 1.21 1.09 1.41 .211 XF >2400 >225 2880 1521 608 n=1,000 UB11111 PP 1.36 1.77 1.84 .894 .064 XF >2400 >225 >2880 >1521 >608 Sample size Estimator type RE for Mean RE for L-scale RE for L-CV RE for L-skew RE for L-kurtosis n 50= 124 probabilities is not uniform. Simulation experiments were conducted with a nonuniform distribution of . There is an infinite variety of nonuniform distributions that could be constructed but as a starting experiment it was decided to redraw a value for if the initially generated was . As a result, three-quarters of the time and only one quarter of the time is in the generation of a single realization from the Kappa distribution. This provides a similar non-uniformity that might be seen in streamflow or runoff hydrographs or rainfall hyetographs in which more observations are reported early in cumulative time than later because the hydrograph is steeper and requiring greater resolution for accurate representation of shape. The same Kappa distribution and parameters, sample sizes, and simulation count used in the simulations reported in tables 18, 19, and 20 were used with the nonuniform distribution just described. The biases for the mean, L-scale, L-CV, L-skew, and L-kurtosis are listed in table 21. From the table it is apparent that only the p-PWM estimator continues to be an unbiased estimator of all the L-moments for sufficiently large samples. The bias of the unbiased estimator—the unbiased estimator is biased under the circumstances of nonuniform probability that are considered here—and the plotting-position estimators rapidly approaches a constant for each L-moment or L-moment ratio as sample size increases. This is not particularly surprising given than a significant fraction of the parent distribution is not sampled as often as it would be under random conditions. The p-PWM estimator does have F FF0.5> F 0.5≤ F 0.5> 125 considerably larger bias for L-kurtosis for samples sizes less than about 50. The sampling standard deviations for the nonuniform simulations are listed in table 22 and followed by the relative efficiencies in table 23. Table 21. Comparison of sample biases for a simulated Kappa distribution using a non-uniform distribution of probability by redrawing F if initial F was greater than 0.5 [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] The p-PWM estimators of the mean, L-scale, and L-CV are more efficient than the unbiased estimators and the plotting-position estimators by the fact that the REs are Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias n=5 UB -0.0276 -0.0012 0.2173 0.1322 0.0669 PP -.0276 -.0030 .1226 .2504 .0244 XF -.0087 -.0070 -.0595 .0918 .2620 n=10 UB -.0281 -.001 .1252 .1280 .0546 PP -.0281 -.0021 .1120 .2074 .0464 XF -.0036 -.0063 -.0530 .0044 .1716 n=20 UB -.0278 -.0010 .1074 .1277 .0487 PP -.0277 -.0016 .1001 .1739 .0500 XF -.0011 -.0031 -.0257 -.0420 .0844 n=50 UB -.0276 -.0010 .0984 .1272 .0445 PP -.0273 -.0012 .0943 .1452 .0434 XF -.0001 -.0006 -.0046 -.0219 .0025 n=100 UB -.0285 -.001 .0997 .1286 .0411 PP -.0280 -.0011 .0970 .1365 .0434 XF .0 -.0002 -.0013 -.0034 -.0067 n=200 UB -.0269 -.0009 .0927 .1307 .0400 PP -.0272 -.0010 .0927 .1314 .0423 XF .0 -.0001 -.0004 -.0025 -.0048 n=500 UB -.0277 -.0010 .0946 .1274 .0415 PP -.0277 -.0010 .0945 .1280 .0426 XF .0 .0 .0 -.0003 .0 n=1,000 UB -.0273 -.0009 .0933 .1279 .0394 PP -.0278 -.0009 .0966 .1295 .0386 XF .0 .0 .0 .0 .0 126 often very much greater than one. The p-PWM estimators are not as efficient for small sample sizes as they were for the uniform simulations reported in the previous section. The p-PWM seem to perform poorly in terms of efficiency for L-skew and L-kurtosis for sample sizes of 200 or less. The p-PWM estimator for L-skew and L-kurtosis had the least bias for sample sizes greater than about 50. It appears as though the p-PWM estimators can have trouble making reliable estimates of L-skew and likely all higher L-moment ratios for small to moderately small samples. p-PWM estimators perform well for the mean, L-scale, and L-CV for all sample sizes. Table 22. Comparison of sample standard deviations for a simulated Kappa distribution using a non-uniform distribution of probability by redrawing F if initial F was greater than 0.5 [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. The sample standard deviation (SD) is defined as the square root of the sampling variance of the estimator. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] Sample size Estimator type SD of Mean SD of L-scale SD of L-CV SD of L-skew SD of L-kurtosis n=5 UB 0.0283 0.0121 4.5813 0.2939 0.3627 PP .0285 .0097 .7668 .1479 .1218 XF .0151 .0141 .1269 .4576 .3200 n=10 UB .0201 .0078 .1667 .1514 .1417 PP .0206 .0071 .1647 .1126 .0886 XF .0077 .0105 .0908 .3472 .2662 n=20 UB .0139 .0053 .1025 .0880 .0773 PP .0144 .0050 .1011 .0794 .0687 XF .0030 .0064 .0539 .2400 .1876 n=50 UB .0090 .0031 .0611 .0519 .0398 PP .0089 .0032 .0607 .0471 .0487 XF .0005 .0014 .0113 .0933 .0763 n=100 UB .0064 .0024 .0434 .0353 .0288 PP .0064 .0024 .0450 .0329 .0418 XF .0001 .0003 .0026 .0213 .0410 n=200 UB .0048 .0015 .0318 .0246 .0197 PP .0048 .0016 .0313 .0232 .0389 XF <.0001 .0001 .0012 .0089 .0220 127 Table 23. Comparison of relative efficiencies for a simulated Kappa distribution using a non- uniform distribution of probability by redrawing F if initial F was greater than 0.5 [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Relative efficiency (RE) is defined as the variance of the unbiased estimator to the variance of the plotting position or prior-probability weighted moment estimator. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments. Number of simulations per estimator was 50,000.] n=500 UB .0029 .0010 .0180 .0154 .0118 PP .0029 .0010 .0193 .0161 .0370 XF <.0001 <.0001 .0001 .0022 .0065 n=1,000 UB .0020 .0008 .0140 .0110 .0091 PP .0023 .0008 .0155 .0135 .0356 XF <.0001 <.0001 <.0001 <.0001 <.0001 Sample size Estimator type RE of Mean RE of L-scale RE of L-CV RE of L-skew RE of L-kurtosis n=5 UB11111 PP .986 1.56 3570 3.95 8.87 XF 3.51 .736 1300 .413 1.28 n=10 UB11111 PP .952 1.21 1.02 1.81 2.56 XF 6.81 .552 3.37 .190 .283 n=20 UB11111 PP .932 1.12 1.03 1.23 1.27 XF 21.5 .686 3.62 .134 .170 n=50 UB11111 PP 1.02 .938 1.01 1.21 .668 XF 3.24 4.90 29.2 .309 .272 n=100 UB11111 PP 1 1 .930 1.15 .475 XF 4100 64 279 2.75 .493 n=200 UB11111 PP 1 .879 1.03 1.12 .256 XF >4100 225 702 7.64 .802 n=500 UB11111 PP 1 1 .870 .915 .102 XF >4100 >225 32400 49 330 n=1,000 UB11111 PP .756 1 .816 .664 .0653 XF >4100 >225 >32400 >49 >330 Sample size Estimator type SD of Mean SD of L-scale SD of L-CV SD of L-skew SD of L-kurtosis 128 Supplemental simulations showing the bias using a variety of non-uniform distribution of constructions are provided in Appendix C. Similar conclusions concerning p-PWMs can be made from the tables in Appendix C. Chapter Conclusions Intuitive argument, example computations, and limited simulation experiments strongly indicate that prior-Probability Weighted Moments (p-PWMs) and a proposed estimator (eq. 46) are attractive for L-moment estimation on previously incompatible data sets. These data sets have two properties that make traditional L-moment computation unsuitable. The first property is that the cumulative probability of each observation of the distribution is known (estimated), and the second property is that there exists prior knowledge or reasonable expectation that the distribution of the probabilities is not uniformly distributed or equally likely. In the uniform simulations, the p-PWMs estimators might have substantial bias and sampling variance in very small samples , but as sample size increases, the p-PWMs rapidly become unbiased and exhibit minimal sampling variance. For nonuniform simulation, the p-PWMs outperform in terms of bias and outperform in terms of efficiency both of unbiased and plotting-position L-moment estimators. For small samples under nonuniform simulation, the p-PWM can exhibit particularly large sampling variance, but the variance drops and the efficiency rapidly increases with sample size. Caution with p-PWM estimators in small samples is advised. F n 10< 129 Two obvious candidate data sets for p-PWM application are rainfall hyetographs and streamflow hydrographs, and examples of each were provided in the chapter. The rainfall hyetograph or streamflow hydrograph data can possess cumulative percent time passed and the cumulative runoff; cumulative percent time is naturally considered as a probability. Finally, a particularly intriguing application would be p-PWM based L-moment estimation of distributions shown in historical reports in which portions, such as the middle third, of the distribution is missing because of media degradation or reproduction induced fading. The p-PWMs thus can be applied to data sets containing missing portions of the distribution. Some types of missing data can be thought of as censored data. Comparison (if one could be made) of the p-PWMs to Partial-PWMs, which are used for censored data applications (Wang, 1996a), is beyond the scope of this dissertation. For the remainder of this dissertation, the p-PWMs will be used without notification for L-moment computation of rainfall hyetographs; chapter 3 is included. Unbiased L-moment estimators will be used in all other circumstances. 130 CHAPTER 5 L-MOMENTS OF DIMENSIONLESS RAINFALL HYETOGRAPHS KNOWN TO PRODUCE RUNOFF IN TEXAS Introduction Using a variety of perspectives this chapter documents research into L-moments of runoff-producing dimensionless rainfall hyetographs in Texas. The L-moments include the mean, L-scale, L-skew, and L-kurtosis, and statistics; the nonparametric median also is included. If the L-moments of the hyetograph distribution are predictable using information such as the total depth of the storm, the time of year that the storm occurred, and geographic location, then progress towards reliable estimation of the hyetograph is achieved. Three principle research components in this chapter are: 1. An investigation of potential dependency or graphical correlations between storm depth and the mean, median, L-scale, L-skew, and L-kurtosis, and of dimensionless hyetographs. The double one-percent hyetograph tail trimming preprocessing method is investigated as well as a no-tail-trimming method. 2. An investigation of the influence of month or season of storm occurrence on the hyetograph mean, median, L-scale, and L-CV statistics. The investigation focuses on the monthly mean values of these four statistics. 3. An investigation of the relation L-scale to the mean and median of dimensionless hyetographs for each of the five hyetograph data base modules (c68c88c86c87c76c81, c71c68c79c79c68c86, c73c82c85c87c90c82c85c87c75, and c86c68c81c68c81c87c82c81c76c82 and the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 modules). Both hyetograph tail trimming methods are considered. Because the τ 5 τ 5 131 urban hyetograph data bases are spatially limited and geographically distinct, the comparison also considers the influence of geographic location on the dimensionless hyetograph. A subcomponent is a modal analysis of the mean, median, and L-scale statistics; the objective of the modal analysis is to estimate the most likely value for each statistic. The sample L-moment statistics presented here are computed by prior-Probability Weighted Moments, which are described in the previous chapter. The median of the hyetograph is computed by linear interpolation between the samples bracketing the mean. The mean summary statistics of the hyetograph L-moments computed by p-PWMs or the median are not computed using p-PWMs but by the common arithmetic average. Relations between Storm Depth and Hyetograph Statistics For the storm depth dependency or correlation investigation (component no. 1), if the L-moments and the nonparametric median are independent of—that is not a substantial function of—the storm magnitude as measured by depth, the logical conclusion is that the shape of L-moment-fitted expected hyetographs are independent of the frequency level (nonexceedance probability) or recurrence interval of the storm. The frequency independence would greatly simplify the use of expected hyetographs for design applications because a separate hyetograph for each level of risk (nonexceedance probability) is not needed. A conclusion drawn within the earlier triangular model chapter (chapter 3) for a given duration is that a substantial relation 132 does not exist between the mean and the storm depth; therefore, a frequency independence assumption is possible. The dependency investigation centers on the same three durations originally considered in the triangular hyetograph model chapter. Specifically, the 0–12 hr, 12–24 hr, and 24 hr and greater (up to about 3 days) storm durations are considered. Separate, but parallel, analysis for each duration is provided in three sections. Within each analysis, box plots (Helsel and Hirsch, 1992, pp. 24–26) of the distributions for a given statistic (such as L-skew) for each storm depth category (1 in. through 10 in. by 1 in. (25.4 mm) increments) are drawn. These same categories were used in the triangular hyetograph chapter. From graphs of the box plots it is possible to make visual inferences about whether one or more aspects of the distribution of the statistic are dependent on storm depth. Detailed explanations of the box plots shown on figures 26–46 (introduced shortly) are shown on figure 25. Multiple explanations are necessary for the figures because rendering of box plots for small sample sizes is not feasible. If the sample size is greater than four, then a traditional box plot showing the mean, median, quartiles, and range of the data is drawn. If the sample size is equal to three, then the mean, median, and the upper and lower data values are drawn; the remaining data point corresponds to the median so it is not plotted. If the sample size is equal to two, then the mean and the two data points are drawn; the median corresponds to the mean in this case. If the sample size is equal to one, then the mean is drawn; the median and the 133 single data point each correspond to the mean. If no data were available for a depth category, then “no data” is indicated in the figures. A few general comments about the interpretation of box plots in this section are required before their introduction. Regarding the stems that indicate the data range by extending to the maximum and minimum values of the data from the upper and lower quartiles, respectively, it is natural for the range of identically distributed data to increase with increasing sample size. Regarding the quartiles, the quartiles were computed according to USGS policy regarding box plot construction (Dennis Helsel, written commun., 1989). This policy advocates linear interpolation between data points bracketing the quartiles; each quartile estimate is prorated. Regarding distribution skewness, if the median plots above the mean, the distribution is said to be negatively skewed; whereas, if the median plots below the mean, the distribution is said to be positively skewed. Helsel and Hirsch (1992, p. 9–10, fig. 1.1) provide an excellent description of distribution skewness. A final introductory note concerning figures 26, 27, 33, 34, 40, and 41 is that the loadedness of the storm is annotated on the ordinate (vertical) axis to assist readers in visualization the influence of the mean or median statistic on the general shape of the hyetograph. A storm with a large mean or median is front-loaded, and a storm with a small mean or median is back-loaded. This concept is illustrated on figures 9 and 54. 134 Figure 25. Explanation of box plots and ancillary glyphs shown on figures 26–46 Storm Durations of 0 to 12 Hours A graph of box plots summarizing the distribution of the mean statistics of the 0–12 hr hyetographs for each depth category is shown on figure 26. The overall mean (plus glyph) of each category and sample size values match the averages and sample sizes listed in columns 3 and 5 of table 11. Neither the category means or medians show substantial dependency on the storm depth. A similar observation using the same (41) No. of storms (sample size) Stem indicates maximum value of the data Upper quartile, 75 percent of data is less than this value Lower quartile, 25 percent of data is less than this value Median value (horizontal bar), 50 percent of data is less than this value Mean value (plus glyph)—number indicates value. Stem indicates minimum value of the data (3) No. of storms (sample size) Data point Median value (horizontal bar), 50 percent of data is less than this value Mean value (plus glyph)—number indicates value. (1) (2) Boxplot explanation for samples sizes of four or more Boxplot explanation for samples sizes equal to three Boxplot explanation for sample sizes equal to two No. of storms (sample size) Data point Mean and median (values correspond), 50 percent of data is less than this Boxplot explanation for sample sizes equal to one No. of storms (sample size) Mean and median (values correspond)—number indicates value. 58 . 8 58 . 8 58 . 8 58 . 8 value—number indicates value. 135 data was made in the triangular hyetograph chapter. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The 0–12 hr hyetograph mean and its distribution are not substantially dependent on storm depth. Figure 26. Box plots showing distribution of hyetograph mean for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the median statistics of the 0–12 hr hyetographs for each depth category is shown on figure 27. The sample sizes match those on figure 26. Neither the category means or medians show substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The 0–12 hr hyetograph median and its distribution are not substantially dependent on storm depth. DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN INCHES MEAN, I N PERCENT 12345678910 0 10 20 30 40 50 60 70 80 90 100 (290) (253) (108) (41) (11) (3) (1) (1) (1) no data 57 . 9 61 . 0 58 . 5 58 . 8 58 . 4 49 . 6 74 . 0 79 . 8 22 . 9 Weighted average of means is 59.1 percent. Front- loaded Back- loaded 136 Figure 27. Box plots showing distribution of hyetograph median for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-scale statistics of the 0–12 hr hyetographs for each depth category is shown on figure 28. The sample sizes match those on figures 26 and 27. The sample size values are slightly different for the 1 in. and 2 in. categories owing to four poorly digitized hyetographs from which no L-scale could be determined—a hyetograph defined by a single data point. Neither the category means or medians show substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The maximum values of L-scale for the 1 in. and 3 in. depth categories are dissimilar from the maximum values for the other categories; DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN INCHES M E D I A N , IN P E R C E N T 12345678910 0 10 20 30 40 50 60 70 80 90 100 (290) (253) (108) (41) (11) (3) (1) (1) (1) no da t a 59 . 0 64 . 3 59 . 8 64 . 3 57 . 0 58 . 0 84 . 4 93 . 3 11 . 9 Weighted average of means is 61.3 percent. Front- loaded Back- loaded 137 this is attributed to just three spurious events (two for the 1 in. category). The 0–12 hr hyetograph L-scale and its distribution are not substantially dependent on storm depth. Figure 28. Box plots showing distribution of hyetograph L-scale for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the coefficient of L-variation (L-CV) statistics of the 0–12 hr hyetographs for each depth category is shown on figure 29. The sample sizes for the 1 in. and 2 in. depth categories are different than those seen on figures 26 and 27; this is attributable to four spurious events with zero values L-scale. Neither the category means or medians show substantial dependency on the storm depth. This necessarily is consistent with the discussion of figures 26 and 28 because L-CV is a ratio of the L-scale to the mean ( ). DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN PERCENT L- SCAL E, I N PERCENT 12345678910 0 10 20 30 40 50 60 (287) (252) (108) (41) (11) (3) (1) (1) (1) no data Weighted average of means is 15.6 percent. 16 . 0 15 . 0 15 . 7 16 . 5 16 . 3 16 . 2 13 . 2 13 . 3 9. 85 λ 2 λ 1 ⁄ 138 Figure 29. Box plots showing distribution of hyetograph coefficient of L-variation for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-skew statistics of the 0–12 hr hyetographs for each depth category is shown on figure 30. The sample sizes are slightly different than those seen on figures 26 and 27 because theoretically impossible values occurred; this is due to poor sampling of the hyetograph ordinates (see chapter on p-PWMs and comments concerning p-PWM sample L-moment estimators producing theoretically impossible values). Both the category means or medians seem to decrease with increasing storm depth. The quartile values also seem to decrease with increasing storm depth. However, the inter-quartile range appears independent of storm depth. The maximum value of L-skew for the 1 in. and 2 in. depth category are dissimilar from the maximum values for the other categories. This is attributed to the substantially larger sample size for these depth categories and a few DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN PERCENT COEFFI CI ENT OF L- V A RI A T I O N 12345678910 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (287) (252) (108) (41) (11) (3) (1) (1) (1) no data Weighted average of means is 0.310. 0. 33 2 0. 28 5 0. 3 1 4 0. 3 0 9 0. 30 6 0. 32 4 0. 17 9 0 . 166 0. 43 2 139 spurious events. The L-skew of the 0–12 hr hyetograph is substantially dependent on storm depth. Hence the skewness of the hyetograph is influenced by the storm depth. The hyetograph becomes slightly negatively skewed with large storm depths. Figure 30. Box plots showing distribution of hyetograph L-skew for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-kurtosis statistics of the 0–12 hr hyetographs for each depth category is shown on figure 31. The sample sizes compared to L-skew (figure 30) are decreased further because even more theoretically impossible values occurred; again this is due to poor sampling of the hyetograph ordinates. Both the category means or medians seem to decrease with increasing storm depth. The quartile values also seem to decrease with increasing storm depth. However, the inter-quartile range appears independent of storm depth. The maximum DEPTH CATEGORY FOR 0 TO 12 HOUR, IN PERCENT L- SKEW 12345678910 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (274) (246) (104) (41) (11) (3) (1) (1) (1) no data Weighted average of means is -0.0003. 0 . 03 51 - 0 . 0 159 -0 . 0 4 2 7 -0 . 0 5 9 9 0 . 01 17 0. 2 3 0 -0 . 2 6 7 -0 . 3 6 3 0. 7 9 4 140 value of L-kurtosis for the 1 in. and 2 in. depth categories are dissimilar from the maximum values for the other categories. This is attributed to the substantially larger sample sizes for these depth categories and a few spurious events. The L-kurtosis of the 0–12 hr hyetograph is substantially dependent on storm depth. Hence the kurtosis of the hyetograph is influenced by the storm depth. The hyetograph becomes less kurtotic with large storm depths. Figure 31. Box plots showing distribution of hyetograph L-kurtosis for 0–12 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-moment statistics of the 0–12 hr hyetographs for each depth category is shown on figure 32. The sample sizes compared to L-kurtosis (figure 31) are decreased further because even more theoretically impossible values occurred; again this is due to poor sampling of the DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN INCHES L- KURT OSI S 12345678910 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (249) (215) (99) (39) (10) (2) (1) (1) no datano da ta Weighted average of means is 0.189. 0. 2 0 1 0. 21 2 0. 15 8 0. 09 19 0. 08 16 0 . 071 7 0. 2 2 5 0. 4 6 9 τ 5 141 hyetograph ordinates. Both the category means or medians seem to decrease with increasing storm depth. The quartile values also seem to decrease with increasing storm depth. However, the inter-quartile range appears independent of storm depth. The maximum value of for the 1 in. through 3 in. depth categories are dissimilar from the maximum values for the other categories. This is attributed to the substantially larger sample size for these depth categories and a few spurious events. The of the 0–12 hr hyetograph is substantially dependent on storm depth. Hence the of the hyetograph is influenced by the storm depth. Whether the influence is substantial is difficult to assess. Figure 32. Box plots showing distribution of hyetograph Tau5 for 0–12 hr storm durations for integer storm depth categories τ 5 τ 5 τ 5 DEPTH CATEGORY FOR 0 TO 12 HOUR STORMS, IN INCHES TA U 5 12345678910 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 (241) (207) (98) (39) (10) (2) (1) (1) -1.25 1.25 no datano data Weighted average of means is 0.116. 0. 13 8 0. 1 1 3 0. 09 70 0. 04 60 0 . 02 18 0. 36 4 0 . 07 38 0 . 406 142 Storm Durations of 12 to 24 Hours A graph of box plots summarizing the distribution of the mean statistics of the 12–24 hr hyetographs for each depth category is shown on figure 33. The overall mean (plus glyph) of each category and sample size values match the averages and sample sizes listed in columns 6 and 8 of table 11. A slight increase in both the category mean and median with increasing storm depth is seen, but the increase is considered unsubstantial. A similar observation was made (using the same data) in the triangular hyetograph chapter. The inter-quartile range seems to contract with increasing storm depth. The contraction is due to the increasing lower quartile; the upper quartile remains essentially constant. The 12–24 hr hyetograph mean and its distribution are not substantially dependent on storm depth. Figure 33. Box plots showing distribution of hyetograph mean for 12–24 hr storm durations for integer storm depth categories DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES MEAN, I N PERCENT 12345678910 0 10 20 30 40 50 60 70 80 90 100 (68) (180) (119) (80) (30) (10) (2) (1) (1) no data 56 . 4 57 . 8 58 . 0 61 . 2 59 . 3 70 . 7 57 . 0 38 . 2 48 . 3 Weighted average of means is 58.5 percent. Front- loaded Back- loaded 143 A graph of box plots summarizing the distribution of the median statistics of the 12–24 hr hyetographs for each depth category is shown on figure 34. The sample sizes match those on figure 33. An increase in both the category mean and median with increasing storm depth is seen, but the increase could be considered unsubstantial. The inter-quartile range seems to contract abruptly between the 1 in. and 2 in. depth categories; this might be attributable to sample size or other factors. The 12–24 hr hyetograph median and its distribution are not substantially dependent on storm depth. Figure 34. Box plots showing distribution of hyetograph median for 12–24 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-scale statistics of the 12–24 hr hyetographs for each depth category is shown on figure 35. The sample sizes match those on figures 33 and 34. Neither the category means or medians show DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN PERCENT MEDI AN, I N PERCENT 12345678910 0 10 20 30 40 50 60 70 80 90 100 (68) (180) (119) (80) (30) (10) (2) (1) (1) no data 56 . 9 60 . 2 59 . 6 64 . 3 63 . 1 84 . 0 68 . 3 54 . 0 62 . 1 Weighted average of means is 61.0 percent. Front- loaded Back- loaded 144 substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The 12–24 hr hyetograph L-scale and its distribution are not substantially dependent on storm depth. Figure 35. Box plots showing distribution of hyetograph L-scale for 12–24 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the coefficient of L-variation (L-CV) statistics of the 12–24 hr hyetographs for each depth category is shown on figure 36. Neither the category means or medians show substantial dependency on the storm depth. This necessarily is consistent with the discussion of figures 33–35 because L-CV is a ratio of the L-scale to the mean ( ). DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES L - SCALE, I N PERCENT 12345678910 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 (68) (180) (119) (80) (30) (10) (2) (1) (1) no data Weighted average of means is 14.2 percent. 14 . 0 13 . 0 15 . 0 15 . 4 15 . 3 14 . 3 19 . 0 15 . 9 22 . 9 λ 2 λ 1 ⁄ 145 Figure 36. Box plots showing distribution of hyetograph coefficient of L-variation for 12–24 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-skew statistics of the 12–24 hr hyetographs for each depth category is shown on figure 37. The sample sizes are different than those seen on figures 33–36 because theoretically impossible values occurred. Both the category means or medians seem to decrease with increasing storm depth—using the large sampled 1 in. through 4 in. categories. Visual inferences on the inter-quartile range and quartile values themselves is difficult. The L-skew of the 12–24 hr hyetograph is weakly dependent on storm depth. Hence the skewness of the hyetograph is influenced by the storm depth. The hyetograph becomes slightly negatively skewed with large storm depths. DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES COEFF I CI ENT OF L- V A RI A T I O N 12345678910 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (68) (180) (119) (80) (30) (10) (2) (1) (1) no data Weighted average of means is 0.292. 0. 3 1 8 0. 28 1 0. 2 9 9 0. 29 1 0. 29 2 0 . 216 0. 36 9 0. 41 6 0. 47 4 146 Figure 37. Box plots showing distribution of hyetograph L-skew for 12–24 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-kurtosis statistics of the 12–24 hr hyetographs for each depth category is shown on figure 38. The sample sizes compared to L-skew (figure 37) are further decreased from those seen on figures 33– 36 because theoretically impossible values occurred; again this is due to poor sampling of the hyetograph ordinates. Both the category means or medians show decrease with increasing storm depth for the 2 in. and greater categories. Furthermore, the inter-quartile range and the quartile values themselves also appear slightly dependent on the storm depth. The L-kurtosis of the 12–24 hr hyetograph is substantially dependent on storm depth. Storms become less kurtotic with increasing storm depth, but the L-kurtosis remains positive (greater than zero). DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES L- SKEW 12345678910 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (63) (172) (118) (79) (30) (10) (2) (1) (1) no data Weighted average of means is -0.0468. -0 . 0 3 3 9 -0 . 0 4 7 5 -0 . 0 9 6 7 -0 . 0 3 4 5 -0 . 2 3 2 -0 . 1 3 8 -0 . 0 8 0 8 -0 . 0 0 1 3 0. 00 84 3 147 Figure 38. Box plots showing distribution of hyetograph L-kurtosis for 12–24 hr storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the statistics of the 12–24 hr hyetographs for each depth category is shown on figure 39. The sample sizes are further decreased, compared to L-kurtosis (figure 38), from those seen on figures 32– 35 because theoretically impossible values occurred; again this is due to poor sampling of the hyetograph ordinates. Neither the category means or medians show substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The of the 12–24 hr hyetograph is not substantially dependent on storm depth. DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES L- K U RTO S I S 12345678910 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (58) (158) (114) (75) (29) (9) (2) (1) (1) no data Weighted average of means is 0.184. 0. 1 8 5 0. 22 8 0. 17 9 0. 1 4 3 0. 1 2 2 0. 1 2 6 -0 . 0 1 5 7 -0 . 0 9 6 7 -0 . 1 6 2 τ 5 τ 5 148 Figure 39. Box plots showing distribution of hyetograph Tau5 for 12–24 hr storm durations for integer storm depth categories Storm Durations of 24 Hours and Greater A graph of box plots summarizing the distribution of the mean statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 40. The overall mean (plus glyph) of each category and sample size values match the averages and sample sizes listed in columns 3 and 5 of table 12. Both the category means or medians show a decrease with increasing storm depth; a decrease of about 10 percentage points. A similar observation is made (using the same data) in the triangular hyetograph chapter. The inter-quartile range does not appreciably change with storm depth, but the quartile values themselves appear to be weakly dependent on the storm depth by decreasing with increasing storm depth. The 24 hr and greater hyetograph mean and its distribution are dependent on storm depth. DEPTH CATEGORY FOR 12 TO 24 HOUR STORMS, IN INCHES TA U 5 12345678910 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 (57) (154) (113) (74) (29) (9) (2) (1) (1) no data 0. 08 58 0. 08 55 0 . 06 37 0. 0 763 0 . 06 69 0. 09 12 0 . 03 42 0. 1 5 4 0 . 06 20 Weighted average of means is 0.0772. 149 Figure 40. Box plots showing distribution of hyetograph mean for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the median statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 41. The sample sizes match those on figure 40. A substantial decrease (about 25 percentage points) in both the category mean and median with increasing storm depth is seen. The inter- quartile range is large and appears to increase slightly with increasing storm depth. The quartile change accordingly. The decrease in the median and mean (figure 39) might be attributable to a tendency for the larger depth storms to simply last longer— that is to have longer periods of substantial rainfall—than the smaller depth storms. Thus distribution location measures such as the median or mean would be smaller. The 24 hr and greater hyetograph median is substantially dependent on storm depth. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES MEAN, I N P E R C ENT 12345678910 0 10 20 30 40 50 60 70 80 90 100 (23) (123) (131) (72) (43) (41) (12) (11) (4) (2) 65 . 0 56 . 8 55 . 7 55 . 8 53 . 7 50 . 6 51 . 6 50 . 2 42 . 2 57 . 0 Weighted average of means is 55.5 percent. Front- loaded Back- loaded 150 Figure 41. Box plots showing distribution of hyetograph median for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-scale statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 42. The sample sizes match those on figures 40 and 41. Both the category means or medians show a slight increase with increasing storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The maximum value of L-scale for the 2 in. depth category is dissimilar from the maximum values for the other categories; this is attributed to just one spurious event. The 24 hr and greater hyetograph L-scale and its distribution is substantially dependent on storm depth. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES MED I AN, I N PERCENT 12345678910 0 10 20 30 40 50 60 70 80 90 100 (23) (123) (131) (72) (43) (41) (12) (11) (4) (2)69 . 7 57 . 4 56 . 3 55 . 5 56 . 0 49 . 1 42 . 9 49 . 0 42 . 5 59 . 8 Weighted average of means is 55.8 percent. Front- loaded Back- loaded 151 Figure 42. Box plots showing distribution of hyetograph L-scale for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the coefficient of L-variation (L-CV) statistics of the 24 hr and hyetographs for each depth category is shown on figure 43. Both the category means or medians show substantial dependency on the storm depth. This necessarily is consistent with the discussion of figures 40–42 because L-CV is a ratio of the L-scale to the mean. As the mean decreases (figure 40), the L-CV must increase. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES L- SCALE, I N PERCENT 12345678910 0 10 20 30 40 50 60 (23) (123) (131) (72) (43) (41) (12) (11) (4) (2) Weighted average of means is 13.8 percent. 11 . 4 13 . 0 13 . 6 14. 6 13 . 5 16 . 4 15 . 4 12 . 7 14 . 7 15 . 1 152 Figure 43. Box plots showing distribution of hyetograph coefficient of L-variation for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-skew statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 44. The sample sizes are different than those seen on figures 40–43 because theoretically impossible values occurred; this is due to poor sampling of the hyetograph ordinates. Both the category means or medians increase with increasing storm depth. This is opposite of the observation made on the 12–24 hr duration L-skew on figure 37. Visual inferences on the inter-quartile range and quartile values themselves is difficult. The L-skew of the 24 hr and greater hyetograph is substantially dependent on storm depth. Hence the skewness of the hyetograph is influenced by the storm depth. The hyetograph becomes slightly positively skewed with large storm depths. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES COEFFI CI ENT OF L - V A RI A T I O N 12345678910 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (23) (123) (131) (72) (43) (41) (12) (11) (4) (2) Weighted average of means is 0.297. 0. 28 2 0. 30 0 0. 29 6 0. 2 8 8 0. 36 1 0. 3 2 9 0. 3 0 4 0. 4 0 8 0. 26 6 0. 2 2 4 153 Figure 44. Box plots showing distribution of hyetograph L-skew for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the L-kurtosis statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 45. The sample sizes compared to L-skew (figure 43) are further decreased from those seen on figures 40–42 because theoretically impossible values occurred; again this is due to poor sampling of the hyetograph ordinates. Neither the category means or medians show substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The possible contraction of the inter-quartile range for the depth categories 4–6 in. is curious; a possible reason for the contraction are unknown. The L-kurtosis of the 24 hr and greater hyetograph is independent of storm depth. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES L- SKEW 12345678910 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (20) (117) (128) (71) (43) (41) (12) (11) (4) (2) Weighted average of means is 0.0207. - 0 . 0 07 52 0. 00 01 45 0. 02 80 - 0 . 0 03 03 - 0 . 0 02 96 0 . 081 3 0 . 131 0. 0 707 0. 13 2 -0 . 0 0 2 5 154 Figure 45. Box plots showing distribution of hyetograph L-kurtosis for 24 hr and greater storm durations for integer storm depth categories A graph of box plots summarizing the distribution of the statistics of the 24 hr and greater hyetographs for each depth category is shown on figure 46. The sample sizes are further decreased, compared to L-kurtosis (figure 45), from those seen on figures 40–42 because theoretically impossible values occurred; again this is due to poor sampling of the hyetograph ordinates. Neither the category means or medians show substantial dependency on the storm depth. Furthermore, the inter-quartile range and the quartile values themselves also do not appear dependent on the storm depth. The of the 24 hr and greater hyetograph is not substantially dependent on storm depth. DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS, IN INCHES L- KU R T O S I S 12345678910 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (15) (108) (114) (68) (42) (41) (12) (11) (4) (2) Weighted average of means is 0.182. 0. 20 4 0. 23 2 0. 2 1 1 0. 1 2 3 0. 17 8 0. 06 75 0. 1 6 5 0. 2 5 0 0 . 07 18 0. 12 3 τ 5 τ 5 155 Figure 46. Box plots showing distribution of hyetograph Tau5 for 24 hr and greater storm durations for integer storm depth categories Section Conclusions A summary of the dependency observations made for figures 26–46 is shown in table 24. The mean, median, and L-scale statistics and influences on their values are most critical for defining the expected hyetograph because the lowest order moments contain the greatest degree of information about the hyetograph distribution. In a loose interpretation, it appears for the 0–12 hr and 12–24 hr duration that only the high L-moments (L-skew, L-kurtosis, and ) are substantially dependent on the storm depth. For the 24 hr and greater storm duration, the lower order statistics (mean, median, L-scale, L-CV, L-skew) appear influenced by storm depth. L-kurtosis and do not. A factor that could contribute to these statistics changing with depth is that the storms lumped into the 24 hr and greater duration—some durations might be many DEPTH CATEGORY FOR 24 HOUR AND GREATER STORMS TA U 5 12345678910 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 (15) (107) (111) (68) (42) (41) (11) (11) (4) (2) Weighted average of means is 0.0489. 0. 03 32 0. 06 73 0 . 04 46 0. 06 91 0 . 04 76 0 . 00 009 51 -0 . 0 1 0 4 0. 06 80 0. 02 74 0. 03 14 τ 5 τ 5 156 days—are more heterogeneous than those having shorter durations. Even the 12–24 hr duration might have excessive heterogeneity expressed because of the slight to moderate upward trend in the mean and the median. The weighted average values for the category means for each statistic for each duration also are listed in table 24. All of the weighted averages, except perhaps , for the 0–12 hr and 12–24 hr durations are close in value, which suggests that these durations can be combined. It is therefore concluded that the distribution of the sub- daily hyetographs (0–24 hr) might be combinable. This affirms a similar conclusion using just the mean statistic in the triangular hyetograph chapter. The remainder of this dissertation however continues to consider the 0–12 hr and 12–24 hr durations distinct. The weighted average values for the category means for each statistic for the 24 hr and greater duration are similar in value to those for the two other durations. However, the mean, median, L-scale, and L-CV values appear different enough from those for the other two durations that the 24 hr and greater duration should be considered separate. The values for L-skew and L-kurtosis appear comparable to the corresponding values for the other two durations. The values for appear less comparable between the three durations, but each is greater than zero. τ 5 τ 5 157 Table 24. Summary of observations made and weighted average values for statistics from the box plot graphs on figures 26–46 [hr, hour; --, no dependency between statistic and storm depth observed; up, an increasing statistic value with increasing storm depth observed; down, a decreasing statistic value with increasing storm depth observed; na, not available] The three L-skew means appear nearly equal to zero; hence a near symmetry in the expected hyetograph distribution is implied. Hosking (1990, p. 112) reports that the L-skew of the Normal distribution (a symmetrical distribution) and the Logistic distribution [ ] (also a symmetrical distribution) is . The near symmetry in the expected hyetograph distribution is unexpected by Statistic name Statistic symbol 0–12 hr duration 12–24 hr duration 24 hr and greater duration Storm dependency summary Mean -- up (not sub- stantial) down Median -- up (not sub- stantial) down L-scale -- -- up L-CV -- -- up L-skew down down up L-kurtosis down down -- Tau5 down -- -- Weighted average values of statistics Mean 59.1 58.5 55.5 Median 61.3 61.0 55.8 L-scale 15.6 14.2 13.8 L-CV .310 .292 .297 L-skew -.0003 -.0468 .0207 L-kurtosis .189 .184 .182 Tau5 .116 .0772 .0489 λ 1 M λ 2 τ τ 3 τ 4 τ 5 λ 1 M λ 2 τ τ 3 τ 4 τ 5 QF() ab F1 F–()⁄()ln×+= τ 3 0= 158 the author considering visual inspection of individual hyetographs seen on figures such as figure 9. The three L-kurtosis means appear nearly equal to a value of about 0.18. Visualization of the L-kurtosis on individual hyetographs seen on figures such as figure 9 is difficult. Hosking (1990, p. 112) also reports that L-kurtosis for the Normal distribution is . Also reported by Hosking is that the Logistic distribution has an L-kurtosis of . The L-kurtosis of the hyetographs is greater than the Normal distribution or the Logistic distribution. The L-kurtosis of the Kappa distribution described in the previous chapter can not acquire a value greater than 0.1667 for an L-skew value of zero. The hyetographs are more peaked as reflected by the L-kurtosis than well documented distributions. Monthly and Seasonal Influences on Hyetograph Statistics Analysis of monthly and seasonal differences of the hyetograph statistics (component no. 2) contributes to documentation of the influence of time of year on dimensionless hyetographs. Specifically, the statistics considered are the mean, median, L-scale, and L-CV of the hyetograph distributions. As used in previous analyses, the same 0–12 hr, 12–24 hr, and 24 hr and greater storm durations are considered. All storms having a depth equal to or greater than 1 in. (25.4 mm) within a given duration category were combined for the analysis. Note that for the 1 in. depth category considered in the previous section depths as small as 0.5 in. (12.7 mm) were τ 4 30π 1– 2()atan 9– 0.1226== τ 4 6 1– 0.1667== 159 included; depths as small as this are not considered in this section. For each duration, the mean of each statistic (a monthly mean) for each of the twelve months of the year was computed. Monthly mean values of the mean, median, L-scale, and L-CV of the dimensionless hyetographs are shown on figure 47. Graph A of the figure compares the monthly means of the hyetograph means to the month of the year; graph A also shows the sample size for the mean. From the graph, it is evident that substantial month to month variation exists for a given duration as well as substantial variation between the durations. However, some possible subtle trends are interpreted. From inspection of graph A it appears that the hyetograph mean increases from an annual low in January to a maximum between May and July. By about August through the end of the year, the hyetograph mean appears unrelated to the month. A larger mean is indicative—albeit not as strongly as the median—of a dimensionless hyetograph that peaks earlier and is steeper than those having smaller values for the mean (see figure 54, referenced out of sequence). The 24 hr and greater duration shows the smoothest trend or with suppressed variation. The sample sizes for each of the three durations indicates that most events occur in May, and April has the second largest number of occurrences. Graph B of the figure compares the monthly means of the hyetograph median to the month of the year. From the graph, it is evident that substantial month to month variation exists for a given duration as well as substantial variation between the durations. More variation in the medians than in the means exists as evidenced by a 160 comparison of graph A and graph B; each graph is plotted on the same scale to facilitate the comparison. The median is known to have a larger sampling variation than the mean, so the greater variation in graph B than in graph A is anticipated. The dashed line representing the 24 hr and greater duration shows the smoothest patterning on the graph. The dashed line also indicates that the two minimums in the hyetograph median are seen about February and again about September, and two maximums are seen between April to June and between October and November. The month intervals for the maximums are coincident with the most frontal activity. The storms during the “frontal seasons” (April, May, and June and then September(?), October, and November) might exhibit faster advection rates and hence are more front-loaded (higher median values)—see discussion in Previous Studies chapter for more discussion of storm advection on hyetograph shape. It is possible that similar behavior is exhibited in the 0–12 hr and 12–24 hr duration lines, but the results are vague and likely inconclusive. 161 Figure 47. Comparison of monthly mean values for the mean, median, L-scale, and L-CV of dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations for storms having greater than one inch of precipitation Graph C of the figure compares both the monthly means of the hyetograph L-scale and the monthly means of L-CV values to the month of the year. By inspecting the 80/35 No. indicates sample size for mean Graphs for L-CV Graphs for L-scale M EAN, I N PERCENT 40 50 60 70 80 8 9 45 96 176 54 44 45 53 59 19 13 9 28 30 86 110 36 20 16 48 53 25 20 24 14 46 57 105 38 14 22 43 32 21 23 Monthly statistics of dimensionless hyetograph for 0 to 12 hr storm duration Monthly statistics of dimensionless hyetograph for 12 to 24 hr storm duration Monthly statistics of dimensionless hyetograph for 24 hr and greater storm duration EXPLANATION MEDI AN, I N PERCENT 35 40 50 60 70 L- SCALE, I N PERCENT L- CV JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC 10 12 16 20 24 28 30 0 0.1 0.2 0.3 0.4 0.5 0.6 Total samples: 621 (0–12 hr), 481 (12–24 hr), 439 (24 hr and greater) A. B. C. 162 graph it is evident that L-scale diminishes from January through about June and then gradually increases to December. The coefficient of L-variation (L-CV) as expected exhibits similar behavior. Interpretation of the L-scale and L-CV is difficult owing to its coupling with the mean or median. Highly front-loaded or alternatively back- loaded storms leave little room within the bounds of the dimensionless hyetograph space for large distribution spread. Therefore, the minimums in distribution spread coincide with the maximums in the hyetograph mean and median values. The geometry of this reasoning is illustrated on figure 54 (again referenced out of sequence). Geographic Influences and Data Base Differences on Hyetograph Statistics Analysis of the potential influences of geographic location on the dimensionless hyetograph mean, median, and L-scale statistics (component no. 3) contributes to the statistical understanding of the hyetograph. If limited geographic patterning is evident or if geographic patterning is absent in the hyetograph statistics, then a logical conclusion is that the five data bases can be combined. Resulting expected hyetographs from the combination hence will have broad geographic applicability in Texas. A further benefit of the analysis is that potential systematic differences between the data bases as expressed in the dimensionless hyetograph statistics can be evaluated. In the first subsection, the analysis is for the dimensionless hyetographs generated from a double one-percent trimming of the hyetograph tails. In the second subsection, the analysis is for the dimensionless hyetographs with no trimming of the 163 hyetograph tails. Readers are reminded that the double one-percent trimming is performed in all statistical processing of hyetographs in this dissertation unless otherwise stated or when two different trimming methods are juxtaposed as in this section. As with the previous section for all analysis in this section, only hyetographs with precipitation depths equal to or greater than 1 in. (25.4 mm) were used. Double One-Percent Trimming of Hyetograph Tails The relation between values for the mean and median of 0–12 hr dimensionless hyetographs computed with double one-percent tail trimming and values for L-scale are shown on figure 48. Summary statistics for the this figure are listed in table 25. These statistics reflect the aggregation of each of the five data bases. For example, the arithmetic mean of the hyetograph means (graph A), medians (graph B), and L-scale values (graphs A and B) are 59.6 percent, 62.0 percent, and 15.5 percent, respectively. The medians of the hyetograph means, medians, and L-scale values are 61.0 percent, 71.0 percent, and 16.0 percent, respectively. The table also list other statistics for other durations indicated on figures in this subsection and the next subsection. 164 Table 25. Summary mean and median statistics for the mean, median, and L-scale values of dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and both tail trimming methods for one inch and greater storm depths [hr, hour; Number in parenthesis beside the entries in the two columns for the 24 hr and greater duration does not include the Dallas hyetograph data base.] From graph A on the figure it is evident that a large variation in the mean exists— the mean appears capable of acquiring almost any value between about 10 and about 95 percent. However, a pronounced joint clustering of mean and L-scale between mean values of about 55 percent to about 90 percent is evident. The clustering is indicated in the figure by the light grey region labeled “cluster”. Summary statistic Dimensionless hyetograph statistic type Double one-percent tail trimming No tail trimming 0–12 hr duration 12–24 hr duration 24 hr and greater duration 0–12 hr duration 12–24 hr duration 24 hr and greater duration (percent) (percent) (percent) (percent) (percent) (percent) Mean of values for the values of the hyetograph statistics Mean 59.6 58.2 54.8 (56.0) 62.0 61.8 58.2 (59.4) Median 62.0 60.6 54.7 (56.3) 64.4 67.0 59.4 (61.4) L-scale 15.5 14.3 13.8 (13.6) 16.0 15.0 15.4 (15.2) Median of values for the values of the hyetograph statistics Mean 61.0 59.5 53.2 (54.8) 64.1 65.1 55.7 (58.6) Median 71.0 66.5 57.3 (59.0) 79.0 77.6 62.4 (66.4) L-scale 16.0 14.7 14.1 (13.9) 16.6 15.4 15.8 (15.7) Graphically estimated modes for the values of the hyetograph statistics Mean 67 66 54. (54) 78 74 54. (54) Median indeterminate L-scale 17 15 14. (14) 19 17 19. (17) 165 Figure 48. Relation between hyetograph mean or median and L-scale for 0–12 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed L-scale also exhibits much variation, but unlike the mean, L-scale seems to approach a limiting value for a given mean. The limiting characteristic is seen in the inverted “U” shape of graph A. This behavior between the mean and L-scale on figure MEAN, IN PERCENT L- SC A L E, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 hyetographs EXPLANATION Four spurious events with L-scale = 50 are not shown on figure. Discordant values A. B. cluster 166 48 is repeatedly seen in pending figures 49–53. An unpublished theoretical or mathematical proof of a limiting L-scale value has been made by Hosking (written commun., 2002). The limits are . A reason by partial description is that as the mean of the dimensionless hyetograph gets large (or small), there is little “room” in the 0 to 100 percent range for large hyetograph slope (large L-scale) near the middle regions of the duration. A graphic further representing a geometric description of this reasoning is shown on figure 54. From graph B on figure 48, it also is evident that the median exhibits much variation and appears to acquire almost any value between 0 and 100 percent. This range is slightly larger than that for the mean. In contrast with the mean (graphs A) the L-scale values do not appear influenced by the median and distinct limiting values for a given median are not evident. Although moderate joint clustering of median and L-scale values at median values between about 65 percent to 100 percent is seen, the cluster is not nearly as pronounced as it was for the mean. From both graphs on figure 48, little visual difference between the four urban data bases is apparent. However, neither the Austin or c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 (accounting for its much larger sample size) data bases have L-scale values as low as those computed in the remaining three urban data bases. All data bases exhibit similar ranges for both the mean and the median. The appearance of a few very small mean and L-scale values could be attributable to sampling biases, poor digitizing of the data, an artifact (desirable or not) of the double one-percent trimming or other unidentified factors. It is 0 λ 2 λ 1 1 λ 1 –()≤≤ 167 concluded that for the 0–12 hr duration that all five data bases can be aggregated for purposes of estimation of the mean, median, and L-scale statistics. Similar behavior as seen for the 0–12 hr duration with the mean, median, and L-scale values is apparent on figures 49 and 50 for the 12–24 hr and 24 hr and greater durations, respectively. Similar conclusions as those for the 0–12 hr duration thus are made including a conclusion that the five data bases are compatible for the 12–24 hr duration. For the 24 hr and greater duration, the Dallas data points are unexpectedly few for mean values between about 60 and 80 percent and for median values between about 65 and 95 percent. There appears a bias towards smaller values of both the mean and median. The result is that the summary mean or median statistics (table 25) are smaller when the Dallas data is included. Alternative statistics without the Dallas data also are listed in table 25. It is unknown whether or not the Dallas data for the 24 hr and greater duration should be used. Reasons for the discrepancy of this particular group of Dallas data are not apparent and inspection of figure 5 provides no insight. A couple of other notable observations of figures 49 and 50 are made. First, for the 12–24 hr duration (fig. 49), less pronounced clustering is evident than that identified for the 0–12 hr duration (fig. 48). No identified clustering is seen for the 24 hr and greater duration (fig. 50). Second, for the 24 hr and greater duration (fig. 50), the values for L-scale do not appear to approach the limiting value as frequently as seen in graphs A of figures 48 and 49. 168 Figure 49. Relation between hyetograph mean or median and L-scale for 12–24 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed MEAN, IN PERCENT L- SC A L E, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 hyetographs EXPLANATION A. B. 169 Figure 50. Relation between hyetograph mean or median and L-scale for 24 hr and greater storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when double one-percent trimming of the hyetograph tails is performed MEAN, IN PERCENT L- SC A L E, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 hyetographs EXPLANATION A. B. 170 No Trimming of Hyetograph Tails The relation between values for the mean and median of 0–12 hr dimensionless hyetographs computed with no tail trimming and values for L-scale are shown on figure 51. The relation between mean, median, and L-scale for the 12–24 hr and 24 hr and greater duration are shown on figures 52 and 53, respectively. Summary statistics for each of these figures are listed in table 25. For example, the arithmetic mean of the 0–12 hr dimensionless hyetograph means (graph A), medians (graph B), and L-scale values (graphs A and B) are 62.0 percent, 66.4 percent, and 16.0 percent, respectively, and the medians of the hyetograph means, medians, and L-scale values are 64.1 percent, 79 percent, and 16.6 percent, respectively. Similar discussion of the characteristics of figures 48–50 can be made for figures 51–53. The 0–12 hr duration graphs (fig. 51) show similar pronounced clustering of the mean and medians. The 12–24 hr duration graphs (fig. 52) also show pronounced clustering—more than that visible on figure 49. The L-scale values for the untrimmed hyetographs visible to more tightly approach a limiting value for a given mean than seen on figures 48–50. This is attributable to the lack of trimming for the figures 51– 53. It is unknown which tail trimming method is preferable when judged by the degree of approach to the limiting L-scale values. 171 Figure 51. Relation between hyetograph mean or median and L-scale for 0–12 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed An inspection of table 25 shows that the mean and median summary statistics increase slightly when no trimming is used. However, numerically the statistics are generally comparable. Finally, it is concluded that few visual differences between the MEAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 hyetographs EXPLANATION A. B. 172 data bases can be seen; hence, the five data bases should be combined if no tail trimming method is used. Figure 52. Relation between hyetograph mean or median and L-scale for 12–24 hr storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed MEAN, IN PERCENT L- SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L- SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 hyetographs EXPLANATION A. B. 173 Figure 53. Relation between hyetograph mean or median and L-scale for 24 hr and greater storm durations and depths greater than or equal to one inch for each of the five hyetograph data base modules when no trimming of the hyetograph tails is performed MEAN, IN PERCENT L - SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 MEDIAN, IN PERCENT L- SCALE, I N PERCENT 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Austin hyetographs Dallas hyetographs Fort Worth hyetographs San Antonio hyetographs smallruralsheds hyetographs EXPLANATION A. B. 174 Figure 54. Illustration of geometric reasoning behind the limiting of L-scale values with the hyetograph mean as seen in graphs A of figures 48–53 Modal Analysis of Hyetograph Statistics The relation between the mean or median and L-scale of dimensionless hyetographs for storm depths greater than or equal to 1 in. (25.4 mm) was documented on figures 48–53 of the previous section. The expected hyetograph could be defined by the summary statistics (table 25) of the figures. For example, an expected hyetograph for the 0–12 hr duration could be defined as a distribution having a mean of 59.6 and L-scale of 15.5 (see fig. 48 and row 1, column 1 and row 3, column 1 of table 25). However, the coordinate (59.6, 15.5) when plotted on graph A of figure 48 would lay just outside of the generalized cluster indicated on the figure. In the author’s opinion, however, the most “likely” hyetograph—the expected hyetograph—should be defined somewhere within the cluster. There is a strong tendency for the joint values of mean and L-scale of a random hyetograph to occur within the approximate space of the cluster. Pronounced clusters also were visible on figures 51 and 52. Less PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 100 0 100 0 100 0 100 large mean moderate mean small mean small L-scale Representative observed dimensionless hyetograph large small L-scale Shallow central slopes are consistent with little spread in the data and hence small L-scale values. Steep central slopes are consistent with large spread in the data and hence large L-scale values. Representative slope near the central regions of the distribution EXPLANATION L-scale FRONT- LOADED CENTER- LOADED BACK- LOADED 175 pronounced clusters are visible on figure 49. No obvious clustering is visible on figures 50 and 53. In the opinion of the author, it is possible that not all of the summary statistics in table 25 reliably represent the joint distribution of the mean and L-scale values. The author concludes that when joint clustering of the values is visible that the mode or “most likely values” of the mean and L-scale might be preferable to define the expected hyetograph. The graphical modal analysis of the data on figures 48–53 is presented in this subsection. The modal analysis for the three durations considered (0–12 hr, 12–24 hr, and 24 hr and greater) is shown on figures 55–60. The double one-percent tail trimming method is represented on figures 55–57, and the no-tail-trimming method is represented on figures 58–60. Because the mean, median, and L-scale statistics are not integers but are real numbers, aggregation of the values within intervals is required to define the general distributional of the data and hence the mode. For the mean and median 4-percent wide interval was used and a 2-percent wide interval was used for the L-scale values. Within each interval the number or count of values was determined. For purposes of illustration the count is plotted on the right-hand side of each interval—an offset of half the interval width to the left is needed to more accurately compute the mode. The symbol plotted for statistics of zero (far left of the figures) is for a zero-width interval. The interval widths and the right-hand side plotting styles prudently are indicated on figures 55–60. 176 Figure 55. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 0–12 hr and depths equal to or greater than one inch The most likely value for the hyetograph means (graphs A on figures 55–60) and the L-scale values (graphs C) is indicated by the labeled thin vertical line. The modes for the medians (graphs B) explicitly are not identified because the counts generally 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESS HYETOGRAPH MEAN, IN PERCENT COUNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT COUNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT COUNT 0 4 8 12162024283236404448525660 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is Symbol is plotted at right- Symbol is plotted at right-hand side of interval. 2 percent. Symbol is plotted on right-hand hand side of interval. A. B. C. Estimated mode Estimated mode (18-1) = 17 percent (69-2) = 67 percent side of interval. 177 are similar across nearly all intervals. It is noted that the mode of the medians are generally greater than about 88 percent. Figure 56. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 12–24 hr and depths equal to or greater than one inch 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESSS HYETOGRAPH MEAN, IN PERCENT C O UNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT CO UNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT COU N T 0 4 8 12162024283236404448525660 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is 2 percent. Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. A. B. C. Estimated mode Estimated mode (16-1) = 15 percent (68-2) = 66 percent 178 Figure 57. Modal analysis of mean, median, and L-scale statistics of double one-percent trimmed dimensionless hyetographs having storm durations of 24 hr and greater and depths equal to or greater than one inch 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESS HYETOGRAPH MEAN, IN PERCENT COUNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT COUNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT COUNT 0 4 8 12162024283236404448525660 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is 2 percent. Symbol is plotted at right- hand side of interval. Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. A. B. C. Estimated mode Estimated mode (15-1) = 14 percent (56-2) = 54 percent 179 Figure 58. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 0–12 hr and depths equal to or greater than one inch 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESS HYETOGRAPH MEAN, IN PERCENT COUNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT CO UNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT CO UNT 0 4 8 12162024283236404448525660 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is 2 percent. Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. A. B. C. Estimated mode (20-1) = 19 percent Estimated mode (80-2) = 78 percent 180 Figure 59. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 12–24 hr and depths equal to or greater than one inch 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESS HYETOGRAPH MEAN, IN PERCENT COUNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT CO UNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT CO UNT 0 4 8 12162024283236404448525660 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is 2 percent. Symbol is plotted at right- Symbol plotted at right- Symbol is plotted at right-hand side of interval. hand side of interval. hand side of interval. A. B. C. Estimated mode (76-2) = 74 percent Estimated mode (18-1) = 17 percent 181 Figure 60. Modal analysis of mean, median, and L-scale statistics of untrimmed dimensionless hyetographs having storm durations of 24 hr and greater and depths equal to or greater than one inch The graphically estimated mode indicated on the figures is an interpretation of the peak of the overall distributional shape of the plotted points and is not always coincident with the point with the largest count. Further, the half-interval offset to the left of the vertical line on the graphs is indicated on the figures for clarity. The author 0 4 12 20 28 36 44 52 60 68 76 84 92 100 DIMENSIONLESS HYETOGRAPH MEAN, IN PERCENT COUNT 0 4 12 20 28 36 44 52 60 68 76 84 92 100 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH MEDIAN, IN PERCENT CO UNT 0 10 20 30 40 50 60 70 DIMENSIONLESS HYETOGRAPH L-SCALE, IN PERCENT CO UNT 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 0 40 80 120 150 Group interval is 4 percent. Group interval is 4 percent. Group interval is 2 percent. Symbol is plotted at right- Symbol is plotted at right-hand side of interval. Symbol is plotted at right-hand side of interval. hand side of interval. A. B. C. Estimated mode Estimated mode (20-1) = 19 percent (56-2) = 54 percent 182 explicitly acknowledges that the graphically estimated mode is contrary to the strict interpretation of an empirical or sample mode. The author concludes that it is necessary and prudent to visually consider the trends in the plotted points when judging and hence estimating the most likely value for the mean and L-scale statistics. The author also acknowledges that other analysts might determine alternative values. Finally, graphical modes of the statistics for the 24 hr and greater storm duration without the Dallas data were separately determined (graphs not presented here), and the modal values also reported in table 25. Chapter Conclusions The follow conclusions drawn from the analysis in this chapter include: 1. The 0–12 hr and 12–24 hr durations dimensionless hyetographs appear compatible. When overall means and medians are compared, these statistics for each duration are very similar. These two durations will continue to remain separate for the remainder of this dissertation. 2. The 24 hr and greater duration dimensionless hyetographs have statistics that appear dissimilar from the other two durations. Further, consultation of storm depth dependency of the hyetograph median and L-moments suggests that the 24 hr and greater duration is more heterogeneous than the shorter durations. 3. The dimensionless hyetographs for the 0–12 hr duration appears independent of storm depth—hence storm frequency—by the independence of the mean, median, and L-scale statistics on the storm depth. The hyetographs for the 183 12–24 hr duration are slightly dependent but it is assumed independent for subsequent analysis in this dissertation. 4. The dimensionless hyetograph distribution as a whole have near zero values for L-skew, which suggests that the distributions are nearly symmetrical. 5. The L-kurtosis of the dimensionless hyetographs as a whole have L-kurtosis values larger than existing L-moment compatible distributions, such as the Normal or the Logistic. Albeit, these distributions are not similarly bounded as the hyetograph distributions are. 6. The dimensionless hyetographs appear affected by the month or season of occurrence. Although the relations between the monthly mean of the mean, median, and L-scale statistics and the month of the calendar are vague, it seems that storms occurring in April through June and again in October and November are more front-loaded than at other times of the year. For purposes of application, it is concluded that annual patterns of the hyetograph distribution are minor and the season or month of storm occurrence is not a critical factor in expected hyetograph estimation. 7. Little geographic differences are seen through graphical analysis of the mean, median, and L-scale statistics. Further, each of the data base modules appears compatible with the other four modules. The lone exception appears to be storms have 24 hr and greater durations for the Dallas data. Separate statistics without consideration of the Dallas data are reported. 184 8. The modal analysis suggests that mean and L-scale values for the 0–12 hr and 12–24 hr durations are different. The analysis also suggests that the graphically estimated modes might be more reliable predicators of most likely mean and L-scale of the dimensionless hyetographs than either the mean or median statistics. 185 CHAPTER 6 L-GAMMA DISTRIBUTION Introduction A promising two-parameter distribution for modeling dimensionless hyetographs is referred to by the author as the L-gamma distribution. The distribution does not appear to have prior description. Written communications to other researchers with interests in quantile functions or L-moments has provided no prior reference (Gilchrist, 2002; Hosking, 2002; Kroll, 2002; Serfling, 2002; Vogel, 2002; Wallis, 2002). It is important to stress that the L-gamma distribution as applied here provides a functional form of the hyetograph and is not intended to represent a random variable whose distribution has a quantile function represented by the hyetograph. The linkage between a quantile function and the hyetograph is useful in the context of the hyetograph modeling objective here but is artificial in a statistical sense. This linkage has precedence in the hydrologic sciences and applications similar to dimensionless hyetograph modeling (Yue and others, 2002, and references therein). The quantile function, , of the L-gamma distribution is , (58) where is the quantile for a nonexceedance probability and . The variables and are shape parameters that require estimation. The is shown separately on the right-hand side of the equation to facilitate subsequent derivations. QF() QF() ΛbcF,,()F b e c 1 F–() e c F b e cF– === Λ bcF,,() F 0 F 1≤≤ bc c 186 The range of the function is for . The curves on figure 61 provide examples of shapes for the L-gamma distribution. The motivation for the L-gamma distribution is that it has an explicit quantile function form, an algebraic first derivative, and is compatible with the theory of L-moments. Further, the L-gamma distribution is attractive for dimensionless hyetograph modeling because the quantiles are bounded by 0 and 1 regardless of the values of the parameters. The boundedness of the distribution is critical in this modeling because, by definition, the dimensionless hyetograph has 0 and 1 lower and upper bounds, respectively. The presence of the quantile function form facilitates practical application of the distribution in hyetograph computations because the distribution is compact and algebraically simple. A simple first derivative permits direct computation of rainfall rates. Finally, the compatibility with the theory of L-moments is needed for this dissertation. It is interesting to note that the L-gamma, with minor adjustments, normally is considered the probability density function (PDF) of the Gamma distribution (Evans and others, 2000, p. 98). The “L-gamma” name given to eq. 58 reflects the Gamma distribution heritage and the “L-” is added to reflect the applicability to L-moment statistical theory. 0 QF() 1≤≤0 F 1≤≤ 187 Figure 61. Example shapes of L-gamma distribution with selected pairs of (b, c) parameters The Gamma has a prominent place in statistics and hydrology (Cunnane, 1989, table 3.1; Stedinger and others, 1992, pp. 18.19–20; Ross, 1994, p. 231–232; Wilks, 1995, pp. 86–93; Hald, 1998; Karian and Dudewicz, 2000, pp. 75–76) and it has historical use in modeling of streamflow hydrographs (Nash, 1959; Croley, 1980; Jin, 1992; and Haan and others, 1994, and references therein). A Log-gamma distribution, which is explicitly based on the Gamma distribution, is described in the literature (Prentice, 1974; Lawless, 1980; and Balakrishnan and Chan, 1995a,b). NONEXCEEDANCE PROBABILITY, IN PERCENT QUANTI L E , I N PERCENT 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 ( 1 . 5 6 1 , - 2 . 4 4 1 ) ( 8 . 3 0 8 , 8 . 2 9 8 ) ( 0 . 2 1 9 , - 0 . 4 0 9 ) ( 2 . 8 1 9 , 2 . 8 1 8 ) ( 0 . 8 6 2 , - 0 . 1 9 2 ) 188 The L-gamma distribution contains two shape parameters and, therefore, is more flexible. The distribution and can better fit the hyetograph data than single parameter distributions. The greater flexibility is the principal motivation for using the L-gamma distribution in this dissertation. Other quantile functions, which obtain similar bounds, are the one-parameter Power (Evans and others, 2000, chap. 3; Gilchrist, 2000, p. 120–121). (59) and the Govindarajulu (Gilchrist, 2000, p. 139) (60) functions. The parameter controls the shape of both distributions. Besides noting that the L-gamma is the product of the Power distribution and (the cumulative distribution function of the Log distribution). The Log distribution (Serfling, written commun., 2002) is attributable to Tocher (1954), Pinkham (1961), and references therein, and Serfling reports that the Log distribution in practice is quite uncommon. The author was not familiar with the Log distribution at the time that the L-gamma was formulated. The noncontinuous triangular distribution used in a previous chapter is another example of a one-parameter distribution. Various parameterization styles of the four (eq. 95, referenced out of sequence) and five parameter Generalized Lambda distributions have quantile functions that are QF() F β = QF() β 1+()F β βF β 1+() –= β e c 1 F–() 189 bounded by 0 and 1. Parameter estimation for these distributions is complex (Karian and Dudewicz, 2000; Gilchrist 2000). The so-called symmetric Lambda or Tukey- Lambda distribution (Gilchrist, 2000, p. 157–159), in which parameter estimation is slightly more tractable, is not suitable for hyetograph modeling because an asymmetry of the hyetograph distribution is anticipated. Parameter Constraints The first and second derivatives of the L-gamma distribution are (61.1) (61.2) If the parameters and of eq. 58 meet two restrictions then the function is monotonically increasing on , and therefore a valid quantile function (Gilchrist, 2000, chap. 1). To meet the monotonic condition, the first derivative, a quantile density function, must be greater than zero. Hence, for a monotonic increase on , then and (62.1) . (62.2) For , is a degenerate case, but for , , which in order to have , implies that and that . Therefore, eq. 58 is a quantile function if and only if both and . Fd d Λ F() Λ′ e c 1 F–() bF b 1– cF b –[]== d dF ------ Λ′ Λ″ e c 1 F–() bb 1–()F b 2– 2bcF b 1– c 2 F b +–== bc 0 F 1≤≤ 0 F 1≤≤ Λ'0≥ 0 e c 1 F–() ≤ bF b 1– cF b –[] 0 bF b 1– cF b –≤ F 0= Λ′ 0= F 1=0bc–≤ Λ′ 0≥ bc≥ b 0≥ bc≥ b 0≥ 190 L-moments of the Distribution To fit the L-gamma to a sample, the parameters and require estimation. A natural estimation procedure is the Method of L-moments (MLM) (Hosking, 1990, p. 119). The method equates the sample L-moments and expressions for the theoretical L-moments of a distribution; the parameter values then are solved. The expressions for the theoretical L-moments of the L-gamma distribution are derived in this section. The first L-moment is the familiar mean and is the quantity . (63) Evaluation of the right hand side of eq. 63 requires the Incomplete Gamma function ( ) . (64) The complete Gamma function is (eq. 57). It is useful to multiply eq. 64 by and solve for the integral . (65) A useful property of the Gamma function is . (66) bc λ 1 Λ F()Fd 0 1 ∫ e c F b e cF– Fd 0 1 ∫ == P n m– Pmnx(, ) nγ mnx,() Γ m() ----------------------- 1 Γ m() ------------- F m 1– 0 x ∫ e nF– dF== Γ x() e c e c F m 1– e nF– Fd 0 1 ∫ e c n m– Γ m()Pmn,()= Γ m 2+() Γ m 1+() ---------------------- m 1+= 191 The definitions, identities, and properties of the Gamma and related functions are described in Abramowitz and Stegun (1964). Continuation of the first L-moment definition in eq. 63 coupled with the Incomplete Gamma function in which and provides . (67) The second L-moment, L-scale, measures the spread of a distribution and is and (68.1) . (68.2) The standard deviation is related to L-scale by (Hosking, 1990, p. 120). Applying the Incomplete Gamma function to eq. 68.2 with and results in . (69) A useful identity from eq. 69 is . (70) The coefficient of L-variation, L-CV, is mb1+= nc= λ 1 e c c b 1+ ------------ Γ b 1+()Pb 1 c,+()= λ 2 Λ F()2F 1–()Fd 0 1 ∫ = λ 2 2e c F b 1+ e cF– Fd 0 1 ∫ e c F b e cF– Fd 0 1 ∫ –= σλ 2 π= mb2+= nc= λ 2 2e c c b 2+ ------------ Γ b 2+()Pb 2 c,+()λ 1 –= λ 2 λ 1 + 2 ----------------- e c F b 1+ e cF– Fd 0 1 ∫ = 192 . (71) It follows that . (72) Which upon simplification, eq. 72 yields . (73) The third L-moment measures the skew of a distribution and is , (74.1) , and (74.2) . (74.3) Substituting the first and second L-moments and applying the Incomplete Gamma function with and as the first integral results in and (75.1) (75.2) A useful identity from eq. 75.2 is τ 2 λ 2 λ 1 ⁄= τ 2 2e c c b–2– Γ b 2+()Pb 2+ c(,)e c c b–1– Γ b 1+()Pb 1+ c(,)– e c c b–1– Γ b 1+()Pb 1+ c(,) ----------------------------------------------------------------------------------------------------------------------------------------------- = τ 2 2 c --- b 1+() Pb 2+ c(,) Pb 1+ c(,) ------------------------ 1–= λ 3 Λ F() 0 1 ∫ 6F 2 6F 1+– dF= λ 3 e c F b e cF– 6F 2 6F–1+()Fd 0 1 ∫ = λ 3 6 e c F b 2+ e cF– Fd 0 1 ∫ 6 e c F b 1+ e cF– Fd 0 1 ∫ e c F b e cF– Fd 0 1 ∫ +–= mb3+= nc= λ 3 6e c c b 3+ ------------ Γ b 3+()Pb 3 c,+()6 λ 2 λ 1 + 2 -----------------   λ 1 +–= λ 3 6e c c b 3+ ------------ Γ b 3+()Pb 3 c,+()3λ 2 2λ 1 ––= 193 . (76) A more conventional expression of skew is L-skew and is . (77) The most convenient means to compute L-skew is through eq. 77 rather than through an algebraic expansion of eq. 75.2 divided by eq. 70. The fourth L-moment measures the kurtosis of a distribution and is and (78.1) . . . (78.2) . (78.3) Substituting the first, second, and third L-moments and applying the Incomplete Gamma function with and for the first integral in eq. 78.2 results in . . . (79.1) . (79.2) Expanding and collecting the L-moment terms results in λ 3 3λ 2 2λ 1 ++ 6 ------------------------------------ e c F b 2+ e cF– Fd 0 1 ∫ = τ 3 λ 3 λ 2 ⁄= λ 4 Λ F()20F 3 30F 2 –12F 1–+ Fd 0 1 ∫ = λ 4 20 e c F b 3+ e cF– Fd 0 1 ∫ 30 e c F b 2+ e cF– F +d 0 1 ∫ –= 12+ e c F b 1+ e cF– Fd 0 1 ∫ e c F b e cF– Fd 0 1 ∫ – mb4+= nc= λ 4 20e c c b 4+ ------------ Γ b 4+()Pb 4+ c(,)30 λ 3 3λ 2 2λ 1 ++ 6 ------------------------------------   +–= 12+ λ 2 λ 1 + 2 -----------------   λ 1 – 194 . (80) A more conventional expression of L-moment kurtosis ( ) is the quantity . (81) The most convenient means to compute L-kurtosis is through eq. 81 rather than through an algebraic expansion of eq. 80 divided by eq. 70. Equations were derived for each of the first four L-moments and L-CV in terms of the parameters and . Any two equations can be used to estimate and . Convention is to fit a distribution to the lowest order moments as the higher moments increasingly reflect less information about the population. Parameter estimation by MLM using eqs. 64 and 70 is difficult because there are no explicit solutions for the parameters in terms of the L-moments. Evaluation of the Gamma and Incomplete Gamma functions require numerical techniques. Although computer algorithms for the Gamma and Incomplete Gamma functions are described by Press and others (1992, pp. 206–213) and root solution schemes seem possible, it was decided to implement an alternative method. Another reason to pursue an alternative method is that the parameter space of the L-gamma distribution based on and is restricted. From the Incomplete Gamma function integral definition and numerical recipes (Press and others, 1993), the λ 4 20e c c b 4+ ------------ Γ b 4+()Pb 4+ c(,)5λ 3 –9λ 2 –5λ 1 –= τ 4 τ 4 λ 4 λ 2 ⁄= bc bc λ 1 λ 2 195 Incomplete Gamma function is difficult to solve for (complex numbers are involved); the theoretical L-moments of the L-gamma therefore are difficult to formulate for . The L-gamma, however, remains a valid quantile function for . For example, three of the five quantile curves on figure 25 have . The alternative method avoids this impediment. c36c3c79c68c87c72c3c81c82c87c72c3c11c39c72c70c17c3c21c19c19c21c12c3c85c72c74c68c85c71c76c81c74c3c86c82c79c88c87c76c82c81c3c82c73c3c87c75c72c3c44c81c70c82c80c83c79c72c87c72c3 c42c68c80c80c68c3c73c88c81c70c87c76c82c81c3c73c82c85c3 c3c76c86c3c81c72c72c71c72c71c17c3c36c70c70c82c85c71c76c81c74c3c87c82c3c70c82c80c80c88c81c76c70c68c87c76c82c81c3 c90c76c87c75c3c43c82c86c78c76c81c74c3c11c21c19c19c21c12c15c3c87c75c72c3c85c72c70c76c83c72c86c3c69c92c3c51c85c72c86c86c3c68c81c71c3c82c87c75c72c85c86c3c11c20c28c28c22c12c3c68c85c72c3 c76c81c70c82c80c83c79c72c87c72c17c3c43c82c86c78c76c81c74c3c85c72c83c82c85c87c86c3c87c75c68c87c3c87c75c72c3c44c81c70c82c80c83c79c72c87c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c3 c51c11c69c15c70c12c3c73c82c85c3 c3c76c86 c15c3c90c75c72c85c72 c3c68c81c71 c17 c55c75c72c3c73c88c81c70c87c76c82c81c3 c3c76c86c3c87c75c72c3c70c82c81c73c79c88c72c81c87c3c75c92c83c72c85c74c72c82c80c72c87c85c76c70c3c73c88c81c70c87c76c82c81c3 c11c36c69c85c68c80c82c90c76c87c93c15c3c83c17c3c24c19c23c12c15c3c68c81c71c3c51c82c70c75c75c68c80c80c72c85c3c54c92c80c69c82c79c86c3c68c85c72c3c71c72c86c70c85c76c69c72c71c3c76c81c3 c36c69c85c68c80c82c90c76c87c93c3c68c81c71c3c54c87c72c74c88c81c3c11c20c28c25c23c15c3c83c17c3c21c24c25c12c17c3c55c75c72c86c72c3c72c84c88c68c87c76c82c81c86c3c90c72c85c72c3c81c82c87c3 c88c86c72c71c3c87c82c3c71c72c73c76c81c72c3c86c82c79c88c87c76c82c81c3c86c83c68c70c72c3c82c73c3c87c75c72c3c47c16c74c68c80c80c68c3c73c82c85c3 c17 Parameter Estimation The alternative parameter estimation method, called the modified Method of L-moments (M-MLM), is based on the median ( ), inter-tercile range ( ), the mean ( ), and L-scale ( ) and is described in this section. Not only is the method straightforward to implement but also is applicable for . The theoretical median of a distribution is the quantile value for which . The sample median can be computed from the sample order statistics of the sample by the c 0< c 0< c 0< c 0< c 0< c 0< P α x,() x α α ----- Φαα 1 x–,+,()= Φ abx,,() a() k b() k --------- x k k! ---- × k 0= n ∑ = x() k Γ xk+() Γ x() --------------------= Φ c 0< MITR λ 1 λ 2 c 0< F 0.5= x 1:n x 2:n … x n:n ≤≤≤ 196 following (Hollander and Wolfe, 1973, p. 453; David, 1981, p. 5): if is even ( , for some integer ) then and if is odd ( for some integer ) then . Alternatively, if estimates of the cumulative probabilities of each data value are not regularly spaced then the median can be estimated using linear interpolation. Following the definition of the theoretical median, the median of the L-gamma quantile function is . (82) The parameters or can be directly solved for, using the natural logarithm ( ), as or (83.1) . (83.2) The ability to explicitly express either or on the left-hand side of the equations is important. A comparison and contrast of eqs. 64 and 70 with eqs. 82 and 83 indicates that the later equations are considerably easier to use. To continue with the M-MLM development, the definition of the inter-tercile range is used. The is defined as the difference between the upper tercile and the lower tercile . The can be expressed in terms of the expectations of order statistics and the quantile function n n 2k= kMx k:n x k 1:n+ +()2⁄= n n 2k 1+= kMx k 1:n+ = M Λ 0.5() 0.5() b e c 10.5–() == bc ln b M()ln 0.5c– 0.5()ln ------------------------------- = c 2 M()ln b 0.5()ln× --------------------------- = bc ITR ITR UT LT ITR 197 . (84) Expansion of the in eq. 84, followed by algebraic manipulation results in . (85) Substitution of eq. 83.2 for expresses the as a function of only. . (86) Thus eqs. 82 and 86 can be used to estimate and in terms of the median and . Numerical methods are still required to solve for in eq. 86, but such root solutions are reasonably straightforward. Other “inter-cile ranges” such as the inter-quartile range ( ) can be defined in a similar fashion. The inter-cile range of order is denoted by . The for example has and the previously shown has . (87.1) (87.2) (87.3) ITR UT LT– Λ 2 3 ---   Λ 1 3 ---   –== Λ F() ITR 1 3 b ----- e 2 3 ---c 2 b e 1 3 ---– c 1–    = cITR b ITR 1 3 b ----- e 4 M()ln× 3b 0.5()ln× ------------------------------ 2 b e 2– M()ln× 3b 0.5()ln× ------------------------------ 1–    = bc ITR b IQR rIcR r IQR r 4= ITR r 3= IcR r upper lower–= IcR r Q r 1– r -----------   Q 1 r ---   –= IcR r e c r 1– r -----------   b e r 1– r -----------c– e c 1 r ---   b e 1 r ---c– –= 198 Root solution by numerical methods is needed to convert the median and into the parameters and of the L-gamma distribution. Once this is performed, then it is possible to convert the parameters into the mean and L-scale of the distribution. If lookup tables mapping all of these six variables with one another are available, then the M-MLM comprises the following steps. 1. Using the value for L-scale, consult a table to determine a range of median and values of the L-gamma that approximate the L-scale value. L-scale varies closely with so more attention to the and L-scale is suggested in this step. L-scale values generally identify which table group to use. 2. Using the range of suitable median and values from step 1, determine the most suitable pairing of these values that simultaneously most approximate both the mean and L-scale values of the data. 3. Using the most approximate median and values from step 2, perform a table lookup of the and parameters of the L-gamma distribution. In general, the M-MLM determines the pairing of the median and that produce an L-gamma distribution, which has a mean and L-scale best approximating those of the data. Then the “best” median and values are converted to the and parameters of the L-gamma distribution. Therefore, in order to provide simple and portable framework for parameter estimation, conversion maps (tables) of the solution space of the distribution were created (Appendix D). ITR bc ITR ITR ITR ITR ITR bc ITR ITR b c 199 Appendix D contains a series of six table groups. A table group is composed of four tables. Each table has columns for constant values of and rows for constant values of the median. Tables with cells containing the parameter are enumerated by D#.1; the parameter tables are enumerated by D#.2; the mean tables are enumerated by D#.3; and the L-scale tables are enumerated by D#.4. The tables were generated by a Perl (http://www.perl.org) computer program called c86c82c79c88c87c76c82c81c86c83c68c70c72c47c42c68c80c80c68c17c83c79 and is listed in Appendix E. In order for the program to operate, it was necessary to develop a Perl module to support a port of FORTRAN Gamma functions of Press and others (1992) and another module specifically for the L-gamma distribution. These modules are c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c80 and c47c42c68c80c80c68c39c76c86c87c85c76c69c88c87c76c82c81c17c83c80 and also are listed in Appendix E. The mean and L-scale of the L-gamma distribution can not be solved for using the Incomplete Gamma function for ; for the entries corresponding to a , direct simulation of the L-gamma distribution and subsequent computation of the L-moments was performed. The simulated values in tables D#.3 and D#.4 are denoted by a leading “s” on the table entries. Comparison of the entries with the simulated values to neighboring non-simulated cells suggests a smooth and anticipated transition from one cell to another. The performance of the simulation appears acceptable. ITR b c c 0< c 0< 200 Parameter Estimation Example To illustrate the use of the parameter space tables, consider the solution for a data set with a median of and inter-tercile range of . From table D5.1, is equal to 2.504 and from table D5.2, is equal to 2.239. The L-gamma distribution corresponding to these parameter values is shown on figure 62. The mean and L-scale values of a are available in tables D5.3 and D5.4, respectively. From table D5.3 and the entry for and , the mean is 0.513. From table D5.4 and the entry for and , the L-scale is 0.199. Finally, the value for the upper tercile is , and the value for the lower tercile is . Dashed vertical lines along the median and the terciles have been included in the figure for clarity and intersecting horizontal dashed lines correspond to the values for the median and the upper and lower terciles. Experiments (not presented here) suggest that the Govindarajulu distribution often mimics the shape of the L-gamma distribution when and are greater than one and similar in value. The Power distribution is considerably different from either of the other two distributions shown in the figure. M 0.54= ITR 0.48= bc Λ 2.504 2.239,() M 0.54= ITR 0.48= M 0.54= ITR 0.48= UT Q 23⁄() 23⁄() 2.504 e 2.239 1 2 3⁄–() 0.764== = LT Q 13⁄() 13⁄() 2.504 e 2.239 1 1 3⁄–() 0.284== = bc 201 Figure 62. Comparison between the L-gamma distribution fit to a median of 0.54 and L-scale of 0.24 and Govindarajulu and Power distributions fit to the same median and the Beta distribution fit to L-scale and the mean of the L-gamma distribution Although, it does not have a quantile function form, the Beta distribution (Ross, 1994, pp. 235–236; Evans and others, 2000, pp. 34–42) is numerically integrated quantile bounds of 0 and 1. Because the Beta distribution is well known it also is shown on figure 62. The probability density function of the Beta distribution is for (88) where (89) NONEXCEEDANCE PROBABILITY IN PERCENT Q U ANTI LE*1 00 00 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 L-GAMMA (2.504, 2.239) G O V I N D A R A J U L U P O W E R P O W E R median = 54 upper tercile = 76.4 lower tercile = 28.4 B E T A ( t h i n n e s t l i n e ) f x() 1 Bab,() ----------------- x a 1– 1 x–() b 1– =0x 1<< Bab,() x a 1– 1 x–() b 1– xd 0 1 ∫ = 202 and and are shape parameters. Parameter estimation for the Beta distribution is straightforward using the product moments (mean and variance) in terms of the parameters as follows. , and (90) . (91) The parameters in terms of the “moments” (Evans and others, 2000, p. 40) are , and (92) . (93) The product moment variance ( ) can be estimated using L-scale as ( ) according to the preference of the “true devotee of L-moments” (Hosking, 1990, p. 117). Using a mean and L-scale of 0.513 and 0.199, respectively, Beta parameter estimates are 0.517 and 0.491 for and , respectively. The quantiles for this function also are shown on figure 62. The Beta distribution is used for comparisons to the L-gamma in the next chapter. The steepest portion of the L-gamma distribution on figure 62 exists where the second derivative of the function is zero. Using eq. 61.2 and substituting ab EX[] µ λ 1 a ab+ ------------== = Var X[] EX 2 [] σ 2 λ 2 π() 2 ab ab+() 2 ab1++() ---------------------------------------------- === = a λ 1 λ 1 1 λ 1 –() λ 2 π() 2 ------------------------- 1–    = b 1 λ 1 –() λ 1 1 λ 1 –() λ 2 π() 2 ------------------------- 1–    = σ 2 λ 2 π() 2 ab b 2.504= 203 and , setting the equation to zero, and solving for , a value of is determined. Thus the L-gamma distribution is steepest at this nonexceedance probability; this is confirmed by inspection of figure 62 confirms. The rate of change is computed using the first derivative (eq. 61.1), substituting the values for , , and , the first derivative is 1.55 percent per percent. A second and more involved parameter estimation example is needed to fully illustrate the M-MLM. Suppose a data set has a mean of 66 percent (0.66) and an L-scale of 17 percent (0.17). After browsing the tables D#.3 and D#.4, it is determined that tables D4.3 and D4.4 contain cells with matching column and row locations that contain entries whose values approximate the mean and L-scale of the data. Specifically, the entries of 0.659 (table D4.3) and 0.168 (table D4.4) for mean and L-scale, respectively, are located on for the median row of 0.75 and column of 0.33 and are closest to the 0.66 (mean) and 0.17 (L-scale) values of the data. Hence, it is estimated the an L-gamma distribution with a mean of 0.66 and L-scale of 0.17 has a median of 0.75 and of 0.33. It follows from table D4.1 that the parameter is 1.195 and from table D4.2 that the parameter is 1.081. Verification of Theoretical L-moments Validation of the derivations of the L-moments for the L-gamma distribution is needed. Example computations presented in this section verify eqs. 63–81. Consider an L-gamma distribution having parameter values of 3 and 1 for and , respectively. c 2.239= FF0.412= bc F ITR ITR b c bc 204 The sample L-moment statistics of the L-gamma computed from unbiased estimators for the distribution are listed in table 26. The sample comprised 200 quantile values computed between at 0.05 nonexceedance probability spacing intervals. The computer program c88c86c72c47c74c68c80c80c68c39c76c86c87c85c76c69c88c87c76c82c81c17c83l was used to compute the quantiles. The program is available in Appendix E. In order to evaluate the theoretical L-moments of the L-gamma, values for the Gamma and Incomplete gamma functions are required. A computer program, c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79 was written based on algorithms provided by Press and others (1992) to compute the values. The program is available in Appendix E. The output of the program for and various values of in which is an integer are shown on figure 63. The values for each function seen in the figure are used in the following computations. Table 26. Comparison of theoretical L-moment and sample L-moment statistics for L-gamma distribution chosen for the verification example [L-CV, coefficient of L-variation. Percent difference is computed by theoretical value minus sample estimate divided by theoretical value. The L-gamma distribution has values for b and c of 3 and 1, respectively. Supporting computation are described in the text.] The theoretical mean is computed by eq. 67, and the computation steps are Moment type Moment symbol Sample estimates Theoretical values Percent difference Mean 31.1 31.0 -0.32 L-scale 17.0 16.8 -1.2 L-CV .546 .542 -.74 L-skew .242 .238 -1.7 L-kurtosis -.0047 -.0046 2.2 Λ 31,() 0 F 1≤≤ b 3= bi+ i λ 1 λ 2 τ τ 3 τ 4 205 , , and . c62c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3 c68c3c68c81c71c3c91c3c73c82c85c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c15c3c42c11c68c12c3c68c81c71 c44c81c70c82c80c83c79c72c87c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c15c3c51c11c68c15c91c12c3c68c85c72c3c85c72c84c88c76c85c72c71c3c82c81c3c70c82c80c80c68c81c71c3c79c76c81c72 c62c68c86c84c88c76c87c75c35c85c72c71c20c68c86c87c64c7c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3c22c3c20 c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c22c3c68c81c71c3c91c32c20 c6c3c3c3c42c11c68c12c29c42c11c22c12c32c20c17c28c28c28c28c28c28c28c28c28c28c28c28c28c27 c6c3c51c11c68c15c91c12c29c51c11c22c15c20c12c32c19c17c19c27c19c22c19c20c22c28c25c21c22c28c26c27c23c21 c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c19c17c20c25c19c25c19c21c26c28c21c23c26c28c24c25c26 c62c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3c23c3c20 c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c23c3c68c81c71c3c91c32c20 c6c3c3c3c42c11c68c12c29c42c11c23c12c32c24c17c28c28c28c28c28c28c28c28c28c28c28c28c28c23 c6c3c51c11c68c15c91c12c29c51c11c23c15c20c12c32c19c17c19c20c27c28c27c27c20c24c25c27c20c21c24c24c25c21 c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c19c17c20c20c22c28c21c27c28c23c19c27c26c24c22c22c25 c62c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3c24c3c20 c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c24c3c68c81c71c3c91c32c20 c6c3c3c3c42c11c68c12c29c42c11c24c12c32c21c22c17c28c28c28c28c28c28c28c28c28c28c28c28c26 c6c3c51c11c68c15c91c12c29c51c11c24c15c20c12c32c19c17c19c19c22c25c24c28c27c23c25c26c25c22c26c23c24c28c27 c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c19c17c19c27c26c27c22c25c22c21c21c22c21c28c28c19c21c25 c62c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3c25c3c20 c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c25c3c68c81c71c3c91c32c20 c6c3c3c3c42c11c68c12c29c42c11c25c12c32c20c20c28c17c28c28c28c28c28c28c28c28c28c28c28c27 c6c3c51c11c68c15c91c12c29c51c11c25c15c20c12c32c19c17c19c19c19c24c28c23c20c27c23c27c20c22c19c25c20c27c23c27 c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c19c17c19c26c20c22c19c21c20c26c26c24c25c26c23c21c19c27 c62c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c3c88c86c72c42c68c80c80c68c41c88c81c70c87c76c82c81c86c17c83c79c3c26c3c20 c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c26c3c68c81c71c3c91c32c20 c6c3c3c3c42c11c68c12c29c42c11c26c12c32c26c21c19c17c19c19c19c19c19c19c19c19c19c19c19c24 c6c3c51c11c68c15c91c12c29c51c11c26c15c20c12c32c27c17c22c21c23c20c20c23c27c28c27c27c19c21c20c24c72c16c19c24 c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c19c17c19c24c28c28c22c22c25c21c26c21c26c20c22c26c24c28 Figure 63. Gamma and Incomplete Gamma function computation results for theoretical L-moments of L-gamma distribution verification example—G(a) is the Gamma function and P(a,x) is the Incomplete Gamma function The theoretical L-scale of the distribution is computed by eq. 69. Steps of the computation are shown below. λ 1 e c c b 1+ ------------ Γ b 1+()Pb 1 c,+()= λ 1 e 1 1 4 ----- Γ 4()P 41,()= λ 1 2.7182818 6 0.018988157×× 0.30969097== 206 , , and . It follows that L-CV from eq. 71 is . The theoretical third L-moment of the distribution is computed by eq. 75.2. Steps of the computation are shown below. , , , and . It follows that L-skew from eq. 77 is . λ 2 2e c c b 2+ ----------- Γ b 2+()Pb 2 c,+()λ 1 –= λ 2 2e 1 1 5 -------- Γ 5()P 51,()0.3097–= λ 2 5.4365637 24 0.0036598× 468 0.30969097–× 0.16783680== τλ 2 λ 1 ⁄ 0.16783680 0.30969097⁄ 0.54194929== = λ 3 6e c c b 3+ ------------ Γ b 3+()Pb 3 c,+()3λ 2 2λ 1 ––= λ 3 6e 1 1 6 -------- Γ 6()P 61,()3 0.16783680()2 0.30969097()––= λ 3 16.309691 120× 0.00059418481 1.1228923–×= λ 3 0.040024136= τ 3 λ 3 λ 2 ⁄ 0.040024136 0.16783680⁄ 0.23847056== = 207 The theoretical fourth L-moment of the distribution is computed by eq. 80. Steps of the computation are shown below. , , , and . It follows that L-kurtosis from eq. 81 is . The theoretical values for the mean, L-scale, L-CV, L-skew, and L-kurtosis also are listed in table 26. In general, there is agreement between the sample estimates and the theoretical values. The percent differences are all small. Therefore, it is concluded that the derivations shown in eqs. 63–81 are correct. It is important to note that careful track of a substantial number of decimal places in the above computations is required. Eight significant figures were propagated through all of the above computations. Fewer significant figures rapidly cause divergence in the sample estimates and the theoretical values. λ 4 20e c c b 4+ ------------ Γ b 4+()Pb 4+ c(,)5λ 3 –9λ 2 –5λ 1 –= λ 4 20e 1 1 7 ----------- Γ 7()P 71( , ) 5 0.040024136()– 9 0.16783680()– 5 0.30969097()–= λ 4 54.365637 720× 0.000083241149 3.2591067–×= λ 4 0.00077690102–= τ 4 λ 4 λ 2 ⁄ 0.00077690102– 0.16783680⁄ 0.0046289075–== = 208 Chapter Conclusions The two-parameter L-gamma distribution is expressible in an algebraically simple quantile function. The distribution has bounds of zero and one. The bounds of the distribution are attractive from the standpoint of modeling data sets expressed in percentage such as dimensionless hyetographs. The distribution is symbolically related to the probability density function of the well-known Gamma distribution. When restrictions on parameter values are made, the function is monotonically increasing with nonexceedance probability, which is a requirement of a quantile distribution function. The L-moments of the L-gamma can be defined, but parameter estimation by the Method of L-moments using the L-moments alone is difficult; complex numerical techniques are required. A modified Method of L-moments for parameter estimation was presented instead. The modified Method of L-moments is suggested and is based on the median and inter-tercile range. Although numerical techniques are still required with the modified method, the problem of parameter estimation is considerably more tractable. Using the modified method fortuitously increases the usable parameter space of the distribution into a region. The parameter space of the L-gamma distribution in terms of the median and inter- tercile range was mapped and lookup tables for parameter estimation are provided in Appendix D. Examples of table usage are provided. Illustrative computations of the c 0< 209 theoretical L-moments of a specified L-gamma distribution agree with sample estimates—the derivations are verified. Finally, a generalized L-gamma distribution can be formulated , (94) where and are location and scale parameters, respectively, and and remain shape parameters. Λ ABbc,,,()ABF b e c 1 F–() ×+= AB bc 210 CHAPTER 7 L-GAMMA MODEL OF DIMENSIONLESS RAINFALL HYETOGRAPHS KNOWN TO PRODUCE RUNOFF IN TEXAS Introduction In this chapter, the L-gamma distribution is used to model dimensionless runoff- producing hyetographs in Texas. Much of the analysis is an extension of chapters 3 and 5 and application of chapter 6. L-gamma distribution hyetograph models are fit in this chapter to the estimates of the mean and L-scale values of the hyetograph distributions. Statistics from the chapter 5 provide the basis for fitting the model. The L-gamma models are compared to those from chapter 3. The comparison is enhanced by inclusion of Beta distribution modeled hyetographs. The Beta distribution has been used to model dimensionless streamflow hydrographs in an analogous fashion to the hyetograph analysis here (Yue and others, 2002). The suitability of the L-gamma hyetograph model also is considered, and an alternative hyetograph analysis technique is used in the suitability assessment. L-gamma Model Each pair of mean and L-scale statistics (a statistic set) in table 25 can be used to estimate the expected hyetograph. The L-gamma distribution can be fit to a statistic set and the expected hyetograph defined. However, some of the statistics in table 25 are more favorable than others. The estimated parameters of the L-gamma distribution for the corresponding favorable statistic sets and duration are listed in table 27. The Dallas data was not used for the 24 hr and greater (up to about 3 days) storm duration; the mean and L-scale values listed within the parenthesis in table 25 were used. The tables 211 of the L-gamma distribution solution space in Appendix D were used as a first approximation of the parameters. Subsequently, modified versions of the c86c82c79c88c87c76c82c81c86c83c68c70c72c47c42c68c80c80c68c17c83c79 program with restricted but higher resolution iterations were used to improve the parameter estimates to four significant figures. The parameters were then verified by computation of sample L-moments directly from the distribution. Table 27. L-gamma distribution parameter estimates for modeling dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths [The statistics without the Dallas hyetographs were used for the 24 hr and greater duration.] Not all of the mean and L-scale statistics in table 25 are favorable and used for parameter estimates reported in tables 27 and 28 for reasons now described. The author strongly believes that the double one-percent tail trimming method produces more reliable statistics of the hyetograph than the absence of a tail trimming method. Figure 23 and the supporting discussion of the figure provides firm justification that Parameter of L-gamma distribution, Double one-percent tail trimming 0–12 hr duration 12–24 hr duration 24 hr and greater duration (percent) (percent) (percent) Parameter estimates from mean of values for the values of the hyetograph statistics b 0.7679 0.4775 0.2945 c .1436 -.3866 -.7907 Parameter estimates from median of values for the values of the hyetograph statistics b .9105 .5879 .3277 c .4297 -.1475 -.7929 Parameter estimates from graphically estimated modes for the values of the hyetograph statistics b 1.262 .7830 .3388 c 1.227 .4368 -.8152 Λ bc,() 212 some tail trimming is required. The right-most columns in table 25 thus explicitly are not considered here. Since the no-tail-trimmed statistics are prudently reported in table 25, other researchers could fit the L-gamma or Beta distributions to the data. The author has concluded that the modal analysis (see subsection Modal Analysis of Hyetograph Statistics in chapter 5) produces the most favorable mean and L-scale statistics for the 0–12 hr and 12–24 hr storm durations. And finally, the author considers reliable estimation of the 24 hr and greater duration hyetograph is problematic because of the perceived greater heterogeneity of the long-duration hyetographs. However, fitted distribution models for all of the statistic sets, including the 24 hr and greater duration and not just the graphical mode statistic sets, are presented in this chapter for completeness and for the benefit of other researchers. The expected hyetographs defined by the L-gamma distribution fit to each parameter pair in table 27 are shown on figures 64 and 65. Each graph on figure 64 shows a comparison between the three statistic estimation methods for a specific duration. Each graph on figure 65 shows a comparison between durations for each of the three statistic estimation methods. Figures 64 and 65 compliment each other. 213 Figure 64. Comparison between statistic estimation method for L-gamma distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERC E N T OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Parameters estimated from mean of hyetograph mean and L-scale values Parameters estimated from median of hyetograph mean and L-scale values Parameters estimated from graphical mode of hyetograph mean and L-scale values EXPLANATION 0–12 hr duration 12–24 hr duration 24 hr and greater duration A. B. C. 214 Figure 65. Comparison between L-gamma distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths for each statistic estimation method PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERC E N T OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Hyetograph for 0–12 hr duration Hyetograph for 12–24 hr duration Hyetograph for 24 hr and greater duration EXPLANATION Parameters estimated by means Parameters estimated by medians Parameters estimated by graphical mode A. B. C. 215 In general the curves for each statistic estimation procedure (fig. 64) are similar. The modal curves (thinnest lines) for the 0–12 hr and 12–24 hr duration are the most distinct from the other two curves. The largest differences are exhibited by the 0–12 hr modal curve. For the other durations the method of statistic estimation has limited influence on the defined expected hyetograph. Alternative curve groups are presented on figure 65 because the relative differences between the expected hyetograph for the three durations are easily visualized within each graph. For each statistic estimation method from figure 65, it is obvious that the 0–12 hr expected hyetograph has the smallest rainfall rates (as measured in percent and not absolute magnitude) early in the duration, but throughout the bulk of the storm duration larger rates are exhibited. Both the expected hyetographs for the 12–24 hr and 24 hr and greater durations have relatively large percentage rainfall rates at the storm beginning, which is followed by a period of reduced rates until the end of the storm. The L-gamma model suggests that the 24 hr and greater expected hyetograph exhibits a second period of increased rainfall rates near the end of the storm. This behavior is evidenced by the 24 hr and greater curve bending up as the end of the storm is approached. Finally, both the expected hyetographs for 12–24 hr and 24 hr and greater durations have large rainfall rates at very small duration percentages; a method to estimate more reasonable rainfall rates is described later. 216 Beta Model Each pair of mean and L-scale statistics (a statistic set) in table 25 can be used to estimate the expected hyetograph using the Beta distribution. The Beta distribution can be fit to a favorable statistic set (see previous section for discussion of favorable statistics). The estimated parameters of the Beta distribution are listed in table 28. The Dallas data was not used for the 24 hr and greater duration; the mean and L-scale values listed within the parenthesis in table 25 were used. Equations 92 and 93 were used to estimate the parameters. The Beta distributions corresponding to each parameter in table 28 are shown on figures 66 and 67. Table 28. Beta distribution parameter estimates for modeling dimensionless hyetographs for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths [The statistics without the Dallas hyetographs were used for the 24 hr and greater duration.] Parameter of Beta distribution Double one-percent tail trimming 0–12 hr duration 12–24 hr duration 24 hr and greater duration (percent) (percent) (percent) Parameter estimates from mean of values for the values of the hyetograph statistics a 1.305 1.622 1.815 b .8848 1.165 1.426 Parameter estimates from median of values for the values of the hyetograph statistics a 1.194 1.517 1.688 b .7636 1.033 1.392 Parameter estimates from graphically estimated modes for the values of the hyetograph statistics a .9616 1.435 1.638 b .4736 .7394 1.396 217 Figure 66. Comparison between statistic estimation method for Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths PERCENT OF ST ORM DEP T H 0 10 20 30 40 50 60 70 80 90 100 PERCENT O F ST ORM DEP T H 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Parameters estimated from mean of hyetograph mean and L-scale values Parameters estimated from median of hyetograph mean and L-scale values Parameters estimated from graphical mode of hyetograph mean and L-scale values EXPLANATION 0–12 hr duration 12–24 hr duration 24 hr and greater duration A. B. C. 218 Figure 67. Comparison between Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths for each statistic estimation method PERCENT OF ST ORM DEP T H 0 10 20 30 40 50 60 70 80 90 100 PERCENT O F ST ORM DEP T H 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Hyetograph for 0–12 hr duration Hyetograph for 12–24 hr duration Hyetograph for 24 hr and greater duration EXPLANATION Parameters estimated by means Parameters estimated by medians Parameters estimated by graphical mode A. B. C. 219 The c37c40c55c36c44c49c57 function of the well known Excel spreadsheet software package by Microsoft (http://www.microsoft.com) was used to compute the ordinates. Each graph on figure 66 shows a comparison between the three statistic estimation methods (mean, median, graphical mode) for a specific duration. Each graph on figure 67 shows a comparison between durations for each of the three statistic estimation methods. Figures 66 and 67 compliment each other. Similar observations as made for the expected hyetographs using the L-gamma distribution in the previous subsection (figs. 64 and 65) are made for the expected hyetographs modeled using the Beta distribution. Some notable differences are described in the next section. Model Comparison The L-gamma distribution models of the expected hyetograph for the three durations based on the graphical modes of the mean and L-scale statistics are presented on figure 68. The author has judged that the graphical modes of the mean and L-scale provide the most reliable values for the expected hyetographs. Therefore, only these modal-based expected hyetographs are considered on the figure. The other statistics were prudently presented so that other researchers could fit the L-gamma or Beta distributions. The triangular models (see chapter 3) of the hyetograph also are shown on figure 68. It is apparent that the L-gamma models are distinctively different in shape than those of the triangular models. The shape differences are due to differing functional 220 forms and the number of parameters. Since the L-gamma distribution was fit to both the mean and L-scale instead of solely the mean as is done for the triangular model, the L-gamma distribution is statistically preferable as it better mimics observed hyetographs. The peak rainfall rates for the triangular model occur at about 30 to 40 percent of the duration. All three L-gamma hyetographs exhibit their peak rainfall rates either at the beginning of the storm (12–24 hr and 24 hr and greater duration) or at about 12 percent (0–12 hr duration). In general, the L-gamma models predict more uniform rainfall rates than the triangular models do. This is evidenced by the more constant shape of the L-gamma curves. The Beta distribution also is a two-parameter distribution and therefore would arguably mimic the data in a fashion equivalent to the L-gamma because both distributions are fit to the same statistics. Because the distributions are not fit to high- order L-moments such as L-skew and L-kurtosis, differences in the model tails or specifically near the beginning and ending of the storm are expected. Comparisons showing differences between the Beta and the L-gamma hyetograph models for each duration for each statistic estimation method (mean, median, or mode) are shown on figures 69–71. The mean statistic estimation method is represented on figure 69 (derived from graphs A on figures 65 and 67), the median method on figure 70 (derived from graphs B on figures 65 and 67), and the graphical mode method on figure 71 (derived from graphs C on figures 65 and 67). 221 Figure 68. Comparison of L-gamma distribution and triangular hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths Inspection of the figures suggests the Beta distribution mimics the L-gamma or alternatively the L-gamma distribution mimics the Beta. One notable difference between the curves is that expected hyetographs for the 12–24 hr and 24 hr and greater durations do not have as high rainfall rates as the corresponding curves for the L-gamma distribution. Although the two distributions provide very similar fits, the L-gamma distribution model is preferable because it is expressed in a quantile function form. This makes the model simple to use and construction of the hyetograph straightforward as the analyst PERCENT OF STORM DURATION PERCENT O F ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 L-gamma distribution hyetograph model for 0–12 hr storm duration using graphical mode statistics for parameter estimation L-gamma distribution hyetograph model for 12–24 hr storm duration using graphical mode statistics for parameter estimation L-gamma distribution hyetograph model for 24 hr and greater storm duration using graphical mode statistics for parameter estimation Triangular hyetograph model for 0–24 hr storm duration Triangular hyetograph model for 24 hr and greater storm duration EXPLANATION Λ(0.3388, -0.8152) Λ(0.7830, 0.4368) Λ(1.262, 1.227) 222 is not encumbered with numerical integration and inversion necessary to use the Beta distribution. Furthermore, the first and second derivatives of the L-gamma are simple and permit direct computation of percentage rainfall rates. Example computation of rainfall rates are now presented. Computation of the maximum or incremental rainfall rates of expected hyetographs as measured by percent of storm depth can be important in practice. The maximum instantaneous rate for the 0–12 hr duration is straightforward. The second derivative (eq. 61.2) with and is used to solve for the percent of storm duration with the largest rate; the largest rate is at percent. The first derivative (eq. 61.1) with , , and then provides a maximum rate of 1.88 percent storm depth per percent storm duration. Suppose that an application requires a 12 hr duration and a rainfall depth of 6.9 in. (175.3 mm). The duration and depth values match those in the Example Application section in chapter 3. The largest rainfall rate using the 0–12 hr expected hyetograph thus is 1.08 in./hr (1.88×6.9 in. / 12 hr) or 27.5 mm/hr. The 1.08 in./hr rate is considerably larger than the 0.748 in./hr (19.0 mm/hr) rate predicted by the 0–24 hr triangular hyetograph (see the previously referenced Example Application section). The maximum rainfall rates for the 12–24 hr and 24 hr and greater durations are more problematic because the first derivative (eq. 61.1) is very large for very small values of . Therefore, it is suggested that the maximum rainfall rate be defined at the b 1.262= c 1.227= F 11.3= b 1.262= c 1.227= F 0.113= F 223 first increment of the hyetograph. Discrete hyetographs are often used in practice. For example, suppose an application requires a hyetograph duration of 48 hr and 15-minute time steps for the hyetograph are to be defined. Fifteen minutes (min) represents about 0.521 percent of the duration [100 × 15 min / (48 hr × 60 min)]. The first derivative at is or about 4.93 percent storm depth per percent storm duration. If the 48 hr rainfall depth for the application is 10 in. (254 mm), then the rainfall rate at the first 15-minute increment is 1.03 in./hr (4.93 × 10 in. / 48 hr) or 26.1 mm/hr. For comparison, the rainfall rate at the second 15-minute increment ( ) is 3.17 percent of storm depth per percent storm duration or 0.661 in./hr (16.8 mm/hr). F 0.00521= e 0.8152 1 0.00521–()– 0.3388 0.00521() 0.3388 1– 0.8152 0.00521() 0.3388 ––[] F 0.0104= 224 Figure 69. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the mean statistics for parameter estimation PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERC E N T OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 L-gamma distribution hyetograph model Beta distribution hyetograph model EXPLANATION 0–12 hr duration 12–24 hr duration 24 hr and greater duration A. B. C. 225 Figure 70. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the median statistics for parameter estimation PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERC E N T OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 L-gamma distribution hyetograph model Beta distribution hyetograph model EXPLANATION 0–12 hr duration 12–24 hr duration 24 hr and greater duration A. B. C. 226 Figure 71. Comparison between L-gamma and Beta distribution hyetograph models for 0–12 hr, 12–24 hr, and 24 hr and greater storm durations and one inch and greater storm depths using the graphical mode statistics for parameter estimation PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DEPTH 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERC E N T OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 L-gamma distribution hyetograph model Beta distribution hyetograph model EXPLANATION 0–12 hr duration 12–24 hr duration 24 hr and greater duration A. B. C. 227 Model Suitability The three L-gamma hyetograph models (best illustrated on figure 68) are fit to the most representative values of the dimensionless hyetograph mean and L-scale. The author is concerned that though each model is statistically defensible (acknowledging a weaker defense for the previously discussed heterogeneity for the 24 hr and greater duration) the models might be unsuitable because the models are based on statistics of statistics. It is difficult to visualize how the models are fit to individual hyetographs. The models might be inconsistent with results derived from alternative hyetograph analysis techniques. To assess suitability of the L-gamma model an alternative analysis of hyetographs that might rely less heavily on complex statistics of statistics, statistical inference, parametric models, and parameter estimation. The analysis is based on duration- interval hyetograph-ordinate statistics and is presented in the following subsection. Empirical Hyetograph Analysis The foundation of the alternative hyetograph analysis uses the concept of empirical hyetographs. Empirical hyetographs are produced from statistical analysis of the distribution of dimensionless hyetograph ordinates (percent storm depth) for evenly spaced intervals of storm duration. An n-th percentile of each interval defines the n-th percentile empirical hyetograph. The construction of empirical hyetographs is performed by decomposing each observed hyetograph (with dimension removed) into 2.5-percent wide intervals of percent storm duration. (A larger and lower resolution 228 5-percent interval also was investigated.) For the hyetograph ordinates within each interval, percentiles and other statistics are computed. The distribution of hyetograph ordinates is investigated using these interval statistics. Empirical hyetographs provide visualizations of the expected hyetograph without the need for a parametric model such as the triangular model or the Beta and L-gamma distributions. The median of the distribution of hyetograph ordinates within each 2.5-percent interval of percent storm duration defines the 50th percentile empirical hyetograph. Empirical hyetograph analysis also documents the uncertainties in the expected hyetograph—a feature not provided by the parametric models. It is important to consider that visualization of the influence of individual hyetographs on the model is difficult in the context of previous parametric models because the models are fit to the statistics of other statistics. An excessive degree of smoothing might have occurred. For example, the graphical display of the models on figures such as figure 68 convey little information about the underlying structure of individual hyetographs. An example of the alternative hyetograph analysis is provided through graphical display on figure 72. The empirical hyetograph (defined by the median statistic) and ancillary statistics for the 0–12 hr duration, and depth greater than 1 in. (25.4 mm) hyetograph is shown in the figure. On graph A of figure 72, each of the 621 events meeting the duration and depth criteria are plotted in a light shade of grey. As 229 previously done, double one-percent trimming of the leading and trailing tails was performed. Superimposed on the individual hyetographs is the empirical statistical analysis. On the figure, the median, lower and upper quartiles (25th or 75th percentile), and lower and upper deciles (10th and 90th percentiles) within each 2.5-percent wide interval of percent storm duration are plotted with a star and the line and whisker combinations. A heavy line connecting the medians also has been drawn to help visualize what the expected hyetograph might look like. The mean is plotted as an open circle; early in the storm duration the mean is slightly greater than the median, but generally, the mean is less than the median. Each grouping of statistics are plotted at the mid point or center of the interval. The means are repeated on graph B of figure 72 with the addition of the sample size within each interval. The sample sizes are reasonably large and therefore the statistics within each interval are expected to be reliable. The reliability is partially evident in that each statistic (median, quartile, decile, and mean) is monotonically or nearly monotonically increasing with percent duration. Some of the sample sizes (801 and 937) for the first and last intervals are each greater than 621 (the total number of hyetographs analyzed). This occurs because of the distribution of data points defining each hyetograph; many hyetographs have multiple points near the 0 percent and 100 percent duration. 230 Figure 72. Empirical hyetograph analysis for 0–12 hr storm duration and one inch and greater storm depths PERCENT O F ST ORM DEP T H 0 10 20 30 40 50 60 70 80 90 100 PERCENT OF STORM DURATION PERCENT O F ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 937 344 335 396 322 321 333 323 328 289 304 287 341 232 257 257 244 252 231 252 246 255 232 254 246 202 310 244 274 249 280 290 272 259 264 297 292 213 801 Individual dimensionless hyetographs Median, quartiles, and deciles of samples within each 2.5-percent wide interval of duration 937 Mean and sample size within each 2.5-percent wide interval of duration 621 trimmed events are shown A. B. 231 Finally, the empirical hyetographs seem to start after and stop before exactly 0 percent and exactly 100 percent duration. This occurs because of the use of intervals for the analysis and more importantly, the double one-percent tail trimming still leaves repeated 0 and 100 percent values of depth on the tails after the trimmed data was re- expressed in 0–100 percent values. Data fitted hyetographs would necessarily start and stop at 0 and 100 percent. The empirical hyetographs (10th, 25th, median, 75th, and 90th percentile) for the 0–12 hr duration are repeated on figure 73. Also shown on the figure is the 0–12 hr expected hyetograph estimated by the L-gamma model. It is apparent that both the median empirical hyetograph and the L-gamma expected hyetograph are similar in shape and magnitude for most of the storm duration. It is important to note that because the L-gamma model is estimated in a fundamentally different fashion than the median empirical hyetograph, the L-gamma curve is not expected to represent a best fit or even a fit specifically to the median empirical hyetograph. This statement applies for the other two durations (figs. 74 and 75) presented shortly. 232 Figure 73. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 0–12 hr storm duration and one inch and greater storm depths The L-gamma hyetograph is necessarily smooth and the empirical hyetograph is not. The general slope of the median empirical hyetograph and the L-gamma hyetograph are similar to about 40 percent of the duration—the two models predict similar rainfall rates. For larger duration percentages the median empirical hyetograph indicates more uniform rainfall rates right to the end of the storm. The L-gamma PERCENT OF STORM DURATION PERCENT OF ST OR M DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 90th percentile 25th percentile Median (50th percentile) empirical hyetograph for 0–12 hr storm duration L-gamma distribution hyetograph model for 0–12 hr storm duration using graphical mode statistics for parameter estimation EXPLANATION 75th percentile 10th percentile 233 model is considered in agreement with the less statistically rigorous empirical hyetographs. Therefore, a conclusion is that the L-gamma model estimates an appropriate runoff-producing the 0–12 hr expected hyetograph for storm depths of one or more inches. The empirical hyetographs (10th, 25th, median, 75th, and 90th percentile) for the 12–24 hr duration are shown on figure 74. Also shown on the figure is the 12–24 hr expected hyetograph estimated by the L-gamma model. It is apparent that both the median empirical hyetograph and the L-gamma expected hyetograph are similar in shape and magnitude for most of the storm duration. The slopes of the median empirical hyetograph and the L-gamma hyetographs are similar except in the first 15 percent or so of the storm duration. Hence, early in the storm the median empirical hyetograph predicts higher percentage rainfall rates. The higher rates of the median empirical hyetograph are followed by a period of reduced rates between about 30 to 55 percent. Comparable percentage rainfall rates are exhibited from about 55 percent of the storm duration to the end of the storm. The L-gamma model is considered to agree with the less statistically rigorous empirical hyetographs. Therefore, a conclusion is that the L-gamma model estimates an appropriate runoff-producing 12–24 hr expected hyetograph for storm depths of one or more inches. 234 Figure 74. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 12–24 hr storm duration and one inch and greater storm depths PERCENT OF STORM DURATION PERCENT OF ST OR M DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 90th percentile 25th percentile Median (50th percentile) empirical hyetograph for 12–24 hr storm duration L-gamma distribution hyetograph model for 12–24 hr storm duration using graphical mode statistics for parameter estimation EXPLANATION 75th percentile 10th percentile 235 The empirical hyetographs (10th, 25th, median, 75th, and 90th percentile) for the 12–24 hr duration are shown on figure 75. Also shown on the figure is the 24 hr and greater expected hyetograph estimated by the L-gamma model. It is apparent that both the median empirical hyetograph and the L-gamma expected hyetograph are similar in shape and magnitude for most of the storm duration. The general slope of the median empirical hyetograph and the L-gamma hyetograph are similar throughout most of the storm duration. The exception being around 10–20 percent; the differences in this duration range are believed to be a vagary of sampling and estimation of the medians within the intervals. The fact that both the median empirical hyetograph and the L-gamma expected hyetograph bend upwards as they approach the end of the storm duration enhances the credibility of the oddly shaped L-gamma model for the 24 hr and greater duration compared to the general shapes of the other two durations. The L-gamma model is considered to agree with the less statistically rigorous empirical hyetographs. Therefore, a conclusion is that the L-gamma model estimates an appropriate runoff-producing 24 hr and greater expected hyetograph for storm depths of one or more inches. 236 Figure 75. Comparison between empirical hyetographs and the L-gamma distribution hyetograph distribution model for 24 hr and greater storm duration and one inch and greater storm depths The percentiles used to produce figures 73–75 are listed in tables F1, F3, and F5 of Appendix F for the 0–12 hr, 12–24 hr, and 24 hr and greater durations, respectively. The L-moments of the distribution of ordinates within each interval are listed in tables F2, F4, and F6 of Appendix F for the 0–12 hr, 12–24 hr, and 24 hr and greater durations, respectively. The L-moments in the tables were computed by unbiased PERCENT OF STORM DURATION PERCENT OF ST OR M DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 90th percentile 25th percentile Median (50th percentile) empirical hyetograph for 24 hr and greater storm duration L-gamma distribution hyetograph model for 24 hr and greater storm duration using graphical mode statistics for parameter estimation EXPLANATION 75th percentile 10th percentile 237 estimators. The tables are provided so that complete and comprehensive documentation of the ordinate distribution within each interval is available for other analysts to research stochastic hyetograph modeling or other research objectives. For example, Karian and Dudewicz (2000, chap. 4, Appendix D) provide an approach and extensive tables for parameter estimation of the generalized Lambda distribution using the Method of Percentiles. The distribution as a quantile function is . (95) The percentiles shown in tables F1, F3, and F5 could provide estimates for the , , , and parameters. The fitted distribution would greatly facilitate simulation of the hyetograph ordinate for a specific 2.5 increment of the percent of storm duration. For example, at the mid point (the median) of the 0–12 hr storm duration (table F1), the median, lower quartile, upper quartile, lower decile, and upper decile of the hyetograph ordinate values are 71.75, 43.99, 86.28, 24.82, and 93.84, respectively. Using the procedures outlines by Karian and Dudewicz (2000, pp. 154–166, 381), an estimate of the parameters is 0.9304, 1.191, 36.91, and 1.98, for , , , and , respectively. The generalized Lambda modeling (no truncation for or shown) the distribution of hyetograph ordinates half way through a 0–12 hr storm is . (96) QF() ξ αF κ 1 F–() h –()+= ξα κ h ξακ h Q 0< Q 1> QF() 0.9304 F 36.91 1 F–() 1.98 – 1.191 ---------------------------------------------+= 238 Additional Empirical Hyetographs The three durations ranges analyzed thus far in this dissertation are considered large relative to the shorter durations commonly used in hyetograph applications (3–6 hr). Although statistics from chapter 5 of the 0–12 hr and 12–24 hr duration indicate that these durations are combinable, it is informative to consider some additional median empirical hyetographs with other durations. The influence of smaller durations and duration ranges is documented by median empirical hyetograph comparisons such as the curves plotted on figure 76 for storms having one or more inches of rainfall. The very short duration storms (0–3 hr) appear to start proportionally slower (less steep implies slower rainfall rate) or are not consistent than the longer duration storms and then eventually parallel the slightly longer 3–6 hr storms when about 50 percent of the duration has passed. The points defining the 0–3 hr storms are considerably more erratic relative to the other duration ranges; this is likely the result of smaller storm sample sizes of 125 verses 203 for the 0–3 hr and 3–6 hr storms respectively. Another source for the slower starting and erratic behavior of the 0–3 hr storms might be related to the resolution of the time increments in the raw data. The hyetograph generally is less well represented for the short duration events because fewer points were recorded. The most common time intervals defining the data for the short duration (few hours) events was 15 to 30 minutes. Some 5-minute data is present in the hyetograph data base, however. It appears that as the duration increases, the median empirical hyetograph plot lower for percent of storm duration values greater than 239 about 20 percent. The peak percentage rainfall rates between 0 and 10 percent of the storm duration are equivalent. Figure 76. Median empirical hyetographs for storm depths of one inch and greater The influence of storm magnitude is documented by median empirical hyetograph comparisons on figure 77. Slightly larger storm durations than those considered on figure 76 are used on figure 77 because of smaller numbers of storms meeting the minimum depth criteria for each duration range. In general, the storms all have similar percentage rainfall rates for the first 20 percent of storm duration. As the duration percentage increases separation in the two median empirical hyetographs is exhibited. PERCENT OF STORM DURATION PERCENT O F ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Storm durations between 0 and 3 hrs Storm durations between 3 and 6 hrs Storm durations between 6 and 12 hrs 240 Figure 77. Median empirical hyetographs for storm depths of three inches and greater Model Criticisms A major criticism of these expected hyetographs is that the hyetographs might not reflect the typical rainfall rates associated with periods of peak rainfall rates. From an hydrologic design consideration, the peak rainfall rates are important because the peak rainfall rates commonly drive the peak runoff response of a watershed. Specifically, the concern is that a simple few-parameter model, like the L-gamma distribution, is not well suited for reproducing the bursting behavior of many real-world hyetographs. The rainfall bursts are significant contributors to peak streamflow of the runoff hydrograph. Graphical illustration and supporting discussion follows. PERCENT OF STORM DURATION P E R C ENT OF ST O R M DEPT H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 Storm durations between 0 and 6 hrs Storm durations between 6 and 12 hrs 241 For example, consider the dimensionless hyetographs used to produce the third box plot from the left on figures 26–32. For the 3 in. category and the 0–12 hr duration the 108 hyetographs meeting these criteria are illustrated on figure 78. The visual appearance of the hyetographs in the figure are similar for the other categories and durations considered in chapter 5. By inspection of the figure, it is clear that large variations in the hyetograph shape exists. Some hyetographs are remarkably front loaded, whereas, a few are back loaded. Further, it seems unusual for a particular hyetograph to gradually transition from 0 to 100 percent like the L-gamma model implies; more commonly the hyetograph makes relatively distinct jumps. These jumps are indicative of rainfall bursts. Additional observations of the hyetographs are annotated on the figure. Finally, several hyetographs are distinctively un-smooth—the second derivative of the hyetograph is often zero for substantial fractions of the duration. The occurrence of un-smooth hyetographs is undoubtedly due to factors including: poor data quality or missing values for the original data, poor digitizing of the raw data into tables during the preparation of the data reports, original transcription errors in final production of the data reports, and transcription or other errors on part of the modern research projects. The now-computerized hyetograph data base passes many quality assurance tests, which include test on the monotonic criteria for all cumulating values. A method by which to objectively (algorithmic and not visual inspection) identify and to remove individual undesirably un-smooth hyetographs from analysis is 242 difficult to envision. So such removal was not performed at all for this dissertation. It has been implicitly assumed for this dissertation that the errors or irregularities in individual hyetographs are “averaged out” when a sufficiently large number of hyetographs are used to compute statistics. Figure 78. Hyetographs for storms having depths between 2.5 and 3.5 inches and 0–12 hr duration To continue, there is a concern that the broad rainfall rate uniformity predicted by the L-gamma model is not consistent with the majority of individual hyetographs— that is real storms—as evidenced by visual inspection of figures such as figure 78. (Graphs C on figures 4–8 also provide evidence.) The uniformity of rainfall rate implied by the L-gamma distribution might be attributable to an over degree of smoothing by using “averages” (specifically, means, medians, or graphical modes) of PERCENT OF STORM DURATION PERCENT OF ST ORM DEP T H 0 10203040506070809010 0 10 20 30 40 50 60 70 80 90 100 front loaded hyetographs—this cluster is due to a single storm smooth un-smooth b u r s t producing simultaneous runoff at several streamflow-gaging stations. ba ck l o ade d 108 events are shown 243 the mean and L-scale values of observed hyetographs. In a sense a statistic of an ensemble of statistics predicts rainfall rates less than the bursts because rainfall can occur at any time in the duration. To clarify, the rainfall bursts front-loaded storms are approximately balanced in number by the rainfall bursts of back-loaded storms so that on a whole the general rainfall rate of the storm is reduced from the individual bursts. A bi-variate statistical analysis of the probability of rainfall for a given increment of the storm duration coupled with an analysis of the distribution of the rainfall burst magnitude for the storm duration increment might prove insightful. Such an analysis would be encumbered by the fact that the original hyetograph data is not preserved in constant increments—the motivation for the prior-Probability Weighted Moments. Example Application of L-gamma Hyetograph It is informative to show an application of the L-gamma hyetograph for the generation of a streamflow hydrograph. The most common technique is to couple the hyetograph with a unit hydrograph (Chow and others, 1988, pp. 218–221). The unit hydrograph is the hydrograph generated by a unit depth of excess precipitation (runoff) on a watershed. Each pulse of precipitation initiates a separate hydrograph. The total streamflow hydrograph is constructed by adding the hydrograph of each pulse of precipitation for each discrete increment of time. This process is known as convolution and is mathematically expressed in discrete space as 244 , (97) where is the streamflow for the -th hydrograph ordinate, is a precipitation pulse, is an ordinate of the unit hydrograph, and is the number of pulses of precipitation. For the application example, consider the 3-hr 25-year storm considered in section Example of the Balanced Storm Hyetograph Method for Austin, Texas in chapter 2. The total depth of precipitation for the storm is 4.55 in. (116 mm). It is assumed for the example here that all of the precipitation in converted to runoff. No abstraction of precipitation is considered. The dimensionless hyetograph (balanced storm hyetograph) derived in table 9 is used to compute a streamflow hydrograph in table 29. The 30-minute unit hydrograph is from Chow and others (1998, example 7.5.1). The 0–12 hr L-gamma hydrograph model is used to compute the streamflow hydrograph in table 30. Subtle differences between in the totals in tables 29 and 30 are due to rounding errors. The hydrograph corresponding to each precipitation input is placed along the diagonals of the matrix. For example, the 0.219 precipitation in table 30 produces a hydrograph with 30-minute ordinates of 88.5, 236, . . ., 60.0, and 37.9 ft 3 /s. Q n P m U nm–1+ m 1= nM≤ ∑ = Q n nP i U i M 245 Table 29. Example computation of streamflow hydrograph convolution of a balanced storm hyetograph and a unit hydrograph [ft 3 /s, cubic feet per inch of excess precipitation; hr, hour; in, inches; ft 3 /s, cubic feet per second. Total excess precipitation is about 4.55 inches.] A comparison between the streamflow hydrographs computed using the balanced storm hyetograph (table 29) and the L-gamma hyetograph (table 30) is provided on figure 79. The L-gamma hyetograph produces a peak streamflow that is substantially less than the peak produced by the balanced storm hyetograph. The downward percent change between the hydrographs for the balanced storm and L-gamma hyetograph is about 16 percent. Accordingly, there is a concomitant increase in the spread of the streamflow hydrograph derived from the L-gamma hyetograph for the volumes to remain equivalent. Time Dimen- sion- less hyeto- graph Excess precip- itation hyeto- graph Stream- flow hydro- graph 30-minute unit hydrograph ordinates (ft 3 /in) 123456789 (1/2 hr) ( ) (in) 404 1,079 2,343 2,506 1,460 453 381 274 173 (ft 3 /s) 1 0.4820.2198.5---------------- 8.5 2 .21.74129236--------------53 3 .737 2.39 966 800 513 -- -- -- -- -- -- 2,279 4 .934 .896 362 2,579 1,736 549 -- -- -- -- -- 5,226 5 1.00 .300 121 967 5,600 1,857 320 -- -- -- -- 8,865 6 1.00 .00 0 324 2,099 5,989 1,082 99.2 -- -- -- 9,593 7 Total 4.55 -- 0 703 2,245 3,489 336 83.4 -- -- 6,856 8 -- -- 0 752 1,308 1,083 282 60.0 -- 3,485 9 -- -- -- 0 438 406 911 203 37.9 1,996 10 -- -- -- -- 0 136 341 655 128 1,260 11 -- -- -- -- -- 0 114 246 413 773 12 -- -- -- -- -- -- 0 82.2 155 237 13 -- -- -- -- -- -- -- 0 51.9 51.9 14 -- -- -- -- -- -- -- -- 0 0 Total 41,245 246 Table 30. Example computation of streamflow hydrograph convolution of 0–12 hr L-gamma hyetograph and a unit hydrograph [ft 3 /s, cubic feet per inch of excess precipitation; hr, hour; in, inches; ft 3 /s, cubic feet per second. Total excess precipitation is about 4.55 inches.] Figure 79. Comparison of streamflow hydrographs derived from L-gamma and balanced storm hyetographs shown in tables 29 and 30 Time Dimen- sion- less hyeto- graph Excess precip- itation hyeto- graph Stream- flow hydro- graph 30-minute unit hydrograph ordinates (ft 3 /in) 123456789 (1/2 hr) ( ) (in) 404 1,079 2,343 2,506 1,460 453 381 274 173 (ft 3 /s) 1 0.290 1.32 533 -- -- -- -- -- -- -- -- 533 2 .561.265091,424--------------1,93 3 .770 .928 375 1,360 3,093 -- -- -- -- -- -- 4,828 4 .903 .605 244 1,001 2,952 3,308 -- -- -- -- -- 7,505 5 .975 .328 133 653 2,174 3,158 1,927 -- -- -- -- 8,045 6 1.00 .114 46.1 354 1,418 2,326 1,840 598 -- -- -- 6,581 7 Total 4.56 -- 123 769 1,516 1,355 571 503 -- -- 4,836 8 -- -- 267 822 883 420 480 362 -- 3,234 9 -- -- -- 286 479 274 355 345 228 1,966 10 -- -- -- -- 166 149 231 254 218 1,018 11 -- -- -- -- -- 51.6 125 166 161 503 12 -- -- -- -- -- -- 43.4 89.9 105 238 13 -- -- -- -- -- -- -- 31.2 56.7 87.9 14 -- -- -- -- -- -- -- -- 19.7 19.7 Total 41,322 HOURS PASSED SINCE BEGINNING OF EVENT STREAMFLOW , I N CUBI C FEET PER SECOND 012345678 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 Streamflow hydrograph derived from balanced storm hyetograph (table 35) Streamflow hydrograph derived from L-gamma hyetograph (table 36) EXPLANATION 247 CHAPTER 8 MODIFICATION OF THE CARMAN-KOZENY EQUATION FOR APPLICATION OF L-MOMENT STATISTICS FOR ESTIMATION OF THE INTRINSIC PERMEABILITY OF POROUS MEDIA Introduction This chapter is unrelated to all other chapters of this dissertation in that no topics related to rainfall hyetographs are considered. This chapter has several purposes, and these are to provide discussion of a potential application of L-moment statistics for estimation of intrinsic permeability (permeability) of porous media, to suggest future research directions for researchers of L-moment statistics, and to encourage L-moment adoption and research by chemical engineers, geologists, hydrogeologists, petroleum engineers, petrophysicists, water and waste-water engineers, and other specialties that consider fluid flow in porous media. Examples of porous media are sandstone aquifers and water-treatment filtration systems. The estimation of permeability in general is of great socio-economic importance for topics such as chemical processing, energy recovery, subsurface contaminant transport, and water supply. This chapter does not provide either a specific review of permeability or a general review of the principles of fluid flow in porous media, but instead suggests and demonstrates an application of L-moment statistics following an analytical lead of Panda and Lake (1994). Panda and Lake use the product moments of a grain-size (particle-size) distribution for enhanced permeability estimation using the Carman- Kozeny equation. Panda and Lake provide important background references on 248 permeability and estimation of permeability. Readers are directed to the cited references of that paper for background information. From Panda and Lake (1994), intrinsic permeability, , can be estimated using the Carman-Kozeny (CK) equation. The equation is , (98) where is porosity, is tortuosity, and is the ratio of the internal surface area of the media to the volume of the media solids. The units for are length 2 . Throughout this chapter is it assumed that and are constant. If it is assumed that the media is composed of spherical uncemented particles of diameter then . (99) (Panda and Lake do not explicitly show the middle terms of eq. 99.) Upon substitution of eq. 99 into the CK equation, the equation becomes . (100) k k φ 3 2ϒ 1 φ+() 2 a 2 v × ----------------------------------------= φϒ a v k φ ϒ D a v πD 2 no. of spheres× π 6 --- D 3 no. of spheres× -------------------------------------------------- 6D 2 D 3 --------- 6 D ---=== k D 2 φ 3 72ϒ 1 φ+() 2 ------------------------------= 249 Panda and Lake then acknowledge that particle diameters have a distribution. The distribution can be defined by the relative number of grains ( ) for a given diameter. The distribution is characterized by a particle-size distribution (psd) and is , and (101) . (102) The psd is analogous to a probability density function (PDF). c36c3c79c68c87c72c3c81c82c87c72c3c11c45c68c81c17c3c21c19c19c22c12c3c85c72c74c68c85c71c76c81c74c3c87c75c72c3c80c82c71c72c79c3c73c82c85c3c87c75c72c3c83c86c71c3c86c75c82c90c81c3c76c81c3 c72c84c17c3c20c19c20c3c76c86c3c81c72c72c71c72c71c17c3c36c70c70c82c85c71c76c81c74c3c87c82c3c83c72c85c86c82c81c68c79c3c70c82c80c80c88c81c76c70c68c87c76c82c81c3c90c76c87c75c3c47c68c78c72c3 c11c21c19c19c22c12c15c3c87c75c72c3c47c82c74c16c81c82c85c80c68c79c3c71c76c86c87c85c76c69c88c87c76c82c81c3c90c68c86c3c88c86c72c71c3c73c82c85c3 c17c3c55c75c72c3c88c86c72c3 c82c73c3c87c75c72c3c47c82c74c16c81c82c85c80c68c79c3c71c76c86c87c85c76c69c88c87c76c82c81c3c76c86c3c68c83c83c68c85c72c81c87c79c92c3c81c82c87c3c85c72c73c72c85c72c81c70c72c71c3c76c81c3 c87c75c72c3c51c68c81c71c68c3c68c81c71c3c47c68c78c72c3c11c20c28c28c23c12c3c83c68c83c72c85c17 Using eq. 101, Panda and Lake indicate or assume without showing a proof that the average or expected value for is , (103) where , and (104) . (105) n i fD i () n i n total ------------ = fD i ()Dd 0 ∞ ∫ 1= f D i () a v a v 6 E 2 × E 3 -------------- = E 2 2nd noncentral product moment D 2 fD i ()Dd 0 ∞ ∫ == E 3 3rd noncentral product moment D 3 fD i ()Dd 0 ∞ ∫ 250 In terms of the central product moments the noncentral product moments are , (106) , (107) (mean grain size), and (108) (coefficient of variation). (109) Using standard notation, is the standard deviation and is the skew of the psd. Panda and Lake then go on to develop the “modified CK equation” , (110) where the quantity in the square brackets is a correction factor to account for the psd. The correction factor is a function of the mean, coefficient of variation, and skewness of the distribution. No other moments are represented. Panda and Lake (1994, p. 1031) conclude that the modified CK equation is valid for any distribution. c36c3c79c68c87c72c3c81c82c87c72c3c11c45c68c81c17c3c21c19c19c22c12c3c76c86c3c81c72c72c71c72c71c3c85c72c74c68c85c71c76c81c74c3c87c75c72c3c51c68c81c71c68c3c68c81c71c3c47c68c78c72c3 c70c82c81c70c79c88c86c76c82c81c3c87c75c68c87c3c87c75c72c3c80c82c71c76c73c76c72c71c3c38c46c3c72c84c88c68c87c76c82c81c3c76c86c3c89c68c79c76c71c3c73c82c85c3c68c81c92c3 c71c76c86c87c85c76c69c88c87c76c82c81c17c3c36c70c70c82c85c71c76c81c74c3c87c82c3c47c68c78c72c3c11c21c19c19c22c12c15c3c87c75c72c3c70c82c81c70c79c88c86c76c82c81c3c86c72c72c80c86c3 c80c82c86c87c3c68c83c83c85c82c83c85c76c68c87c72c3c73c82c85c3c68c81c92c3c180c73c76c87c87c72c71c181c3c47c82c74c16c81c82c85c80c68c79c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3 c87c75c72c3c47c82c74c16c81c82c85c80c68c79c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c83c83c68c85c72c81c87c79c92c3c90c68c86c3c88c86c72c71c3c68c86c3c68c3c69c68c86c76c86c3c73c82c85c3 c87c75c72c3c68c81c68c79c92c87c76c70c68c79c3c71c72c85c76c89c68c87c76c82c81c86c3c86c75c82c90c81c3c76c81c3c51c68c81c71c68c3c68c81c71c3c47c68c78c72c3c11c20c28c28c23c12c17 The author is concerned that the distribution independence conclusion is actually an assumption, which might not be supportable. This concern is a partial motivation E 2 D 2 σ 2 += E 3 D 3 3σ 2 D γσ 3 ++= DDfD i ()Dd 0 ∞ ∫ = C σ D ---= σγ k D 2 φ 3 72ϒ 1 φ+() 2 ------------------------------ γC 2 3C 2 1++() 2 1 C 2 +() 2 ------------------------------------------ = 251 for this chapter. Panda and Lake do not indicate that the conclusion of distribution independence is an assumption; neither a proof or further discussion is provided. The remainder of the Panda and Lake paper explores the enhanced predictive capabilities of the modified CK equation. The author has another concern about the Panda and Lake paper, and this is whether or not an assumption was made for the expected or average value of being equal to . The units (length -1 ) of certainly are consistent for , and squaring or cubing of the diameter is seen in eqs. 104 and 105 as in eq. 99. However, the appropriateness of the equality is not intuitive to the author. For example, any noncentral moment or order divided by the noncentral moment of order would yield consistent units for . The author suggests an alternative analytical track with derivations following the lead established by Panda and Lake. The derivations clarify the distribution independence, the definition of the average , and demonstrate applicability of L-moment statistics to the CK equation. If has a distribution, then a quantile function (either algebraically or numerically solved) of the distribution must exist. The particle-size quantile function is for , (111) a v 6 E 2 × E 3 ⁄ E 2 E 3 ⁄ a v nn1+ a v a v D DF() 0 F 1≤≤ 252 where is the particle size (quantile) for a given nonexceedance probability . Because of the ability to perform algebraic manipulations on quantile functions (see Gilchrist, 2000, chapter 3), most surely (for uncemented spherical grains) follows the quantile function . (112) The compliment ( ) is needed to make an increasing function according to the reciprocal rule of quantile function algebra (Gilchrist, 2000, p. 65). By substituting eq. 112 into the CK equation, also becomes a quantile function . (113) The is recovered in the left-hand side because of the reciprocal rule. The expected value of is , (114.1) , where (114.2) . (114.3) DF() F a v a v 1 F–() 6D 2 F() D 3 F() ------------------- 6 DF() ------------== 1 F– a v k kF() φ 3 2ϒ 1 φ+() 2 a 2 v 1 F–()× -----------------------------------------------------------= F kF() k φ 3 2ϒ 1 φ+() 2 a 2 v 1 F–()× ----------------------------------------------------------- Fd 0 1 ∫ = k φ 3 72ϒ 1 φ+() 2 ------------------------------ D 2 F()Fd 0 1 ∫ φ 3 72ϒ 1 φ+() 2 ------------------------------ ζ== ζ D 2 F()Fd 0 1 ∫ = 253 Note that the is recovered in eq. 114.2 from because of the reciprocal rule. It is natural to consider various forms of . It is evident from eq. 114.3 that different parametric forms of the quantile function will produce different formulas for the correction factor. Different correction factors imply that the CK equation is influenced by the form (model or type) of the psd and the values of the L-moments and the product moments by generality. For example, if , a constant quantile function, then the usual CK equation results because ; the correction to the CK equation hence is unity. Clearly the particle-size quantile function is not always a constant so this is a degenerate example. Other forms are described in the following sections. Linear Model of the Particle-Size Distribution Suppose that the distribution of particle diameter is , (115) where and are location and scale parameters, respectively. The first two L-moments of the linear model are and (116.1) . (116.2) DF() a v 1 F–() DF() DF() D= ζ D 2 = DF() ε αF+= εα λ 1 DF()Fd 0 1 ∫ εαF+()Fd 0 1 ∫ εα2⁄+== = λ 2 DF()2F 1–()Fd 0 1 ∫ εαF+()2F 1–()Fd 0 1 ∫ α 6⁄== = 254 The parameters in terms of the L-moments are and . In terms of the L-moments then . (117) However, a more convenient analytical direction is , (118.1) , and (118.2) . (118.3) Hence upon substitution into the CK equation and incorporation of the coefficient of L-variation , the result is . (119) Since is the expected particle size, , a symbolically consistent expression for the CK equation is . (120) ελ 1 3λ 2 –= α 6λ 2 = ζλ 1 3λ 2 6λ 2 F+–() 2 Fd 0 1 ∫ = ζεαF+() 2 Fd 0 1 ∫ ε 2 2εαF α 2 F 2 ++()Fd 0 1 ∫ == ζε 2 εα α 2 3⁄++= ζλ 2 1 1 3λ 2 2 λ 2 1 ----------- +    λ 2 1 13τ 2 +()== τλ 2 λ 1 ⁄= k λ 2 1 φ 3 72ϒ 1 φ+() 2 ------------------------------ 13τ 2 +()= λ 1 D k D 2 φ 3 72ϒ 1 φ+() 2 ------------------------------ 13τ 2 +()= 255 The is the correction necessary if the psd is linear. The correction is proportional to the square of the coefficient of L-variation ( ), which is a measure of particle sorting. For example, suppose the mean ( ) and L-scale ( ) of the psd are 50 mm and 15 mm, respectively. The coefficient of L-variation = 0.3 (15/50). Hence the CK equation with this linear model of the psd is . (121) Two-Parameter Power Model of the Particle-Size Distribution As another example, suppose that the distribution of particle diameter is . (122) This equation is a two-parameter Power distribution similar to the Power distribution shown in eq. 59. Following the lead established for the linear model, the first two L-moments of this model are , (123.1) , and (123.2) . (123.3) The parameters in terms of the L-moments are 13τ 2 +() τ λ 1 λ 2 τ k 2500φ 3 72ϒ 1 φ+() 2 ------------------------------ 1.27()= DF() αF β = λ 1 αF β ()Fd 0 1 ∫ α β 1+ ------------== λ 2 αF β ()2F 1–()Fd 0 1 ∫ αβ β 2+()β1+() ----------------------------------== τ λ 2 λ 1 ----- β β 2+() -----------------== 256 and (124.1) . (124.2) Hence, is . (125) Therefore the CK equation when the particle size is distributed according to a Power distribution is . (126) Thus the correction factor on the CK equation for a Power model of the psd is more complicated than that for the linear model. This factor is different than that derived for a linear psd. For example, suppose a psd has a mean of 50 mm and an L-scale ( ) of 15 mm; hence, is equal to 0.3. Hence the CK equation with this linear model of the psd is . (127) The correction of 1.27 for the two-parameter Power model with round off is the same as the correction for the linear model fit to the same L-moments. However, if αλ 1 τ–1– τ 1– ----------------   = β 2τ– τ 1– -----------= ζ ζαF β () 2 Fd 0 1 ∫ α 2 β 1+ ------------ λ 2 1 τ 1+() 2 1 τ–()3τ 1+() -------------------------------------   === k D 2 φ 3 72ϒ 1 φ+() 2 ------------------------------ τ 1+() 2 1 τ–()3τ 1+() -------------------------------------   = λ 2 τ k 2500φ 3 72ϒ 1 φ+() 2 ------------------------------ 1.27()= 257 and the psd a Power distribution, then the correction factor is 4.76—the correction for the linear model is just 2.92. The much larger correction for the Power distributed psd is attributed to the heavier upper tail of the Power distribution. Since is proportional to the square of the diameter, the larger diameters from the Power model necessarily increase the . This is exhibited in the larger correction factors as increases. Three-Parameter Power Model of the Particle-Size Distribution For another example, suppose that the distribution of particle diameter is . (128) This model is a three-parameter Power distribution. The L-moments (mean, L-scale, and L-skew) of this distribution are , (129.1) , and (129.2) . (129.3) The parameters in terms of the L-moments are , (130.1) , and (130.2) τ 0.8= k k τ DF() ε αF β += λ 1 ε α β 1+ ------------+= λ 2 αβ β 2+()β1+() ----------------------------------= τ 3 β 1– β 1+ ------------ = β 3τ 3 1+ 1 τ 3 – ----------------- = α λ 2 β 2+()β1+() β -----------------------------------------= 258 . (130.3) Hence, is . (131) Substitution of eqs. 129.1–129.3 into the right-hand side of eq. 130 yields a complicated expression. Therefore, it is more convenient to compute the parameters and substitute their numerical values into eq. 130. For example, suppose that a psd has values for the mean ( ), L-scale ( ), and L-skew ( ) of 50 mm, 15 mm, and 0.1, respectively. The corresponding parameters for , , and are 21.5 mm, 91.8 mm, and 2.22, respectively. These parameters and provide a = 3237. Dividing this by the square of the mean grain size (50 2 = 2500), yields a correction on the CK equation of 1.29 (3237/2500). Hence the full CK equation for this example is . (132) Four-Parameter Kappa Model of the Particle-Size Distribution As another example, consider a four-parameter Kappa distribution (Hosking, 1994) model of the psd , (133) ελ 1 α β 1+ ------------–= ζ ζεαF β +() 2 Fd 0 1 ∫ ε 2 2εα β 1+ ------------ α 2 2β 1+ --------------- ++== λ 1 λ 2 τ 3 εα β ζζ k 2500φ 3 72ϒ 1 φ+() 2 ------------------------------ 1.29()= DF() ξ α κ --- 1 1 F h – h ---------------   κ –    += 259 where , , , and are location, scale, and shape1 and shape2 parameters of the distribution. The Kappa distribution was considered in chapter 4. The L-moments are shown in eqs. 55.1–55.4. Assuming the parameters of the distribution are known, then becomes and (134.1) . (134.2) An important identity (Hosking, 1994, eq. 9) is . (135) where and in eq. 134.2 are and (136.1) (136.2) ξακ h ζ ζξ α κ --- 1 1 F h – h --------------   κ –    +    2 Fd 0 1 ∫ = ζξ 2 2ξα κ --------- 1 H 1 –() α 2 κ 2 ----- 12H 1 – H 2 +()++= H r h 1– 1 F h –()[] rκ Fd 0 1 ∫ = H 1 H 2 H 1 1 h 1 κ+ ------------ Γ 1 κ+()Γ1 h⁄() Γ 1 k 1 h⁄++ ----------------------------------------   , h 0> 1 h–() 1 k+ ------------------- Γ 1 κ+()Γκ–1h⁄–() Γ 11h⁄–() -----------------------------------------------------   , h 0<        = H 2 1 h 12κ+ --------------- Γ 12κ+()Γ1 h⁄() Γ 12κ 1 h⁄++ --------------------------------------------   , h 0> 1 h–() 12κ+ ---------------------- Γ 12κ+()Γ2κ–1h⁄–() Γ 11h⁄–() ------------------------------------------------------------- , h 0<        = 260 For a computational example, suppose the L-moments of the psd are 50 mm, 15 mm, 0 (zero), and 0.1226, for the mean ( ), L-scale ( ), L-skew ( ), and L-kurtosis ( ), respectively. An L-skew of zero and L-kurtosis of 0.1226 corresponds to the Normal distribution. So a Kappa distribution fitted to these L-moments approximates the Normal distribution that has a mean of 50 mm and standard deviation of 26.59 mm ( ). The parameters corresponding to these L-moments are 42.89, 23.3, 0.2138, and -0.1613 for , , , and , respectively using algorithms derived from algorithms by Hosking (1996). Substituting these values into eq. 133.2 yields . Dividing this by the square of the mean grain size (50 2 = 2500), yields a correction on the CK equation of 1.29 (3223/2500). Hence the full CK equation this example is . (137) The coefficient of variation ( ) for this example is 0.5318 (26.59/50), and the skew is zero. Upon substitution of and into eq. 110, the Panda and Lake correction is 2.08. So the Panda and Lake correction is about 1.6 times larger than the correction from the Kappa model described here; the expected values for the intrinsic permeability necessarily differ. However, the Kappa or even the Normal distribution fit to the L-moments in this example produce negative quantiles for the particle diameter for very small values of —a physically impossible situation. Thus, care in λ 1 λ 2 τ 3 τ 4 πλ 2 ξακ h ζ 2620= ζ k 2500φ 3 72ϒ 1 φ+() 2 ------------------------------ 1.29()= C γ C γ F 261 selecting a suitable distribution to model the psd is necessary. Movement of the analysis into log-space of the diameter or a bounded parameter fitting scheme for the Kappa would solve this dilemma. Four-Parameter Generalized Lambda Model of the Particle-Size Distribution The four-parameter Generalized Lambda distribution (GLD, eq. 95) is another promising distribution for modeling psd’s. It is therefore interesting to consider the multiplier for a psd having a GLD; a slightly more compact form of the distribution is listed at the bottom of table 31. The Method of Percentiles (Karian and Dudewicz, 2000, chapter 4) for parameter estimation is readily used with graphical curves showing the percentiles of the psd—neither complex numerical methods or the manually difficult L-moment computations are required. Karian and Dudewicz (2000) provide a complete discussion and tables for parameter estimation for the GLD using the 10th, 50th, and 90th percentiles. The GLD-based requires the following integral (Abramowitz and Stegun, 1964, p. 258) for and . (138) To conclude the GLD discussion, consider a GLD approximation to the quantile function of the Normal distribution—the Normal distribution has no explicit quantile functional form. The GLD-Normal distribution is , (139) ζ ζ F κ 1 F–() h Fd 0 1 ∫ Γκ 1+()Γh 1+() Γκ h 2++ -------------------------------------------= κ 1–> h 1–> DF() D σ 5.0633 F 0.135 1 F–() 0.135 –[]{}+= 262 where the quantity within the braces {} is an approximation to the standard normal distribution. Using the derivation for the GLD in table 31 and eq. 139, it can be shown that and (2nd noncentral product moment, eq. 104). Finally, using the coefficient of variation . (140) Therefore the correction on the CK equation when the particles are distributed according to a Normal distribution is . This correction also indicates that permeability goes up as the degree of sorting goes down. Additional Remarks on Models of the Particle-Size Distribution A summary of the for the distributions considered in this chapter is provided in tables 31 and 32. Numerous other distributions could be considered to model the psd, although explicit analytical solutions for the correction factor might not exist. The availability of an explicit analytical solution rapidly diminishes as the number of parameters and hence L-moments are involved. Fortunately, numerical integration of eq. 114.3 is not difficult. Distributions that are bounded below by zero are favorable ζ ζ D 2 25.64σ 2 1.575()51.27σ 2 0.7679()–+= ζ D 2 σ 2 + E 2 == C ζ D 2 1 σ 2 D 2 ------+    D 2 1 C 2 +()== 1 C 2 + ζ 263 because of the physical limitation that particle diameter be greater than zero. Further, four and five parameter distributions are promising as these distributions contain two additional parameters and hence moments—more information concerning the psd is retained. Panda and Lake (1994) only show that the first three moments influence the CK equation. However, the derivations based on quantile functions indicate that as many moments as present in distribution model are involved in the correction on the CK equation. This is an intuitively satisfactory observation because a distribution is uniquely defined by all of its moments (parameters); hence the correction factor should also be uniquely defined by all of the distribution moments (parameters). 264 Table 31. Examples of the particle-diameter multiplier on the Carman-Kozeny equation based on quantile function models of the particle-size distribution [D(F), Quantile function of particle-size distribution; D, diameter; F, nonexceedance probability; ε, α, β, ξ, κ, and h, quantile function parameters; τ, coefficient of L-variation; , mean particle diameter; , standard deviation; , coefficient of variation. Note that is multiplied to the CK equation in lieu of the square of the mean particle diameter.] Emphasis is needed on the fact that when is divided by the square of the mean particle diameter , the resulting value is the correction on the usual CK equation (eq. 100). Quantile model of particle- size distribution No. of parameters and moments Equation for the quantile model, D(F) Particle-diameter multiplier for the Carman-Kozeny equation in terms of L-moments or parameters Constant 1 Two-parameter linear 2 Two-parameter Power 2 Three-parameter Power 3 Generalized Lambda approximation to the Normal distribution 2 D σ C ζ ζ D D 2 εαF+ D 2 13τ 2 +() αF β D 2 τ 1+() 2 1 τ–()3τ 1+() ------------------------------------   εαF β + ε 2 2εα β 1+ ------------ α 2 2β 1+ --------------- ++   D 5.0633σ F 0.135 1 F–() 0.135 –[]+ D 2 1 C 2 +() ζ D 2 265 Table 32. Complex examples of the particle-diameter multiplier on the Carman-Kozeny equation based on four-parameter quantile function models of the particle-size distribution [D(F), Quantile function of particle-size distribution; D, diameter; F, nonexceedance probability; ε, α, ξ, κ, and h, quantile function parameters; is the Gamma function. Note that is multiplied to the CK equation in lieu of the square of the mean particle diameter.] It is illustrative to compare the correction factors from the equation (eq. 110) suggested by Panda and Lake (1994), the linear distribution (eq. 115), and two- parameter Power distribution models (eqs. 122) described here. A skew of zero was used for the Panda and Lake correction. The comparison is made on figure 80. Quantile model of particle- size distribution No. of parameters and moments Equation for the quantile model, D(F) Particle-diameter multiplier for the Carman-Kozeny equation in terms of L-moments or parameters Four-parameter Kappa 4 see next line Multiplier where and Four- parameter Generalized Lambda 4 see next line Γ () ζ ζ ξ α κ --- 1 1 F h – h ---------------   κ –    + ξ 2 2ξα κ --------- 1 H 1 –() α 2 κ 2 ----- 12H 1 – H 2 +()++  H 1 1 h 1 κ+ ------------ Γ 1 κ+()Γ1 h⁄() Γ 1 κ 1 h⁄++ ----------------------------------------   , h 0> 1 h–() 1 κ+ -------------------- Γ 1 κ+()Γκ–1h⁄–() Γ 11h⁄–() -----------------------------------------------------   , h 0<        = H 2 1 h 12κ+ --------------- Γ 12κ+()Γ1 h⁄() Γ 12κ 1 h⁄++ --------------------------------------------   , h 0> 1 h–() 12κ+ ---------------------- Γ 12κ+()Γ2κ–1h⁄–() Γ 11h⁄–() ------------------------------------------------------------- , h 0<        = ξαF κ 1 F–() h –[]+ ξ 2 2αξ 1 κ 1+ ------------ 1 h 1+ ------------–   α 2 1 2κ 1+ ---------------- 1 2h 1+ ---------------+   2α 2 Γκ 1+()Γh 1+() Γκ h 2++ ----------------------------------------------------–++ 266 Figure 80. Comparison of correction factors for the Carman-Kozeny equation as a function of the coefficient of L-variation (L-CV) of the particle-size distribution Chapter Conclusions The sampling properties described in chapter 4 make L-moments particularly attractive for characterization of particle-size distribution (psd), which are well known to have large ranges, exhibit large variabilities, and are often highly non-Normally distributed. The prior-Probability Weighted Moments for sample L-moment estimation have a natural application to particle or grain size data as suggested in chapter 4. The straightforward derivations and four extended examples in this chapter suggest that L-moment statistics could play a role in estimation of intrinsic permeability using the Carman-Kozeny equation following the product moment-based lead of Panda and Lake (1994). The derivations also indicate that the distribution form COEFFICIENT OF L-VARIATION, DIMENSIONLESS OR PRODUCT MOMENT COEFFICIENT OF VARIATION DIVIDED BY π 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 20 22 Correction factor suggested by Panda and Lake (1994) for a symmetrical (skew, γ = 0) particle-size distribution Correction factor from L-moment based derivations of a linear distribution model of the particle-size distribution Correction factor from L-moment based derivations of a Power distribution model of the particle-size distribution CORRECTI ON F A CT OR FO R CARMAN- K OZENY EQUA T I ON, DI M E N S I O NLESS 267 or type influences the intrinsic permeability estimation from the Carman-Kozeny equation; this conclusion is counter to a conclusion reached by Panda and Lake. Naturally, it is unknown whether application of L-moment statistics and particle-size quantile functions would in practice provide a measurable improvement in permeability estimation. A substantial field of research can be envisioned. For example, laboratory research in which porous media of particle diameters following a specific multiple-parameter distribution are constructed would permit direct testing of the hypothesized derivations shown in this chapter. Finally for the derivations shown here, the porosity and tortuosity of the porous media are assumed independent of the psd. A more important application of the derivations provided here could be for estimation of tortuosity (a difficult phenomena to measure) because porosity and intrinsic permeability often can be estimated with simple laboratory experiments. 268 CHAPTER 9 CONCLUSIONS The major conclusions of this dissertation are enumerated below. Subordinate conclusions or analysis summaries are provided in Chapter Conclusions sections at the end of some chapters. 1. Primary hypothesis. Expected hyetographs of runoff-producing storms in Texas were successfully defined using L-moment statistics. This confirms the primary hypothesis of this dissertation. However, there are other model forms could be used as a basis for expected hyetograph definition, or other L-gamma model parameters could be estimated that differ from the preferred models (figure 68 and others). Further, the use of classical product moment statistics could result in subtle analytical differences because of alternative sample statistic estimates. The sole use of product moments in this dissertation are for the computation of coefficient of variation values in tables 11 and 12 of chapter 2. 2. Secondary hypothesis. The parameter estimates for the L-gamma model are deemed reliable. The parameters are based on defensible statistical predictions of the mean and L-scale values of observed dimensionless hyetographs. Substantial content of chapters 6 and 8 provides the defense of the statistics. The graphical modes of the mean and L-scale values (figs. 55–57 and table 25) are preferred but values for other statistics are provided to aid the interpretations of other researchers. The empirical hyetograph analysis (figs. 269 73–75) provides median hyetographs that are consistent with the preferred L-gamma models; the use of the graphical modes is defended. 3. On the applicability of the L-gamma distribution. The L-gamma distribution is a viable tool for hyetograph modeling. The Beta distribution and L-gamma distribution when fit to the same hyetograph statistics are similar (see figs. 69– 71 in chapter 8). The Beta is a well known distribution so the agreement between the distributions is encouraging. The L-gamma distribution in the context of hyetograph modeling is easier to use, algebraically more compact, and more compatible with L-moments. 4. On the suitability of the L-gamma model. The L-gamma model is a suitable tool for modeling the expected hyetograph. Comparison of the three L-gamma models to empirical hyetographs demonstrates a degree of consistency that supports the predicted hyetograph ordinates. The 24 hr and greater (up to about 3 days) expected hyetograph estimated by the L-gamma model likely is not as reliable as the models for 0–12 hr and 12–24 hr durations because of greater perceived heterogeneity of the hyetograph statistics (see chapter 6). 5. On the double one-percent tail trimming method. The double one-percent hyetograph tail trimming method was used for almost all analysis and figure construction. The tail trimming is vital. The trimming is symmetrical so substantial distortion of a given hyetograph is avoided. Precise sensitivity analysis of the trimming percentages is difficult to envision. Firm justification 270 of a need for tail trimming is graphically illustrated by the storm seen on figures 1, 3, and 23 and table 17 of chapters 1 and 4. However, to mitigate against potential interpretive problems by universal use of the trimming method, analyses of untrimmed mean, median, and L-scale values of the observed dimensionless hyetographs are presented in chapter 6 and figures 51– 53 and table 25. 6. On the influence of storm magnitude (frequency) on hyetograph statistics. Contrary to prior expectation, the storm magnitude or depth of the storm, hence its frequency, appears to have relatively little influence on the low-order L-moment statistics (mean and L-scale) and median of the 0–12 hr and 12–24 hr storm durations. This is important because it suggests that all storm recurrence intervals (such as the 25-year storm) have the hyetographs of the same shape even though the depths of the rainfall for the recurrence intervals can be quite different. The higher-order L-moment statistics (L-skew, L-kurtosis, and ) appear influenced, but these statistics are not as critical for hyetograph definition as are the mean, median, and L-scale statistics. The Beta and L-gamma distributions are fit only to the mean and L-scale statistics. For the 24 hr and greater duration storms there is a notable graphical correlation between the storm depth and the hyetograph statistics. A substantial number of figures and concomitant discussion of box plots showing the distribution of each statistic for specific with storm depth categories are provided for each τ 5 271 duration in chapter 6 on figures 26–46. Table 24 of chapter 6 summarizes the conclusions. Further details supporting this conclusion are provided in chapter 3—particularly tables 11 and 12. 7. On the influence of storm duration on hyetograph statistics. The storm duration certainly influences the shape of the dimensionless hyetograph. Three storm durations of 0–12 hr, 12–24 hr, and 24 hr and greater principally are considered throughout this dissertation. The shorter durations exhibit more front- loadedness of the rainfall—higher rainfall rates occur early in the storm duration. Contrary to prior expectation, the statistics of the 0–12 hr and 12–24 hr duration dimensionless hyetographs appear comparable. The empirical hyetograph analysis in chapter 8 considers shorter and narrower duration ranges such as 0–3 hr and 3–6 hr. In general, rainfall rates, as expressed in percentage of depth, are equivalent for the first 20 percent or so of the storm duration independent of duration length up to about 12 hr. 8. On the influence of season or month of storm occurrence on hyetograph statistics. Very subtle relations between the mean, median, and L-scale statistics of observed dimensionless hyetographs and the month of occurrence were seen, but the relations are difficult to define. It is concluded that any seasonal differences in the hyetograph shape are insignificant compared to other inherent uncertainties in the analysis. The monthly analysis is centered on figure 47 of chapter 6. 272 9. On the compatibility of the five data bases. Five data base modules (c68c88c86c87c76c81, c71c68c79c79c68c86, c73c82c85c87c90c82c85c87c75, c86c68c81c68c81c87c82c81c76c82, c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86) of observed rainfall hyetographs are considered. Four of the modules represent the Austin, Dallas, Fort Worth, and San Antonio urban areas and the c86c80c68c79c79c85c88c85c68c79c86c75c72c71c86 module represents more disperse rural regions of Texas. The modules are deemed compatible with one exception. The c71c68c79c79c68c86 module hyetographs having 24 hr and greater durations appear to lack the large mean and median values seen in the other modules for reasons that are not understood. The analysis is graphically illustrated on figures 48–53 of chapter 6. In a few instances the 24 hr and greater hyetographs for the c71c68c79c79c68c86 module were not considered (see table 25). The text explicitly identifies the exceptions. 10. On the prior-Probability Weighted Moments. The use of the prior-Probability Weighted Moments for the computation of the L-moments was required because of the uneven and nonrandom distribution of the data points defining the observed hyetographs. The method requires a simple numerical approximation to a derivative in eq. 46 of chapter 4. The method is useful in the context of the hyetograph analysis but likely has rather limited potential for other applications. The method or similar analogs have not been previously described in the literature. Unbiased estimators of the L-moments were used in all other components of this dissertation outside of the computation of hyetograph L-moments. 273 11. On the Modification of the Carman-Kozeny Equation for Application of L-moment Statistics for Estimation of the Intrinsic Permeability of Porous Media. The Carman-Kozeny equation is important for estimation of the intrinsic permeability of porous media based on physical properties of the media such as grain size (diameter) and porosity. The derivations and supporting discussion in chapter 8 indicate that a promising new application of L-moments for estimation of permeability is possible through a. Characterizing the distribution of grain size (diameter) data by computing prior-Probability Weighted Moment based L-moment statistics. b. Selecting a suitable quantile function to model the distribution. c. Fitting of the suitable quantile function to the L-moments. d. Solving an integral either analytically or numerically to estimate an correction factor to the Carman-Kozeny equation. Based on a few simple analytical examples, the correction factor appears dependent on the model of the grain- or particle-size distribution as well as the “moments” of the distribution itself. This conclusion is contrary to a conclusion of distribution independence reached by previous researchers as discussed in chapter 8. 274 APPENDIX A Reference list of U.S. Geological Survey reports used for development of the hyetograph data base 275 AUSTIN URBAN STUDIES U.S. Geological Survey, 1967, Basic data for urban hydrology study, Austin, Texas, 1967: U.S. Geological Survey, Texas District, 59 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Austin, Texas, 1968: U.S. Geological Survey, Texas District, 68 p. Robbins, W.D., 1969, Annual compilation and analysis of hydrologic data for urban studies in the Austin, Texas metropolitan area, 1969: U.S. Geological Survey, Texas District, 46 p. VanZandt, J.K., 1972, Annual compilation and analysis of hydrologic data for urban studies in the Austin, Texas metropolitan area, 1970: U.S. Geological Survey Open-File Report, 70 p. Tovar, F.H., 1973, Annual compilation and analysis of hydrologic data for urban studies in the Austin, Texas metropolitan area, 1971: U.S. Geological Survey Open-File Report, 73 p. Wehmeyer, E.E., 1974, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1972: U.S. Geological Survey Open-File Report, 49 p. Mitchell, R.N., 1975, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1973: U.S. Geological Survey Open-File Report, 61 p. Mitchell, R.N., 1976, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1974: U.S. Geological Survey Open-File Report, 60 p. Choffel, K.L., Roddy, W.R., and Mitchell, R.N., 1977, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1975: U.S. Geological Survey Open-File Report 76–885, 126 p. Maderak, M.L., Gordon, J.D., and Mitchell, R.N., 1978, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1976: U.S. Geological Survey Open-File Report 78–457, 263 p. Slade, R.M., Jr., Gordon, J.D., and Mitchell, R.N., 1979, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1977: U.S. Geological Survey Open-File Report 79–271, 192 p. Slade, R.M., Jr., Dorsey, M.E., Gordon, J.D., and Mitchell, R.N., 1980, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1978: U.S. Geological Survey Open-File Report 80–728, 227 p. Slade, R.M., Jr., Dorsey, M.E., Gordon, J.D., Mitchell, R.N., and Gaylord, J.L., 1981, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1979: U.S. Geological Survey Open-File Report 81–628, 281 p. 276 Slade, R.M., Jr., Gaylord, J.L., Dorsey, M.E., Mitchell, R.N., and Gordon, J.D., 1982, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1980: U.S. Geological Survey Open-File Report 82–506, 264 p. Massey, B.C., Reeves, W.E., and Lear, W.A., 1982, Flood of May 24-25, 1981, in the Austin, Texas metropolitan area: U.S. Geological Survey Hydrologic Investigations Atlas HA–656, 2 p. Slade, R.M., Jr., Veenhuis, J.E., Dorsey, M.E., Gardiner, H., and Smith, A.E., 1983, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1981: U.S. Geological Survey Open-File Report 83–44, 293 p. Slade, R.M., Jr., Veenhuis, J.E., Dorsey, M.E., Stewart, S.L., and Ruiz, L.M., 1984, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1982: U.S. Geological Survey Open-File Report 84–061, 196 p. Gordon, J.D., Pate, D.L., and Dorsey, M.E., 1985, Hydrologic data for urban studies in the Austin, Texas metropolitan area, 1983: U.S. Geological Survey Open-File Report 85–172, 154 p. Gordon, J.D., Pate, D.L., and Dorsey, M.E., 1986, Hydrologic data for urban studies in the Austin metropolitan area, Texas, 1984: U.S. Geological Survey Open-File Report 85–676, 92 p. Veenhuis, J.E. and Gannett, D.G., 1986, The effects of urbanization on floods in the Austin metropolitan area, Texas: U.S. Geological Survey Water-Resources Investigations Report 86–4069, 65 p. Gordon, J.D., Pate, D.L., and Dorsey, M.E., 1987, Hydrologic data for urban studies in the Austin metropolitan area, Texas, 1985: U.S. Geological Survey Open-File Report 87–224, 170 p. Gordon, J.D., Pate, D.L., and Slagle, D.L., 1988, Hydrologic data for urban studies in the Austin metropolitan area, Texas, 1986: U.S. Geological Survey Open-File Report 87–768, 144 p. 277 DALLAS URBAN STUDIES U.S. Geological Survey, 1965, Basic data for urban hydrology study, Dallas, Texas, 1965: U.S. Geological Survey, Texas District, 80 p. U.S. Geological Survey, 1966, Basic data for urban hydrology study, Dallas, Texas, 1966: U.S. Geological Survey, Texas District, 198 p. U.S. Geological Survey, 1967, Basic data for urban hydrology study, Dallas, Texas, 1967: U.S. Geological Survey, Texas District, 80 p. U.S. Geological Survey, 1968, Basic data for urban hydrology study, Dallas, Texas, 1968: U.S. Geological Survey, Texas District, 103 p. Dempster, G.R., Jr. and Massey, B.C., 1971, Annual compilation and analysis of hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1969: U.S. Geological Survey, Texas District, 136 p. Dempster, G.R., Jr. and Massey, B.C., 1972, Annual compilation and analysis of hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1970: U.S. Geological Survey, Texas District, 122 p. Massey, B.C., 1973, Annual compilation and analysis of hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1971: U.S. Geological Survey, Texas District, 84 p. Massey, B.C. and Wood, C.M., 1974, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1972: U.S. Geological Survey Open-File Report, 136 p. Dempster, G.R., Jr., 1974, Effects of urbanization on floods in the Dallas, Texas metropolitan area: U.S. Geological Survey Water-Resources Investigations Report 60–73, 51 p. Hampton, B.B., 1975, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1973: U.S. Geological Survey Open-File Report, 146 p. Hampton, B.B., 1976, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1974: U.S. Geological Survey Open-File Report, 182 p. Hampton, B.B. and Wood, C.M.,1977, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1975: U.S. Geological Survey Open-File Report 77–381, 209 p. Hampton, B.B. and Wood, C.M., 1978, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1976: U.S. Geological Survey Open-File Report 78–437, 161 p. 278 Hampton, B.B. and Wood, C.M., 1979, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1977: U.S. Geological Survey Open-File Report 79–562, 146 p. Hampton, B.B. and Wood, C.M., 1980, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1978: U.S. Geological Survey Open-File Report 80–1004, 114 p. Wood, C.M., Butler, H.S., and Benton, J.D., 1981, Hydrologic data for urban studies in the Dallas, Texas metropolitan area, 1979: U.S. Geological Survey Open-File Report 81–491, 147 p. Land, L.F., Schroeder, E.E., and Hampton, B.B., 1982, Techniques for estimating the magnitude and frequency of floods in the Dallas-Fort Worth metropolitan area, Texas: U.S. Geological Survey Water-Resources Investigations Report 82–18, 55 p. 279 FORT WORTH URBAN STUDIES Dempster, G.R., Jr., and Massey, B.C., 1969, Annual compilation and analysis of hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1969: U.S. Geological Survey Open-File Report, 50 p. Dempster, G.R., Jr., and Massey, B.C., 1972, Annual compilation and analysis of hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1970: U.S. Geological Survey Open-File Report, 89 p. Hampton, B.B., 1973, Annual compilation and analysis of hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1971: U.S. Geological Survey Open-File Report, 77 p. Hampton, B.B., 1974, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1972: U.S. Geological Survey Open-File Report, 123 p. Hampton, B.B., 1975, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1973: U.S. Geological Survey Open-File Report, 129 p. Slade, R.M., Jr., and Taylor, J.M., 1976, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1974: U.S. Geological Survey Open-File Report, 100 p. Slade, R.M., Jr., and Taylor, J.M., 1977, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1975: U.S. Geological Survey Open-File Report 77–266, 96 p. Slade, R.M., Jr., and Taylor, J.M., 1978, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1976: U.S. Geological Survey Open-File Report 77–770, 84 p. Slade, R.M., Jr., Taylor, J.M., and Haynes, D.L., 1979, Hydrologic data for urban studies in the Fort Worth, Texas metropolitan area, 1977: U.S. Geological Survey Open-File Report 79–216, 39 p. Land, L.F., Schroeder, E.E., and Hampton, B.B., 1982, Techniques for estimating the magnitude and frequency of floods in the Dallas-Fort Worth metropolitan area, Texas: U.S. Geological Survey Water-Resources Investigations Report 82–18, 55 p. [also listed in DALLAS URBAN STUDIES section] 280 SAN ANTONIO URBAN STUDIES Land, L.F., 1971, Annual compilation and analysis of hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1969: U.S. Geological Survey 109 p. Land, L.F., 1972, Annual compilation and analysis of hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1970: U.S. Geological Survey, Texas District, 178 p. Steger, R.D., 1973, Annual compilation and analysis of hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1971: U.S. Geological Survey, Texas District, 109 p. Steger, R.D., 1974, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1972: U.S. Geological Survey Open-File Report, 102 p. Steger, R.D., 1975, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1973: U.S. Geological Survey Open-File Report, 127 p. Gonzalez, V., 1976, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1974: U.S. Geological Survey Open-File Report, 109 p. Harmsen, L., 1977, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1975: U.S. Geological Survey Open-File Report 77–221, 91 p. Harmsen, L., 1978, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1976: U.S. Geological Survey Open-File Report 78–164, 132 p. Perez, R., 1980, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1977: U.S. Geological Survey Open-File Report 80–743, 100 p. Perez, R., 1981, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1978: U.S. Geological Survey Open-File Report 81–922, 91 p. Perez, R., 1982, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1979–80: U.S. Geological Survey Open-File Report 82–158, 125 p. Perez, R., 1983, Hydrologic data for urban studies in the San Antonio, Texas metropolitan area, 1981: U.S. Geological Survey Open-File Report 83–35, 58 p. 281 BRAZOS BASIN: Cow Bayou Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Cow Bayou Basin, Texas: U.S. Geological Survey, Texas District, 52 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Cow Bayou Basin, Texas: U.S. Geological Survey, Texas District, 17 p. U.S. Geological Survey, 1962, Hydrologic data of Cow Bayou, Brazos River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 68 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 40 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 55 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 64 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 72 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 62 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Cow Bayou, Brazos River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 80 p. U.S. Geological Survey, 1969, Hydrologic studies of small watersheds, Cow Bayou, Brazos River Basin, Texas, 1955–64: U.S. Geological Survey, Texas District, 66 p. Sansom, J.N., 1969, Annual compilation and analysis of hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 76 p. Watson, J.A., 1970, Annual compilation and analysis of hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1970: U.S. Geological Survey, Texas District, 85 p. Watson, J.A., 1971, Annual compilation and analysis of hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1971: U.S. Geological Survey, Texas District, 63 p. VanZandt, J.K., 1972, Hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1972: U.S.Geological Survey Open-File Report, 72 p. VanZandt, J.K., 1973, Hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1973: U.S.Geological Survey Open-File Report, 74 p. 282 VanZandt, J.K., 1974, Hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1974: U.S.Geological Survey Open-File Report, 63 p. Mitchell, R.N. and Wehmeyer, E.E., 1977, Hydrologic data for Cow Bayou, Brazos River Basin, Texas, 1975: U.S.Geological Survey Open-File Report 76–723, 81 p. BRAZOS RIVER BASIN: Green Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Green Creek Basin, Texas: U.S. Geological Survey, Texas District, 42 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Green Creek study area, Texas, 1961: U.S. Geological Survey, Texas District, [variously paged]. U.S. Geological Survey, 1962, Hydrologic data of Green Creek, Brazos River Basin, Texas, 1962: U.S. Geological Survey, Texas District, [variously paged]. U.S. Geological Survey, 1963, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 52 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 58 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 44 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 109 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 34 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Green Creek, Brazos River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 66 p. Massey, B.C., 1969, Annual compilation and analysis of hydrologic data for Green Creek, Brazos River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 44 p. U.S. Geological Survey, 1969, Preparation of hydrologic data small watersheds project: U.S. Geological Survey, Texas District, 43 p. Hampton, B.B., 1970, Annual compilation and analysis of hydrologic data for Green Creek, Brazos River Basin, Texas,1970: U.S. Geological Survey Open-File Report, 42 p. 283 Hampton, B.B., 1971, Annual compilation and analysis of hydrologic data for Green Creek, Brazos River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 30 p. U.S. Geological Survey, 1972, Hydrologic studies of small watersheds, Green Creek, Brazos River Basin, Texas, 1955–66: U.S. Geological Survey, Texas District, 55 p. BRAZOS RIVER BASIN: Little Pond Creek and North Elm Creek Watershed Studies Slade, R.M., Jr., 1970, Annual compilation and analysis of hydrologic data for Little Pond Creek and North Elm Creek, Brazos River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 44 p. Hawkinson, R.O., 1972, Hydrologic studies of Little Pond Creek and North Elm Creek watersheds, Brazos River Basin, Texas, 1963-69: U.S. Geological Survey, Texas District, 137 p. VanZandt, J.K., 1973, Annual compilation and analysis of hydrologic data for Little Pond Creek and North Elm Creek, Brazos River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 52 p. Mitchell, R.N., 1974, Hydrologic data for Little Pond Creek and North Elm Creek, Brazos River Basin, Texas, 1972: U.S. Geological Survey Open-File Report, 37 p. 284 COLORADO RIVER BASIN: Deep Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Deep Creek Basin, Texas: U.S. Geological Survey, Texas District, 58 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Deep Creek study area, Texas, 1961: U.S. Geological Survey, Texas District, 111 p. U.S. Geological Survey, 1962, Hydrologic data of Deep Creek, Colorado River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 51 p. U.S. Geological Survey, 1963, Hydrologic data of Deep Creek, Colorado River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 153 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Deep Creek, Colorado River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 78 p. U.S. Geological Survey, 1965, Hydrologic studies of small watersheds, Deep Creek, Colorado River Basin, Texas, 1951–61: U.S. Geological Survey, Texas District, 123 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Deep Creek, Colorado River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 72 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Deep Creek, Colorado River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 55 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Deep Creek, Colorado River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 87 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Deep Creek, Colorado River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 86 p. Hejl, H.R., 1971, Annual compilation and analysis of hydrologic data for Deep Creek, Colorado River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 73 p. Hejl, H.R., 1972, Annual compilation and analysis of hydrologic data for Deep Creek, Colorado River Basin, Texas, 1970: U.S. Geological Survey, Texas District, 35 p. Lee, J.N., 1973, Annual compilation and analysis of hydrologic data for Deep Creek, Colorado River Basin, Texas, 1971: U.S. Geological Survey, Texas District, 76 p. COLORADO RIVER BASIN: Mukewater Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds Mukewater Creek Basin, Texas: U.S. Geological Survey, Texas District, 30 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds Mukewater Creek study area, Texas, 1961: U.S. Geological Survey, Texas District, 47 p. 285 U.S. Geological Survey, 1962, Hydrologic data of Mukewater Creek, Colorado River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 60 p. U.S. Geological Survey, 1963, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 80 p. U.S. Geological Survey, 1964, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 56 p. U.S. Geological Survey, 1965, Hydrologic studies of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1952–60: U.S. Geological Survey, Texas District, 70 p. U.S. Geological Survey, 1965, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 67 p. U.S. Geological Survey, 1966, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 56 p. U.S. Geological Survey, 1967, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 87 p. U.S. Geological Survey, 1968, Compilation of hydrologic data of small watersheds Mukewater Creek, Colorado River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 81 p. Hejl, H.R., 1969, Annual compilation and analysis of hydrologic data for Mukewater Creek, Colorado River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 84 p. Hejl, H.R., 1972, Annual compilation and analysis of hydrologic data for Mukewater Creek, Colorado River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 69 p. Lee, J.N., 1973, Annual compilation and analysis of hydrologic data for Mukewater Creek, Colorado River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 89 p. Lee, J.N., 1974, Hydrologic data for Mukewater Creek, Colorado River Basin, Texas, 1972: U.S. Geological Survey Open-File Report, 67 p. Lee, J.N., 1975, Hydrologic data for Mukewater Creek, Colorado River Basin, Texas, 1973: U.S. Geological Survey Open-File Report, 65 p. 286 SAN ANTONIO RIVER BASIN: Calaveras Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Calaveras Creek Basin, Texas, 1960: U.S. Geological Survey, Texas District, 36 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Calaveras Creek study area, Texas, 1961: U.S. Geological Survey, Texas District, [variously paged]. U.S. Geological Survey, 1962, Hydrologic data of Calaveras Creek, San Antonio River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 41 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 41 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 50 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 62 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 34 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 47 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Calaveras Creek, San Antonio River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 66 p. Alexander, J.M., 1969, Annual compilation and analysis of hydrologic data for Calaveras Creek, San Antonio River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 56 p. Reddy, D.R., 1971, Annual compilation and analysis of hydrologic data for Calaveras Creek, San Antonio River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 63 p. U.S. Geological Survey, 1972, Hydrologic studies of small watersheds, Calaveras Creek, San Antonio River Basin, Texas, 1955–68: U.S. Geological Survey, Texas District, 109 p. Steger, R.D., 1973, Annual compilation and analysis of hydrologic data for Calaveras and Escondido Creeks, San Antonio River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 74 p. 287 SAN ANTONIO RIVER BASIN: Escondido Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Escondido Creek Basin, Texas, 1960: U.S. Geological Survey, Texas District, 45 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Escondido Creek Basin, Texas, 1961: U.S. Geological Survey, Texas District, 70 p. U.S. Geological Survey, 1962, Hydrologic data of Escondido Creek, San Antonio River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 43 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 85 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 86 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 72 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 44 p. U.S. Geological Survey, 1967, Hydrologic studies of small watersheds, Escondido Creek, San Antonio River Basin, Texas, 1955–63: U.S. Geological Survey, Texas District, 123 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 73 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Escondido Creek, San Antonio River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 73 p. Reddy, D.R., 1971, Annual compilation and analysis of hydrologic data for Escondido Creek, San Antonio River Basin Texas, 1969: U.S. Geological Survey, Texas District, 62 p. Reddy, D.R., 1971, Annual compilation and analysis of hydrologic data for Escondido Creek, San Antonio River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 65 p. Steger, R.D., 1973, Annual compilation and analysis of hydrologic data for Calaveras and Escondido Creeks, San Antonio River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 74 p. [Also listed in SAN ANTONIO RIVER BASIN Calaveras Creek Watershed Studies Section] 288 TRINITY RIVER BASIN: Elm Fork Watershed Studies U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Elm Fork Trinity River study area, Texas, 1961: U.S. Geological Survey, Texas District, 43 p. U.S. Geological Survey, 1962, Hydrologic studies of small watersheds, Elm Fork Trinity River Basin, Montague and Cooke Counties, Texas 1956–60: U.S. Geological Survey, Texas District, 77 p. U.S. Geological Survey, 1962, Hydrologic data of Elm Fork Trinity River, Trinity River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 60 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Elm Fork Trinity River, Trinity River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 60 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Elm Fork Trinity River, Trinity River Basin, Texas, 1964–65: U.S. Geological Survey, Texas District, 135 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Elm Fork Trinity River, Trinity River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 60 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Elm Fork Trinity River, Trinity River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 54 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Elm Fork Trinity River, Trinity River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 84 p. Sansom, J.N., 1969, Annual compilation and analysis of hydrologic data for Elm Fork Trinity River, Trinity River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 47 p. Sansom, J.N., 1972, Annual compilation and analysis of hydrologic data for Elm Fork Trinity River, Trinity River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 47 p. Lucero, E.D., 1973, Annual compilation and analysis of hydrologic data for Elm Fork Trinity River, Trinity River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 27 p. 289 TRINITY RIVER BASIN: Honey Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Honey Creek Basin, Texas 1960: U.S. Geological Survey, Texas District, 89 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Honey Creek Basin, Texas 1961: U.S. Geological Survey, Texas District, 73 p. U.S. Geological Survey, 1962, Hydrologic studies of small watersheds, Honey Creek Basin, Collin and Grayson Counties, Texas 1953–59: U.S. Geological Survey, Texas District, 102 p. U.S. Geological Survey, 1962, Hydrologic data of Honey Creek, Trinity River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 148 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 83 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 77 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Honey Creek, Trinity River basin Texas, 1965: U.S. Geological Survey, Texas District, 79 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 126 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 68 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 78 p. Sansom, J.N., 1969, Annual compilation and analysis of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1969: U.S. Geological Survey Open-File Report, 85 p. Sansom, J.N., 1972, Annual compilation and analysis of hydrologic data, Honey Creek, Trinity River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 66 p. Hampton, B.B., 1973, Annual compilation and analysis of hydrologic data for Honey Creek, Trinity River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 28 p. 290 TRINITY RIVER BASIN: Little Elm Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Little Elm Creek Basin, Texas, 1960: U.S. Geological Survey, Texas District, 27 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Little Elm Creek study area, Trinity River Basin, Texas, 1961: U.S. Geological Survey, Texas District, 23 p. U.S. Geological Survey, 1962, Hydrologic data of Little Elm Creek, Trinity River Basin, Texas, 1962: U.S. Geological Survey, Texas District, 35 p. U.S. Geological Survey, 1963, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1963: U.S. Geological Survey, Texas District, 34 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 28 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 28 p. U.S. Geological Survey, 1966, Hydrologic studies of small watersheds, Little Elm Creek, Trinity River Basin, Texas, 1956–62: U.S. Geological Survey, Texas District, 59 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 69 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 81 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Little Elm Creek, Trinity River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 83 p. Hampton, B.B., 1971, Annual compilation and analysis of hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 68 p. Hampton, B.B., 1972, Annual compilation and analysis of hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1970: U.S. Geological Survey, Texas District, 100 p. Hampton, B.B., 1973, Annual compilation and analysis of hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1971: U.S. Geological Survey, Texas District, 47 p. Hampton, B.B., 1974, Hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1972: U.S. Geological Survey Open-File Report, 74 p. 291 Slade, R.M., Jr., and Taylor, J.M., 1975, Hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1973: U.S. Geological Survey Open-File Report, 74 p. Slade, R.M., Jr., and Taylor, J.M., 1976, Hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1974: U.S. Geological Survey Open-File Report, 73 p. Slade, R.M., Jr., and Taylor, J.M., 1977, Hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1975: U.S. Geological Survey Open-File Report 77–83, 108 p. Slade, R.M., Jr., and Taylor, J.M., 1978, Hydrologic data for Little Elm Creek, Trinity River Basin, Texas, 1976: U.S. Geological Survey Open-File Report 78–100, 78 p. TRINITY RIVER BASIN: North Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, North Creek Basin, Texas: U.S. Geological Survey, Texas District, 19 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, North Creek study area, Texas: U.S. Geological Survey, Texas District, [variously paged]. U.S. Geological Survey, 1963, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1962–63: U.S. Geological Survey, Texas District, 47 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1964: U.S. Geological Survey, Texas District, 19 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 30 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 28 p. U.S. Geological Survey, 1967, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 27 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, North Creek, Trinity River Basin, Texas, 1968: U.S. Geological Survey, Texas District, 36 p. Kidwell, C.C., 1969, Annual compilation and analysis of hydrologic data for North Creek, Trinity River Basin, Texas, 1969: U.S. Geological Survey, Texas District, 32 p. Maderak, M.L. and Lucero, E.D., 1971, Annual compilation and analysis of hydrologic data for North Creek, Trinity River Basin, Texas, 1970: U.S. Geological Survey Open-File Report, 32 p. 292 Lucero, E.D., 1973, Annual compilation and analysis of hydrologic data for North Creek, Trinity River Basin, Texas, 1971: U.S. Geological Survey Open-File Report, 23 p. Slade, R.M., Jr., 1974, Hydrologic data for North Creek Trinity River Basin, Texas, 1972: U.S. Geological Survey Open-File Report, 31 p. Slade, R.M., Jr., 1975, Hydrologic data for North Creek Trinity River Basin, Texas, 1973: U.S. Geological Survey Open-File Report, 44 p. Slade, R.M., Jr., 1976, Hydrologic data for North Creek Trinity River Basin, Texas, 1974: U.S. Geological Survey Open-File Report, 40 p. Kidwell, C.C., 1977, Hydrologic data for North Creek Trinity River Basin, Texas, 1975: U.S. Geological Survey Open-File Report 76–724, 50 p. Kidwell, C.C., 1978, Hydrologic data for North Creek Trinity River Basin, Texas, 1976: U.S. Geological Survey Open-File Report 77–732, 42 p. Kidwell, C.C., 1979, Hydrologic data for North Creek Trinity River Basin, Texas, 1977: U.S. Geological Survey Open-File Report 79–335, 39 p. Kidwell, C.C., 1980, Hydrologic data for North Creek Trinity River Basin, Texas, 1978: U.S. Geological Survey Open-File Report 80–573, 44 p. Kidwell, C.C., 1981, Hydrologic data for North Creek Trinity River Basin, Texas, 1979: U.S. Geological Survey Open-File Report 81–823, 38 p. TRINITY RIVER BASIN: Pin Oak Creek Watershed Studies U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Pin Oak Creek, Trinity River Basin, Texas, 1956–62: U.S. Geological Survey, Texas District, 69 p. U.S. Geological Survey, 1960, Hydrologic studies of small watersheds, Pin Oak Creek Basin, Texas, 1960: U.S. Geological Survey, Texas District, 44 p. U.S. Geological Survey, 1961, Hydrologic studies of small watersheds, Pin Oak Creek study area, Texas 1961: U.S. Geological Survey, Texas District, 38 p. U.S. Geological Survey, 1964, Compilation of hydrologic data, Pin Oak Creek, Trinity River Basin, Texas, 1962–64: U.S. Geological Survey, Texas District, 36 p. U.S. Geological Survey, 1965, Compilation of hydrologic data, Pin Oak Creek, Trinity River Basin, Texas, 1965: U.S. Geological Survey, Texas District, 30 p. U.S. Geological Survey, 1966, Compilation of hydrologic data, Pin Oak Creek, Trinity River Basin, Texas, 1966: U.S. Geological Survey, Texas District, 27 p. 293 U.S. Geological Survey, 1967, Compilation of hydrologic data, Pin Oak Creek, Trinity River Basin, Texas, 1967: U.S. Geological Survey, Texas District, 25 p. U.S. Geological Survey, 1968, Compilation of hydrologic data, Pin Oak Creek, Trinity River Basin Texas, 1968: U.S. Geological Survey, Texas District, 44 p. Hampton, B.B. and Myers, D.R., 1969, Annual compilation and analysis of hydrologic data, Pin Oak Creek, Trinity River Basin Texas, 1969: U.S. Geological Survey, Texas District, 28 p. Hampton, B.B. and Myers, D.R., 1972, Annual compilation and analysis of hydrologic data, Pin Oak Creek, Trinity River Basin Texas, 1970: U.S. Geological Survey Open-File Report, 50 p. Hampton, B.B., 1973, Annual compilation and analysis of hydrologic data for Pin Oak Creek, Trinity River Basin Texas, 1971: U.S. Geological Survey Open-File Report, 18 p. Hampton, B.B., 1974, Hydrologic data for Pin Oak Creek, Trinity River Basin Texas, 1972: U.S. Geological Survey Open-File Report, 26 p. 294 APPENDIX B Background on L-moment Statistical Theory The L-moments and probability weighted moments are relatively unknown features of univariate statistical theory. Because of the widespread unfamiliarity with these statistics, it is necessary to provide a basic, albeit brief, review in this appendix. References are included in the References section of this dissertation. 295 MOMENTS OF A DISTRIBUTION Distribution description is an important component of much statistical analysis and is conducted by the statistical summarization of sample observations of a random variable . Traditionally, the data are statistically summarized by the product moments of the data (arithmetic mean, standard deviation or variance, skew, and kurtosis). The mathematical definitions and the sampling properties of the product moments are widely known, but are reviewed here. The theoretical product moments of random variable are , (B-1.1) , (B-1.2) , and (B-1.3) (B-1.4) where , , , and denote the theoretical mean, variance, third moment, and fourth moment, respectively. In general, the higher moments are .(B-2) The is the expectation operator and in terms of the probability density function is ,(B-3) and in terms of the quantile function is .(B-4) X X µ EX[]= σ 2 Var X[] EXµ–() 2 []== µ 3 EXµ–() 3 []= µ 4 EXµ–() 4 []= µσ 2 µ 3 µ 4 r 2≥ µ r EXµ–() r []= E [] f x() EX r [] x r fx()xd ∞– ∞ ∫ = xF() EX r [] xF() r Fd 0 1 ∫ = 296 Note that the term in eq. B-4 is analogous to the incremental probability for each observation in a finite sample. A special treatment of the term plays a central role in the p-PWMs described in chapter 4 of this dissertation. The mean locates the center of the distribution along the real-number line whereas the spread or width of the distribution is measured by the standard deviation, . An often used dimensionless representation of the distribution spread is the coefficient of variation, . The higher moments are almost always expressed as dimensionless quantities and where and denote the skewness and kurtosis, respectively. The product moments also are defined for finite samples. The first sample product moment is the mean and is .(B-5) The quantity is a minimum variance unbiased estimator (MVUE) of the theoretical mean . The higher product moments are .(B-6) are not unbiased. Unbiased estimates are obtained by , (B-7.1) , and (B-7.2) . (B-7.3) dF dF σσ 2 = CV σµ⁄= γµ 3 σ 3 ⁄= K µ 4 σ 4 ⁄= γ K m 1 n --- x j j 1= n ∑ = m µ r 2≥ m r 1 n --- x j m–() r j 1= n ∑ = m r m 2 s 2 n n 1–() ---------------- m 2 == m 3 n 2 n 1–()n 2–() ---------------------------------- m 3 = m 4 n 2 n 2–()n 3–() ---------------------------------- n 1+ n 1– ------------   m 4 3m 2 2 –    = 297 The sample standard deviation is estimated by the square root of the sample variance, . However, though is unbiased, is not. The sample CV, skew, and kurtosis are estimated, but are not unbiased, by , , and , respectively. The uniformly minimum variance unbiased estimate (David, 1981, p. 185) of the standard deviation, , is ,(B-8) where the complete Gamma function is .(B-9) Finally, it is important to note that sample estimates of and can not exceed certain values. The values depend on the size of the sample regardless of the true values of the statistic (Kirby, 1974). These limits are and (B-10.1) . (B-10.2) For example, if then and regardless of the true values of or . These bounds are especially problematic during the analysis of highly variable or highly skewed data. Such data sets in the natural sciences might include annual peak flood time series, grain-size distributions, and permeability measurements. ss 2 = s 2 s cv s m⁄= gm 3 s 3 ⁄= kk 4 s 4 ⁄ 3+= s UMVU s UMVU Γ n 1–()2⁄[] Γ n 2⁄[]2 -------------------------------- x j m–() 2 j 1= n ∑ = Γ k() Γ k() u k 1– e u– ud 0 ∞ ∫ = cv g cv n 1–()< g n 2– n 1–() --------------------- < n 30= cv 5.39< g 5.20< CV γ 298 L-MOMENTS The majority of this review is derived from Hosking (1990), Hosking and Wallis (1997), and references therein. When the need arises, other references are provided for specifics of L-moment theory development. Consider a real-valued random variable with a cumulative distribution function and a quantile function . As before, is a cumulative probability or nonexceedance probability and . If a random sample of size is drawn from the distribution of , and the sample is arranged in ascending order, the values become the order statistics of . The mathematical framework of order statistics is well described by David (1981). The expectation of an order statistic can be expressed as .(B-1) The L-moments are the expectations of specific linear combinations of the order statistic expectations. In general, the L-moments are defined by . (B-12) The formula for the combinations of distinct items taken at a time is , (B-13) and by definition . The first four L-moments from eq. B-12 are , (B-14.1) , (B-14.2) X Fx() xF() F 0 F 1≤≤ n X X 1:n X 2:n … X n:n ≤≤≤ X EX j:r [] r! j 1–()! rj–()! ---------------------------------- xF()F j 1– 1 F–() rj– Fd 0 1 ∫ = λ r 1 r --- 1– k r 1– k   EX rk:r– [] k 0= r 1– ∑ ≡ n r a b   C ab a! b! ab–()! ------------------------ == a 0   1= λ 1 EX 1:1 []= λ 2 1 2 --- EX 2:2 X 1:2 –[]= 299 , and (B-14.3) . (B-14.4) The first L-moment is the mean. The mean is the expected value of a single observation of . The second L-moment is a measure of the dispersion or spread of much like the usual standard deviation. is read as one half the expected difference between the two order statistics of a sample of size , and is referred to as L-scale or L-variation. The difference in order statistic expectations, , is known as Gini’s Mean Difference (Serfling, 1980, p. 174, 263–264; David, 1981, p. 192; Kaigh and Driscoll, 1987, p. 26; and Hosking, 1990, p. 110 and 114). Gini’s Mean Difference can be computed in a variety of ways; David (1981) provides three variations. The algebraically simplest variation is , (B-15) in which is one of sample observations. The relation between the and is . (B-16) Upon substituting the expression into the L-moment definition, the L-moments are λ 3 1 3 --- EX 3:3 2X 2:3 – X 1:3 +[]= λ 4 1 4 --- EX 4:4 3X 3:4 –3X 2:4 X 1:4 –+= λ 1 X λ 2 X λ 2 n 2= E X 2:2 []E X 1:2 []– G E X 2:2 []E X 1:2 []– 1 nn 1–() -------------------- x i x j – ji= n ∑ i 1= n ∑ == x i nGλ 2 λ 2 1 2 --- G()= EX j:r [] 300 for (B-17) where is known as the shifted Legendre polynomial with the explicit form (B-18) in which . (B-19) It is possible to define an L-moment ratio , which is analogous to the usual coefficient of variation. is known as L-CV and is referred to as the coefficient of L-variation. satisfies (Hosking, 1989). It is often convenient to standardize the higher L-moments for so that they are independent of the measurement units of . This is accomplished through division by the second L-moment . The L-moment ratios are defined as for (B-20) The L-moment ratios and can be interpreted as standardized measures of skew and kurtosis, respectively. Hence, and are known as L-skew and L-kurtosis. The bounds on each are such that and . If a distribution can only acquire positive values, then . The boundedness λ r xF()P r 1– * F()Fd 0 1 ∫ = r 12… n,, ,= P r 1– * F() r 1– P r * F() p r,k * F k k 0= r ∑ = p r,k * 1– rk– r k  rk+ k  1– rk– rk+()! k!() 2 rk–()! ---------------------------------== τλ 2 λ 1 ⁄≡ τ τ 0 τ 1<< λ r r 3≥ X λ 2 τ r λ r λ 2 ----- ≡ r 34…,,= τ 3 τ 4 τ 3 τ 4 1– τ 3 1<< 1 4 --- 5τ 3 2 1–()τ 4 ≤ 1< 2τ 1– τ 3 1<≤ 301 on L-skew and L-kurtosis is a very convenient property and is a property that is not shared by the unbounded product moment skew and kurtosis. PROBABILITY WEIGHTED MOMENTS The probability weighted moments (PWMs) defined by Greenwood and others (1979) for quantile function and cumulative probability are . (B-21) The PWM is the mean. The higher PWMs and are not easily interpreted. However, the L-moments and PWMs can be expressed as linear combinations of each other. Because of linearity, procedures based on L-moments or the PWMs are equivalent. The PWMs actually predate the L-moments, but the L-moments are far more convenient and directly interpretable as measures of distributions. The PWMs thus are generally considered as a means to compute the L-moments. Particularly useful PWMs for L-moment theory are and . The focus is on here with a note that and can be shown as linear combinations of each other , (B-22.1) . (B-22.2) One can contrast with the usual product moment definition (eq. B-4), which is repeated here . (B-23) The L-moments defined in linear terms of the PWMs are the quantities xF() F M prs,, ExF() p F r 1 F–() s []= M 100,, ExF()[]= r 0> s 0> α r M 10r,, = β r M 1 r 0,, = β r α r β r α r xF()1 F–() r Fd 0 1 ∫ = β r xF()F r Fd 0 1 ∫ xFx()[] r fx()xd ∞– ∞ ∫ == β r EX r [] xF() r Fd 0 1 ∫ = 302 for , (B-24) where was defined earlier. The first four L-moments are , (B-25.1) , (B-25.2) , and (B-25.3) , (B-25.4) or equivalently in terms of the quantile function as , (B-26.1) , (B-26.2) , and (B-26.3) . (B-26.4) SAMPLE L-MOMENTS AND PROBABILITY WEIGHTED MOMENTS L-moments and PWMs are defined for the quantile function , but in general are estimated for finite samples of size by arranging the sample in ascending order to acquire the sample order statistics of random variable . Presently there are two classes of L-moment estimators for finite samples, the unbiased estimators and the plotting-position estimators. An additional class of estimator based on p-PWMs is the subject of chapter 4 of this dissertation. λ r 1+ p rk, * β k k 0= r ∑ = r 01… n 1–,, ,= p rk, * λ 1 β 0 = λ 2 2β 1 β 0 –= λ 3 6β 2 6β 1 – β 0 += λ 4 20β 3 30β 2 –12β 1 β 0 –+= λ 1 xF()Fd 0 1 ∫ = λ 2 xF()2F 1–()Fd 0 1 ∫ = λ 3 xF()6F 2 6F–1+ Fd 0 1 ∫ = λ 4 xF()20F 3 30F 2 –12F 1–+ Fd 0 1 ∫ = xF() n x 1:n x 2:n … x n:n ≤≤≤ X 303 Unbiased Estimators Unbiased estimates of , hence , can be made by for . (B-27) The “unbiased” weight factor on a specific is and is given by . (B-28) The multiplication to the left of the summation has been included in the unbiased weight factor to facilitate later analysis. The first four unbiased PWM estimators are , (B-29.1) , (B-29.2) , and (B-29.3) . (B-29.4) Unbiased estimates of the first four L-moments in terms of the unbiased PWM estimates are , (B-30.1) , (B-30.2) β r λ r b r 1 n --- n 1– r   1– j 1– r   x j:n jr1+= n ∑ = r 01… n 1–,, ,= x j:n w jr, w jr, j 1– r   n n 1– r   --------------------= b 0 1 n --- x j:n j 1= n ∑ = b 1 1 n --- j 1– n 1–   x j:n j 1= n ∑ = b 2 1 n --- j 1– n 1–   j 2– n 2–   x j:n j 1= n ∑ = b 3 1 n --- j 1– n 1–  j 2– n 2–  j 3– n 3–  x j:n j 1= n ∑ = l 1 b 0 = l 2 2b 1 b 0 –= 304 , and (B-30.3) . (B-30.4) Hence, in general the unbiased L-moment estimator is for . (B-31) Wang (1996b) described an algorithm for unbiased L-moment estimation that does not require the prior computation of . The direct sample L-moment algorithm by Wang is numerically equivalent to eq. B-31. Natural estimators of (L-CV) and the other L-moment ratios are , and (B-32.1) for . (B-32.2) These and estimators are not unbiased, but their biases typically are small in moderate and larger samples. The biases are distribution dependent. Fortunately for many distributions, these biases are operationally negligible for sample sizes of or more. Plotting-Position Estimators Estimates of and also can be made with a second class of estimator called plotting-position estimators. A plotting position is a distribution free or nonparametric estimator of cumulative probability . Historically, plotting positions have been commonly used for graphical display of random samples of . Often recognized reasonable choices include for , where is the plotting position for the th ascending order observation of a random sample of size . Hence, the PWMs are estimated as l 3 6b 2 6b 1 – b 0 += l 4 20b 3 30b 2 –12b 1 b 0 –+= l r 1+ p rk, * b k k 0= r ∑ = r 01… n 1–,, ,= b r τ tl 2 l 1 ⁄= t r l r l 2 ⁄= r 3≥ tt r n 20= β r λ r Fx j:n () X p j:n j δ+()n ε+()⁄= δε 1–>> p j:n j n 305 . (B-33) The plotting-position weight factor on a specific is and is given by . (B-34) The constant is included in the plotting-position weight factor to facilitate later analysis. The plotting-position L-moments and L-moment ratios are estimated as , (B-35.1) , and (B-35.2) for . (B-35.3) In general, is not an unbiased estimator of , but the bias tends to zero in large samples. The biases of are generally larger than biases of . The unbiased L-moments and operationally unbiased L-moment ratios generally are preferred over plotting-position estimators (Hosking and Wallis, 1997, pp. 31–34). However, under special circumstances such as when a specific probability distribution is fitted to the data, plotting-position estimators might provide more accurate parameter and quantile estimates. For example, the plotting position performs well for the generalized Pareto (Hosking and Wallis, 1987), generalized extreme value (GEV) (Hosking and others, 1985), and Wakeby (Landwehr and others, 1979b) distributions. These distributions are common in L-moment based magnitude and frequency analysis of environmental data such as flood peak discharge (stream flow) and extreme precipitation. In rare cases, plotting-position estimators can produce L-moments with β ˜ r 1 n --- p j:n () r x j:n i 1= n ∑ = x j:n w ˜ jr, w ˜ jr, p j:n () r n ---------------= n 1– λ ˜ r 1+ p rk, * β r ˜ k 0= n ∑ = τ ˜ λ ˜ 2 λ ˜ 1 ⁄= τ ˜ r λ ˜ r λ ˜ 2 ⁄= r 3≥ λ ˜ r λ r τ ˜ r t r p j:n j 0.35–()n⁄= 306 theoretically impossible values such as or ; such occurrences are not possible with unbiased estimators. This is a very important consideration for algorithm development involving plotting-position PWM estimation. Confirmation that the bounds of the L-moments are satisfied is absolutely required. EXAMPLE OF MANUAL COMPUTATION OF UNBIASED L-MOMENTS Compute the unbiased L-moments for the following sample ( ) that is arranged in ascending order The unbiased PWMs are required for . (B-36) For the first PWM ( ) (B-37.1) , (B-37.2) (B-37.3) , (B-37.4) , and (B-37.5) . (B-37.6) The first PWM is equal to the arithmetic mean , (B-38.1) , (B-38.2) λ 2 0<τ 3 1> n 10= 0 0.1 0.2 0.25 0.3 0.3 1.0 6.0 10.0 25.0,,, ,,,,, , b r 1 n --- n 1– r  1– j 1– r  x j:n jr1+= n ∑ = r 01… n 1–,, ,= r 0= b 0 1 10 ------ 9 0  1– 0 0  0[] 1 0  0.1[] 2 0  0.2[] 3 0  0.25[] 4 0  0.3[] 5 0  0.3[]more+++ +++= more 6 0  1.0[] 7 0  6.0[] 8 0  10.0[] 9 0  25.0[]++ += b 0 1 10 ------ 1 1 --- 10[] 10.1[]10.2[]10.25[]10.3[]10.3[]more+++ +++= more 1 1.0[]16.0[]1 10.0[]125.0[]++ += b 0 1 10 ------ 43.15[]= b 0 4.315= m 1 n --- x j j 1= n ∑ = m 1 10 ------ 00.10.20.250.30.3161025+ + + + + +++ +[]= 307 , and (B-38.3) (a far simpler computation compared to ) (B-38.4) For the second PWM ( ) (B-39.1) , (B-39.2) (B-39.3) , (B-39.4) , and (B-39.5) . (B-39.6) For the third PWM ( ) (B-40.1) , (B-40.2) (B-40.3) , (B-40.4) , and (B-40.5) . (B-40.6) For the fourth PWM ( ) (B-41.1) , (B-41.2) m 1 10 ------ 4.315[]= m 4.315= b o r 1= b 1 1 10 ------ 9 1  1– 1 1  0.1[] 2 1  0.2[] 3 1  0.25[] 4 1  0.3[] 5 1  0.3[]more++ +++= more 6 1  1.0[] 7 1  6.0[] 8 1  10.0[] 9 1  25.0[]++ += b 1 1 10 ------ 1 9 --- 10.1[]20.2[]30.25[]40.3[]50.3[]more++ +++= more 6 1.0[]76.0[]8 10.0[]925.0[]++ += b 1 1 90 ------ 356.95[]= b 1 3.966= r 2= b 2 1 10 ------ 9 2  1– 2 2  0.2[] 3 2  0.25[] 4 2  0.3[] 5 2  0.3[]more+ +++= more 6 2  1.0[] 7 2  6.0[] 8 2  10.0[] 9 2  25.0[]++ += b 2 1 10 ------ 1 36 ------ 10.2[]30.25[]60.3[]10 0.3[]more++++= more 15 1.0[]21 6.0[]28 10.0[]36 25.0[]++ += b 2 1 360 --------- 1326.75[]= b 2 3.685= r 3= b 3 1 10 ------ 9 3  1– 3 3  0.25[] 4 3  0.3[] 5 3  0.3[]more+++= more 6 3  1.0[] 7 3  6.0[] 8 3  10.0[] 9 3  25.0[]++ += 308 (B-41.3) , (B-41.4) , and (B-41.5) . (B-41.6) For the first L-moment . (B-42) For the second L-moment and (B-43.1) . (B-43.2) For the third L-moment and (B-44.1) . (B-45.1) For the fourth L-moment , (B-46.1) , and (B-46.2) . (B-46.3) For the coefficient of L-variation (L-CV) . (B-47) For L-skew , (B-48) which is a positive or right-tail skewness and is a number . b 3 1 10 ------ 1 36 ------ 10.25[]40.3[]10 0.3[]more++ += more 20 1.0[]35 6.0[]56 10.0[]84 25.0[]++ += b 3 1 840 --------- 2894.25[]= b 3 3.446= λ 1 b 0 4.315== λ 2 2b 1 b 0 – 2 3.966()4.315– 3.617== = λ 2 3.617= λ 3 6b 2 6b 1 b 0 +– 6 3.685()6 3.966()4.315+–== λ 3 2.629= λ 4 20b 3 30b 2 12b 1 b 0 –+–= λ 4 20 3.446()30 3.685()12 3.966()4.315–+–= λ 4 1.647= τ λ 2 λ 1 ----- 3.617 4.315 ------------ 0.8382== = τ 3 λ 3 λ 2 ----- 2.629 3.617 ------------- 0.7268== = 1– τ 3 1<< 309 For L-kurtosis , (B-49) which is a number or . The standard deviation, , is related to the second L-moment . (B-50) The standard deviation, , using an unbiased variance computation through the product moments is and (B-51.1) . (B-51.2) The standard deviation, , using an biased variance computation through the product moments is and (B-52.1) . (B-53.1) The uniformly minimum variance unbiased estimate (David, 1981, p. 185) of the standard deviation, , is and (B-54.1) τ 4 λ 4 λ 2 ----- 1.647 3.617 ------------- 0.4553== = 1 4 --- 5τ 3 2 1–()τ 4 ≤ 1< 0.4103 τ 4 1<≤ σ′ σ′ πλ 2 π 3.617()6.411== = s unbvar s unbvar 1 n 1– ----------- x j m–() 2 j 1= n ∑ 1 9 --- x j 4.315–() 2 j 1= 10 ∑ == s unbvar 576.1 9 -------------8.0== s bvar s bvar 1 n --- x j m–() 2 j 1= n ∑ 1 10 ----- x j 4.315–() 2 j 1= 10 ∑ == s bvar 576.1 10 -------------7.59== s UMVU s UMVU Γ n 1–()2⁄[] Γ n 2⁄[]2 -------------------------------- x j m–() 2 j 1= n ∑ = 310 . (B-55.1) The computer program, c79c80c82c80c72c81c87c86c17c83c79, (Appendix E) can be used to verify the L-moments and PWMs in the above computations. The output from the program is shown below. (B-56.1) c62c90c68c86c84c88c76c87c75c35c79c76c81c88c91c75c82c86c87c64c7c3c79c80c82c80c72c81c87c86c17c83c79c3c16c88c69 c6c3c47c16c48c50c48c40c49c55c54c3c50c41c3c36c3c38c56c48c56c47c36c55c44c57c40c3c51c40c53c38c40c49c55c36c42c40c3c43c60c39c53c50c42c53c36c51c43 c6c3c40c81c87c72c85c3c86c83c68c70c72c3c71c72c79c76c80c76c87c72c71c3c70c88c80c88c79c68c87c76c89c72c3c83c85c82c69c68c69c76c79c76c87c92c3c68c81c71c3c71c68c87c68c3c89c68c79c88c72c86c17 c6c3c58c82c85c78c76c81c74c3c82c81c3c56c37c3c11c51c51c15c3c83c79c82c87c87c76c81c74c3c83c82c86c76c87c76c82c81c30c3c56c37c15c3c88c81c69c76c68c86c72c71c30c3c51c48c15c3c83c85c82c71c88c70c87 c6c3c80c82c80c72c81c87c12c17c3c44c73c3c82c81c72c3c89c68c79c88c72c3c76c86c3c74c76c89c72c81c15c3c87c75c72c81c3c76c87c3c76c86c3c88c86c72c71c17c3c44c73c3c87c90c82c3c89c68c79c88c72c86 c6c3c68c85c72c3c74c76c89c72c81c15c3c87c75c72c81c3c87c75c72c3c86c72c70c82c81c71c3c82c81c72c3c76c86c3c88c86c72c71c17c3c55c75c76c86c3c73c72c68c87c88c85c72c3c83c72c85c80c76c87c86c3c86c90c76c87c70c75c76c81c74 c6c3c69c72c87c90c72c72c81c3c87c75c72c3c83c85c76c82c85c16c51c58c48c3c80c72c87c75c82c71c3c68c81c71c3c87c75c72c3c82c87c75c72c85c3c80c72c87c75c82c71c86c3c90c76c87c75c82c88c87c3c68c3c70c75c68c81c74c72c3c76c81 c6c3c87c75c72c3c76c81c83c88c87c3c86c87c85c72c68c80c17 c6c3c50c81c72c3c89c68c79c88c72c3c82c85c3c41c3c3c3c59c3c83c68c76c85c3c83c72c85c3c79c76c81c72 c6c3c59c3c3c82c85c3c3c41c11c76c74c81c82c85c72c71c12c3c3c3c59 c19 c17c20 c17c21 c17c21c24 c17c22 c17c22 c20 c25 c20c19 c21c24 c6c3c51c85c82c69c68c69c76c79c76c87c76c72c86c3c68c81c71c3c39c68c87c68c3c75c68c89c72c3c69c72c72c81c3c72c81c87c72c85c72c71c17c3c17c3c17 c6c3c47c16c80c82c80c72c81c87c3c38c82c80c83c88c87c68c87c76c82c81 c6c3c51c85c82c68c69c76c79c76c87c92c3c58c72c76c74c75c87c72c71c3c48c82c80c72c81c87c86c3c68c85c72c29 c6c3c3c3c37c72c87c68c19c3c3c32c3c23c17c22c20c24c19 c6c3c3c3c37c72c87c68c20c3c3c32c3c22c17c28c25c25c20 c6c3c3c3c37c72c87c68c21c3c3c32c3c22c17c25c27c24c23 c6c3c3c3c37c72c87c68c22c3c3c32c3c22c17c23c23c24c27 c6c3c3c3c37c72c87c68c23c3c3c32c3c22c17c21c22c24c25 c6c3c47c16c80c82c80c72c81c87c86c3c68c85c72c29 c6c3c3c3c48c72c68c81c3c3c3c3c3c3c3c3c32c3c23c17c22c20c24c19 c6c3c3c3c47c16c54c70c68c79c72c3c3c3c3c3c32c3c22c17c25c20c26c21c3c11c54c87c39c72c89c32c25c17c23c20c20c22c12 c6c3c3c3c47c16c38c57c3c3c3c3c3c3c3c3c32c3c19c17c27c22c27c22 c6c3c3c3c47c16c86c78c72c90c3c3c3c3c3c3c32c3c19c17c26c21c26c22 c6c3c3c3c47c16c78c88c85c87c82c86c76c86c3c3c32c3c19c17c23c24c20c19 c6c3c3c3c55c68c88c24c3c3c3c3c3c3c3c3c32c3c19c17c21c20c19c22 c6c3c54c68c80c83c79c72c86c3c32c3c20c19 c6c3c48c72c68c81c3c3c3c47c16c86c70c68c79c72c3c3c3c47c16c38c57c3c3c3c3c47c16c86c78c72c90c3c3c3c47c16c78c88c85c87c82c86c76c86c3c3c3c55c68c88c24c3c3c3c48c72c71c76c68c81 c23c17c22c20c24c19c3c3c3c22c17c25c20c26c21c3c3c3c19c17c27c22c27c22c3c3c3c19c17c26c21c26c22c3c3c3c19c17c23c24c20c19c3c3c3c19c17c21c20c19c22c3c3c3c19c17c22 s UMVU 11.6317 24 2 ------------------ 576.1 8.23== 311 APPENDIX C Supplemental non-uniform simulations showing biases in the unbiased, plotting-position, and prior-Probability Weighted Moment L-moment estimators 312 Table C1. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was less than 0.5. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB 0.0280 -0.0079 -0.0960 -0.0932 0.1047 n=5 PP .0274 -.0051 -.0779 .0904 .1374 XF .0145 .0043 -.0043 -.1307 .1355 UB .0275 -.0077 -.1067 -.1296 .1139 n=10 PP .0276 -.0064 -.0991 .0147 .1226 XF .0072 .0002 -.0152 -.1057 .0141 UB .0278 -.0076 -.1131 -.1397 .1084 n=20 PP .0279 -.0072 -.1106 -.0544 .1147 XF .0027 -.0005 -.0103 -.0269 -.0260 UB .0275 -.0076 -.1153 -.1441 .1064 n=50 PP .0285 -.0078 -.1183 -.1090 .1109 XF .0006 -.0003 -.0043 .0040 -.0176 UB .0280 -.0079 -.1191 -.1442 .1074 n=100 PP .0270 -.0074 -.1142 -.1222 .1027 XF .0002 -.0002 -.0022 .0036 -.0100 UB .0275 -.0076 -.1175 -.1440 .1038 n=1,000 PP .0277 -.0078 -.1193 -.1397 .1012 XF .0 .0 .0 .0 .0 313 Table C2. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was less than 0.2. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB 0.0214 -0.0095 -0.1093 0.0291 0.0273 n=5 PP .0210 -.0070 -.0934 .1603 .1272 XF .0125 -.0043 -.0701 -.1060 .2236 UB .0208 -.0094 -.1141 .0299 .0128 n=10 PP .0208 -.0083 -.1066 .1265 .0847 XF .0090 -.0049 -.0630 -.1045 .1281 UB .0211 -.0095 -.1189 .0201 .0126 n=20 PP .0210 -.0088 -.1134 .0793 .0506 XF .0059 -.0037 -.0452 -.0424 .0411 UB .0207 -.0094 -.1188 .0200 .0094 n=50 PP .0209 -.0094 -.1191 .0482 .0256 XF .0028 -.0019 -.0233 .0020 .0055 UB .0208 -.0095 -.1206 .0202 .0111 n=100 PP .0207 -.0093 -.1188 .0323 .0193 XF .0011 -.0008 -.0095 .0041 -.0028 UB .0207 -.0094 -.1198 .0173 .0097 n=1,000 PP .0213 -.0095 -.1215 .0143 .0149 XF .0 .0 -.0003 .0006 -.0007 314 Table C3. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was greater than 0.8. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB -0.0159 -0.0029 0.07799 0.0284 0.0391 n=5 PP -.0157 -.0039 .0418 .1880 .0347 XF -.0039 -.0138 -.1195 -.2717 .2128 UB -.0159 -.0025 .0519 .0168 .0221 n=10 PP -.0158 -.0031 .0438 .1248 .0348 XF -.0026 -.0117 -.0993 -.3113 .1250 UB -.0160 -.0028 .0368 .0163 .0159 n=20 PP -.0155 -.0031 .0313 .0764 .0271 XF -.0019 -.0078 -.0649 -.3035 .0412 UB -.0158 -.0027 .0307 .0149 .0141 n=50 PP -.0159 -.0029 .0287 .0429 .0181 XF -.0008 -.0031 -.0250 -.1682 .0039 UB -.0156 -.0027 .0273 .0143 .0118 n=100 PP -.0164 -.0025 .0321 .0309 .0142 XF -.0003 -.0012 -.0094 -.0676 -.0148 UB -.0158 -.0026 .0262 .0155 .0125 n=1,000 PP -.0156 -.0026 .0260 .0188 .0107 XF .0 .0 -.0001 -.0003 -.0017 315 Table C4. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was less than 0.1. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB 0.0135 -.0065 -0.0688 0.0377 0.0042 n=5 PP .0133 -.0052 -.0622 .1754 .0919 XF .0083 -.0051 -.0679 -.0277 .2170 UB .0132 -.0065 -.0774 .0324 -.0014 n=10 PP .0132 -.0059 -.0733 .1345 .0568 XF .0058 -.0052 -.0584 -.0470 .0720 UB .0128 -.0064 -.0797 .0333 -.0121 n=20 PP .0129 -.0063 -.0792 .0913 .0259 XF .0041 -.0036 -.0412 .0063 -.0128 UB .0128 -.0065 -.0827 .0380 -.0174 n=50 PP .0081 -.0065 -.0831 .0586 .0034 XF .0026 -.0023 -.0268 .0293 -.0308 UB .0130 -.0064 -.0833 .0320 -.0164 n=100 PP .0132 -.0065 -.0844 .0470 -.0077 XF .0017 -.0015 -.0177 .0222 -.0182 UB .0132 -.0064 -.0843 .0345 -.0197 n=1,000 PP .0132 -.0065 -.0849 .0340 -.0104 XF .0001 -.0001 -.0007 .0013 -.0014 316 Table C5. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was greater than 0.9. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB -0.0092 -0.0015 0.0757 0.0113 0.0254 n=5 PP -.0089 -.0027 .0457 .1819 .0375 XF -.0007 -.0091 -.0822 -.2280 .0564 UB -.0088 -.0016 .0353 .0062 .0111 n=10 PP -.0089 -.0023 .0264 .1185 .0352 XF -.0007 -.0076 -.0655 -.2800 -.0082 UB -.0084 -.0016 .0217 .0019 .0075 n=20 PP -.0084 -.0019 .0184 .0636 .0222 XF -.0006 -.0048 -.0407 -.2450 -.0472 UB -.0085 -.0017 .0145 .0030 .0029 n=50 PP -.0090 -.0017 .0163 .0313 .0078 XF -.0005 -.0026 -.0215 -.1564 -.0569 UB -.0087 -.0014 .0155 .0022 .0012 n=100 PP -.0087 -.0018 .0124 .0171 .0044 XF -.0004 -.0016 -.0131 -.0996 -.0474 UB -.0083 -.0019 .0006 .0033 .0040 n=1,000 PP -.0008 -.0017 .0115 .0067 -.0031 XF .0 .0 -.0004 -.0028 -.0052 317 Table C6. Comparison of biases for a simulated Kappa distribution using a non-uniform probability distribution by redrawing F if initial F was greater than 0.25 and less than 0.75. [The Kappa distribution had specified L-moments of 0.114, 0.0378, -0.148, and 0.0476 for the mean, L-scale, L-skew, and L-kurtosis, respectively. These L-moments correspond to estimated Kappa parameters of 0.0669, 0.1439, 1.022, and 0.5045 for the location, scale, shape 1, and shape 2 parameters, respectively. Bias is defined as the simulated mean statistic minus the true value. UB, unbiased; PP, plotting position based probability weighted moments; XF, prior-probability weighted moments.] Sample size Estimator type Mean bias L-scale bias L-CV bias L-skew bias L-kurtosis bias UB -0.0047 0.0068 1.6945 0.0039 -0.0457 n=5 PP -.0038 .0041 .0962 .1919 -.0017 XF -.0007 .0062 .0545 .1930 .1344 UB -.0048 .0069 .1113 .0068 -.0829 n=10 PP -.0046 .0055 .0797 .0401 -.0790 XF -.0004 .0020 .0192 .1081 .0116 UB -.0052 .0068 .0936 .0155 -.0915 n=20 PP -.0048 .0063 .0866 .0719 -.0585 XF -.0001 .0007 .0066 .0489 -.0277 UB -.0052 .0068 .0839 .0179 -.0918 n=50 PP -.0047 .0065 .0797 .0401 -.0790 XF .0 .0001 .0009 .0114 -.0095 UB -.0046 .0067 .0782 .0152 -.0944 n=100 PP -.0045 .0066 .0762 .0261 -.0853 XF .0 .0 .0 .0044 -.0036 UB -.0044 .0066 .0739 .0152 -.0948 n=1,000 PP -.0050 .0068 .0773 .0203 -.0955 XF .0 .0 .0 .0 .0 318 APPENDIX D Parameter Space Maps of L-gamma Distribution 319 c55c68c69c79c72c3c39c20c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c19c20c19c3c87c82c3c19c17c20c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c19c20c19c3c3c19c17c19c21c19c3c3c19c17c19c22c19c3c3c19c17c19c23c19c3c3c19c17c19c24c19c3c3c19c17c19c25c19c3c3c19c17c19c26c19c3c3c19c17c19c27c19c3c3c19c17c19c28c19c3c3c19c17c20c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c20c17c27c24c24c3c3c23c17c27c19c24c3c3c26c17c23c19c22c3c3c28c17c26c19c19c3c3c20c20c17c26c23c3c3c20c22c17c24c27 c19c17c19c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c22c23c20c3c3c21c17c21c23c24c3c3c23c17c19c24c20c3c3c24c17c26c23c21 c19c17c19c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c22c28c20c3c3c20c17c26c27c25 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c27c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c21 c19c17c27c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c20 c19c17c27c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c27c3c3c19c17c19c26c19 c19c17c27c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c26c3c3c19c17c19c28c27 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c24c3c3c19c17c19c26c25c3c3c19c17c20c21c25 c19c17c27c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c23c3c3c19c17c19c24c23c3c3c19c17c20c19c23c3c3c19c17c20c24c23 c19c17c28c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c22c3c3c19c17c19c27c21c3c3c19c17c20c22c21c3c3c19c17c20c27c21 c19c17c28c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c22c3c3c19c17c19c25c21c3c3c19c17c20c20c20c3c3c19c17c20c25c19c3c3c19c17c21c19c28 c19c17c28c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c21c3c3c19c17c19c28c19c3c3c19c17c20c22c28c3c3c19c17c20c27c26c3c3c19c17c21c22c25 c19c17c28c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c23c3c3c19c17c19c26c20c3c3c19c17c20c20c28c3c3c19c17c20c25c25c3c3c19c17c21c20c24c3c3c19c17c21c25c22 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c25c3c3c19c17c19c24c21c3c3c19c17c19c28c28c3c3c19c17c20c23c25c3c3c19c17c20c28c23c3c3c19c17c21c23c21c3c3c19c17c21c28c19 c19c17c28c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c24c3c3c19c17c19c27c20c3c3c19c17c20c21c26c3c3c19c17c20c26c23c3c3c19c17c21c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c27c3c3c19c17c19c25c22c3c3c19c17c20c19c28c3c3c19c17c20c24c24c3c3c19c17c21c19c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c26c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c19c20c3c3c19c17c19c23c25c3c3c19c17c19c28c21c3c3c19c17c20c22c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c27c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c22c19c3c3c19c17c19c26c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c28c19c3c95c3c3c19c17c19c20c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c20c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c55c68c69c79c72c3c39c20c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c19c20c19c3c87c82c3c19c17c20c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c19c20c19c3c3c19c17c19c21c19c3c3c19c17c19c22c19c3c3c19c17c19c23c19c3c3c19c17c19c24c19c3c3c19c17c19c25c19c3c3c19c17c19c26c19c3c3c19c17c19c27c19c3c3c19c17c19c28c19c3c3c19c17c20c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c25c17c25c22c27c3c16c21c17c24c23c28c3c14c20c17c19c24c22c3c14c23c17c21c22c26c3c14c26c17c19c26c23c3c14c28c17c25c21c26 c19c17c19c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c26c17c22c24c21c3c16c23c17c26c20c21c3c16c21c17c21c19c28c3c14c19c17c20c22c25 c19c17c19c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c25c17c23c26c21c3c16c23c17c24c22c26 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c27c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c22c22c21 c19c17c27c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c21c25c27 c19c17c27c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c21c26c26c3c16c19c17c21c19c24 c19c17c27c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c21c20c22c3c16c19c17c20c23c21 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c21c21c20c3c16c19c17c20c24c20c3c16c19c17c19c27c19 c19c17c27c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c21c21c26c3c16c19c17c20c24c27c3c16c19c17c19c27c28c3c16c19c17c19c20c28 c19c17c28c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c20c25c24c3c16c19c17c19c28c25c3c16c19c17c19c21c27c3c14c19c17c19c23c20 c19c17c28c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c20c26c19c3c16c19c17c20c19c22c3c16c19c17c19c22c24c3c14c19c17c19c22c22c3c14c19c17c20c19c20 c19c17c28c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c20c19c27c3c16c19c17c19c23c20c3c14c19c17c19c21c25c3c14c19c17c19c28c22c3c14c19c17c20c25c20 c19c17c28c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c20c20c21c3c16c19c17c19c23c26c3c14c19c17c19c20c28c3c14c19c17c19c27c25c3c14c19c17c20c24c21c3c14c19c17c21c20c28 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c20c20c25c3c16c19c17c19c24c20c3c14c19c17c19c20c23c3c14c19c17c19c26c28c3c14c19c17c20c23c24c3c14c19c17c21c20c20c3c14c19c17c21c26c27 c19c17c28c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c19c24c24c3c14c19c17c19c19c28c3c14c19c17c19c26c23c3c14c19c17c20c22c28c3c14c19c17c21c19c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c19c24c26c3c14c19c17c19c19c25c3c14c19c17c19c26c19c3c14c19c17c20c22c22c3c14c19c17c20c28c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c26c19c3c95c3c3c3c16c16c3c3c3c16c19c17c19c24c28c3c14c19c17c19c19c22c3c14c19c17c19c25c25c3c14c19c17c20c21c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c27c19c3c95c3c3c3c16c16c3c3c3c14c19c17c19c19c20c3c14c19c17c19c25c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c28c19c3c95c3 c19c17c19c19c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c20c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 320 c55c68c69c79c72c3c39c20c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c19c20c19c3c87c82c3c19c17c20c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c19c20c19c3c3c19c17c19c21c19c3c3c19c17c19c22c19c3c3c19c17c19c23c19c3c3c19c17c19c24c19c3c3c19c17c19c25c19c3c3c19c17c19c26c19c3c3c19c17c19c27c19c3c3c19c17c19c28c19c3c3c19c17c20c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c20c23c3c86c19c17c20c21c23c3c3c19c17c20c22c23c3c3c19c17c20c23c22c3c3c19c17c20c24c22c3c3c19c17c20c25c21 c19c17c19c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c28c3c86c19c17c20c22c25c3c86c19c17c20c23c23c3c3c19c17c20c24c20 c19c17c19c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c23c3c86c19c17c20c24c19 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c27c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c23c20 c19c17c27c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c23c24 c19c17c27c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c24c28c3c86c19c17c27c23c27 c19c17c27c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c25c21c3c86c19c17c27c24c20 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c26c25c3c86c19c17c27c25c24c3c86c19c17c27c24c24 c19c17c27c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c28c20c3c86c19c17c27c26c28c3c86c19c17c27c25c27c3c86c19c17c27c24c28 c19c17c28c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c27c28c22c3c86c19c17c27c27c21c3c86c19c17c27c26c21c3c3c19c17c27c25c21 c19c17c28c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c19c27c3c86c19c17c27c28c25c3c86c19c17c27c27c24c3c3c19c17c27c26c24c3c3c19c17c27c25c25 c19c17c28c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c20c19c3c86c19c17c27c28c28c3c3c19c17c27c27c28c3c3c19c17c27c26c28c3c3c19c17c27c26c19 c19c17c28c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c21c23c3c86c19c17c28c20c22c3c3c19c17c28c19c21c3c3c19c17c27c28c21c3c3c19c17c27c27c22c3c3c19c17c27c26c23 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c22c28c3c86c19c17c28c21c26c3c3c19c17c28c20c25c3c3c19c17c28c19c24c3c3c19c17c27c28c25c3c3c19c17c27c27c25c3c3c19c17c27c26c27 c19c17c28c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c23c20c3c3c19c17c28c21c28c3c3c19c17c28c20c28c3c3c19c17c28c19c28c3c3c19c17c27c28c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c28c24c24c3c3c19c17c28c23c22c3c3c19c17c28c22c21c3c3c19c17c28c21c21c3c3c19c17c28c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c28c25c28c3c3c19c17c28c24c26c3c3c19c17c28c23c25c3c3c19c17c28c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c27c19c3c95c3c3c3c16c16c3c3c3c3c19c17c28c26c20c3c3c19c17c28c24c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c28c19c3c95c3c3c19c17c28c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c20c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c55c68c69c79c72c3c39c20c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c19c20c19c3c87c82c3c19c17c20c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c19c20c19c3c3c19c17c19c21c19c3c3c19c17c19c22c19c3c3c19c17c19c23c19c3c3c19c17c19c24c19c3c3c19c17c19c25c19c3c3c19c17c19c26c19c3c3c19c17c19c27c19c3c3c19c17c19c28c19c3c3c19c17c20c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c27c28c3c86c19c17c19c28c25c3c3c19c17c20c19c22c3c3c19c17c20c19c28c3c3c19c17c20c20c24c3c3c19c17c20c21c20 c19c17c19c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c25c3c86c19c17c20c19c20c3c86c19c17c20c19c25c3c3c19c17c20c20c20 c19c17c19c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c19c22c3c86c19c17c20c19c27 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c27c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c24c21 c19c17c27c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c24c23 c19c17c27c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c23c26c3c86c19c17c19c24c26 c19c17c27c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c24c19c3c86c19c17c19c25c19 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c23c22c3c86c19c17c19c24c22c3c86c19c17c19c25c21 c19c17c27c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c22c25c3c86c19c17c19c23c25c3c86c19c17c19c24c25c3c86c19c17c19c25c23 c19c17c28c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c22c28c3c86c19c17c19c23c28c3c86c19c17c19c24c27c3c3c19c17c19c25c25 c19c17c28c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c22c21c3c86c19c17c19c23c21c3c86c19c17c19c24c21c3c3c19c17c19c25c19c3c3c19c17c19c25c27 c19c17c28c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c22c24c3c86c19c17c19c23c24c3c3c19c17c19c24c23c3c3c19c17c19c25c22c3c3c19c17c19c26c19 c19c17c28c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c21c27c3c86c19c17c19c22c27c3c3c19c17c19c23c27c3c3c19c17c19c24c26c3c3c19c17c19c25c24c3c3c19c17c19c26c21 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c21c20c3c86c19c17c19c22c21c3c3c19c17c19c23c20c3c3c19c17c19c24c19c3c3c19c17c19c24c28c3c3c19c17c19c25c26c3c3c19c17c19c26c23 c19c17c28c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c21c24c3c3c19c17c19c22c24c3c3c19c17c19c23c23c3c3c19c17c19c24c22c3c3c19c17c19c25c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c20c27c3c3c19c17c19c21c27c3c3c19c17c19c22c27c3c3c19c17c19c23c26c3c3c19c17c19c24c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c19c20c20c3c3c19c17c19c21c20c3c3c19c17c19c22c20c3c3c19c17c19c23c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c27c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c20c23c3c3c19c17c19c21c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c28c19c3c95c3c3c19c17c19c19c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c20c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 321 c55c68c69c79c72c3c39c21c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c20c20c3c87c82c3c19c17c21c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c20c20c19c3c3c19c17c20c21c19c3c3c19c17c20c22c19c3c3c19c17c20c23c19c3c3c19c17c20c24c19c3c3c19c17c20c25c19c3c3c19c17c20c26c19c3c3c19c17c20c27c19c3c3c19c17c20c28c19c3c3c19c17c21c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c20c24c17c21c25c3c3c20c25c17c26c27c3c3c20c27c17c20c28c3c3c20c28c17c24c19c3c3c21c19c17c26c20c3c3c21c20c17c27c24c3c3c21c21c17c28c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c26c17c22c20c27c3c3c27c17c26c27c23c3c3c20c19c17c20c24c3c3c20c20c17c23c21c3c3c20c21c17c25c21c3c3c20c22c17c26c23c3c3c20c23c17c27c19c3c3c20c24c17c27c19c3c3c20c25c17c26c24c3c3c20c26c17c25c24 c19c17c19c22c19c3c95c3c3c22c17c20c23c26c3c3c23c17c23c24c28c3c3c24c17c26c20c23c3c3c25c17c28c20c19c3c3c27c17c19c23c26c3c3c28c17c20c21c26c3c3c20c19c17c20c24c3c3c20c20c17c20c22c3c3c20c21c17c19c24c3c3c20c21c17c28c23 c19c17c19c23c19c3c95c3c3c19c17c26c23c25c3c3c20c17c27c24c22c3c3c21c17c28c23c24c3c3c23c17c19c20c21c3c3c24c17c19c23c27c3c3c25c17c19c24c20c3c3c26c17c19c20c25c3c3c26c17c28c23c24c3c3c27c17c27c22c26c3c3c28c17c25c28c23 c19c17c19c24c19c3c95c3c3c3c16c16c3c3c3c3c19c17c21c23c27c3c3c20c17c20c26c19c3c3c21c17c19c28c21c3c3c22c17c19c19c25c3c3c22c17c28c19c25c3c3c23c17c26c27c27c3c3c24c17c25c23c27c3c3c25c17c23c27c23c3c3c26c17c21c28c25 c19c17c19c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c27c3c3c19c17c27c19c24c3c3c20c17c24c28c27c3c3c21c17c22c28c20c3c3c22c17c20c26c28c3c3c22c17c28c24c28c3c3c23c17c26c21c25c3c3c24c17c23c27c19 c19c17c19c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c25c20c27c3c3c20c17c22c20c21c3c3c21c17c19c19c28c3c3c21c17c26c19c26c3c3c22c17c23c19c21c3c3c23c17c19c28c20 c19c17c19c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c24c22c26c3c3c20c17c20c24c22c3c3c20c17c26c26c24c3c3c21c17c22c28c28c3c3c22c17c19c21c23 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c24c21c19c3c3c20c17c19c26c24c3c3c20c17c25c22c25c3c3c21c17c21c19c19 c19c17c20c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c27c3c3c19c17c24c23c25c3c3c20c17c19c24c19c3c3c20c17c24c25c20 c19c17c20c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c23c20c3c3c19c17c24c28c27c3c3c20c17c19c25c20 c19c17c20c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c21c23c25c3c3c19c17c25c25c27 c19c17c20c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c22c24c27 c19c17c20c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c20c19 c19c17c20c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16 c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c25c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c21 c19c17c25c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c27 c19c17c25c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c25c23 c19c17c25c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c28c3c3c19c17c19c28c20 c19c17c25c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c25c3c3c19c17c20c20c26 c19c17c25c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c23c3c3c19c17c19c26c22c3c3c19c17c20c23c21 c19c17c25c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c21c3c3c19c17c20c19c19c3c3c19c17c20c25c27 c19c17c25c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c24c28c3c3c19c17c20c21c25c3c3c19c17c20c28c23 c19c17c26c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c20c3c3c19c17c19c27c25c3c3c19c17c20c24c22c3c3c19c17c21c20c28 c19c17c26c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c28c3c3c19c17c20c20c22c3c3c19c17c20c26c28c3c3c19c17c21c23c23 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c22c3c3c19c17c19c26c25c3c3c19c17c20c23c19c3c3c19c17c21c19c24c3c3c19c17c21c26c19 c19c17c26c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c20c3c3c19c17c20c19c23c3c3c19c17c20c25c26c3c3c19c17c21c22c20c3c3c19c17c21c28c24 c19c17c26c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c26c3c3c19c17c19c25c28c3c3c19c17c20c22c20c3c3c19c17c20c28c22c3c3c19c17c21c24c25c3c3c19c17c22c20c28 c19c17c26c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c25c3c3c19c17c19c28c26c3c3c19c17c20c24c27c3c3c19c17c21c21c19c3c3c19c17c21c27c21c3c3c19c17c22c23c23 c19c17c26c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c24c3c3c19c17c19c25c23c3c3c19c17c20c21c23c3c3c19c17c20c27c24c3c3c19c17c21c23c25c3c3c19c17c22c19c26c3c3c19c17c22c25c28 c19c17c26c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c22c3c3c19c17c19c28c21c3c3c19c17c20c24c21c3c3c19c17c21c20c20c3c3c19c17c21c26c21c3c3c19c17c22c22c21c3c3c19c17c22c28c22 c19c17c26c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c23c3c3c19c17c19c25c21c3c3c19c17c20c21c19c3c3c19c17c20c26c28c3c3c19c17c21c22c27c3c3c19c17c21c28c26c3c3c19c17c22c24c26c3c3c19c17c23c20c26 c19c17c26c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c22c3c3c19c17c19c28c19c3c3c19c17c20c23c27c3c3c19c17c21c19c25c3c3c19c17c21c25c23c3c3c19c17c22c21c22c3c3c19c17c22c27c21c3c3c19c17c23c23c20 c19c17c27c19c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c19c24c3c3c19c17c19c25c20c3c3c19c17c20c20c27c3c3c19c17c20c26c24c3c3c19c17c21c22c21c3c3c19c17c21c28c19c3c3c19c17c22c23c27c3c3c19c17c23c19c25c3c3c19c17c23c25c24 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c22c23c3c3c19c17c19c28c19c3c3c19c17c20c23c25c3c3c19c17c21c19c21c3c3c19c17c21c24c28c3c3c19c17c22c20c25c3c3c19c17c22c26c22c3c3c19c17c23c22c20c3c3c19c17c23c27c28 c19c17c27c21c19c3c95c3c3c19c17c19c19c27c3c3c19c17c19c25c22c3c3c19c17c20c20c27c3c3c19c17c20c26c22c3c3c19c17c21c21c28c3c3c19c17c21c27c24c3c3c19c17c22c23c20c3c3c19c17c22c28c27c3c3c19c17c23c24c24c3c3c19c17c24c20c22 c19c17c27c22c19c3c95c3c3c19c17c19c22c26c3c3c19c17c19c28c20c3c3c19c17c20c23c24c3c3c19c17c21c19c19c3c3c19c17c21c24c24c3c3c19c17c22c20c20c3c3c19c17c22c25c26c3c3c19c17c23c21c22c3c3c19c17c23c26c28c3c3c19c17c24c22c25 c19c17c27c23c19c3c95c3c3c19c17c19c25c24c3c3c19c17c20c20c28c3c3c19c17c20c26c22c3c3c19c17c21c21c26c3c3c19c17c21c27c21c3c3c19c17c22c22c26c3c3c19c17c22c28c21c3c3c19c17c23c23c26c3c3c19c17c24c19c22c3c3c19c17c24c24c28 c19c17c27c24c19c3c95c3c3c19c17c19c28c23c3c3c19c17c20c23c26c3c3c19c17c21c19c19c3c3c19c17c21c24c23c3c3c19c17c22c19c27c3c3c19c17c22c25c21c3c3c19c17c23c20c26c3c3c19c17c23c26c21c3c3c19c17c24c21c26c3c3c19c17c24c27c21 c19c17c27c25c19c3c95c3c3c19c17c20c21c21c3c3c19c17c20c26c23c3c3c19c17c21c21c26c3c3c19c17c21c27c19c3c3c19c17c22c22c23c3c3c19c17c22c27c26c3c3c19c17c23c23c20c3c3c19c17c23c28c25c3c3c19c17c24c24c19c3c3c19c17c25c19c24 c19c17c27c26c19c3c95c3c3c19c17c20c24c19c3c3c19c17c21c19c21c3c3c19c17c21c24c23c3c3c19c17c22c19c26c3c3c19c17c22c24c28c3c3c19c17c23c20c22c3c3c19c17c23c25c25c3c3c19c17c24c21c19c3c3c19c17c24c26c23c3c3c19c17c25c21c27 c19c17c27c27c19c3c95c3c3c19c17c20c26c26c3c3c19c17c21c21c28c3c3c19c17c21c27c20c3c3c19c17c22c22c22c3c3c19c17c22c27c24c3c3c19c17c23c22c26c3c3c19c17c23c28c19c3c3c19c17c24c23c23c3c3c19c17c24c28c26c3c3c19c17c25c24c20 c19c17c27c28c19c3c95c3c3c19c17c21c19c24c3c3c19c17c21c24c25c3c3c19c17c22c19c26c3c3c19c17c22c24c27c3c3c19c17c23c20c19c3c3c19c17c23c25c21c3c3c19c17c24c20c24c3c3c19c17c24c25c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c19c19c3c95c3c3c19c17c21c22c21c3c3c19c17c21c27c21c3c3c19c17c22c22c22c3c3c19c17c22c27c23c3c3c19c17c23c22c24c3c3c19c17c23c27c26c3c3c19c17c24c22c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c20c19c3c95c3c3c19c17c21c24c28c3c3c19c17c22c19c28c3c3c19c17c22c24c28c3c3c19c17c23c19c28c3c3c19c17c23c25c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c21c19c3c95c3c3c19c17c21c27c24c3c3c19c17c22c22c24c3c3c19c17c22c27c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c22c19c3c95c3c3c19c17c22c20c21c3c3c19c17c22c25c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 322 c55c68c69c79c72c3c39c21c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c20c20c3c87c82c3c19c17c21c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c20c20c19c3c3c19c17c20c21c19c3c3c19c17c20c22c19c3c3c19c17c20c23c19c3c3c19c17c20c24c19c3c3c19c17c20c25c19c3c3c19c17c20c26c19c3c3c19c17c20c27c19c3c3c19c17c20c28c19c3c3c19c17c21c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c14c20c20c17c28c23c3c14c20c23c17c19c25c3c14c20c25c17c19c20c3c14c20c26c17c27c21c3c14c20c28c17c24c20c3c14c21c20c17c19c28c3c14c21c21c17c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c14c21c17c22c21c19c3c14c23c17c22c24c23c3c14c25c17c21c23c27c3c14c27c17c19c20c27c3c14c28c17c25c26c24c3c14c20c20c17c21c22c3c14c20c21c17c26c19c3c14c20c23c17c19c27c3c14c20c24c17c23c19c3c14c20c25c17c25c24 c19c17c19c22c19c3c95c3c16c21c17c25c24c20c3c16c19c17c27c22c21c3c14c19c17c28c19c27c3c14c21c17c24c25c25c3c14c23c17c20c23c22c3c14c24c17c25c23c19c3c14c26c17c19c25c22c3c14c27c17c23c20c25c3c14c28c17c26c19c23c3c14c20c19c17c28c22 c19c17c19c23c19c3c95c3c16c24c17c23c19c22c3c16c22c17c27c25c28c3c16c21c17c22c24c25c3c16c19c17c27c26c25c3c14c19c17c24c25c20c3c14c20c17c28c24c19c3c14c22c17c21c27c28c3c14c23c17c24c26c25c3c14c24c17c27c20c22c3c14c26c17c19c19c20 c19c17c19c24c19c3c95c3c3c3c16c16c3c3c3c16c24c17c25c23c27c3c16c23c17c22c25c28c3c16c22c17c19c28c21c3c16c20c17c27c21c24c3c16c19c17c24c26c26c3c14c19c17c25c23c25c3c14c20c17c27c22c27c3c14c21c17c28c28c27c3c14c23c17c20c21c22 c19c17c19c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c24c17c25c19c21c3c16c23c17c24c20c20c3c16c22c17c23c20c21c3c16c21c17c22c20c22c3c16c20c17c21c21c19c3c16c19c17c20c22c28c3c14c19c17c28c21c24c3c14c20c17c28c26c19 c19c17c19c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c23c17c23c25c21c3c16c22c17c24c19c19c3c16c21c17c24c22c22c3c16c20c17c24c25c25c3c16c19c17c25c19c21c3c14c19c17c22c24c22 c19c17c19c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c23c17c22c19c27c3c16c22c17c23c24c22c3c16c21c17c24c28c20c3c16c20c17c26c21c25c3c16c19c17c27c25c19 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c23c17c19c28c24c3c16c22c17c22c21c25c3c16c21c17c24c23c27c3c16c20c17c26c25c25 c19c17c20c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c23c17c24c22c28c3c16c22c17c27c23c28c3c16c22c17c20c23c28c3c16c21c17c23c23c20 c19c17c20c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c23c17c21c20c28c3c16c22c17c24c27c25c3c16c21c17c28c23c22 c19c17c20c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c22c17c27c28c28c3c16c22c17c22c20c23 c19c17c20c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c22c17c24c27c24 c19c17c20c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c22c17c26c26c28 c19c17c20c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c25c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c28c23c19 c19c17c25c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c27c26c20 c19c17c25c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c27c19c22 c19c17c25c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c27c22c24c3c16c19c17c26c22c25 c19c17c25c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c26c25c26c3c16c19c17c25c25c28 c19c17c25c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c26c28c24c3c16c19c17c26c19c19c3c16c19c17c25c19c22 c19c17c25c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c26c21c26c3c16c19c17c25c22c22c3c16c19c17c24c22c27 c19c17c25c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c25c25c19c3c16c19c17c24c25c26c3c16c19c17c23c26c23 c19c17c26c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c25c27c24c3c16c19c17c24c28c23c3c16c19c17c24c19c21c3c16c19c17c23c19c28 c19c17c26c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c25c20c27c3c16c19c17c24c21c27c3c16c19c17c23c22c26c3c16c19c17c22c23c25 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c25c22c28c3c16c19c17c24c24c20c3c16c19c17c23c25c22c3c16c19c17c22c26c22c3c16c19c17c21c27c22 c19c17c26c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c24c26c22c3c16c19c17c23c27c25c3c16c19c17c22c28c27c3c16c19c17c22c20c19c3c16c19c17c21c21c20 c19c17c26c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c24c28c21c3c16c19c17c24c19c26c3c16c19c17c23c21c20c3c16c19c17c22c22c23c3c16c19c17c21c23c26c3c16c19c17c20c24c28 c19c17c26c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c24c21c25c3c16c19c17c23c23c20c3c16c19c17c22c24c25c3c16c19c17c21c26c20c3c16c19c17c20c27c24c3c16c19c17c19c28c27 c19c17c26c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c24c23c22c3c16c19c17c23c25c19c3c16c19c17c22c26c26c3c16c19c17c21c28c22c3c16c19c17c21c19c27c3c16c19c17c20c21c22c3c16c19c17c19c22c27 c19c17c26c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c23c26c26c3c16c19c17c22c28c24c3c16c19c17c22c20c22c3c16c19c17c21c22c19c3c16c19c17c20c23c25c3c16c19c17c19c25c21c3c14c19c17c19c21c21 c19c17c26c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c23c28c21c3c16c19c17c23c20c20c3c16c19c17c22c22c20c3c16c19c17c21c23c28c3c16c19c17c20c25c26c3c16c19c17c19c27c24c3c16c19c17c19c19c21c3c14c19c17c19c27c20 c19c17c26c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c19c17c23c21c25c3c16c19c17c22c23c26c3c16c19c17c21c25c26c3c16c19c17c20c27c25c3c16c19c17c20c19c25c3c16c19c17c19c21c23c3c14c19c17c19c24c27c3c14c19c17c20c23c19 c19c17c27c19c19c3c95c3c3c3c16c16c3c3c3c16c19c17c23c22c28c3c16c19c17c22c25c20c3c16c19c17c21c27c22c3c16c19c17c21c19c23c3c16c19c17c20c21c23c3c16c19c17c19c23c23c3c14c19c17c19c22c25c3c14c19c17c20c20c26c3c14c19c17c20c28c28 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c16c19c17c22c26c23c3c16c19c17c21c28c26c3c16c19c17c21c21c19c3c16c19c17c20c23c21c3c16c19c17c19c25c22c3c14c19c17c19c20c25c3c14c19c17c19c28c25c3c14c19c17c20c26c25c3c14c19c17c21c24c25 c19c17c27c21c19c3c95c3c16c19c17c22c27c25c3c16c19c17c22c20c19c3c16c19c17c21c22c23c3c16c19c17c20c24c26c3c16c19c17c19c27c19c3c16c19c17c19c19c21c3c14c19c17c19c26c25c3c14c19c17c20c24c24c3c14c19c17c21c22c23c3c14c19c17c22c20c23 c19c17c27c22c19c3c95c3c16c19c17c22c21c21c3c16c19c17c21c23c26c3c16c19c17c20c26c20c3c16c19c17c19c28c24c3c16c19c17c19c20c28c3c14c19c17c19c24c27c3c14c19c17c20c22c24c3c14c19c17c21c20c22c3c14c19c17c21c28c21c3c14c19c17c22c26c19 c19c17c27c23c19c3c95c3c16c19c17c21c24c27c3c16c19c17c20c27c23c3c16c19c17c20c19c28c3c16c19c17c19c22c23c3c14c19c17c19c23c21c3c14c19c17c20c20c27c3c14c19c17c20c28c23c3c14c19c17c21c26c20c3c14c19c17c22c23c28c3c14c19c17c23c21c26 c19c17c27c24c19c3c95c3c16c19c17c20c28c24c3c16c19c17c20c21c20c3c16c19c17c19c23c27c3c14c19c17c19c21c26c3c14c19c17c20c19c21c3c14c19c17c20c26c26c3c14c19c17c21c24c22c3c14c19c17c22c21c28c3c14c19c17c23c19c24c3c14c19c17c23c27c21 c19c17c27c25c19c3c95c3c16c19c17c20c22c21c3c16c19c17c19c25c19c3c14c19c17c19c20c22c3c14c19c17c19c27c26c3c14c19c17c20c25c20c3c14c19c17c21c22c24c3c14c19c17c22c20c19c3c14c19c17c22c27c25c3c14c19c17c23c25c20c3c14c19c17c24c22c27 c19c17c27c26c19c3c95c3c16c19c17c19c26c20c3c14c19c17c19c19c20c3c14c19c17c19c26c23c3c14c19c17c20c23c25c3c14c19c17c21c21c19c3c14c19c17c21c28c22c3c14c19c17c22c25c26c3c14c19c17c23c23c21c3c14c19c17c24c20c26c3c14c19c17c24c28c21 c19c17c27c27c19c3c95c3c16c19c17c19c20c19c3c14c19c17c19c25c21c3c14c19c17c20c22c22c3c14c19c17c21c19c24c3c14c19c17c21c26c27c3c14c19c17c22c24c20c3c14c19c17c23c21c23c3c14c19c17c23c28c27c3c14c19c17c24c26c21c3c14c19c17c25c23c26 c19c17c27c28c19c3c95c3c14c19c17c19c24c20c3c14c19c17c20c21c20c3c14c19c17c20c28c21c3c14c19c17c21c25c23c3c14c19c17c22c22c24c3c14c19c17c23c19c27c3c14c19c17c23c27c19c3c14c19c17c24c24c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c19c19c3c95c3c14c19c17c20c20c20c3c14c19c17c20c27c20c3c14c19c17c21c24c20c3c14c19c17c22c21c21c3c14c19c17c22c28c22c3c14c19c17c23c25c23c3c14c19c17c24c22c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c20c19c3c95c3c14c19c17c20c26c19c3c14c19c17c21c22c28c3c14c19c17c22c19c28c3c14c19c17c22c26c28c3c14c19c17c23c23c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c21c19c3c95c3c14c19c17c21c21c28c3c14c19c17c21c28c26c3c14c19c17c22c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c22c19c3c95c3c14c19c17c21c27c26c3c14c19c17c22c24c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 323 c55c68c69c79c72c3c39c21c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c20c20c3c87c82c3c19c17c21c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c20c20c19c3c3c19c17c20c21c19c3c3c19c17c20c22c19c3c3c19c17c20c23c19c3c3c19c17c20c24c19c3c3c19c17c20c25c19c3c3c19c17c20c26c19c3c3c19c17c20c27c19c3c3c19c17c20c28c19c3c3c19c17c21c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c19c17c20c26c20c3c3c19c17c20c27c19c3c3c19c17c20c27c28c3c3c19c17c20c28c26c3c3c19c17c21c19c25c3c3c19c17c21c20c23c3c3c19c17c21c21c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c19c17c20c24c28c3c3c19c17c20c25c25c3c3c19c17c20c26c23c3c3c19c17c20c27c20c3c3c19c17c20c27c28c3c3c19c17c20c28c25c3c3c19c17c21c19c22c3c3c19c17c21c20c19c3c3c19c17c21c20c27c3c3c19c17c21c21c24 c19c17c19c22c19c3c95c3c86c19c17c20c24c25c3c86c19c17c20c25c21c3c3c19c17c20c25c27c3c3c19c17c20c26c24c3c3c19c17c20c27c21c3c3c19c17c20c27c27c3c3c19c17c20c28c24c3c3c19c17c21c19c21c3c3c19c17c21c19c27c3c3c19c17c21c20c24 c19c17c19c23c19c3c95c3c86c19c17c20c24c27c3c86c19c17c20c25c22c3c86c19c17c20c25c27c3c86c19c17c20c26c23c3c3c19c17c20c26c28c3c3c19c17c20c27c24c3c3c19c17c20c28c20c3c3c19c17c20c28c26c3c3c19c17c21c19c22c3c3c19c17c21c19c28 c19c17c19c24c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c25c26c3c86c19c17c20c26c20c3c86c19c17c20c26c25c3c86c19c17c20c27c20c3c86c19c17c20c27c25c3c3c19c17c20c28c20c3c3c19c17c20c28c25c3c3c19c17c21c19c20c3c3c19c17c21c19c26 c19c17c19c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c26c26c3c86c19c17c20c27c19c3c86c19c17c20c27c23c3c86c19c17c20c27c27c3c86c19c17c20c28c22c3c86c19c17c20c28c26c3c3c19c17c21c19c21c3c3c19c17c21c19c26 c19c17c19c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c27c28c3c86c19c17c20c28c21c3c86c19c17c20c28c25c3c86c19c17c21c19c19c3c86c19c17c21c19c23c3c3c19c17c21c19c27 c19c17c19c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c28c27c3c86c19c17c21c19c20c3c86c19c17c21c19c23c3c86c19c17c21c19c27c3c86c19c17c21c20c20 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c19c26c3c86c19c17c21c20c19c3c86c19c17c21c20c22c3c86c19c17c21c20c25 c19c17c20c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c20c24c3c86c19c17c21c20c25c3c86c19c17c21c20c27c3c86c19c17c21c21c20 c19c17c20c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c21c23c3c86c19c17c21c21c24c3c86c19c17c21c21c26 c19c17c20c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c22c22c3c86c19c17c21c22c23 c19c17c20c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c23c20 c19c17c20c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c23c28 c19c17c20c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c25c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c23c21 c19c17c25c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c23c27 c19c17c25c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c24c22 c19c17c25c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c25c27c3c86c19c17c25c24c28 c19c17c25c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c26c22c3c86c19c17c25c25c23 c19c17c25c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c27c26c3c86c19c17c25c26c27c3c86c19c17c25c26c19 c19c17c25c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c28c21c3c86c19c17c25c27c22c3c86c19c17c25c26c24 c19c17c25c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c25c28c26c3c86c19c17c25c27c28c3c86c19c17c25c27c20 c19c17c26c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c20c21c3c86c19c17c26c19c21c3c86c19c17c25c28c23c3c86c19c17c25c27c25 c19c17c26c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c20c25c3c86c19c17c26c19c26c3c86c19c17c25c28c28c3c86c19c17c25c28c21 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c22c20c3c86c19c17c26c21c20c3c86c19c17c26c20c21c3c86c19c17c26c19c23c3c86c19c17c25c28c26 c19c17c26c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c22c24c3c86c19c17c26c21c25c3c86c19c17c26c20c26c3c86c19c17c26c20c19c3c86c19c17c26c19c22 c19c17c26c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c24c19c3c86c19c17c26c23c19c3c86c19c17c26c22c20c3c86c19c17c26c21c22c3c86c19c17c26c20c24c3c86c19c17c26c19c27 c19c17c26c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c24c23c3c86c19c17c26c23c23c3c86c19c17c26c22c25c3c86c19c17c26c21c27c3c86c19c17c26c21c19c3c86c19c17c26c20c23 c19c17c26c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c25c28c3c86c19c17c26c24c27c3c86c19c17c26c23c28c3c86c19c17c26c23c20c3c86c19c17c26c22c22c3c86c19c17c26c21c25c3c86c19c17c26c20c28 c19c17c26c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c26c22c3c86c19c17c26c25c22c3c86c19c17c26c24c23c3c86c19c17c26c23c24c3c86c19c17c26c22c27c3c86c19c17c26c22c20c3c3c19c17c26c21c24 c19c17c26c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c27c26c3c86c19c17c26c26c26c3c86c19c17c26c25c26c3c86c19c17c26c24c27c3c86c19c17c26c24c19c3c86c19c17c26c23c22c3c86c19c17c26c22c25c3c3c19c17c26c22c19 c19c17c26c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c26c28c20c3c86c19c17c26c27c20c3c86c19c17c26c26c21c3c86c19c17c26c25c22c3c86c19c17c26c24c24c3c86c19c17c26c23c27c3c3c19c17c26c23c21c3c3c19c17c26c22c24 c19c17c27c19c19c3c95c3c3c3c16c16c3c3c3c86c19c17c27c19c24c3c86c19c17c26c28c24c3c86c19c17c26c27c24c3c86c19c17c26c26c25c3c86c19c17c26c25c27c3c86c19c17c26c25c19c3c3c19c17c26c24c22c3c3c19c17c26c23c26c3c3c19c17c26c23c20 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c86c19c17c27c19c28c3c86c19c17c26c28c28c3c86c19c17c26c27c28c3c86c19c17c26c27c20c3c86c19c17c26c26c22c3c3c19c17c26c25c24c3c3c19c17c26c24c28c3c3c19c17c26c24c21c3c3c19c17c26c23c25 c19c17c27c21c19c3c95c3c86c19c17c27c21c23c3c86c19c17c27c20c22c3c86c19c17c27c19c22c3c86c19c17c26c28c23c3c86c19c17c26c27c24c3c86c19c17c26c26c27c3c3c19c17c26c26c19c3c3c19c17c26c25c23c3c3c19c17c26c24c26c3c3c19c17c26c24c21 c19c17c27c22c19c3c95c3c86c19c17c27c21c26c3c86c19c17c27c20c26c3c86c19c17c27c19c26c3c86c19c17c26c28c27c3c86c19c17c26c28c19c3c3c19c17c26c27c21c3c3c19c17c26c26c24c3c3c19c17c26c25c28c3c3c19c17c26c25c22c3c3c19c17c26c24c26 c19c17c27c23c19c3c95c3c86c19c17c27c22c19c3c86c19c17c27c21c19c3c86c19c17c27c20c20c3c86c19c17c27c19c22c3c3c19c17c26c28c24c3c3c19c17c26c27c26c3c3c19c17c26c27c19c3c3c19c17c26c26c23c3c3c19c17c26c25c27c3c3c19c17c26c25c21 c19c17c27c24c19c3c95c3c86c19c17c27c22c23c3c86c19c17c27c21c23c3c86c19c17c27c20c24c3c3c19c17c27c19c26c3c3c19c17c26c28c28c3c3c19c17c26c28c21c3c3c19c17c26c27c24c3c3c19c17c26c26c28c3c3c19c17c26c26c22c3c3c19c17c26c25c27 c19c17c27c25c19c3c95c3c86c19c17c27c22c27c3c86c19c17c27c21c27c3c3c19c17c27c21c19c3c3c19c17c27c20c21c3c3c19c17c27c19c23c3c3c19c17c26c28c26c3c3c19c17c26c28c19c3c3c19c17c26c27c23c3c3c19c17c26c26c28c3c3c19c17c26c26c22 c19c17c27c26c19c3c95c3c86c19c17c27c23c21c3c3c19c17c27c22c22c3c3c19c17c27c21c23c3c3c19c17c27c20c25c3c3c19c17c27c19c28c3c3c19c17c27c19c21c3c3c19c17c26c28c25c3c3c19c17c26c27c28c3c3c19c17c26c27c23c3c3c19c17c26c26c27 c19c17c27c27c19c3c95c3c86c19c17c27c23c24c3c3c19c17c27c22c26c3c3c19c17c27c21c27c3c3c19c17c27c21c20c3c3c19c17c27c20c23c3c3c19c17c27c19c26c3c3c19c17c27c19c20c3c3c19c17c26c28c24c3c3c19c17c26c27c28c3c3c19c17c26c27c23 c19c17c27c28c19c3c95c3c3c19c17c27c23c28c3c3c19c17c27c23c20c3c3c19c17c27c22c22c3c3c19c17c27c21c24c3c3c19c17c27c20c27c3c3c19c17c27c20c21c3c3c19c17c27c19c25c3c3c19c17c27c19c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c19c19c3c95c3c3c19c17c27c24c22c3c3c19c17c27c23c24c3c3c19c17c27c22c26c3c3c19c17c27c22c19c3c3c19c17c27c21c22c3c3c19c17c27c20c26c3c3c19c17c27c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c20c19c3c95c3c3c19c17c27c24c27c3c3c19c17c27c23c28c3c3c19c17c27c23c21c3c3c19c17c27c22c24c3c3c19c17c27c21c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c21c19c3c95c3c3c19c17c27c25c21c3c3c19c17c27c24c23c3c3c19c17c27c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c22c19c3c95c3c3c19c17c27c25c25c3c3c19c17c27c24c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 324 c55c68c69c79c72c3c39c21c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c20c20c3c87c82c3c19c17c21c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c20c20c19c3c3c19c17c20c21c19c3c3c19c17c20c22c19c3c3c19c17c20c23c19c3c3c19c17c20c24c19c3c3c19c17c20c25c19c3c3c19c17c20c26c19c3c3c19c17c20c27c19c3c3c19c17c20c28c19c3c3c19c17c21c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c20c19c3c95c3c3c19c17c20c21c26c3c3c19c17c20c22c22c3c3c19c17c20c22c28c3c3c19c17c20c23c23c3c3c19c17c20c24c19c3c3c19c17c20c24c24c3c3c19c17c20c25c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c19c17c20c20c25c3c3c19c17c20c21c20c3c3c19c17c20c21c25c3c3c19c17c20c22c20c3c3c19c17c20c22c25c3c3c19c17c20c23c20c3c3c19c17c20c23c24c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c28 c19c17c19c22c19c3c95c3c86c19c17c20c20c21c3c86c19c17c20c20c25c3c3c19c17c20c21c20c3c3c19c17c20c21c24c3c3c19c17c20c21c28c3c3c19c17c20c22c22c3c3c19c17c20c22c27c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19 c19c17c19c23c19c3c95c3c86c19c17c20c20c19c3c86c19c17c20c20c23c3c86c19c17c20c20c27c3c86c19c17c20c21c21c3c3c19c17c20c21c24c3c3c19c17c20c21c28c3c3c19c17c20c22c22c3c3c19c17c20c22c26c3c3c19c17c20c23c20c3c3c19c17c20c23c23 c19c17c19c24c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c20c22c3c86c19c17c20c20c26c3c86c19c17c20c21c19c3c86c19c17c20c21c22c3c86c19c17c20c21c26c3c3c19c17c20c22c19c3c3c19c17c20c22c23c3c3c19c17c20c22c26c3c3c19c17c20c23c20 c19c17c19c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c20c25c3c86c19c17c20c20c28c3c86c19c17c20c21c21c3c86c19c17c20c21c25c3c86c19c17c20c21c28c3c86c19c17c20c22c21c3c3c19c17c20c22c24c3c3c19c17c20c22c28 c19c17c19c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c21c3c86c19c17c20c21c24c3c86c19c17c20c21c27c3c86c19c17c20c22c20c3c86c19c17c20c22c23c3c3c19c17c20c22c26 c19c17c19c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c24c3c86c19c17c20c21c27c3c86c19c17c20c22c20c3c86c19c17c20c22c22c3c86c19c17c20c22c25 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c26c3c86c19c17c20c22c19c3c86c19c17c20c22c22c3c86c19c17c20c22c25 c19c17c20c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c19c3c86c19c17c20c22c22c3c86c19c17c20c22c25 c19c17c20c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c28c3c86c19c17c20c22c21c3c86c19c17c20c22c24 c19c17c20c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c21c3c86c19c17c20c22c24 c19c17c20c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c24 c19c17c20c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c23 c19c17c20c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c3c37c85c72c68c78c3c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16c16 c19c17c25c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c19c22 c19c17c25c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c19c23 c19c17c25c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c19c24 c19c17c25c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c27c3c86c19c17c20c19c25 c19c17c25c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c28c3c86c19c17c20c19c26 c19c17c25c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c21c3c86c19c17c20c19c19c3c86c19c17c20c19c27 c19c17c25c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c22c3c86c19c17c20c19c20c3c86c19c17c20c19c28 c19c17c25c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c28c24c3c86c19c17c20c19c21c3c86c19c17c20c20c19 c19c17c26c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c27c26c3c86c19c17c19c28c25c3c86c19c17c20c19c23c3c86c19c17c20c20c20 c19c17c26c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c27c28c3c86c19c17c19c28c26c3c86c19c17c20c19c24c3c86c19c17c20c20c21 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c27c21c3c86c19c17c19c28c20c3c86c19c17c19c28c28c3c86c19c17c20c19c25c3c86c19c17c20c20c21 c19c17c26c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c27c23c3c86c19c17c19c28c21c3c86c19c17c20c19c19c3c86c19c17c20c19c26c3c86c19c17c20c20c22 c19c17c26c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c26c25c3c86c19c17c19c27c24c3c86c19c17c19c28c23c3c86c19c17c20c19c20c3c86c19c17c20c19c27c3c86c19c17c20c20c23 c19c17c26c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c26c27c3c86c19c17c19c27c26c3c86c19c17c19c28c24c3c86c19c17c20c19c21c3c86c19c17c20c19c28c3c86c19c17c20c20c24 c19c17c26c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c26c20c3c86c19c17c19c27c19c3c86c19c17c19c27c28c3c86c19c17c19c28c25c3c86c19c17c20c19c22c3c86c19c17c20c20c19c3c86c19c17c20c20c25 c19c17c26c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c26c22c3c86c19c17c19c27c21c3c86c19c17c19c28c19c3c86c19c17c19c28c27c3c86c19c17c20c19c24c3c86c19c17c20c20c20c3c3c19c17c20c20c26 c19c17c26c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c25c25c3c86c19c17c19c26c24c3c86c19c17c19c27c23c3c86c19c17c19c28c21c3c86c19c17c19c28c28c3c86c19c17c20c19c25c3c86c19c17c20c20c21c3c3c19c17c20c20c27 c19c17c26c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c19c25c27c3c86c19c17c19c26c26c3c86c19c17c19c27c25c3c86c19c17c19c28c22c3c86c19c17c20c19c19c3c86c19c17c20c19c26c3c3c19c17c20c20c22c3c3c19c17c20c20c27 c19c17c27c19c19c3c95c3c3c3c16c16c3c3c3c86c19c17c19c25c20c3c86c19c17c19c26c19c3c86c19c17c19c26c28c3c86c19c17c19c27c26c3c86c19c17c19c28c24c3c86c19c17c20c19c20c3c3c19c17c20c19c27c3c3c19c17c20c20c23c3c3c19c17c20c20c28 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c86c19c17c19c25c22c3c86c19c17c19c26c22c3c86c19c17c19c27c20c3c86c19c17c19c27c28c3c86c19c17c19c28c25c3c3c19c17c20c19c22c3c3c19c17c20c19c28c3c3c19c17c20c20c24c3c3c19c17c20c21c19 c19c17c27c21c19c3c95c3c86c19c17c19c24c25c3c86c19c17c19c25c25c3c86c19c17c19c26c24c3c86c19c17c19c27c22c3c86c19c17c19c28c19c3c86c19c17c19c28c26c3c3c19c17c20c19c23c3c3c19c17c20c20c19c3c3c19c17c20c20c24c3c3c19c17c20c21c20 c19c17c27c22c19c3c95c3c86c19c17c19c24c28c3c86c19c17c19c25c27c3c86c19c17c19c26c26c3c86c19c17c19c27c24c3c86c19c17c19c28c21c3c3c19c17c19c28c27c3c3c19c17c20c19c24c3c3c19c17c20c20c20c3c3c19c17c20c20c25c3c3c19c17c20c21c21 c19c17c27c23c19c3c95c3c86c19c17c19c25c20c3c86c19c17c19c26c19c3c86c19c17c19c26c28c3c86c19c17c19c27c25c3c3c19c17c19c28c22c3c3c19c17c20c19c19c3c3c19c17c20c19c25c3c3c19c17c20c20c21c3c3c19c17c20c20c26c3c3c19c17c20c21c21 c19c17c27c24c19c3c95c3c86c19c17c19c25c23c3c86c19c17c19c26c21c3c86c19c17c19c27c19c3c3c19c17c19c27c27c3c3c19c17c19c28c24c3c3c19c17c20c19c20c3c3c19c17c20c19c26c3c3c19c17c20c20c22c3c3c19c17c20c20c27c3c3c19c17c20c21c22 c19c17c27c25c19c3c95c3c86c19c17c19c25c25c3c86c19c17c19c26c23c3c3c19c17c19c27c21c3c3c19c17c19c27c28c3c3c19c17c19c28c25c3c3c19c17c20c19c21c3c3c19c17c20c19c27c3c3c19c17c20c20c23c3c3c19c17c20c20c28c3c3c19c17c20c21c23 c19c17c27c26c19c3c95c3c86c19c17c19c25c27c3c3c19c17c19c26c25c3c3c19c17c19c27c23c3c3c19c17c19c28c20c3c3c19c17c19c28c26c3c3c19c17c20c19c22c3c3c19c17c20c19c28c3c3c19c17c20c20c24c3c3c19c17c20c21c19c3c3c19c17c20c21c24 c19c17c27c27c19c3c95c3c86c19c17c19c26c19c3c3c19c17c19c26c27c3c3c19c17c19c27c24c3c3c19c17c19c28c21c3c3c19c17c19c28c27c3c3c19c17c20c19c23c3c3c19c17c20c20c19c3c3c19c17c20c20c24c3c3c19c17c20c21c19c3c3c19c17c20c21c24 c19c17c27c28c19c3c95c3c3c19c17c19c26c21c3c3c19c17c19c27c19c3c3c19c17c19c27c26c3c3c19c17c19c28c22c3c3c19c17c20c19c19c3c3c19c17c20c19c25c3c3c19c17c20c20c20c3c3c19c17c20c20c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c19c19c3c95c3c3c19c17c19c26c23c3c3c19c17c19c27c21c3c3c19c17c19c27c27c3c3c19c17c19c28c24c3c3c19c17c20c19c20c3c3c19c17c20c19c26c3c3c19c17c20c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c20c19c3c95c3c3c19c17c19c26c25c3c3c19c17c19c27c22c3c3c19c17c19c28c19c3c3c19c17c19c28c25c3c3c19c17c20c19c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c21c19c3c95c3c3c19c17c19c26c27c3c3c19c17c19c27c24c3c3c19c17c19c28c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c22c19c3c95c3c3c19c17c19c27c19c3c3c19c17c19c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c28c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 325 c55c68c69c79c72c3c39c22c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c21c20c3c87c82c3c19c17c22c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c20c27c17c24c20c3c3c20c28c17c22c22c3c3c21c19c17c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c22c19c3c95c3c3c20c22c17c26c28c3c3c20c23c17c25c19c3c3c20c24c17c22c26c3c3c20c25c17c20c21c3c3c20c25c17c27c22c3c3c20c26c17c24c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c23c19c3c95c3c3c20c19c17c24c20c3c3c20c20c17c22c19c3c3c20c21c17c19c25c3c3c20c21c17c27c19c3c3c20c22c17c24c19c3c3c20c23c17c20c27c3c3c20c23c17c27c22c3c3c20c24c17c23c26c3c3c20c25c17c19c27c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c27c17c19c27c21c3c3c27c17c27c23c22c3c3c28c17c24c26c27c3c3c20c19c17c21c27c3c3c20c19c17c28c26c3c3c20c20c17c25c23c3c3c20c21c17c21c27c3c3c20c21c17c28c19c3c3c20c22c17c24c20c3c3c20c23c17c19c28 c19c17c19c25c19c3c95c3c3c25c17c21c20c26c3c3c25c17c28c22c25c3c3c26c17c25c22c26c3c3c27c17c22c21c19c3c3c27c17c28c27c22c3c3c28c17c25c21c27c3c3c20c19c17c21c24c3c3c20c19c17c27c25c3c3c20c20c17c23c24c3c3c20c21c17c19c21 c19c17c19c26c19c3c95c3c3c23c17c26c26c21c3c3c24c17c23c23c21c3c3c25c17c20c19c20c3c3c25c17c26c23c26c3c3c26c17c22c26c28c3c3c26c17c28c28c26c3c3c27c17c25c19c19c3c3c28c17c20c27c28c3c3c28c17c26c25c22c3c3c20c19c17c22c21 c19c17c19c27c19c3c95c3c3c22c17c25c23c25c3c3c23c17c21c25c24c3c3c23c17c27c26c26c3c3c24c17c23c27c21c3c3c25c17c19c26c27c3c3c25c17c25c25c23c3c3c26c17c21c22c28c3c3c26c17c27c19c22c3c3c27c17c22c24c25c3c3c27c17c27c28c27 c19c17c19c28c19c3c95c3c3c21c17c26c25c26c3c3c22c17c22c22c22c3c3c22c17c27c28c27c3c3c23c17c23c24c28c3c3c24c17c19c20c25c3c3c24c17c24c25c27c3c3c25c17c20c20c21c3c3c25c17c25c23c27c3c3c26c17c20c26c26c3c3c26c17c25c28c25 c19c17c20c19c19c3c95c3c3c21c17c19c26c25c3c3c21c17c24c28c22c3c3c22c17c20c20c21c3c3c22c17c25c22c20c3c3c23c17c20c23c28c3c3c23c17c25c25c23c3c3c24c17c20c26c25c3c3c24c17c25c27c21c3c3c25c17c20c27c22c3c3c25c17c25c26c28 c19c17c20c20c19c3c95c3c3c20c17c24c22c19c3c3c21c17c19c19c23c3c3c21c17c23c27c19c3c3c21c17c28c24c28c3c3c22c17c23c22c28c3c3c22c17c28c20c27c3c3c23c17c22c28c25c3c3c23c17c27c26c21c3c3c24c17c22c23c25c3c3c24c17c27c20c24 c19c17c20c21c19c3c95c3c3c20c17c19c28c26c3c3c20c17c24c22c20c3c3c20c17c28c25c28c3c3c21c17c23c20c19c3c3c21c17c27c24c24c3c3c22c17c22c19c19c3c3c22c17c26c23c26c3c3c23c17c20c28c22c3c3c23c17c25c22c27c3c3c24c17c19c27c20 c19c17c20c22c19c3c95c3c3c19c17c26c24c20c3c3c20c17c20c24c19c3c3c20c17c24c24c23c3c3c20c17c28c25c21c3c3c21c17c22c26c22c3c3c21c17c26c27c26c3c3c22c17c21c19c22c3c3c22c17c25c21c19c3c3c23c17c19c22c27c3c3c23c17c23c24c24 c19c17c20c23c19c3c95c3c3c19c17c23c26c22c3c3c19c17c27c23c20c3c3c20c17c21c20c24c3c3c20c17c24c28c21c3c3c20c17c28c26c23c3c3c21c17c22c25c19c3c3c21c17c26c23c27c3c3c22c17c20c22c27c3c3c22c17c24c22c19c3c3c22c17c28c21c21 c19c17c20c24c19c3c95c3c3c19c17c21c23c28c3c3c19c17c24c28c19c3c3c19c17c28c22c26c3c3c20c17c21c27c27c3c3c20c17c25c23c22c3c3c21c17c19c19c21c3c3c21c17c22c25c24c3c3c21c17c26c22c19c3c3c22c17c19c28c26c3c3c22c17c23c25c25 c19c17c20c25c19c3c95c3c3c19c17c19c25c26c3c3c19c17c22c27c24c3c3c19c17c26c19c27c3c3c20c17c19c22c24c3c3c20c17c22c25c26c3c3c20c17c26c19c21c3c3c21c17c19c23c20c3c3c21c17c22c27c23c3c3c21c17c26c21c27c3c3c22c17c19c26c25 c19c17c20c26c19c3c95c3c3c3c16c16c3c3c3c3c19c17c21c20c26c3c3c19c17c24c20c28c3c3c19c17c27c21c24c3c3c20c17c20c22c24c3c3c20c17c23c24c19c3c3c20c17c26c25c27c3c3c21c17c19c27c28c3c3c21c17c23c20c22c3c3c21c17c26c23c19 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c26c28c3c3c19c17c22c25c21c3c3c19c17c25c24c19c3c3c19c17c28c23c20c3c3c20c17c21c22c25c3c3c20c17c24c22c24c3c3c20c17c27c22c27c3c3c21c17c20c23c22c3c3c21c17c23c24c20 c19c17c20c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c21c22c22c3c3c19c17c24c19c22c3c3c19c17c26c26c27c3c3c20c17c19c24c25c3c3c20c17c22c22c26c3c3c20c17c25c21c22c3c3c20c17c28c20c20c3c3c21c17c21c19c21 c19c17c21c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c21c24c3c3c19c17c22c27c20c3c3c19c17c25c23c19c3c3c19c17c28c19c21c3c3c20c17c20c25c28c3c3c20c17c23c22c27c3c3c20c17c26c20c20c3c3c20c17c28c27c25 c19c17c21c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c25c3c3c19c17c21c26c27c3c3c19c17c24c21c22c3c3c19c17c26c26c21c3c3c20c17c19c21c23c3c3c20c17c21c27c19c3c3c20c17c24c22c27c3c3c20c17c26c28c28 c19c17c21c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c28c21c3c3c19c17c23c21c24c3c3c19c17c25c25c20c3c3c19c17c28c19c20c3c3c20c17c20c23c22c3c3c20c17c22c27c28c3c3c20c17c25c22c26 c19c17c21c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c21c20c3c3c19c17c22c23c22c3c3c19c17c24c25c26c3c3c19c17c26c28c24c3c3c20c17c19c21c25c3c3c20c17c21c24c28c3c3c20c17c23c28c24 c19c17c21c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c25c21c3c3c19c17c21c26c22c3c3c19c17c23c27c26c3c3c19c17c26c19c23c3c3c19c17c28c21c23c3c3c20c17c20c23c26c3c3c20c17c22c26c21 c19c17c21c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c22c3c3c19c17c21c20c24c3c3c19c17c23c20c28c3c3c19c17c25c21c26c3c3c19c17c27c22c26c3c3c20c17c19c23c28c3c3c20c17c21c25c24 c19c17c21c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c25c25c3c3c19c17c22c25c21c3c3c19c17c24c25c19c3c3c19c17c26c25c20c3c3c19c17c28c25c24c3c3c20c17c20c26c20 c19c17c21c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c20c21c25c3c3c19c17c22c20c22c3c3c19c17c24c19c23c3c3c19c17c25c28c25c3c3c19c17c27c28c21c3c3c20c17c19c27c28 c19c17c21c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c28c22c3c3c19c17c21c26c22c3c3c19c17c23c24c25c3c3c19c17c25c23c20c3c3c19c17c27c21c27c3c3c20c17c19c20c27 c19c17c21c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c25c25c3c3c19c17c21c22c28c3c3c19c17c23c20c24c3c3c19c17c24c28c22c3c3c19c17c26c26c22c3c3c19c17c28c24c25 c19c17c22c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c24c3c3c19c17c21c20c21c3c3c19c17c22c27c20c3c3c19c17c24c24c21c3c3c19c17c26c21c25c3c3c19c17c28c19c20 c19c17c22c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c27c3c3c19c17c20c27c28c3c3c19c17c22c24c22c3c3c19c17c24c20c27c3c3c19c17c25c27c24c3c3c19c17c27c24c23 c19c17c22c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c25c3c3c19c17c20c26c21c3c3c19c17c22c21c28c3c3c19c17c23c27c28c3c3c19c17c25c24c20c3c3c19c17c27c20c23 c19c17c22c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c27c3c3c19c17c20c24c27c3c3c19c17c22c20c20c3c3c19c17c23c25c24c3c3c19c17c25c21c20c3c3c19c17c26c26c28 c19c17c22c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c21c3c3c19c17c20c23c27c3c3c19c17c21c28c25c3c3c19c17c23c23c24c3c3c19c17c24c28c25c3c3c19c17c26c23c28 c19c17c22c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c19c3c3c19c17c20c23c21c3c3c19c17c21c27c24c3c3c19c17c23c21c28c3c3c19c17c24c26c25c3c3c19c17c26c21c22 c19c17c22c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c20c3c3c19c17c20c22c27c3c3c19c17c21c26c25c3c3c19c17c23c20c26c3c3c19c17c24c24c28c3c3c19c17c26c19c21 c19c17c22c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c22c3c3c19c17c20c22c25c3c3c19c17c21c26c20c3c3c19c17c23c19c26c3c3c19c17c24c23c24c3c3c19c17c25c27c23 c19c17c22c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c27c3c3c19c17c20c22c26c3c3c19c17c21c25c27c3c3c19c17c23c19c20c3c3c19c17c24c22c23c3c3c19c17c25c25c28 c19c17c22c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c23c3c3c19c17c20c23c19c3c3c19c17c21c25c27c3c3c19c17c22c28c25c3c3c19c17c24c21c25c3c3c19c17c25c24c27 c19c17c23c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c21c3c3c19c17c20c23c24c3c3c19c17c21c25c28c3c3c19c17c22c28c23c3c3c19c17c24c21c20c3c3c19c17c25c23c28 c19c17c23c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c21c3c3c19c17c20c24c20c3c3c19c17c21c26c21c3c3c19c17c22c28c23c3c3c19c17c24c20c26c3c3c19c17c25c23c21 c19c17c23c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c22c3c3c19c17c20c24c28c3c3c19c17c21c26c26c3c3c19c17c22c28c25c3c3c19c17c24c20c25c3c3c19c17c25c22c26 c19c17c23c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c24c24c3c3c19c17c20c25c27c3c3c19c17c21c27c22c3c3c19c17c22c28c28c3c3c19c17c24c20c25c3c3c19c17c25c22c24 c19c17c23c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c25c27c3c3c19c17c20c26c28c3c3c19c17c21c28c20c3c3c19c17c23c19c23c3c3c19c17c24c20c27c3c3c19c17c25c22c23 c19c17c23c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c27c21c3c3c19c17c20c28c19c3c3c19c17c22c19c19c3c3c19c17c23c20c19c3c3c19c17c24c21c21c3c3c19c17c25c22c23 c19c17c23c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c28c26c3c3c19c17c21c19c22c3c3c19c17c22c19c28c3c3c19c17c23c20c27c3c3c19c17c24c21c26c3c3c19c17c25c22c26 c19c17c23c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c19c28c3c3c19c17c20c20c21c3c3c19c17c21c20c25c3c3c19c17c22c21c19c3c3c19c17c23c21c25c3c3c19c17c24c22c22c3c3c19c17c25c23c19 c19c17c23c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c21c27c3c3c19c17c20c21c27c3c3c19c17c21c22c19c3c3c19c17c22c22c21c3c3c19c17c23c22c24c3c3c19c17c24c23c19c3c3c19c17c25c23c24 c19c17c23c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c23c26c3c3c19c17c20c23c24c3c3c19c17c21c23c23c3c3c19c17c22c23c23c3c3c19c17c23c23c25c3c3c19c17c24c23c27c3c3c19c17c25c24c20 c19c17c24c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c25c25c3c3c19c17c20c25c21c3c3c19c17c21c25c19c3c3c19c17c22c24c27c3c3c19c17c23c24c26c3c3c19c17c24c24c26c3c3c19c17c25c24c26 c19c17c24c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c27c25c3c3c19c17c20c27c19c3c3c19c17c21c26c24c3c3c19c17c22c26c20c3c3c19c17c23c25c28c3c3c19c17c24c25c25c3c3c19c17c25c25c24 c19c17c24c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c20c23c3c3c19c17c20c19c25c3c3c19c17c20c28c27c3c3c19c17c21c28c21c3c3c19c17c22c27c25c3c3c19c17c23c27c20c3c3c19c17c24c26c26c3c3c19c17c25c26c23 c19c17c24c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c22c25c3c3c19c17c20c21c25c3c3c19c17c21c20c26c3c3c19c17c22c19c27c3c3c19c17c23c19c20c3c3c19c17c23c28c23c3c3c19c17c24c27c27c3c3c19c17c25c27c22 c19c17c24c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c24c27c3c3c19c17c20c23c25c3c3c19c17c21c22c25c3c3c19c17c22c21c25c3c3c19c17c23c20c25c3c3c19c17c24c19c27c3c3c19c17c25c19c19c3c3c19c17c25c28c22 c19c17c24c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c19c17c19c27c19c3c3c19c17c20c25c26c3c3c19c17c21c24c24c3c3c19c17c22c23c22c3c3c19c17c23c22c21c3c3c19c17c24c21c21c3c3c19c17c25c20c21c3c3c19c17c26c19c23 c19c17c24c25c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c20c27c3c3c19c17c20c19c22c3c3c19c17c20c27c27c3c3c19c17c21c26c23c3c3c19c17c22c25c20c3c3c19c17c23c23c27c3c3c19c17c24c22c25c3c3c19c17c25c21c24c3c3c19c17c26c20c24 c19c17c24c26c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c23c21c3c3c19c17c20c21c25c3c3c19c17c21c19c28c3c3c19c17c21c28c23c3c3c19c17c22c26c28c3c3c19c17c23c25c24c3c3c19c17c24c24c20c3c3c19c17c25c22c28c3c3c19c17c26c21c26 c19c17c24c27c19c3c95c3c3c3c16c16c3c3c3c3c19c17c19c25c26c3c3c19c17c20c23c27c3c3c19c17c21c22c20c3c3c19c17c22c20c23c3c3c19c17c22c28c26c3c3c19c17c23c27c21c3c3c19c17c24c25c26c3c3c19c17c25c24c22c3c3c19c17c26c22c28 c19c17c24c28c19c3c95c3c3c19c17c19c20c20c3c3c19c17c19c28c20c3c3c19c17c20c26c20c3c3c19c17c21c24c21c3c3c19c17c22c22c23c3c3c19c17c23c20c25c3c3c19c17c23c28c28c3c3c19c17c24c27c21c3c3c19c17c25c25c26c3c3c19c17c26c24c21 c19c17c25c19c19c3c95c3c3c19c17c19c22c25c3c3c19c17c20c20c24c3c3c19c17c20c28c23c3c3c19c17c21c26c23c3c3c19c17c22c24c23c3c3c19c17c23c22c24c3c3c19c17c24c20c25c3c3c19c17c24c28c27c3c3c19c17c25c27c20c3c3c19c17c26c25c24 c19c17c25c20c19c3c95c3c3c19c17c19c25c21c3c3c19c17c20c22c28c3c3c19c17c21c20c26c3c3c19c17c21c28c24c3c3c19c17c22c26c23c3c3c19c17c23c24c23c3c3c19c17c24c22c23c3c3c19c17c25c20c24c3c3c19c17c25c28c25c3c3c19c17c26c26c27 c19c17c25c21c19c3c95c3c3c19c17c19c27c26c3c3c19c17c20c25c22c3c3c19c17c21c23c19c3c3c19c17c22c20c26c3c3c19c17c22c28c23c3c3c19c17c23c26c22c3c3c19c17c24c24c21c3c3c19c17c25c22c20c3c3c19c17c26c20c20c3c3c19c17c26c28c21 c19c17c25c22c19c3c95c3c3c19c17c20c20c21c3c3c19c17c20c27c26c3c3c19c17c21c25c22c3c3c19c17c22c22c27c3c3c19c17c23c20c24c3c3c19c17c23c28c21c3c3c19c17c24c26c19c3c3c19c17c25c23c27c3c3c19c17c26c21c26c3c3c19c17c27c19c25 c19c17c25c23c19c3c95c3c3c19c17c20c22c27c3c3c19c17c21c20c20c3c3c19c17c21c27c24c3c3c19c17c22c25c19c3c3c19c17c23c22c24c3c3c19c17c24c20c20c3c3c19c17c24c27c27c3c3c19c17c25c25c24c3c3c19c17c26c23c21c3c3c19c17c27c21c20 c19c17c25c24c19c3c95c3c3c19c17c20c25c22c3c3c19c17c21c22c24c3c3c19c17c22c19c27c3c3c19c17c22c27c21c3c3c19c17c23c24c25c3c3c19c17c24c22c20c3c3c19c17c25c19c25c3c3c19c17c25c27c21c3c3c19c17c26c24c27c3c3c19c17c27c22c24 c19c17c25c25c19c3c95c3c3c19c17c20c27c27c3c3c19c17c21c24c28c3c3c19c17c22c22c20c3c3c19c17c23c19c23c3c3c19c17c23c26c26c3c3c19c17c24c24c19c3c3c19c17c25c21c23c3c3c19c17c25c28c28c3c3c19c17c26c26c23c3c3c19c17c27c24c19 c19c17c25c26c19c3c95c3c3c19c17c21c20c21c3c3c19c17c21c27c22c3c3c19c17c22c24c23c3c3c19c17c23c21c24c3c3c19c17c23c28c26c3c3c19c17c24c26c19c3c3c19c17c25c23c22c3c3c19c17c26c20c26c3c3c19c17c26c28c20c3c3c19c17c27c25c24 326 c55c68c69c79c72c3c39c22c17c20c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c25c27c19c3c95c3c3c19c17c21c22c26c3c3c19c17c22c19c26c3c3c19c17c22c26c26c3c3c19c17c23c23c26c3c3c19c17c24c20c27c3c3c19c17c24c28c19c3c3c19c17c25c25c21c3c3c19c17c26c22c23c3c3c19c17c27c19c26c3c3c19c17c27c27c20 c19c17c25c28c19c3c95c3c3c19c17c21c25c21c3c3c19c17c22c22c19c3c3c19c17c22c28c28c3c3c19c17c23c25c28c3c3c19c17c24c22c28c3c3c19c17c25c19c28c3c3c19c17c25c27c19c3c3c19c17c26c24c21c3c3c19c17c27c21c23c3c3c19c17c27c28c25 c19c17c26c19c19c3c95c3c3c19c17c21c27c25c3c3c19c17c22c24c23c3c3c19c17c23c21c21c3c3c19c17c23c28c19c3c3c19c17c24c24c28c3c3c19c17c25c21c28c3c3c19c17c25c28c28c3c3c19c17c26c25c28c3c3c19c17c27c23c19c3c3c19c17c28c20c21 c19c17c26c20c19c3c95c3c3c19c17c22c20c20c3c3c19c17c22c26c26c3c3c19c17c23c23c23c3c3c19c17c24c20c21c3c3c19c17c24c27c19c3c3c19c17c25c23c28c3c3c19c17c26c20c27c3c3c19c17c26c27c26c3c3c19c17c27c24c26c3c3c19c17c28c21c27 c19c17c26c21c19c3c95c3c3c19c17c22c22c24c3c3c19c17c23c19c20c3c3c19c17c23c25c26c3c3c19c17c24c22c23c3c3c19c17c25c19c20c3c3c19c17c25c25c27c3c3c19c17c26c22c25c3c3c19c17c27c19c24c3c3c19c17c27c26c23c3c3c19c17c28c23c22 c19c17c26c22c19c3c95c3c3c19c17c22c24c28c3c3c19c17c23c21c23c3c3c19c17c23c27c28c3c3c19c17c24c24c24c3c3c19c17c25c21c20c3c3c19c17c25c27c27c3c3c19c17c26c24c24c3c3c19c17c27c21c22c3c3c19c17c27c28c20c3c3c19c17c28c24c28 c19c17c26c23c19c3c95c3c3c19c17c22c27c22c3c3c19c17c23c23c26c3c3c19c17c24c20c21c3c3c19c17c24c26c26c3c3c19c17c25c23c21c3c3c19c17c26c19c27c3c3c19c17c26c26c23c3c3c19c17c27c23c20c3c3c19c17c28c19c27c3c3c19c17c28c26c25 c19c17c26c24c19c3c95c3c3c19c17c23c19c26c3c3c19c17c23c26c19c3c3c19c17c24c22c23c3c3c19c17c24c28c27c3c3c19c17c25c25c22c3c3c19c17c26c21c27c3c3c19c17c26c28c22c3c3c19c17c27c24c28c3c3c19c17c28c21c24c3c3c19c17c28c28c21 c19c17c26c25c19c3c95c3c3c19c17c23c22c20c3c3c19c17c23c28c22c3c3c19c17c24c24c25c3c3c19c17c25c20c28c3c3c19c17c25c27c22c3c3c19c17c26c23c26c3c3c19c17c27c20c21c3c3c19c17c27c26c26c3c3c19c17c28c23c21c3c3c20c17c19c19c27 c19c17c26c26c19c3c95c3c3c19c17c23c24c23c3c3c19c17c24c20c25c3c3c19c17c24c26c27c3c3c19c17c25c23c20c3c3c19c17c26c19c23c3c3c19c17c26c25c26c3c3c19c17c27c22c20c3c3c19c17c27c28c24c3c3c19c17c28c24c28c3c3c20c17c19c21c23 c19c17c26c27c19c3c95c3c3c19c17c23c26c27c3c3c19c17c24c22c28c3c3c19c17c25c19c19c3c3c19c17c25c25c21c3c3c19c17c26c21c23c3c3c19c17c26c27c26c3c3c19c17c27c24c19c3c3c19c17c28c20c22c3c3c19c17c28c26c26c3c3c20c17c19c23c20 c19c17c26c28c19c3c95c3c3c19c17c24c19c20c3c3c19c17c24c25c20c3c3c19c17c25c21c21c3c3c19c17c25c27c22c3c3c19c17c26c23c23c3c3c19c17c27c19c25c3c3c19c17c27c25c27c3c3c19c17c28c22c20c3c3c19c17c28c28c23c3c3c20c17c19c24c26 c19c17c27c19c19c3c95c3c3c19c17c24c21c23c3c3c19c17c24c27c23c3c3c19c17c25c23c23c3c3c19c17c26c19c23c3c3c19c17c26c25c24c3c3c19c17c27c21c25c3c3c19c17c27c27c26c3c3c19c17c28c23c28c3c3c20c17c19c20c20c3c3c20c17c19c26c23 c19c17c27c20c19c3c95c3c3c19c17c24c23c26c3c3c19c17c25c19c25c3c3c19c17c25c25c24c3c3c19c17c26c21c24c3c3c19c17c26c27c24c3c3c19c17c27c23c24c3c3c19c17c28c19c25c3c3c19c17c28c25c26c3c3c20c17c19c21c28c3c3c20c17c19c28c19 c19c17c27c21c19c3c95c3c3c19c17c24c26c19c3c3c19c17c25c21c28c3c3c19c17c25c27c26c3c3c19c17c26c23c25c3c3c19c17c27c19c24c3c3c19c17c27c25c24c3c3c19c17c28c21c24c3c3c19c17c28c27c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c22c19c3c95c3c3c19c17c24c28c22c3c3c19c17c25c24c20c3c3c19c17c26c19c28c3c3c19c17c26c25c26c3c3c19c17c27c21c24c3c3c19c17c27c27c23c3c3c19c17c28c23c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c19c17c25c20c25c3c3c19c17c25c26c22c3c3c19c17c26c22c19c3c3c19c17c26c27c26c3c3c19c17c27c23c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c24c19c3c95c3c3c19c17c25c22c27c3c3c19c17c25c28c24c3c3c19c17c26c24c20c3c3c19c17c27c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c25c19c3c95c3c3c19c17c25c25c20c3c3c19c17c26c20c25c3c3c19c17c26c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c26c19c3c95c3c3c19c17c25c27c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 327 c55c68c69c79c72c3c39c22c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c21c20c3c87c82c3c19c17c22c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c14c20c26c17c27c22c3c14c20c27c17c28c26c3c14c21c19c17c19c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c22c19c3c95c3c14c20c21c17c20c19c3c14c20c22c17c21c21c3c14c20c23c17c22c19c3c14c20c24c17c22c22c3c14c20c25c17c22c21c3c14c20c26c17c21c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c23c19c3c95c3c14c27c17c20c23c21c3c14c28c17c21c22c27c3c14c20c19c17c21c28c3c14c20c20c17c22c19c3c14c20c21c17c21c27c3c14c20c22c17c21c21c3c14c20c23c17c20c22c3c14c20c24c17c19c20c3c14c20c24c17c27c25c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c14c24c17c21c20c21c3c14c25c17c21c25c26c3c14c26c17c21c27c26c3c14c27c17c21c26c22c3c14c28c17c21c21c25c3c14c20c19c17c20c23c3c14c20c20c17c19c23c3c14c20c20c17c28c19c3c14c20c21c17c26c23c3c14c20c22c17c24c24 c19c17c19c25c19c3c95c3c14c21c17c28c28c21c3c14c22c17c28c27c28c3c14c23c17c28c25c20c3c14c24c17c28c19c26c3c14c25c17c27c21c25c3c14c26c17c26c21c19c3c14c27c17c24c27c27c3c14c28c17c23c22c21c3c14c20c19c17c21c24c3c14c20c20c17c19c23 c19c17c19c26c19c3c95c3c14c20c17c21c28c26c3c14c21c17c21c21c25c3c14c22c17c20c23c19c3c14c23c17c19c22c24c3c14c23c17c28c20c20c3c14c24c17c26c25c27c3c14c25c17c25c19c23c3c14c26c17c23c21c19c3c14c27c17c21c20c25c3c14c27c17c28c28c21 c19c17c19c27c19c3c95c3c14c19c17c19c19c22c3c14c19c17c27c25c20c3c14c20c17c26c19c28c3c14c21c17c24c23c27c3c14c22c17c22c26c23c3c14c23c17c20c27c25c3c14c23c17c28c27c23c3c14c24c17c26c25c25c3c14c25c17c24c22c22c3c14c26c17c21c27c22 c19c17c19c28c19c3c95c3c16c19c17c28c27c20c3c16c19c17c20c28c24c3c14c19c17c24c27c27c3c14c20c17c22c25c25c3c14c21c17c20c22c27c3c14c21c17c28c19c21c3c14c22c17c25c24c26c3c14c23c17c23c19c20c3c14c24c17c20c22c22c3c14c24c17c27c24c22 c19c17c20c19c19c3c95c3c16c20c17c26c21c27c3c16c20c17c19c20c19c3c16c19c17c21c28c19c3c14c19c17c23c21c28c3c14c20c17c20c23c26c3c14c20c17c27c25c20c3c14c21c17c24c26c19c3c14c22c17c21c26c21c3c14c22c17c28c25c26c3c14c23c17c25c24c22 c19c17c20c20c19c3c95c3c16c21c17c21c28c22c3c16c20c17c25c22c26c3c16c19c17c28c26c25c3c16c19c17c22c20c22c3c14c19c17c22c24c21c3c14c20c17c19c20c26c3c14c20c17c25c27c19c3c14c21c17c22c23c19c3c14c21c17c28c28c25c3c14c22c17c25c23c26 c19c17c20c21c19c3c95c3c16c21c17c26c21c19c3c16c21c17c20c20c27c3c16c20c17c24c20c20c3c16c19c17c27c28c28c3c16c19c17c21c27c22c3c14c19c17c22c22c24c3c14c19c17c28c24c22c3c14c20c17c24c26c21c3c14c21c17c20c27c27c3c14c21c17c27c19c22 c19c17c20c22c19c3c95c3c16c22c17c19c22c28c3c16c21c17c23c27c26c3c16c20c17c28c21c26c3c16c20c17c22c25c20c3c16c19c17c26c28c20c3c16c19c17c21c20c26c3c14c19c17c22c25c19c3c14c19c17c28c22c28c3c14c20c17c24c20c27c3c14c21c17c19c28c25 c19c17c20c23c19c3c95c3c16c22c17c21c26c26c3c16c21c17c26c25c25c3c16c21c17c21c23c28c3c16c20c17c26c21c24c3c16c20c17c20c28c24c3c16c19c17c25c25c20c3c16c19c17c20c21c22c3c14c19c17c23c20c27c3c14c19c17c28c25c20c3c14c20c17c24c19c24 c19c17c20c24c19c3c95c3c16c22c17c23c23c28c3c16c21c17c28c26c25c3c16c21c17c23c28c25c3c16c21c17c19c19c28c3c16c20c17c24c20c26c3c16c20c17c19c20c28c3c16c19c17c24c20c25c3c16c19c17c19c20c19c3c14c19c17c23c28c28c3c14c20c17c19c20c20 c19c17c20c25c19c3c95c3c16c22c17c24c26c21c3c16c22c17c20c22c20c3c16c21c17c25c27c23c3c16c21c17c21c22c19c3c16c20c17c26c26c20c3c16c20c17c22c19c24c3c16c19c17c27c22c24c3c16c19c17c22c25c20c3c14c19c17c20c20c26c3c14c19c17c24c28c28 c19c17c20c26c19c3c95c3c3c3c16c16c3c3c3c16c22c17c21c23c22c3c16c21c17c27c21c24c3c16c21c17c23c19c19c3c16c20c17c28c26c19c3c16c20c17c24c22c23c3c16c20c17c19c28c22c3c16c19c17c25c23c27c3c16c19c17c20c28c27c3c14c19c17c21c24c24 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c16c22c17c22c21c19c3c16c21c17c28c21c26c3c16c21c17c24c21c28c3c16c21c17c20c21c24c3c16c20c17c26c20c25c3c16c20c17c22c19c20c3c16c19c17c27c27c21c3c16c19c17c23c24c28c3c16c19c17c19c22c20 c19c17c20c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c28c28c28c3c16c21c17c25c21c23c3c16c21c17c21c23c23c3c16c20c17c27c24c27c3c16c20c17c23c25c26c3c16c20c17c19c26c21c3c16c19c17c25c26c21c3c16c19c17c21c25c28 c19c17c21c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c22c17c19c23c24c3c16c21c17c25c28c20c3c16c21c17c22c22c21c3c16c20c17c28c25c27c3c16c20c17c24c28c28c3c16c20c17c21c21c24c3c16c19c17c27c23c26c3c16c19c17c23c25c24 c19c17c21c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c22c17c19c26c20c3c16c21c17c26c22c25c3c16c21c17c22c28c25c3c16c21c17c19c24c20c3c16c20c17c26c19c20c3c16c20c17c22c23c26c3c16c19c17c28c27c28c3c16c19c17c25c21c26 c19c17c21c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c26c25c20c3c16c21c17c23c22c28c3c16c21c17c20c20c20c3c16c20c17c26c27c19c3c16c20c17c23c23c23c3c16c20c17c20c19c22c3c16c19c17c26c24c28 c19c17c21c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c26c26c21c3c16c21c17c23c25c23c3c16c21c17c20c24c22c3c16c20c17c27c22c26c3c16c20c17c24c20c27c3c16c20c17c20c28c23c3c16c19c17c27c25c25 c19c17c21c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c26c25c28c3c16c21c17c23c26c25c3c16c21c17c20c26c28c3c16c20c17c27c26c27c3c16c20c17c24c26c22c3c16c20c17c21c25c23c3c16c19c17c28c24c21 c19c17c21c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c26c24c24c3c16c21c17c23c26c24c3c16c21c17c20c28c20c3c16c20c17c28c19c23c3c16c20c17c25c20c22c3c16c20c17c22c20c27c3c16c20c17c19c20c28 c19c17c21c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c23c25c23c3c16c21c17c20c28c22c3c16c20c17c28c20c26c3c16c20c17c25c22c28c3c16c20c17c22c24c26c3c16c20c17c19c26c20 c19c17c21c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c23c23c23c3c16c21c17c20c27c23c3c16c20c17c28c21c19c3c16c20c17c25c24c22c3c16c20c17c22c27c22c3c16c20c17c20c19c28 c19c17c21c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c23c20c27c3c16c21c17c20c25c27c3c16c20c17c28c20c23c3c16c20c17c25c24c27c3c16c20c17c22c28c27c3c16c20c17c20c22c24 c19c17c21c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c22c27c23c3c16c21c17c20c23c23c3c16c20c17c28c19c19c3c16c20c17c25c24c23c3c16c20c17c23c19c23c3c16c20c17c20c24c20 c19c17c22c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c22c23c25c3c16c21c17c20c20c23c3c16c20c17c27c27c19c3c16c20c17c25c23c21c3c16c20c17c23c19c21c3c16c20c17c20c24c27 c19c17c22c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c22c19c22c3c16c21c17c19c27c19c3c16c20c17c27c24c22c3c16c20c17c25c21c23c3c16c20c17c22c28c21c3c16c20c17c20c24c27 c19c17c22c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c21c24c26c3c16c21c17c19c23c20c3c16c20c17c27c21c21c3c16c20c17c25c19c20c3c16c20c17c22c26c26c3c16c20c17c20c24c20 c19c17c22c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c21c19c26c3c16c20c17c28c28c27c3c16c20c17c26c27c26c3c16c20c17c24c26c22c3c16c20c17c22c24c25c3c16c20c17c20c22c27 c19c17c22c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c20c24c23c3c16c20c17c28c24c21c3c16c20c17c26c23c26c3c16c20c17c24c23c19c3c16c20c17c22c22c20c3c16c20c17c20c20c28 c19c17c22c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c19c28c28c3c16c20c17c28c19c22c3c16c20c17c26c19c24c3c16c20c17c24c19c24c3c16c20c17c22c19c21c3c16c20c17c19c28c26 c19c17c22c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c21c17c19c23c22c3c16c20c17c27c24c21c3c16c20c17c25c25c19c3c16c20c17c23c25c25c3c16c20c17c21c25c28c3c16c20c17c19c26c19 c19c17c22c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c28c27c23c3c16c20c17c26c28c28c3c16c20c17c25c20c22c3c16c20c17c23c21c23c3c16c20c17c21c22c22c3c16c20c17c19c23c19 c19c17c22c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c28c21c23c3c16c20c17c26c23c24c3c16c20c17c24c25c22c3c16c20c17c22c27c19c3c16c20c17c20c28c24c3c16c20c17c19c19c26 c19c17c22c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c27c25c22c3c16c20c17c25c27c28c3c16c20c17c24c20c21c3c16c20c17c22c22c23c3c16c20c17c20c24c23c3c16c19c17c28c26c20 c19c17c23c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c27c19c21c3c16c20c17c25c22c21c3c16c20c17c23c25c19c3c16c20c17c21c27c25c3c16c20c17c20c20c20c3c16c19c17c28c22c22 c19c17c23c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c26c22c28c3c16c20c17c24c26c22c3c16c20c17c23c19c25c3c16c20c17c21c22c26c3c16c20c17c19c25c25c3c16c19c17c27c28c22 c19c17c23c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c25c26c25c3c16c20c17c24c20c23c3c16c20c17c22c24c20c3c16c20c17c20c27c25c3c16c20c17c19c21c19c3c16c19c17c27c24c21 c19c17c23c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c25c20c21c3c16c20c17c23c24c23c3c16c20c17c21c28c24c3c16c20c17c20c22c23c3c16c19c17c28c26c21c3c16c19c17c27c19c27 c19c17c23c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c24c23c27c3c16c20c17c22c28c23c3c16c20c17c21c22c28c3c16c20c17c19c27c21c3c16c19c17c28c21c22c3c16c19c17c26c25c22 c19c17c23c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c23c27c23c3c16c20c17c22c22c22c3c16c20c17c20c27c21c3c16c20c17c19c21c27c3c16c19c17c27c26c23c3c16c19c17c26c20c26 c19c17c23c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c23c20c28c3c16c20c17c21c26c21c3c16c20c17c20c21c23c3c16c19c17c28c26c23c3c16c19c17c27c21c22c3c16c19c17c25c26c19 c19c17c23c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c23c28c26c3c16c20c17c22c24c24c3c16c20c17c21c20c20c3c16c20c17c19c25c25c3c16c19c17c28c21c19c3c16c19c17c26c26c21c3c16c19c17c25c21c22 c19c17c23c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c23c21c28c3c16c20c17c21c28c19c3c16c20c17c20c24c19c3c16c20c17c19c19c27c3c16c19c17c27c25c23c3c16c19c17c26c21c19c3c16c19c17c24c26c23 c19c17c23c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c22c25c21c3c16c20c17c21c21c25c3c16c20c17c19c27c27c3c16c19c17c28c23c28c3c16c19c17c27c19c28c3c16c19c17c25c25c27c3c16c19c17c24c21c24 c19c17c24c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c21c28c24c3c16c20c17c20c25c20c3c16c20c17c19c21c26c3c16c19c17c27c28c19c3c16c19c17c26c24c22c3c16c19c17c25c20c24c3c16c19c17c23c26c24 c19c17c24c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c21c21c27c3c16c20c17c19c28c26c3c16c19c17c28c25c24c3c16c19c17c27c22c21c3c16c19c17c25c28c26c3c16c19c17c24c25c21c3c16c19c17c23c21c24 c19c17c24c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c21c27c28c3c16c20c17c20c25c21c3c16c20c17c19c22c22c3c16c19c17c28c19c23c3c16c19c17c26c26c22c3c16c19c17c25c23c20c3c16c19c17c24c19c27c3c16c19c17c22c26c23 c19c17c24c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c21c21c19c3c16c20c17c19c28c24c3c16c19c17c28c25c28c3c16c19c17c27c23c21c3c16c19c17c26c20c23c3c16c19c17c24c27c24c3c16c19c17c23c24c23c3c16c19c17c22c21c22 c19c17c24c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c20c24c21c3c16c20c17c19c21c28c3c16c19c17c28c19c25c3c16c19c17c26c27c20c3c16c19c17c25c24c24c3c16c19c17c24c21c28c3c16c19c17c23c19c20c3c16c19c17c21c26c21 c19c17c24c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c16c20c17c19c27c23c3c16c19c17c28c25c23c3c16c19c17c27c23c22c3c16c19c17c26c21c19c3c16c19c17c24c28c26c3c16c19c17c23c26c21c3c16c19c17c22c23c26c3c16c19c17c21c21c19 c19c17c24c25c19c3c95c3c3c3c16c16c3c3c3c16c20c17c20c22c23c3c16c20c17c19c20c26c3c16c19c17c27c28c28c3c16c19c17c26c27c19c3c16c19c17c25c24c28c3c16c19c17c24c22c27c3c16c19c17c23c20c25c3c16c19c17c21c28c22c3c16c19c17c20c25c28 c19c17c24c26c19c3c95c3c3c3c16c16c3c3c3c16c20c17c19c25c24c3c16c19c17c28c24c19c3c16c19c17c27c22c23c3c16c19c17c26c20c26c3c16c19c17c24c28c28c3c16c19c17c23c27c19c3c16c19c17c22c25c19c3c16c19c17c21c22c28c3c16c19c17c20c20c26 c19c17c24c27c19c3c95c3c3c3c16c16c3c3c3c16c19c17c28c28c26c3c16c19c17c27c27c23c3c16c19c17c26c26c19c3c16c19c17c25c24c24c3c16c19c17c24c22c28c3c16c19c17c23c21c21c3c16c19c17c22c19c23c3c16c19c17c20c27c24c3c16c19c17c19c25c24 c19c17c24c28c19c3c95c3c16c20c17c19c23c19c3c16c19c17c28c21c28c3c16c19c17c27c20c27c3c16c19c17c26c19c25c3c16c19c17c24c28c22c3c16c19c17c23c26c28c3c16c19c17c22c25c23c3c16c19c17c21c23c27c3c16c19c17c20c22c20c3c16c19c17c19c20c22 c19c17c25c19c19c3c95c3c16c19c17c28c26c20c3c16c19c17c27c25c21c3c16c19c17c26c24c22c3c16c19c17c25c23c21c3c16c19c17c24c22c20c3c16c19c17c23c20c28c3c16c19c17c22c19c25c3c16c19c17c20c28c21c3c16c19c17c19c26c26c3c14c19c17c19c22c28 c19c17c25c20c19c3c95c3c16c19c17c28c19c22c3c16c19c17c26c28c25c3c16c19c17c25c27c27c3c16c19c17c24c26c28c3c16c19c17c23c26c19c3c16c19c17c22c25c19c3c16c19c17c21c23c27c3c16c19c17c20c22c25c3c16c19c17c19c21c23c3c14c19c17c19c28c19 c19c17c25c21c19c3c95c3c16c19c17c27c22c24c3c16c19c17c26c22c19c3c16c19c17c25c21c23c3c16c19c17c24c20c26c3c16c19c17c23c19c28c3c16c19c17c22c19c20c3c16c19c17c20c28c20c3c16c19c17c19c27c20c3c14c19c17c19c22c19c3c14c19c17c20c23c21 c19c17c25c22c19c3c95c3c16c19c17c26c25c27c3c16c19c17c25c25c24c3c16c19c17c24c25c19c3c16c19c17c23c24c24c3c16c19c17c22c23c28c3c16c19c17c21c23c21c3c16c19c17c20c22c23c3c16c19c17c19c21c25c3c14c19c17c19c27c22c3c14c19c17c20c28c23 c19c17c25c23c19c3c95c3c16c19c17c26c19c21c3c16c19c17c25c19c19c3c16c19c17c23c28c26c3c16c19c17c22c28c22c3c16c19c17c21c27c28c3c16c19c17c20c27c23c3c16c19c17c19c26c27c3c14c19c17c19c21c28c3c14c19c17c20c22c26c3c14c19c17c21c23c24 c19c17c25c24c19c3c95c3c16c19c17c25c22c25c3c16c19c17c24c22c25c3c16c19c17c23c22c23c3c16c19c17c22c22c21c3c16c19c17c21c21c28c3c16c19c17c20c21c25c3c16c19c17c19c21c20c3c14c19c17c19c27c23c3c14c19c17c20c28c19c3c14c19c17c21c28c25 c19c17c25c25c19c3c95c3c16c19c17c24c26c20c3c16c19c17c23c26c21c3c16c19c17c22c26c21c3c16c19c17c21c26c21c3c16c19c17c20c26c19c3c16c19c17c19c25c27c3c14c19c17c19c22c24c3c14c19c17c20c22c27c3c14c19c17c21c23c22c3c14c19c17c22c23c27 c19c17c25c26c19c3c95c3c16c19c17c24c19c25c3c16c19c17c23c19c28c3c16c19c17c22c20c19c3c16c19c17c21c20c20c3c16c19c17c20c20c20c3c16c19c17c19c20c20c3c14c19c17c19c28c19c3c14c19c17c20c28c21c3c14c19c17c21c28c24c3c14c19c17c22c28c28 328 c55c68c69c79c72c3c39c22c17c21c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c25c27c19c3c95c3c16c19c17c23c23c22c3c16c19c17c22c23c25c3c16c19c17c21c23c28c3c16c19c17c20c24c21c3c16c19c17c19c24c22c3c14c19c17c19c23c25c3c14c19c17c20c23c25c3c14c19c17c21c23c25c3c14c19c17c22c23c27c3c14c19c17c23c23c28 c19c17c25c28c19c3c95c3c16c19c17c22c26c28c3c16c19c17c21c27c23c3c16c19c17c20c27c28c3c16c19c17c19c28c21c3c14c19c17c19c19c24c3c14c19c17c20c19c21c3c14c19c17c21c19c20c3c14c19c17c22c19c19c3c14c19c17c23c19c19c3c14c19c17c24c19c19 c19c17c26c19c19c3c95c3c16c19c17c22c20c25c3c16c19c17c21c21c22c3c16c19c17c20c21c27c3c16c19c17c19c22c22c3c14c19c17c19c25c21c3c14c19c17c20c24c28c3c14c19c17c21c24c25c3c14c19c17c22c24c22c3c14c19c17c23c24c21c3c14c19c17c24c24c20 c19c17c26c20c19c3c95c3c16c19c17c21c24c23c3c16c19c17c20c25c21c3c16c19c17c19c25c28c3c14c19c17c19c21c24c3c14c19c17c20c20c28c3c14c19c17c21c20c23c3c14c19c17c22c20c19c3c14c19c17c23c19c25c3c14c19c17c24c19c22c3c14c19c17c25c19c20 c19c17c26c21c19c3c95c3c16c19c17c20c28c22c3c16c19c17c20c19c20c3c16c19c17c19c20c19c3c14c19c17c19c27c22c3c14c19c17c20c26c25c3c14c19c17c21c26c19c3c14c19c17c22c25c23c3c14c19c17c23c24c28c3c14c19c17c24c24c24c3c14c19c17c25c24c20 c19c17c26c22c19c3c95c3c16c19c17c20c22c21c3c16c19c17c19c23c21c3c14c19c17c19c23c28c3c14c19c17c20c23c19c3c14c19c17c21c22c21c3c14c19c17c22c21c24c3c14c19c17c23c20c27c3c14c19c17c24c20c20c3c14c19c17c25c19c25c3c14c19c17c26c19c20 c19c17c26c23c19c3c95c3c16c19c17c19c26c20c3c14c19c17c19c20c27c3c14c19c17c20c19c26c3c14c19c17c20c28c26c3c14c19c17c21c27c27c3c14c19c17c22c26c28c3c14c19c17c23c26c20c3c14c19c17c24c25c22c3c14c19c17c25c24c26c3c14c19c17c26c24c19 c19c17c26c24c19c3c95c3c16c19c17c19c20c20c3c14c19c17c19c26c26c3c14c19c17c20c25c24c3c14c19c17c21c24c23c3c14c19c17c22c23c22c3c14c19c17c23c22c22c3c14c19c17c24c21c23c3c14c19c17c25c20c24c3c14c19c17c26c19c26c3c14c19c17c27c19c19 c19c17c26c25c19c3c95c3c14c19c17c19c23c27c3c14c19c17c20c22c24c3c14c19c17c21c21c21c3c14c19c17c22c20c19c3c14c19c17c22c28c27c3c14c19c17c23c27c26c3c14c19c17c24c26c26c3c14c19c17c25c25c26c3c14c19c17c26c24c26c3c14c19c17c27c23c28 c19c17c26c26c19c3c95c3c14c19c17c20c19c26c3c14c19c17c20c28c22c3c14c19c17c21c26c28c3c14c19c17c22c25c24c3c14c19c17c23c24c22c3c14c19c17c24c23c20c3c14c19c17c25c21c28c3c14c19c17c26c20c27c3c14c19c17c27c19c26c3c14c19c17c27c28c26 c19c17c26c27c19c3c95c3c14c19c17c20c25c24c3c14c19c17c21c24c19c3c14c19c17c22c22c24c3c14c19c17c23c21c20c3c14c19c17c24c19c26c3c14c19c17c24c28c23c3c14c19c17c25c27c20c3c14c19c17c26c25c28c3c14c19c17c27c24c26c3c14c19c17c28c23c25 c19c17c26c28c19c3c95c3c14c19c17c21c21c22c3c14c19c17c22c19c26c3c14c19c17c22c28c20c3c14c19c17c23c26c24c3c14c19c17c24c25c20c3c14c19c17c25c23c25c3c14c19c17c26c22c21c3c14c19c17c27c20c28c3c14c19c17c28c19c25c3c14c19c17c28c28c23 c19c17c27c19c19c3c95c3c14c19c17c21c27c20c3c14c19c17c22c25c22c3c14c19c17c23c23c25c3c14c19c17c24c22c19c3c14c19c17c25c20c23c3c14c19c17c25c28c28c3c14c19c17c26c27c23c3c14c19c17c27c25c28c3c14c19c17c28c24c25c3c14c20c17c19c23c21 c19c17c27c20c19c3c95c3c14c19c17c22c22c26c3c14c19c17c23c20c28c3c14c19c17c24c19c20c3c14c19c17c24c27c23c3c14c19c17c25c25c26c3c14c19c17c26c24c19c3c14c19c17c27c22c24c3c14c19c17c28c20c28c3c14c20c17c19c19c23c3c14c20c17c19c28c19 c19c17c27c21c19c3c95c3c14c19c17c22c28c23c3c14c19c17c23c26c23c3c14c19c17c24c24c25c3c14c19c17c25c22c26c3c14c19c17c26c20c28c3c14c19c17c27c19c21c3c14c19c17c27c27c24c3c14c19c17c28c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c22c19c3c95c3c14c19c17c23c24c19c3c14c19c17c24c21c28c3c14c19c17c25c20c19c3c14c19c17c25c28c19c3c14c19c17c26c26c20c3c14c19c17c27c24c22c3c14c19c17c28c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c14c19c17c24c19c24c3c14c19c17c24c27c23c3c14c19c17c25c25c22c3c14c19c17c26c23c22c3c14c19c17c27c21c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c24c19c3c95c3c14c19c17c24c25c19c3c14c19c17c25c22c27c3c14c19c17c26c20c25c3c14c19c17c26c28c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c25c19c3c95c3c14c19c17c25c20c23c3c14c19c17c25c28c20c3c14c19c17c26c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c26c19c3c95c3c14c19c17c25c25c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 329 c55c68c69c79c72c3c39c22c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c21c20c3c87c82c3c19c17c22c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c19c17c21c22c21c3c3c19c17c21c22c28c3c3c19c17c21c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c22c19c3c95c3c3c19c17c21c21c20c3c3c19c17c21c21c27c3c3c19c17c21c22c23c3c3c19c17c21c23c19c3c3c19c17c21c23c26c3c3c19c17c21c24c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c23c19c3c95c3c3c19c17c21c20c24c3c3c19c17c21c21c20c3c3c19c17c21c21c26c3c3c19c17c21c22c22c3c3c19c17c21c22c28c3c3c19c17c21c23c24c3c3c19c17c21c24c20c3c3c19c17c21c24c26c3c3c19c17c21c25c22c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c19c17c21c20c22c3c3c19c17c21c20c27c3c3c19c17c21c21c23c3c3c19c17c21c21c28c3c3c19c17c21c22c24c3c3c19c17c21c23c20c3c3c19c17c21c23c25c3c3c19c17c21c24c21c3c3c19c17c21c24c26c3c3c19c17c21c25c22 c19c17c19c25c19c3c95c3c3c19c17c21c20c21c3c3c19c17c21c20c26c3c3c19c17c21c21c21c3c3c19c17c21c21c26c3c3c19c17c21c22c22c3c3c19c17c21c22c27c3c3c19c17c21c23c22c3c3c19c17c21c23c28c3c3c19c17c21c24c23c3c3c19c17c21c24c28 c19c17c19c26c19c3c95c3c3c19c17c21c20c22c3c3c19c17c21c20c27c3c3c19c17c21c21c21c3c3c19c17c21c21c26c3c3c19c17c21c22c21c3c3c19c17c21c22c26c3c3c19c17c21c23c21c3c3c19c17c21c23c26c3c3c19c17c21c24c21c3c3c19c17c21c24c26 c19c17c19c27c19c3c95c3c3c19c17c21c20c24c3c3c19c17c21c20c28c3c3c19c17c21c21c23c3c3c19c17c21c21c27c3c3c19c17c21c22c22c3c3c19c17c21c22c26c3c3c19c17c21c23c21c3c3c19c17c21c23c26c3c3c19c17c21c24c20c3c3c19c17c21c24c25 c19c17c19c28c19c3c95c3c86c19c17c21c20c28c3c86c19c17c21c21c22c3c3c19c17c21c21c25c3c3c19c17c21c22c19c3c3c19c17c21c22c23c3c3c19c17c21c22c28c3c3c19c17c21c23c22c3c3c19c17c21c23c26c3c3c19c17c21c24c21c3c3c19c17c21c24c25 c19c17c20c19c19c3c95c3c86c19c17c21c21c23c3c86c19c17c21c21c26c3c86c19c17c21c22c19c3c3c19c17c21c22c22c3c3c19c17c21c22c26c3c3c19c17c21c23c20c3c3c19c17c21c23c24c3c3c19c17c21c23c28c3c3c19c17c21c24c22c3c3c19c17c21c24c26 c19c17c20c20c19c3c95c3c86c19c17c21c21c28c3c86c19c17c21c22c21c3c86c19c17c21c22c23c3c86c19c17c21c22c27c3c3c19c17c21c23c20c3c3c19c17c21c23c23c3c3c19c17c21c23c27c3c3c19c17c21c24c20c3c3c19c17c21c24c24c3c3c19c17c21c24c28 c19c17c20c21c19c3c95c3c86c19c17c21c22c24c3c86c19c17c21c22c26c3c86c19c17c21c23c19c3c86c19c17c21c23c21c3c86c19c17c21c23c24c3c3c19c17c21c23c27c3c3c19c17c21c24c20c3c3c19c17c21c24c24c3c3c19c17c21c24c27c3c3c19c17c21c25c21 c19c17c20c22c19c3c95c3c86c19c17c21c23c21c3c86c19c17c21c23c22c3c86c19c17c21c23c24c3c86c19c17c21c23c26c3c86c19c17c21c24c19c3c86c19c17c21c24c22c3c3c19c17c21c24c24c3c3c19c17c21c24c27c3c3c19c17c21c25c21c3c3c19c17c21c25c24 c19c17c20c23c19c3c95c3c86c19c17c21c23c28c3c86c19c17c21c24c19c3c86c19c17c21c24c21c3c86c19c17c21c24c22c3c86c19c17c21c24c24c3c86c19c17c21c24c27c3c86c19c17c21c25c19c3c3c19c17c21c25c22c3c3c19c17c21c25c25c3c3c19c17c21c25c28 c19c17c20c24c19c3c95c3c86c19c17c21c24c26c3c86c19c17c21c24c26c3c86c19c17c21c24c27c3c86c19c17c21c25c19c3c86c19c17c21c25c20c3c86c19c17c21c25c22c3c86c19c17c21c25c24c3c86c19c17c21c25c27c3c3c19c17c21c26c19c3c3c19c17c21c26c22 c19c17c20c25c19c3c95c3c86c19c17c21c25c25c3c86c19c17c21c25c24c3c86c19c17c21c25c24c3c86c19c17c21c25c25c3c86c19c17c21c25c26c3c86c19c17c21c25c28c3c86c19c17c21c26c20c3c86c19c17c21c26c22c3c3c19c17c21c26c24c3c3c19c17c21c26c26 c19c17c20c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c21c26c22c3c86c19c17c21c26c22c3c86c19c17c21c26c22c3c86c19c17c21c26c23c3c86c19c17c21c26c24c3c86c19c17c21c26c26c3c86c19c17c21c26c27c3c86c19c17c21c27c19c3c3c19c17c21c27c21 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c86c19c17c21c27c21c3c86c19c17c21c27c20c3c86c19c17c21c27c19c3c86c19c17c21c27c20c3c86c19c17c21c27c21c3c86c19c17c21c27c22c3c86c19c17c21c27c23c3c86c19c17c21c27c25c3c86c19c17c21c27c27 c19c17c20c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c27c28c3c86c19c17c21c27c27c3c86c19c17c21c27c27c3c86c19c17c21c27c27c3c86c19c17c21c27c28c3c86c19c17c21c28c19c3c86c19c17c21c28c21c3c86c19c17c21c28c22 c19c17c21c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c21c28c26c3c86c19c17c21c28c25c3c86c19c17c21c28c24c3c86c19c17c21c28c24c3c86c19c17c21c28c25c3c86c19c17c21c28c26c3c86c19c17c21c28c27c3c86c19c17c21c28c28 c19c17c21c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c19c25c3c86c19c17c22c19c23c3c86c19c17c22c19c22c3c86c19c17c22c19c22c3c86c19c17c22c19c22c3c86c19c17c22c19c22c3c86c19c17c22c19c23c3c86c19c17c22c19c24 c19c17c21c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c20c21c3c86c19c17c22c20c20c3c86c19c17c22c20c19c3c86c19c17c22c20c19c3c86c19c17c22c20c19c3c86c19c17c22c20c20c3c86c19c17c22c20c21 c19c17c21c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c21c20c3c86c19c17c22c20c28c3c86c19c17c22c20c27c3c86c19c17c22c20c26c3c86c19c17c22c20c26c3c86c19c17c22c20c26c3c86c19c17c22c20c27 c19c17c21c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c21c28c3c86c19c17c22c21c26c3c86c19c17c22c21c24c3c86c19c17c22c21c24c3c86c19c17c22c21c23c3c86c19c17c22c21c23c3c86c19c17c22c21c24 c19c17c21c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c22c27c3c86c19c17c22c22c24c3c86c19c17c22c22c22c3c86c19c17c22c22c21c3c86c19c17c22c22c20c3c86c19c17c22c22c20c3c86c19c17c22c22c20 c19c17c21c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c23c22c3c86c19c17c22c23c20c3c86c19c17c22c23c19c3c86c19c17c22c22c28c3c86c19c17c22c22c27c3c86c19c17c22c22c27 c19c17c21c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c24c21c3c86c19c17c22c23c28c3c86c19c17c22c23c26c3c86c19c17c22c23c25c3c86c19c17c22c23c24c3c86c19c17c22c23c24 c19c17c21c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c25c19c3c86c19c17c22c24c26c3c86c19c17c22c24c24c3c86c19c17c22c24c22c3c86c19c17c22c24c21c3c86c19c17c22c24c21 c19c17c21c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c25c27c3c86c19c17c22c25c24c3c86c19c17c22c25c22c3c86c19c17c22c25c20c3c86c19c17c22c25c19c3c86c19c17c22c24c28 c19c17c22c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c26c26c3c86c19c17c22c26c22c3c86c19c17c22c26c19c3c86c19c17c22c25c27c3c86c19c17c22c25c26c3c86c19c17c22c25c25 c19c17c22c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c27c24c3c86c19c17c22c27c20c3c86c19c17c22c26c27c3c86c19c17c22c26c25c3c86c19c17c22c26c23c3c86c19c17c22c26c22 c19c17c22c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c22c28c22c3c86c19c17c22c27c28c3c86c19c17c22c27c25c3c86c19c17c22c27c23c3c86c19c17c22c27c21c3c86c19c17c22c27c19 c19c17c22c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c19c21c3c86c19c17c22c28c26c3c86c19c17c22c28c23c3c86c19c17c22c28c20c3c86c19c17c22c27c28c3c86c19c17c22c27c27 c19c17c22c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c20c19c3c86c19c17c23c19c24c3c86c19c17c23c19c21c3c86c19c17c22c28c28c3c86c19c17c22c28c26c3c86c19c17c22c28c24 c19c17c22c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c20c27c3c86c19c17c23c20c22c3c86c19c17c23c19c28c3c86c19c17c23c19c25c3c86c19c17c23c19c23c3c86c19c17c23c19c21 c19c17c22c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c21c25c3c86c19c17c23c21c20c3c86c19c17c23c20c26c3c86c19c17c23c20c23c3c86c19c17c23c20c20c3c86c19c17c23c19c28 c19c17c22c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c22c23c3c86c19c17c23c21c28c3c86c19c17c23c21c24c3c86c19c17c23c21c20c3c86c19c17c23c20c28c3c86c19c17c23c20c26 c19c17c22c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c23c21c3c86c19c17c23c22c26c3c86c19c17c23c22c21c3c86c19c17c23c21c28c3c86c19c17c23c21c25c3c86c19c17c23c21c23 c19c17c22c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c24c19c3c86c19c17c23c23c23c3c86c19c17c23c23c19c3c86c19c17c23c22c25c3c86c19c17c23c22c22c3c86c19c17c23c22c20 c19c17c23c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c24c26c3c86c19c17c23c24c21c3c86c19c17c23c23c26c3c86c19c17c23c23c23c3c86c19c17c23c23c20c3c86c19c17c23c22c27 c19c17c23c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c25c24c3c86c19c17c23c24c28c3c86c19c17c23c24c24c3c86c19c17c23c24c20c3c86c19c17c23c23c27c3c86c19c17c23c23c24 c19c17c23c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c26c21c3c86c19c17c23c25c26c3c86c19c17c23c25c21c3c86c19c17c23c24c27c3c86c19c17c23c24c24c3c86c19c17c23c24c21 c19c17c23c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c27c19c3c86c19c17c23c26c23c3c86c19c17c23c26c19c3c86c19c17c23c25c25c3c86c19c17c23c25c21c3c86c19c17c23c24c28 c19c17c23c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c27c26c3c86c19c17c23c27c21c3c86c19c17c23c26c26c3c86c19c17c23c26c22c3c86c19c17c23c25c28c3c86c19c17c23c25c26 c19c17c23c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c23c28c23c3c86c19c17c23c27c28c3c86c19c17c23c27c23c3c86c19c17c23c27c19c3c86c19c17c23c26c26c3c86c19c17c23c26c23 c19c17c23c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c19c21c3c86c19c17c23c28c25c3c86c19c17c23c28c20c3c86c19c17c23c27c26c3c86c19c17c23c27c23c3c86c19c17c23c27c20 c19c17c23c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c20c24c3c86c19c17c24c19c28c3c86c19c17c24c19c22c3c86c19c17c23c28c27c3c86c19c17c23c28c23c3c86c19c17c23c28c20c3c86c19c17c23c27c27 c19c17c23c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c21c21c3c86c19c17c24c20c25c3c86c19c17c24c20c19c3c86c19c17c24c19c24c3c86c19c17c24c19c20c3c86c19c17c23c28c27c3c86c19c17c23c28c23 c19c17c23c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c21c28c3c86c19c17c24c21c22c3c86c19c17c24c20c26c3c86c19c17c24c20c21c3c86c19c17c24c19c27c3c86c19c17c24c19c24c3c86c19c17c24c19c20 c19c17c24c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c22c25c3c86c19c17c24c22c19c3c86c19c17c24c21c23c3c86c19c17c24c20c28c3c86c19c17c24c20c24c3c86c19c17c24c20c20c3c86c19c17c24c19c27 c19c17c24c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c23c22c3c86c19c17c24c22c25c3c86c19c17c24c22c20c3c86c19c17c24c21c25c3c86c19c17c24c21c21c3c86c19c17c24c20c27c3c86c19c17c24c20c24 c19c17c24c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c24c26c3c86c19c17c24c23c28c3c86c19c17c24c23c22c3c86c19c17c24c22c27c3c86c19c17c24c22c22c3c86c19c17c24c21c28c3c86c19c17c24c21c24c3c86c19c17c24c21c21 c19c17c24c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c25c22c3c86c19c17c24c24c25c3c86c19c17c24c24c19c3c86c19c17c24c23c23c3c86c19c17c24c23c19c3c86c19c17c24c22c25c3c86c19c17c24c22c21c3c86c19c17c24c21c28 c19c17c24c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c26c19c3c86c19c17c24c25c22c3c86c19c17c24c24c26c3c86c19c17c24c24c20c3c86c19c17c24c23c25c3c86c19c17c24c23c21c3c86c19c17c24c22c28c3c86c19c17c24c22c24 c19c17c24c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c24c26c25c3c86c19c17c24c25c28c3c86c19c17c24c25c22c3c86c19c17c24c24c27c3c86c19c17c24c24c22c3c86c19c17c24c23c28c3c86c19c17c24c23c24c3c86c19c17c24c23c21 c19c17c24c25c19c3c95c3c3c3c16c16c3c3c3c86c19c17c24c28c19c3c86c19c17c24c27c21c3c86c19c17c24c26c25c3c86c19c17c24c26c19c3c86c19c17c24c25c23c3c86c19c17c24c25c19c3c86c19c17c24c24c25c3c86c19c17c24c24c21c3c86c19c17c24c23c27 c19c17c24c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c24c28c25c3c86c19c17c24c27c28c3c86c19c17c24c27c21c3c86c19c17c24c26c25c3c86c19c17c24c26c20c3c86c19c17c24c25c25c3c86c19c17c24c25c21c3c86c19c17c24c24c27c3c86c19c17c24c24c24 c19c17c24c27c19c3c95c3c3c3c16c16c3c3c3c86c19c17c25c19c21c3c86c19c17c24c28c24c3c86c19c17c24c27c27c3c86c19c17c24c27c22c3c86c19c17c24c26c26c3c86c19c17c24c26c22c3c86c19c17c24c25c28c3c86c19c17c24c25c24c3c86c19c17c24c25c21 c19c17c24c28c19c3c95c3c86c19c17c25c20c25c3c86c19c17c25c19c27c3c86c19c17c25c19c20c3c86c19c17c24c28c24c3c86c19c17c24c27c28c3c86c19c17c24c27c23c3c86c19c17c24c26c28c3c86c19c17c24c26c24c3c86c19c17c24c26c21c3c86c19c17c24c25c27 c19c17c25c19c19c3c95c3c86c19c17c25c21c21c3c86c19c17c25c20c23c3c86c19c17c25c19c26c3c86c19c17c25c19c20c3c86c19c17c24c28c24c3c86c19c17c24c28c19c3c86c19c17c24c27c25c3c86c19c17c24c27c21c3c86c19c17c24c26c27c3c3c19c17c24c26c24 c19c17c25c20c19c3c95c3c86c19c17c25c21c27c3c86c19c17c25c21c19c3c86c19c17c25c20c22c3c86c19c17c25c19c26c3c86c19c17c25c19c21c3c86c19c17c24c28c26c3c86c19c17c24c28c21c3c86c19c17c24c27c27c3c86c19c17c24c27c23c3c3c19c17c24c27c20 c19c17c25c21c19c3c95c3c86c19c17c25c22c23c3c86c19c17c25c21c25c3c86c19c17c25c21c19c3c86c19c17c25c20c22c3c86c19c17c25c19c27c3c86c19c17c25c19c22c3c86c19c17c24c28c28c3c86c19c17c24c28c24c3c3c19c17c24c28c20c3c3c19c17c24c27c26 c19c17c25c22c19c3c95c3c86c19c17c25c23c19c3c86c19c17c25c22c21c3c86c19c17c25c21c25c3c86c19c17c25c21c19c3c86c19c17c25c20c23c3c86c19c17c25c19c28c3c86c19c17c25c19c24c3c86c19c17c25c19c20c3c3c19c17c24c28c26c3c3c19c17c24c28c23 c19c17c25c23c19c3c95c3c86c19c17c25c23c24c3c86c19c17c25c22c27c3c86c19c17c25c22c21c3c86c19c17c25c21c25c3c86c19c17c25c21c19c3c86c19c17c25c20c25c3c86c19c17c25c20c20c3c3c19c17c25c19c26c3c3c19c17c25c19c23c3c3c19c17c25c19c19 c19c17c25c24c19c3c95c3c86c19c17c25c24c20c3c86c19c17c25c23c23c3c86c19c17c25c22c27c3c86c19c17c25c22c21c3c86c19c17c25c21c26c3c86c19c17c25c21c21c3c86c19c17c25c20c27c3c3c19c17c25c20c23c3c3c19c17c25c20c19c3c3c19c17c25c19c26 c19c17c25c25c19c3c95c3c86c19c17c25c24c26c3c86c19c17c25c24c19c3c86c19c17c25c23c23c3c86c19c17c25c22c27c3c86c19c17c25c22c22c3c86c19c17c25c21c27c3c3c19c17c25c21c23c3c3c19c17c25c21c19c3c3c19c17c25c20c25c3c3c19c17c25c20c22 c19c17c25c26c19c3c95c3c86c19c17c25c25c21c3c86c19c17c25c24c25c3c86c19c17c25c24c19c3c86c19c17c25c23c23c3c86c19c17c25c22c28c3c86c19c17c25c22c23c3c3c19c17c25c22c19c3c3c19c17c25c21c25c3c3c19c17c25c21c21c3c3c19c17c25c20c28 330 c55c68c69c79c72c3c39c22c17c22c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c25c27c19c3c95c3c86c19c17c25c25c27c3c86c19c17c25c25c21c3c86c19c17c25c24c25c3c86c19c17c25c24c19c3c86c19c17c25c23c24c3c3c19c17c25c23c19c3c3c19c17c25c22c25c3c3c19c17c25c22c21c3c3c19c17c25c21c28c3c3c19c17c25c21c24 c19c17c25c28c19c3c95c3c86c19c17c25c26c23c3c86c19c17c25c25c26c3c86c19c17c25c25c20c3c86c19c17c25c24c25c3c3c19c17c25c24c20c3c3c19c17c25c23c26c3c3c19c17c25c23c21c3c3c19c17c25c22c27c3c3c19c17c25c22c24c3c3c19c17c25c22c20 c19c17c26c19c19c3c95c3c86c19c17c25c26c28c3c86c19c17c25c26c22c3c86c19c17c25c25c26c3c86c19c17c25c25c21c3c3c19c17c25c24c26c3c3c19c17c25c24c22c3c3c19c17c25c23c27c3c3c19c17c25c23c24c3c3c19c17c25c23c20c3c3c19c17c25c22c27 c19c17c26c20c19c3c95c3c86c19c17c25c27c24c3c86c19c17c25c26c28c3c86c19c17c25c26c22c3c3c19c17c25c25c27c3c3c19c17c25c25c22c3c3c19c17c25c24c28c3c3c19c17c25c24c23c3c3c19c17c25c24c20c3c3c19c17c25c23c26c3c3c19c17c25c23c23 c19c17c26c21c19c3c95c3c86c19c17c25c28c20c3c86c19c17c25c27c24c3c86c19c17c25c26c28c3c3c19c17c25c26c23c3c3c19c17c25c25c28c3c3c19c17c25c25c24c3c3c19c17c25c25c20c3c3c19c17c25c24c26c3c3c19c17c25c24c22c3c3c19c17c25c24c19 c19c17c26c22c19c3c95c3c86c19c17c25c28c25c3c86c19c17c25c28c19c3c3c19c17c25c27c24c3c3c19c17c25c27c19c3c3c19c17c25c26c24c3c3c19c17c25c26c20c3c3c19c17c25c25c26c3c3c19c17c25c25c22c3c3c19c17c25c24c28c3c3c19c17c25c24c25 c19c17c26c23c19c3c95c3c86c19c17c26c19c21c3c3c19c17c25c28c25c3c3c19c17c25c28c20c3c3c19c17c25c27c25c3c3c19c17c25c27c20c3c3c19c17c25c26c26c3c3c19c17c25c26c22c3c3c19c17c25c25c28c3c3c19c17c25c25c24c3c3c19c17c25c25c21 c19c17c26c24c19c3c95c3c86c19c17c26c19c26c3c3c19c17c26c19c21c3c3c19c17c25c28c25c3c3c19c17c25c28c20c3c3c19c17c25c27c26c3c3c19c17c25c27c22c3c3c19c17c25c26c28c3c3c19c17c25c26c24c3c3c19c17c25c26c20c3c3c19c17c25c25c27 c19c17c26c25c19c3c95c3c3c19c17c26c20c22c3c3c19c17c26c19c26c3c3c19c17c26c19c21c3c3c19c17c25c28c26c3c3c19c17c25c28c22c3c3c19c17c25c27c27c3c3c19c17c25c27c23c3c3c19c17c25c27c20c3c3c19c17c25c26c26c3c3c19c17c25c26c23 c19c17c26c26c19c3c95c3c3c19c17c26c20c28c3c3c19c17c26c20c22c3c3c19c17c26c19c27c3c3c19c17c26c19c22c3c3c19c17c25c28c28c3c3c19c17c25c28c23c3c3c19c17c25c28c19c3c3c19c17c25c27c26c3c3c19c17c25c27c22c3c3c19c17c25c27c19 c19c17c26c27c19c3c95c3c3c19c17c26c21c23c3c3c19c17c26c20c28c3c3c19c17c26c20c23c3c3c19c17c26c19c28c3c3c19c17c26c19c23c3c3c19c17c26c19c19c3c3c19c17c25c28c25c3c3c19c17c25c28c22c3c3c19c17c25c27c28c3c3c19c17c25c27c25 c19c17c26c28c19c3c95c3c3c19c17c26c22c19c3c3c19c17c26c21c23c3c3c19c17c26c20c28c3c3c19c17c26c20c24c3c3c19c17c26c20c19c3c3c19c17c26c19c25c3c3c19c17c26c19c21c3c3c19c17c25c28c28c3c3c19c17c25c28c24c3c3c19c17c25c28c21 c19c17c27c19c19c3c95c3c3c19c17c26c22c24c3c3c19c17c26c22c19c3c3c19c17c26c21c24c3c3c19c17c26c21c19c3c3c19c17c26c20c25c3c3c19c17c26c20c21c3c3c19c17c26c19c27c3c3c19c17c26c19c23c3c3c19c17c26c19c20c3c3c19c17c25c28c27 c19c17c27c20c19c3c95c3c3c19c17c26c23c20c3c3c19c17c26c22c24c3c3c19c17c26c22c20c3c3c19c17c26c21c25c3c3c19c17c26c21c21c3c3c19c17c26c20c27c3c3c19c17c26c20c23c3c3c19c17c26c20c19c3c3c19c17c26c19c26c3c3c19c17c26c19c23 c19c17c27c21c19c3c95c3c3c19c17c26c23c25c3c3c19c17c26c23c20c3c3c19c17c26c22c25c3c3c19c17c26c22c21c3c3c19c17c26c21c26c3c3c19c17c26c21c22c3c3c19c17c26c21c19c3c3c19c17c26c20c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c22c19c3c95c3c3c19c17c26c24c21c3c3c19c17c26c23c26c3c3c19c17c26c23c21c3c3c19c17c26c22c26c3c3c19c17c26c22c22c3c3c19c17c26c21c28c3c3c19c17c26c21c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c19c17c26c24c26c3c3c19c17c26c24c21c3c3c19c17c26c23c26c3c3c19c17c26c23c22c3c3c19c17c26c22c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c24c19c3c95c3c3c19c17c26c25c22c3c3c19c17c26c24c27c3c3c19c17c26c24c22c3c3c19c17c26c23c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c25c19c3c95c3c3c19c17c26c25c27c3c3c19c17c26c25c22c3c3c19c17c26c24c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c26c19c3c95c3c3c19c17c26c26c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 331 c55c68c69c79c72c3c39c22c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c21c20c3c87c82c3c19c17c22c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c21c19c3c95c3c3c19c17c20c25c22c3c3c19c17c20c25c26c3c3c19c17c20c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c22c19c3c95c3c3c19c17c20c24c23c3c3c19c17c20c24c27c3c3c19c17c20c25c21c3c3c19c17c20c25c25c3c3c19c17c20c26c19c3c3c19c17c20c26c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c23c19c3c95c3c3c19c17c20c23c27c3c3c19c17c20c24c21c3c3c19c17c20c24c25c3c3c19c17c20c24c28c3c3c19c17c20c25c22c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c19c17c20c23c23c3c3c19c17c20c23c27c3c3c19c17c20c24c20c3c3c19c17c20c24c24c3c3c19c17c20c24c27c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24 c19c17c19c25c19c3c95c3c3c19c17c20c23c21c3c3c19c17c20c23c24c3c3c19c17c20c23c27c3c3c19c17c20c24c21c3c3c19c17c20c24c24c3c3c19c17c20c24c27c3c3c19c17c20c25c20c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c20 c19c17c19c26c19c3c95c3c3c19c17c20c23c19c3c3c19c17c20c23c22c3c3c19c17c20c23c25c3c3c19c17c20c23c28c3c3c19c17c20c24c21c3c3c19c17c20c24c25c3c3c19c17c20c24c28c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c27 c19c17c19c27c19c3c95c3c3c19c17c20c22c28c3c3c19c17c20c23c21c3c3c19c17c20c23c24c3c3c19c17c20c23c27c3c3c19c17c20c24c20c3c3c19c17c20c24c23c3c3c19c17c20c24c26c3c3c19c17c20c25c19c3c3c19c17c20c25c21c3c3c19c17c20c25c24 c19c17c19c28c19c3c95c3c86c19c17c20c22c28c3c86c19c17c20c23c20c3c3c19c17c20c23c23c3c3c19c17c20c23c26c3c3c19c17c20c23c28c3c3c19c17c20c24c21c3c3c19c17c20c24c24c3c3c19c17c20c24c27c3c3c19c17c20c25c20c3c3c19c17c20c25c22 c19c17c20c19c19c3c95c3c86c19c17c20c22c27c3c86c19c17c20c23c20c3c86c19c17c20c23c22c3c3c19c17c20c23c25c3c3c19c17c20c23c28c3c3c19c17c20c24c20c3c3c19c17c20c24c23c3c3c19c17c20c24c26c3c3c19c17c20c24c28c3c3c19c17c20c25c21 c19c17c20c20c19c3c95c3c86c19c17c20c22c27c3c86c19c17c20c23c20c3c86c19c17c20c23c22c3c86c19c17c20c23c25c3c3c19c17c20c23c27c3c3c19c17c20c24c20c3c3c19c17c20c24c22c3c3c19c17c20c24c25c3c3c19c17c20c24c27c3c3c19c17c20c25c20 c19c17c20c21c19c3c95c3c86c19c17c20c22c27c3c86c19c17c20c23c20c3c86c19c17c20c23c22c3c86c19c17c20c23c25c3c86c19c17c20c23c27c3c3c19c17c20c24c19c3c3c19c17c20c24c22c3c3c19c17c20c24c24c3c3c19c17c20c24c27c3c3c19c17c20c25c19 c19c17c20c22c19c3c95c3c86c19c17c20c22c27c3c86c19c17c20c23c19c3c86c19c17c20c23c22c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c3c19c17c20c24c21c3c3c19c17c20c24c24c3c3c19c17c20c24c26c3c3c19c17c20c25c19 c19c17c20c23c19c3c95c3c86c19c17c20c22c26c3c86c19c17c20c23c19c3c86c19c17c20c23c22c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c3c19c17c20c24c24c3c3c19c17c20c24c26c3c3c19c17c20c24c28 c19c17c20c24c19c3c95c3c86c19c17c20c22c26c3c86c19c17c20c23c19c3c86c19c17c20c23c22c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c3c19c17c20c24c26c3c3c19c17c20c24c28 c19c17c20c25c19c3c95c3c86c19c17c20c22c25c3c86c19c17c20c22c28c3c86c19c17c20c23c21c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c3c19c17c20c24c26c3c3c19c17c20c24c28 c19c17c20c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c22c28c3c86c19c17c20c23c21c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c3c19c17c20c24c28 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c22c27c3c86c19c17c20c23c21c3c86c19c17c20c23c24c3c86c19c17c20c23c27c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c20c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c20c3c86c19c17c20c23c23c3c86c19c17c20c23c26c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c19c3c86c19c17c20c23c23c3c86c19c17c20c23c26c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c28c3c86c19c17c20c23c22c3c86c19c17c20c23c26c3c86c19c17c20c24c19c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c22c3c86c19c17c20c23c25c3c86c19c17c20c23c28c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c21c3c86c19c17c20c23c25c3c86c19c17c20c23c28c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c20c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c19c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c20c3c86c19c17c20c24c23c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c20c3c86c19c17c20c24c23c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c22c3c86c19c17c20c23c26c3c86c19c17c20c24c20c3c86c19c17c20c24c23c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c21c3c86c19c17c20c23c26c3c86c19c17c20c24c19c3c86c19c17c20c24c23c3c86c19c17c20c24c26c3c86c19c17c20c24c28 c19c17c21c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c20c3c86c19c17c20c23c25c3c86c19c17c20c24c19c3c86c19c17c20c24c22c3c86c19c17c20c24c25c3c86c19c17c20c24c28 c19c17c22c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c25c3c86c19c17c20c24c28 c19c17c22c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c25c3c86c19c17c20c24c28 c19c17c22c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25c3c86c19c17c20c24c28 c19c17c22c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c27c3c86c19c17c20c23c22c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c27 c19c17c22c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c26c3c86c19c17c20c23c22c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c24c3c86c19c17c20c24c27 c19c17c22c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c26c3c86c19c17c20c23c21c3c86c19c17c20c23c26c3c86c19c17c20c24c20c3c86c19c17c20c24c24c3c86c19c17c20c24c27 c19c17c22c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c25c3c86c19c17c20c23c21c3c86c19c17c20c23c26c3c86c19c17c20c24c20c3c86c19c17c20c24c24c3c86c19c17c20c24c27 c19c17c22c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c24c3c86c19c17c20c23c20c3c86c19c17c20c23c25c3c86c19c17c20c24c20c3c86c19c17c20c24c23c3c86c19c17c20c24c27 c19c17c22c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c24c3c86c19c17c20c23c20c3c86c19c17c20c23c25c3c86c19c17c20c24c19c3c86c19c17c20c24c23c3c86c19c17c20c24c27 c19c17c22c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c24c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c24c19c3c86c19c17c20c24c23c3c86c19c17c20c24c26 c19c17c23c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c23c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c24c19c3c86c19c17c20c24c23c3c86c19c17c20c24c26 c19c17c23c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c23c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c26 c19c17c23c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c26 c19c17c23c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c26 c19c17c23c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c25 c19c17c23c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c25 c19c17c23c25c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c22c3c86c19c17c20c24c25 c19c17c23c26c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c23c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c22c3c86c19c17c20c22c27c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c23c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c22c3c86c19c17c20c22c27c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c22c3c86c19c17c20c22c27c3c86c19c17c20c23c22c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c25c3c86c19c17c20c22c22c3c86c19c17c20c22c27c3c86c19c17c20c23c22c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c19c3c86c19c17c20c21c26c3c86c19c17c20c22c22c3c86c19c17c20c22c27c3c86c19c17c20c23c22c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c22c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c19c3c86c19c17c20c21c26c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c19c3c86c19c17c20c21c26c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c24c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c86c19c17c20c21c20c3c86c19c17c20c21c26c3c86c19c17c20c22c22c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c25c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c20c23c3c86c19c17c20c21c20c3c86c19c17c20c21c27c3c86c19c17c20c22c23c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c26c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c20c23c3c86c19c17c20c21c21c3c86c19c17c20c21c27c3c86c19c17c20c22c23c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c27c19c3c95c3c3c3c16c16c3c3c3c86c19c17c20c20c24c3c86c19c17c20c21c21c3c86c19c17c20c21c27c3c86c19c17c20c22c23c3c86c19c17c20c22c28c3c86c19c17c20c23c23c3c86c19c17c20c23c27c3c86c19c17c20c24c21c3c86c19c17c20c24c25 c19c17c24c28c19c3c95c3c86c19c17c20c19c27c3c86c19c17c20c20c25c3c86c19c17c20c21c22c3c86c19c17c20c21c28c3c86c19c17c20c22c23c3c86c19c17c20c23c19c3c86c19c17c20c23c23c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c86c19c17c20c24c25 c19c17c25c19c19c3c95c3c86c19c17c20c19c28c3c86c19c17c20c20c25c3c86c19c17c20c21c22c3c86c19c17c20c21c28c3c86c19c17c20c22c24c3c86c19c17c20c23c19c3c86c19c17c20c23c23c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c3c19c17c20c24c25 c19c17c25c20c19c3c95c3c86c19c17c20c20c19c3c86c19c17c20c20c26c3c86c19c17c20c21c23c3c86c19c17c20c22c19c3c86c19c17c20c22c24c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c86c19c17c20c24c22c3c3c19c17c20c24c25 c19c17c25c21c19c3c95c3c86c19c17c20c20c20c3c86c19c17c20c20c27c3c86c19c17c20c21c23c3c86c19c17c20c22c19c3c86c19c17c20c22c24c3c86c19c17c20c23c19c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c25 c19c17c25c22c19c3c95c3c86c19c17c20c20c20c3c86c19c17c20c20c27c3c86c19c17c20c21c24c3c86c19c17c20c22c19c3c86c19c17c20c22c25c3c86c19c17c20c23c20c3c86c19c17c20c23c24c3c86c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c26 c19c17c25c23c19c3c95c3c86c19c17c20c20c21c3c86c19c17c20c20c28c3c86c19c17c20c21c24c3c86c19c17c20c22c20c3c86c19c17c20c22c25c3c86c19c17c20c23c20c3c86c19c17c20c23c24c3c3c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c26 c19c17c25c24c19c3c95c3c86c19c17c20c20c22c3c86c19c17c20c21c19c3c86c19c17c20c21c25c3c86c19c17c20c22c20c3c86c19c17c20c22c25c3c86c19c17c20c23c20c3c86c19c17c20c23c24c3c3c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c26 c19c17c25c25c19c3c95c3c86c19c17c20c20c23c3c86c19c17c20c21c19c3c86c19c17c20c21c25c3c86c19c17c20c22c21c3c86c19c17c20c22c26c3c86c19c17c20c23c20c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c22c3c3c19c17c20c24c26 c19c17c25c26c19c3c95c3c86c19c17c20c20c24c3c86c19c17c20c21c20c3c86c19c17c20c21c26c3c86c19c17c20c22c21c3c86c19c17c20c22c26c3c86c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c26 332 c55c68c69c79c72c3c39c22c17c23c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c21c20c19c3c3c19c17c21c21c19c3c3c19c17c21c22c19c3c3c19c17c21c23c19c3c3c19c17c21c24c19c3c3c19c17c21c25c19c3c3c19c17c21c26c19c3c3c19c17c21c27c19c3c3c19c17c21c28c19c3c3c19c17c22c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c25c27c19c3c95c3c86c19c17c20c20c25c3c86c19c17c20c21c21c3c86c19c17c20c21c26c3c86c19c17c20c22c22c3c86c19c17c20c22c26c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c26 c19c17c25c28c19c3c95c3c86c19c17c20c20c25c3c86c19c17c20c21c21c3c86c19c17c20c21c27c3c86c19c17c20c22c22c3c3c19c17c20c22c27c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c27 c19c17c26c19c19c3c95c3c86c19c17c20c20c26c3c86c19c17c20c21c22c3c86c19c17c20c21c27c3c86c19c17c20c22c23c3c3c19c17c20c22c27c3c3c19c17c20c23c22c3c3c19c17c20c23c26c3c3c19c17c20c24c20c3c3c19c17c20c24c23c3c3c19c17c20c24c27 c19c17c26c20c19c3c95c3c86c19c17c20c20c27c3c86c19c17c20c21c23c3c86c19c17c20c21c28c3c3c19c17c20c22c23c3c3c19c17c20c22c28c3c3c19c17c20c23c22c3c3c19c17c20c23c26c3c3c19c17c20c24c20c3c3c19c17c20c24c24c3c3c19c17c20c24c27 c19c17c26c21c19c3c95c3c86c19c17c20c20c28c3c86c19c17c20c21c23c3c86c19c17c20c22c19c3c3c19c17c20c22c23c3c3c19c17c20c22c28c3c3c19c17c20c23c22c3c3c19c17c20c23c26c3c3c19c17c20c24c20c3c3c19c17c20c24c24c3c3c19c17c20c24c27 c19c17c26c22c19c3c95c3c86c19c17c20c20c28c3c86c19c17c20c21c24c3c3c19c17c20c22c19c3c3c19c17c20c22c24c3c3c19c17c20c22c28c3c3c19c17c20c23c23c3c3c19c17c20c23c27c3c3c19c17c20c24c20c3c3c19c17c20c24c24c3c3c19c17c20c24c28 c19c17c26c23c19c3c95c3c86c19c17c20c21c19c3c3c19c17c20c21c24c3c3c19c17c20c22c20c3c3c19c17c20c22c24c3c3c19c17c20c23c19c3c3c19c17c20c23c23c3c3c19c17c20c23c27c3c3c19c17c20c24c21c3c3c19c17c20c24c24c3c3c19c17c20c24c28 c19c17c26c24c19c3c95c3c86c19c17c20c21c20c3c3c19c17c20c21c25c3c3c19c17c20c22c20c3c3c19c17c20c22c25c3c3c19c17c20c23c19c3c3c19c17c20c23c23c3c3c19c17c20c23c27c3c3c19c17c20c24c21c3c3c19c17c20c24c25c3c3c19c17c20c24c28 c19c17c26c25c19c3c95c3c3c19c17c20c21c20c3c3c19c17c20c21c26c3c3c19c17c20c22c21c3c3c19c17c20c22c25c3c3c19c17c20c23c20c3c3c19c17c20c23c24c3c3c19c17c20c23c28c3c3c19c17c20c24c21c3c3c19c17c20c24c25c3c3c19c17c20c24c28 c19c17c26c26c19c3c95c3c3c19c17c20c21c21c3c3c19c17c20c21c26c3c3c19c17c20c22c21c3c3c19c17c20c22c26c3c3c19c17c20c23c20c3c3c19c17c20c23c24c3c3c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c25c3c3c19c17c20c25c19 c19c17c26c27c19c3c95c3c3c19c17c20c21c22c3c3c19c17c20c21c27c3c3c19c17c20c22c22c3c3c19c17c20c22c26c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c23c28c3c3c19c17c20c24c22c3c3c19c17c20c24c25c3c3c19c17c20c25c19 c19c17c26c28c19c3c95c3c3c19c17c20c21c23c3c3c19c17c20c21c28c3c3c19c17c20c22c22c3c3c19c17c20c22c27c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c22c3c3c19c17c20c24c26c3c3c19c17c20c25c19 c19c17c27c19c19c3c95c3c3c19c17c20c21c23c3c3c19c17c20c21c28c3c3c19c17c20c22c23c3c3c19c17c20c22c27c3c3c19c17c20c23c21c3c3c19c17c20c23c25c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c26c3c3c19c17c20c25c19 c19c17c27c20c19c3c95c3c3c19c17c20c21c24c3c3c19c17c20c22c19c3c3c19c17c20c22c23c3c3c19c17c20c22c28c3c3c19c17c20c23c22c3c3c19c17c20c23c26c3c3c19c17c20c24c19c3c3c19c17c20c24c23c3c3c19c17c20c24c26c3c3c19c17c20c25c20 c19c17c27c21c19c3c95c3c3c19c17c20c21c25c3c3c19c17c20c22c19c3c3c19c17c20c22c24c3c3c19c17c20c22c28c3c3c19c17c20c23c22c3c3c19c17c20c23c26c3c3c19c17c20c24c20c3c3c19c17c20c24c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c22c19c3c95c3c3c19c17c20c21c25c3c3c19c17c20c22c20c3c3c19c17c20c22c25c3c3c19c17c20c23c19c3c3c19c17c20c23c23c3c3c19c17c20c23c27c3c3c19c17c20c24c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c23c19c3c95c3c3c19c17c20c21c26c3c3c19c17c20c22c21c3c3c19c17c20c22c25c3c3c19c17c20c23c19c3c3c19c17c20c23c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c24c19c3c95c3c3c19c17c20c21c27c3c3c19c17c20c22c21c3c3c19c17c20c22c26c3c3c19c17c20c23c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c25c19c3c95c3c3c19c17c20c21c27c3c3c19c17c20c22c22c3c3c19c17c20c22c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c26c19c3c95c3c3c19c17c20c21c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 333 c55c68c69c79c72c3c39c23c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c22c20c3c87c82c3c19c17c23c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c20c23c17c25c25c3c3c20c24c17c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c25c19c3c95c3c3c20c21c17c24c27c3c3c20c22c17c20c22c3c3c20c22c17c25c24c3c3c20c23c17c20c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c26c19c3c95c3c3c20c19c17c27c25c3c3c20c20c17c23c19c3c3c20c20c17c28c21c3c3c20c21c17c23c21c3c3c20c21c17c28c20c3c3c20c22c17c23c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c27c19c3c95c3c3c28c17c23c21c26c3c3c28c17c28c23c24c3c3c20c19c17c23c24c3c3c20c19c17c28c23c3c3c20c20c17c23c22c3c3c20c20c17c28c19c3c3c20c21c17c22c25c3c3c20c21c17c27c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c28c19c3c95c3c3c27c17c21c19c25c3c3c27c17c26c19c26c3c3c28c17c20c28c27c3c3c28c17c25c26c28c3c3c20c19c17c20c24c3c3c20c19c17c25c20c3c3c20c20c17c19c25c3c3c20c20c17c24c20c3c3c20c20c17c28c23c3c3c20c21c17c22c26 c19c17c20c19c19c3c95c3c3c26c17c20c25c26c3c3c26c17c25c23c27c3c3c27c17c20c21c20c3c3c27c17c24c27c26c3c3c28c17c19c23c24c3c3c28c17c23c28c24c3c3c28c17c28c22c25c3c3c20c19c17c22c26c3c3c20c19c17c26c28c3c3c20c20c17c21c20 c19c17c20c20c19c3c95c3c3c25c17c21c27c19c3c3c25c17c26c22c28c3c3c26c17c20c28c23c3c3c26c17c25c23c21c3c3c27c17c19c27c23c3c3c27c17c24c20c28c3c3c27c17c28c23c27c3c3c28c17c22c26c19c3c3c28c17c26c27c24c3c3c20c19c17c20c28 c19c17c20c21c19c3c95c3c3c24c17c24c21c20c3c3c24c17c28c24c28c3c3c25c17c22c28c21c3c3c25c17c27c21c20c3c3c26c17c21c23c25c3c3c26c17c25c25c25c3c3c27c17c19c27c19c3c3c27c17c23c27c28c3c3c27c17c27c28c21c3c3c28c17c21c27c28 c19c17c20c22c19c3c95c3c3c23c17c27c26c21c3c3c24c17c21c27c25c3c3c24c17c25c28c28c3c3c25c17c20c19c28c3c3c25c17c24c20c24c3c3c25c17c28c20c27c3c3c26c17c22c20c26c3c3c26c17c26c20c21c3c3c27c17c20c19c21c3c3c27c17c23c27c26 c19c17c20c23c19c3c95c3c3c23c17c22c20c24c3c3c23c17c26c19c26c3c3c24c17c19c28c27c3c3c24c17c23c27c27c3c3c24c17c27c26c25c3c3c25c17c21c25c21c3c3c25c17c25c23c24c3c3c26c17c19c21c24c3c3c26c17c23c19c20c3c3c26c17c26c26c22 c19c17c20c24c19c3c95c3c3c22c17c27c22c25c3c3c23c17c21c19c26c3c3c23c17c24c26c27c3c3c23c17c28c23c27c3c3c24c17c22c20c27c3c3c24c17c25c27c25c3c3c25c17c19c24c22c3c3c25c17c23c20c26c3c3c25c17c26c26c28c3c3c26c17c20c22c27 c19c17c20c25c19c3c95c3c3c22c17c23c21c24c3c3c22c17c26c26c24c3c3c23c17c20c21c25c3c3c23c17c23c26c26c3c3c23c17c27c21c28c3c3c24c17c20c27c19c3c3c24c17c24c22c19c3c3c24c17c27c26c28c3c3c25c17c21c21c25c3c3c25c17c24c26c21 c19c17c20c26c19c3c95c3c3c22c17c19c25c28c3c3c22c17c23c19c19c3c3c22c17c26c22c22c3c3c23c17c19c25c25c3c3c23c17c23c19c19c3c3c23c17c26c22c23c3c3c24c17c19c25c27c3c3c24c17c23c19c21c3c3c24c17c26c22c24c3c3c25c17c19c25c26 c19c17c20c27c19c3c95c3c3c21c17c26c25c21c3c3c22c17c19c26c24c3c3c22c17c22c28c19c3c3c22c17c26c19c25c3c3c23c17c19c21c22c3c3c23c17c22c23c21c3c3c23c17c25c25c19c3c3c23c17c28c26c28c3c3c24c17c21c28c27c3c3c24c17c25c20c25 c19c17c20c28c19c3c95c3c3c21c17c23c28c25c3c3c21c17c26c28c21c3c3c22c17c19c28c20c3c3c22c17c22c28c20c3c3c22c17c25c28c21c3c3c22c17c28c28c24c3c3c23c17c21c28c28c3c3c23c17c25c19c22c3c3c23c17c28c19c27c3c3c24c17c21c20c22 c19c17c21c19c19c3c95c3c3c21c17c21c25c24c3c3c21c17c24c23c25c3c3c21c17c27c21c28c3c3c22c17c20c20c23c3c3c22c17c23c19c20c3c3c22c17c25c27c28c3c3c22c17c28c26c28c3c3c23c17c21c26c19c3c3c23c17c24c25c20c3c3c23c17c27c24c22 c19c17c21c20c19c3c95c3c3c21c17c19c25c22c3c3c21c17c22c22c19c3c3c21c17c24c28c28c3c3c21c17c27c26c19c3c3c22c17c20c23c22c3c3c22c17c23c20c27c3c3c22c17c25c28c24c3c3c22c17c28c26c21c3c3c23c17c21c24c20c3c3c23c17c24c22c19 c19c17c21c21c19c3c95c3c3c20c17c27c27c27c3c3c21c17c20c23c20c3c3c21c17c22c28c26c3c3c21c17c25c24c25c3c3c21c17c28c20c25c3c3c22c17c20c26c27c3c3c22c17c23c23c21c3c3c22c17c26c19c26c3c3c22c17c28c26c23c3c3c23c17c21c23c20 c19c17c21c22c19c3c95c3c3c20c17c26c22c23c3c3c20c17c28c26c25c3c3c21c17c21c21c19c3c3c21c17c23c25c25c3c3c21c17c26c20c23c3c3c21c17c28c25c24c3c3c22c17c21c20c26c3c3c22c17c23c26c20c3c3c22c17c26c21c25c3c3c22c17c28c27c21 c19c17c21c23c19c3c95c3c3c20c17c25c19c19c3c3c20c17c27c22c20c3c3c21c17c19c25c22c3c3c21c17c21c28c28c3c3c21c17c24c22c25c3c3c21c17c26c26c24c3c3c22c17c19c20c25c3c3c22c17c21c24c28c3c3c22c17c24c19c22c3c3c22c17c26c23c28 c19c17c21c24c19c3c95c3c3c20c17c23c27c21c3c3c20c17c26c19c22c3c3c20c17c28c21c24c3c3c21c17c20c24c19c3c3c21c17c22c26c26c3c3c21c17c25c19c25c3c3c21c17c27c22c26c3c3c22c17c19c26c19c3c3c22c17c22c19c23c3c3c22c17c24c23c19 c19c17c21c25c19c3c95c3c3c20c17c22c26c28c3c3c20c17c24c28c19c3c3c20c17c27c19c22c3c3c21c17c19c20c28c3c3c21c17c21c22c25c3c3c21c17c23c24c24c3c3c21c17c25c26c26c3c3c21c17c28c19c19c3c3c22c17c20c21c24c3c3c22c17c22c24c20 c19c17c21c26c19c3c95c3c3c20c17c21c27c28c3c3c20c17c23c28c20c3c3c20c17c25c28c24c3c3c20c17c28c19c21c3c3c21c17c20c20c19c3c3c21c17c22c21c20c3c3c21c17c24c22c22c3c3c21c17c26c23c26c3c3c21c17c28c25c22c3c3c22c17c20c27c20 c19c17c21c27c19c3c95c3c3c20c17c21c20c19c3c3c20c17c23c19c23c3c3c20c17c25c19c19c3c3c20c17c26c28c27c3c3c20c17c28c28c27c3c3c21c17c21c19c19c3c3c21c17c23c19c24c3c3c21c17c25c20c19c3c3c21c17c27c20c27c3c3c22c17c19c21c26 c19c17c21c28c19c3c95c3c3c20c17c20c23c19c3c3c20c17c22c21c25c3c3c20c17c24c20c24c3c3c20c17c26c19c25c3c3c20c17c27c28c27c3c3c21c17c19c28c22c3c3c21c17c21c27c28c3c3c21c17c23c27c26c3c3c21c17c25c27c26c3c3c21c17c27c27c27 c19c17c22c19c19c3c95c3c3c20c17c19c26c28c3c3c20c17c21c24c28c3c3c20c17c23c23c19c3c3c20c17c25c21c23c3c3c20c17c27c19c28c3c3c20c17c28c28c25c3c3c21c17c20c27c24c3c3c21c17c22c26c25c3c3c21c17c24c25c28c3c3c21c17c26c25c22 c19c17c22c20c19c3c95c3c3c20c17c19c21c25c3c3c20c17c20c28c28c3c3c20c17c22c26c23c3c3c20c17c24c24c20c3c3c20c17c26c21c28c3c3c20c17c28c20c19c3c3c21c17c19c28c21c3c3c21c17c21c26c25c3c3c21c17c23c25c21c3c3c21c17c25c23c28 c19c17c22c21c19c3c95c3c3c19c17c28c26c28c3c3c20c17c20c23c25c3c3c20c17c22c20c24c3c3c20c17c23c27c25c3c3c20c17c25c24c27c3c3c20c17c27c22c21c3c3c21c17c19c19c27c3c3c21c17c20c27c25c3c3c21c17c22c25c24c3c3c21c17c24c23c25 c19c17c22c22c19c3c95c3c3c19c17c28c22c28c3c3c20c17c20c19c19c3c3c20c17c21c25c22c3c3c20c17c23c21c27c3c3c20c17c24c28c24c3c3c20c17c26c25c22c3c3c20c17c28c22c22c3c3c21c17c20c19c23c3c3c21c17c21c26c27c3c3c21c17c23c24c21 c19c17c22c23c19c3c95c3c3c19c17c28c19c22c3c3c20c17c19c25c19c3c3c20c17c21c20c26c3c3c20c17c22c26c26c3c3c20c17c24c22c27c3c3c20c17c26c19c20c3c3c20c17c27c25c24c3c3c21c17c19c22c20c3c3c21c17c20c28c27c3c3c21c17c22c25c26 c19c17c22c24c19c3c95c3c3c19c17c27c26c22c3c3c20c17c19c21c23c3c3c20c17c20c26c26c3c3c20c17c22c22c20c3c3c20c17c23c27c26c3c3c20c17c25c23c24c3c3c20c17c27c19c23c3c3c20c17c28c25c24c3c3c21c17c20c21c26c3c3c21c17c21c28c19 c19c17c22c25c19c3c95c3c3c19c17c27c23c26c3c3c19c17c28c28c23c3c3c20c17c20c23c21c3c3c20c17c21c28c20c3c3c20c17c23c23c21c3c3c20c17c24c28c24c3c3c20c17c26c23c28c3c3c20c17c28c19c24c3c3c21c17c19c25c21c3c3c21c17c21c21c19 c19c17c22c26c19c3c95c3c3c19c17c27c21c24c3c3c19c17c28c25c26c3c3c20c17c20c20c20c3c3c20c17c21c24c25c3c3c20c17c23c19c21c3c3c20c17c24c24c19c3c3c20c17c26c19c19c3c3c20c17c27c24c20c3c3c21c17c19c19c22c3c3c21c17c20c24c26 c19c17c22c27c19c3c95c3c3c19c17c27c19c25c3c3c19c17c28c23c23c3c3c20c17c19c27c23c3c3c20c17c21c21c24c3c3c20c17c22c25c26c3c3c20c17c24c20c20c3c3c20c17c25c24c25c3c3c20c17c27c19c21c3c3c20c17c28c24c19c3c3c21c17c19c28c28 c19c17c22c28c19c3c95c3c3c19c17c26c28c20c3c3c19c17c28c21c24c3c3c20c17c19c25c19c3c3c20c17c20c28c26c3c3c20c17c22c22c24c3c3c20c17c23c26c24c3c3c20c17c25c20c25c3c3c20c17c26c24c27c3c3c20c17c28c19c21c3c3c21c17c19c23c26 c19c17c23c19c19c3c95c3c3c19c17c26c26c27c3c3c19c17c28c19c27c3c3c20c17c19c23c19c3c3c20c17c20c26c22c3c3c20c17c22c19c27c3c3c20c17c23c23c22c3c3c20c17c24c27c19c3c3c20c17c26c20c28c3c3c20c17c27c24c27c3c3c20c17c28c28c28 c19c17c23c20c19c3c95c3c3c19c17c26c25c27c3c3c19c17c27c28c24c3c3c20c17c19c21c22c3c3c20c17c20c24c22c3c3c20c17c21c27c22c3c3c20c17c23c20c24c3c3c20c17c24c23c28c3c3c20c17c25c27c22c3c3c20c17c27c20c28c3c3c20c17c28c24c25 c19c17c23c21c19c3c95c3c3c19c17c26c25c19c3c3c19c17c27c27c23c3c3c20c17c19c19c28c3c3c20c17c20c22c24c3c3c20c17c21c25c21c3c3c20c17c22c28c20c3c3c20c17c24c21c19c3c3c20c17c25c24c20c3c3c20c17c26c27c23c3c3c20c17c28c20c26 c19c17c23c22c19c3c95c3c3c19c17c26c24c23c3c3c19c17c27c26c24c3c3c19c17c28c28c26c3c3c20c17c20c20c28c3c3c20c17c21c23c23c3c3c20c17c22c25c28c3c3c20c17c23c28c24c3c3c20c17c25c21c22c3c3c20c17c26c24c21c3c3c20c17c27c27c21 c19c17c23c23c19c3c95c3c3c19c17c26c24c19c3c3c19c17c27c25c27c3c3c19c17c28c27c26c3c3c20c17c20c19c26c3c3c20c17c21c21c27c3c3c20c17c22c24c19c3c3c20c17c23c26c22c3c3c20c17c24c28c27c3c3c20c17c26c21c22c3c3c20c17c27c24c19 c19c17c23c24c19c3c95c3c3c19c17c26c23c27c3c3c19c17c27c25c22c3c3c19c17c28c26c28c3c3c20c17c19c28c25c3c3c20c17c21c20c23c3c3c20c17c22c22c22c3c3c20c17c23c24c23c3c3c20c17c24c26c24c3c3c20c17c25c28c26c3c3c20c17c27c21c20 c19c17c23c25c19c3c95c3c3c19c17c26c23c27c3c3c19c17c27c25c19c3c3c19c17c28c26c22c3c3c20c17c19c27c26c3c3c20c17c21c19c22c3c3c20c17c22c20c28c3c3c20c17c23c22c25c3c3c20c17c24c24c24c3c3c20c17c25c26c23c3c3c20c17c26c28c24 c19c17c23c26c19c3c95c3c3c19c17c26c23c28c3c3c19c17c27c24c27c3c3c19c17c28c25c28c3c3c20c17c19c27c20c3c3c20c17c20c28c22c3c3c20c17c22c19c26c3c3c20c17c23c21c21c3c3c20c17c24c22c26c3c3c20c17c25c24c23c3c3c20c17c26c26c20 c19c17c23c27c19c3c95c3c3c19c17c26c24c20c3c3c19c17c27c24c27c3c3c19c17c28c25c25c3c3c20c17c19c26c24c3c3c20c17c20c27c25c3c3c20c17c21c28c26c3c3c20c17c23c19c28c3c3c20c17c24c21c21c3c3c20c17c25c22c25c3c3c20c17c26c24c20 c19c17c23c28c19c3c95c3c3c19c17c26c24c24c3c3c19c17c27c24c28c3c3c19c17c28c25c24c3c3c20c17c19c26c21c3c3c20c17c20c27c19c3c3c20c17c21c27c27c3c3c20c17c22c28c27c3c3c20c17c24c19c27c3c3c20c17c25c21c19c3c3c20c17c26c22c21 c19c17c24c19c19c3c95c3c3c19c17c26c24c28c3c3c19c17c27c25c21c3c3c19c17c28c25c24c3c3c20c17c19c26c19c3c3c20c17c20c26c24c3c3c20c17c21c27c20c3c3c20c17c22c27c27c3c3c20c17c23c28c26c3c3c20c17c25c19c25c3c3c20c17c26c20c24 c19c17c24c20c19c3c95c3c3c19c17c26c25c24c3c3c19c17c27c25c24c3c3c19c17c28c25c26c3c3c20c17c19c25c28c3c3c20c17c20c26c21c3c3c20c17c21c26c25c3c3c20c17c22c27c20c3c3c20c17c23c27c26c3c3c20c17c24c28c22c3c3c20c17c26c19c20 c19c17c24c21c19c3c95c3c3c19c17c26c26c20c3c3c19c17c27c26c19c3c3c19c17c28c25c28c3c3c20c17c19c25c28c3c3c20c17c20c26c19c3c3c20c17c21c26c21c3c3c20c17c22c26c24c3c3c20c17c23c26c27c3c3c20c17c24c27c22c3c3c20c17c25c27c27 c19c17c24c22c19c3c95c3c3c19c17c26c26c28c3c3c19c17c27c26c24c3c3c19c17c28c26c22c3c3c20c17c19c26c20c3c3c20c17c20c26c19c3c3c20c17c21c25c28c3c3c20c17c22c26c19c3c3c20c17c23c26c20c3c3c20c17c24c26c23c3c3c20c17c25c26c26 c19c17c24c23c19c3c95c3c3c19c17c26c27c26c3c3c19c17c27c27c20c3c3c19c17c28c26c26c3c3c20c17c19c26c22c3c3c20c17c20c26c19c3c3c20c17c21c25c27c3c3c20c17c22c25c26c3c3c20c17c23c25c25c3c3c20c17c24c25c25c3c3c20c17c25c25c26 c19c17c24c24c19c3c95c3c3c19c17c26c28c25c3c3c19c17c27c27c28c3c3c19c17c28c27c21c3c3c20c17c19c26c26c3c3c20c17c20c26c21c3c3c20c17c21c25c27c3c3c20c17c22c25c23c3c3c20c17c23c25c21c3c3c20c17c24c25c19c3c3c20c17c25c24c28 c19c17c24c25c19c3c95c3c3c19c17c27c19c24c3c3c19c17c27c28c25c3c3c19c17c28c27c27c3c3c20c17c19c27c20c3c3c20c17c20c26c23c3c3c20c17c21c25c27c3c3c20c17c22c25c22c3c3c20c17c23c24c28c3c3c20c17c24c24c24c3c3c20c17c25c24c21 c19c17c24c26c19c3c95c3c3c19c17c27c20c24c3c3c19c17c28c19c24c3c3c19c17c28c28c24c3c3c20c17c19c27c25c3c3c20c17c20c26c26c3c3c20c17c21c26c19c3c3c20c17c22c25c22c3c3c20c17c23c24c26c3c3c20c17c24c24c20c3c3c20c17c25c23c25 c19c17c24c27c19c3c95c3c3c19c17c27c21c25c3c3c19c17c28c20c23c3c3c20c17c19c19c21c3c3c20c17c19c28c21c3c3c20c17c20c27c21c3c3c20c17c21c26c21c3c3c20c17c22c25c23c3c3c20c17c23c24c25c3c3c20c17c24c23c27c3c3c20c17c25c23c21 c19c17c24c28c19c3c95c3c3c19c17c27c22c26c3c3c19c17c28c21c23c3c3c20c17c19c20c19c3c3c20c17c19c28c27c3c3c20c17c20c27c25c3c3c20c17c21c26c24c3c3c20c17c22c25c24c3c3c20c17c23c24c25c3c3c20c17c24c23c26c3c3c20c17c25c22c27 c19c17c25c19c19c3c95c3c3c19c17c27c23c28c3c3c19c17c28c22c23c3c3c20c17c19c20c28c3c3c20c17c20c19c24c3c3c20c17c20c28c21c3c3c20c17c21c26c28c3c3c20c17c22c25c27c3c3c20c17c23c24c25c3c3c20c17c24c23c25c3c3c20c17c25c22c25 c19c17c25c20c19c3c95c3c3c19c17c27c25c20c3c3c19c17c28c23c23c3c3c20c17c19c21c27c3c3c20c17c20c20c22c3c3c20c17c20c28c27c3c3c20c17c21c27c23c3c3c20c17c22c26c20c3c3c20c17c23c24c27c3c3c20c17c24c23c25c3c3c20c17c25c22c23 c19c17c25c21c19c3c95c3c3c19c17c27c26c22c3c3c19c17c28c24c24c3c3c20c17c19c22c27c3c3c20c17c20c21c20c3c3c20c17c21c19c24c3c3c20c17c21c28c19c3c3c20c17c22c26c24c3c3c20c17c23c25c19c3c3c20c17c24c23c26c3c3c20c17c25c22c23 c19c17c25c22c19c3c95c3c3c19c17c27c27c25c3c3c19c17c28c25c26c3c3c20c17c19c23c27c3c3c20c17c20c22c19c3c3c20c17c21c20c21c3c3c20c17c21c28c25c3c3c20c17c22c26c28c3c3c20c17c23c25c23c3c3c20c17c24c23c27c3c3c20c17c25c22c23 c19c17c25c23c19c3c95c3c3c19c17c27c28c28c3c3c19c17c28c26c28c3c3c20c17c19c24c28c3c3c20c17c20c22c28c3c3c20c17c21c21c19c3c3c20c17c22c19c21c3c3c20c17c22c27c23c3c3c20c17c23c25c26c3c3c20c17c24c24c20c3c3c20c17c25c22c24 c19c17c25c24c19c3c95c3c3c19c17c28c20c22c3c3c19c17c28c28c20c3c3c20c17c19c26c19c3c3c20c17c20c23c28c3c3c20c17c21c21c28c3c3c20c17c22c19c28c3c3c20c17c22c28c19c3c3c20c17c23c26c21c3c3c20c17c24c24c23c3c3c20c17c25c22c26 c19c17c25c25c19c3c95c3c3c19c17c28c21c26c3c3c20c17c19c19c22c3c3c20c17c19c27c20c3c3c20c17c20c24c28c3c3c20c17c21c22c26c3c3c20c17c22c20c26c3c3c20c17c22c28c25c3c3c20c17c23c26c26c3c3c20c17c24c24c26c3c3c20c17c25c22c28 c19c17c25c26c19c3c95c3c3c19c17c28c23c20c3c3c20c17c19c20c25c3c3c20c17c19c28c22c3c3c20c17c20c25c28c3c3c20c17c21c23c26c3c3c20c17c22c21c24c3c3c20c17c23c19c22c3c3c20c17c23c27c21c3c3c20c17c24c25c21c3c3c20c17c25c23c21 c19c17c25c27c19c3c95c3c3c19c17c28c24c24c3c3c20c17c19c21c28c3c3c20c17c20c19c23c3c3c20c17c20c27c19c3c3c20c17c21c24c25c3c3c20c17c22c22c22c3c3c20c17c23c20c19c3c3c20c17c23c27c27c3c3c20c17c24c25c25c3c3c20c17c25c23c24 c19c17c25c28c19c3c95c3c3c19c17c28c25c28c3c3c20c17c19c23c22c3c3c20c17c20c20c26c3c3c20c17c20c28c20c3c3c20c17c21c25c25c3c3c20c17c22c23c21c3c3c20c17c23c20c27c3c3c20c17c23c28c24c3c3c20c17c24c26c21c3c3c20c17c25c23c28 c19c17c26c19c19c3c95c3c3c19c17c28c27c23c3c3c20c17c19c24c25c3c3c20c17c20c21c28c3c3c20c17c21c19c22c3c3c20c17c21c26c25c3c3c20c17c22c24c20c3c3c20c17c23c21c25c3c3c20c17c24c19c20c3c3c20c17c24c26c26c3c3c20c17c25c24c23 334 c55c68c69c79c72c3c39c23c17c20c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c26c20c19c3c95c3c3c19c17c28c28c27c3c3c20c17c19c26c19c3c3c20c17c20c23c21c3c3c20c17c21c20c23c3c3c20c17c21c27c26c3c3c20c17c22c25c19c3c3c20c17c23c22c23c3c3c20c17c24c19c28c3c3c20c17c24c27c23c3c3c20c17c25c24c28 c19c17c26c21c19c3c95c3c3c20c17c19c20c22c3c3c20c17c19c27c23c3c3c20c17c20c24c24c3c3c20c17c21c21c25c3c3c20c17c21c28c27c3c3c20c17c22c26c19c3c3c20c17c23c23c22c3c3c20c17c24c20c25c3c3c20c17c24c28c19c3c3c20c17c25c25c23 c19c17c26c22c19c3c95c3c3c20c17c19c21c27c3c3c20c17c19c28c27c3c3c20c17c20c25c27c3c3c20c17c21c22c27c3c3c20c17c22c19c28c3c3c20c17c22c27c19c3c3c20c17c23c24c21c3c3c20c17c24c21c23c3c3c20c17c24c28c26c3c3c3c16c16c3c3 c19c17c26c23c19c3c95c3c3c20c17c19c23c23c3c3c20c17c20c20c21c3c3c20c17c20c27c20c3c3c20c17c21c24c20c3c3c20c17c22c21c19c3c3c20c17c22c28c20c3c3c20c17c23c25c20c3c3c20c17c24c22c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c24c19c3c95c3c3c20c17c19c24c28c3c3c20c17c20c21c26c3c3c20c17c20c28c24c3c3c20c17c21c25c22c3c3c20c17c22c22c21c3c3c20c17c23c19c20c3c3c20c17c23c26c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c25c19c3c95c3c3c20c17c19c26c23c3c3c20c17c20c23c20c3c3c20c17c21c19c27c3c3c20c17c21c26c25c3c3c20c17c22c23c23c3c3c20c17c23c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c26c19c3c95c3c3c20c17c19c28c19c3c3c20c17c20c24c25c3c3c20c17c21c21c21c3c3c20c17c21c27c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c27c19c3c95c3c3c20c17c20c19c24c3c3c20c17c20c26c19c3c3c20c17c21c22c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c28c19c3c95c3c3c20c17c20c21c20c3c3c20c17c20c27c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c19c19c3c95c3c3c20c17c20c22c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 335 c55c68c69c79c72c3c39c23c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c22c20c3c87c82c3c19c17c23c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c14c20c23c17c22c23c3c14c20c24c17c20c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c25c19c3c95c3c14c20c20c17c27c21c3c14c20c21c17c24c26c3c14c20c22c17c22c19c3c14c20c23c17c19c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c26c19c3c95c3c14c28c17c26c23c28c3c14c20c19c17c23c27c3c14c20c20c17c21c19c3c14c20c20c17c28c19c3c14c20c21c17c24c28c3c14c20c22c17c21c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c27c19c3c95c3c14c27c17c19c20c26c3c14c27c17c26c22c24c3c14c28c17c23c22c26c3c14c20c19c17c20c21c3c14c20c19c17c26c28c3c14c20c20c17c23c24c3c14c20c21c17c19c28c3c14c20c21c17c26c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c28c19c3c95c3c14c25c17c24c25c19c3c14c26c17c21c24c23c3c14c26c17c28c22c24c3c14c27c17c25c19c22c3c14c28c17c21c24c26c3c14c28c17c27c28c27c3c14c20c19c17c24c21c3c14c20c20c17c20c23c3c14c20c20c17c26c23c3c14c20c21c17c22c22 c19c17c20c19c19c3c95c3c14c24c17c22c22c19c3c14c24c17c28c28c26c3c14c25c17c25c24c23c3c14c26c17c21c28c28c3c14c26c17c28c22c23c3c14c27c17c24c24c26c3c14c28c17c20c26c19c3c14c28c17c26c26c20c3c14c20c19c17c22c25c3c14c20c19c17c28c23 c19c17c20c20c19c3c95c3c14c23c17c21c28c20c3c14c23c17c28c21c27c3c14c24c17c24c24c27c3c14c25c17c20c26c28c3c14c25c17c26c28c21c3c14c26c17c22c28c24c3c14c26c17c28c28c19c3c14c27c17c24c26c24c3c14c28c17c20c24c19c3c14c28c17c26c20c25 c19c17c20c21c19c3c95c3c14c22c17c23c20c22c3c14c23c17c19c21c19c3c14c23c17c25c21c20c3c14c24c17c21c20c25c3c14c24c17c27c19c24c3c14c25c17c22c27c25c3c14c25c17c28c25c20c3c14c26c17c24c21c26c3c14c27c17c19c27c25c3c14c27c17c25c22c26 c19c17c20c22c19c3c95c3c14c21c17c25c26c22c3c14c22c17c21c23c27c3c14c22c17c27c21c19c3c14c23c17c22c27c27c3c14c23c17c28c24c20c3c14c24c17c24c20c19c3c14c25c17c19c25c22c3c14c25c17c25c20c19c3c14c26c17c20c24c20c3c14c26c17c25c27c24 c19c17c20c23c19c3c95c3c14c21c17c19c23c28c3c14c21c17c24c28c22c3c14c22c17c20c22c25c3c14c22c17c25c26c25c3c14c23c17c21c20c23c3c14c23c17c26c23c28c3c14c24c17c21c27c19c3c14c24c17c27c19c25c3c14c25c17c22c21c26c3c14c25c17c27c23c23 c19c17c20c24c19c3c95c3c14c20c17c24c21c23c3c14c21c17c19c22c27c3c14c21c17c24c24c21c3c14c22c17c19c25c25c3c14c22c17c24c26c27c3c14c23c17c19c27c27c3c14c23c17c24c28c25c3c14c24c17c20c19c20c3c14c24c17c25c19c22c3c14c25c17c20c19c20 c19c17c20c25c19c3c95c3c14c20c17c19c27c21c3c14c20c17c24c25c27c3c14c21c17c19c24c23c3c14c21c17c24c23c21c3c14c22c17c19c21c28c3c14c22c17c24c20c24c3c14c23c17c19c19c20c3c14c23c17c23c27c24c3c14c23c17c28c25c25c3c14c24c17c23c23c24 c19c17c20c26c19c3c95c3c14c19c17c26c20c20c3c14c20c17c20c26c19c3c14c20c17c25c22c19c3c14c21c17c19c28c22c3c14c21c17c24c24c25c3c14c22c17c19c20c28c3c14c22c17c23c27c21c3c14c22c17c28c23c24c3c14c23c17c23c19c25c3c14c23c17c27c25c25 c19c17c20c27c19c3c95c3c14c19c17c22c28c28c3c14c19c17c27c22c22c3c14c20c17c21c26c19c3c14c20c17c26c19c27c3c14c21c17c20c23c27c3c14c21c17c24c27c28c3c14c22c17c19c22c20c3c14c22c17c23c26c22c3c14c22c17c28c20c23c3c14c23c17c22c24c24 c19c17c20c28c19c3c95c3c14c19c17c20c22c28c3c14c19c17c24c23c28c3c14c19c17c28c25c22c3c14c20c17c22c26c28c3c14c20c17c26c28c26c3c14c21c17c21c20c26c3c14c21c17c25c22c27c3c14c22c17c19c25c19c3c14c22c17c23c27c22c3c14c22c17c28c19c24 c19c17c21c19c19c3c95c3c16c19c17c19c26c28c3c14c19c17c22c20c19c3c14c19c17c26c19c21c3c14c20c17c19c28c27c3c14c20c17c23c28c25c3c14c20c17c27c28c24c3c14c21c17c21c28c26c3c14c21c17c26c19c19c3c14c22c17c20c19c23c3c14c22c17c24c19c28 c19c17c21c20c19c3c95c3c16c19c17c21c25c20c3c14c19c17c20c19c28c3c14c19c17c23c27c21c3c14c19c17c27c24c27c3c14c20c17c21c22c25c3c14c20c17c25c20c26c3c14c21c17c19c19c20c3c14c21c17c22c27c24c3c14c21c17c26c26c21c3c14c22c17c20c24c28 c19c17c21c21c19c3c95c3c16c19c17c23c20c20c3c16c19c17c19c25c19c3c14c19c17c21c28c24c3c14c19c17c25c24c22c3c14c20c17c19c20c23c3c14c20c17c22c26c26c3c14c20c17c26c23c22c3c14c21c17c20c20c20c3c14c21c17c23c27c20c3c14c21c17c27c24c21 c19c17c21c22c19c3c95c3c16c19c17c24c22c24c3c16c19c17c21c19c19c3c14c19c17c20c22c27c3c14c19c17c23c26c28c3c14c19c17c27c21c23c3c14c20c17c20c26c20c3c14c20c17c24c21c19c3c14c20c17c27c26c21c3c14c21c17c21c21c25c3c14c21c17c24c27c20 c19c17c21c23c19c3c95c3c16c19c17c25c22c25c3c16c19c17c22c20c26c3c14c19c17c19c19c25c3c14c19c17c22c22c21c3c14c19c17c25c25c20c3c14c19c17c28c28c22c3c14c20c17c22c21c26c3c14c20c17c25c25c23c3c14c21c17c19c19c22c3c14c21c17c22c23c22 c19c17c21c24c19c3c95c3c16c19c17c26c20c26c3c16c19c17c23c20c21c3c16c19c17c20c19c23c3c14c19c17c21c19c27c3c14c19c17c24c21c22c3c14c19c17c27c23c19c3c14c20c17c20c25c19c3c14c20c17c23c27c22c3c14c20c17c27c19c27c3c14c21c17c20c22c24 c19c17c21c25c19c3c95c3c16c19c17c26c27c21c3c16c19c17c23c28c19c3c16c19c17c20c28c23c3c14c19c17c20c19c23c3c14c19c17c23c19c25c3c14c19c17c26c20c19c3c14c20c17c19c20c26c3c14c20c17c22c21c25c3c14c20c17c25c22c27c3c14c20c17c28c24c20 c19c17c21c26c19c3c95c3c16c19c17c27c22c21c3c16c19c17c24c24c21c3c16c19c17c21c25c28c3c14c19c17c19c20c27c3c14c19c17c22c19c26c3c14c19c17c24c28c28c3c14c19c17c27c28c22c3c14c20c17c20c28c19c3c14c20c17c23c27c28c3c14c20c17c26c28c20 c19c17c21c27c19c3c95c3c16c19c17c27c25c28c3c16c19c17c25c19c19c3c16c19c17c22c21c27c3c16c19c17c19c24c22c3c14c19c17c21c21c23c3c14c19c17c24c19c23c3c14c19c17c26c27c26c3c14c20c17c19c26c22c3c14c20c17c22c25c20c3c14c20c17c25c24c20 c19c17c21c28c19c3c95c3c16c19c17c27c28c24c3c16c19c17c25c22c26c3c16c19c17c22c26c24c3c16c19c17c20c20c20c3c14c19c17c20c24c25c3c14c19c17c23c21c24c3c14c19c17c25c28c27c3c14c19c17c28c26c21c3c14c20c17c21c23c28c3c14c20c17c24c21c27 c19c17c22c19c19c3c95c3c16c19c17c28c20c21c3c16c19c17c25c25c22c3c16c19c17c23c20c21c3c16c19c17c20c24c26c3c14c19c17c20c19c19c3c14c19c17c22c24c28c3c14c19c17c25c21c21c3c14c19c17c27c27c25c3c14c20c17c20c24c22c3c14c20c17c23c21c21 c19c17c22c20c19c3c95c3c16c19c17c28c21c20c3c16c19c17c25c27c20c3c16c19c17c23c22c27c3c16c19c17c20c28c22c3c14c19c17c19c24c24c3c14c19c17c22c19c24c3c14c19c17c24c24c27c3c14c19c17c27c20c22c3c14c20c17c19c26c19c3c14c20c17c22c22c19 c19c17c22c21c19c3c95c3c16c19c17c28c21c21c3c16c19c17c25c28c19c3c16c19c17c23c24c25c3c16c19c17c21c20c28c3c14c19c17c19c21c19c3c14c19c17c21c25c20c3c14c19c17c24c19c24c3c14c19c17c26c24c20c3c14c20c17c19c19c19c3c14c20c17c21c24c19 c19c17c22c22c19c3c95c3c16c19c17c28c20c25c3c16c19c17c25c28c21c3c16c19c17c23c25c25c3c16c19c17c21c22c27c3c16c19c17c19c19c26c3c14c19c17c21c21c26c3c14c19c17c23c25c21c3c14c19c17c26c19c19c3c14c19c17c28c23c19c3c14c20c17c20c27c21 c19c17c22c23c19c3c95c3c16c19c17c28c19c24c3c16c19c17c25c27c28c3c16c19c17c23c26c19c3c16c19c17c21c23c28c3c16c19c17c19c21c25c3c14c19c17c21c19c19c3c14c19c17c23c21c27c3c14c19c17c25c24c27c3c14c19c17c27c28c19c3c14c20c17c20c21c23 c19c17c22c24c19c3c95c3c16c19c17c27c27c28c3c16c19c17c25c27c19c3c16c19c17c23c25c27c3c16c19c17c21c24c23c3c16c19c17c19c22c27c3c14c19c17c20c27c20c3c14c19c17c23c19c20c3c14c19c17c25c21c23c3c14c19c17c27c23c28c3c14c20c17c19c26c24 c19c17c22c25c19c3c95c3c16c19c17c27c25c28c3c16c19c17c25c25c25c3c16c19c17c23c25c20c3c16c19c17c21c24c22c3c16c19c17c19c23c23c3c14c19c17c20c25c27c3c14c19c17c22c27c21c3c14c19c17c24c28c26c3c14c19c17c27c20c24c3c14c20c17c19c22c24 c19c17c22c26c19c3c95c3c16c19c17c27c23c24c3c16c19c17c25c23c27c3c16c19c17c23c23c28c3c16c19c17c21c23c27c3c16c19c17c19c23c23c3c14c19c17c20c25c20c3c14c19c17c22c25c27c3c14c19c17c24c26c26c3c14c19c17c26c27c27c3c14c20c17c19c19c20 c19c17c22c27c19c3c95c3c16c19c17c27c20c27c3c16c19c17c25c21c25c3c16c19c17c23c22c22c3c16c19c17c21c22c27c3c16c19c17c19c23c19c3c14c19c17c20c24c28c3c14c19c17c22c25c19c3c14c19c17c24c25c22c3c14c19c17c26c25c27c3c14c19c17c28c26c24 c19c17c22c28c19c3c95c3c16c19c17c26c27c26c3c16c19c17c25c19c20c3c16c19c17c23c20c22c3c16c19c17c21c21c23c3c16c19c17c19c22c21c3c14c19c17c20c25c21c3c14c19c17c22c24c26c3c14c19c17c24c24c23c3c14c19c17c26c24c22c3c14c19c17c28c24c23 c19c17c23c19c19c3c95c3c16c19c17c26c24c23c3c16c19c17c24c26c22c3c16c19c17c22c28c20c3c16c19c17c21c19c25c3c16c19c17c19c21c19c3c14c19c17c20c25c27c3c14c19c17c22c24c27c3c14c19c17c24c24c19c3c14c19c17c26c23c23c3c14c19c17c28c22c28 c19c17c23c20c19c3c95c3c16c19c17c26c20c28c3c16c19c17c24c23c22c3c16c19c17c22c25c24c3c16c19c17c20c27c24c3c16c19c17c19c19c23c3c14c19c17c20c26c28c3c14c19c17c22c25c23c3c14c19c17c24c24c19c3c14c19c17c26c22c28c3c14c19c17c28c21c27 c19c17c23c21c19c3c95c3c16c19c17c25c27c21c3c16c19c17c24c20c19c3c16c19c17c22c22c26c3c16c19c17c20c25c21c3c14c19c17c19c20c24c3c14c19c17c20c28c22c3c14c19c17c22c26c22c3c14c19c17c24c24c23c3c14c19c17c26c22c27c3c14c19c17c28c21c21 c19c17c23c22c19c3c95c3c16c19c17c25c23c22c3c16c19c17c23c26c24c3c16c19c17c22c19c25c3c16c19c17c20c22c25c3c14c19c17c19c22c25c3c14c19c17c21c20c19c3c14c19c17c22c27c24c3c14c19c17c24c25c21c3c14c19c17c26c23c19c3c14c19c17c28c21c19 c19c17c23c23c19c3c95c3c16c19c17c25c19c21c3c16c19c17c23c22c28c3c16c19c17c21c26c23c3c16c19c17c20c19c27c3c14c19c17c19c25c19c3c14c19c17c21c21c28c3c14c19c17c23c19c19c3c14c19c17c24c26c22c3c14c19c17c26c23c26c3c14c19c17c28c21c21 c19c17c23c24c19c3c95c3c16c19c17c24c25c19c3c16c19c17c23c19c20c3c16c19c17c21c23c19c3c16c19c17c19c26c27c3c14c19c17c19c27c25c3c14c19c17c21c24c20c3c14c19c17c23c20c27c3c14c19c17c24c27c25c3c14c19c17c26c24c25c3c14c19c17c28c21c26 c19c17c23c25c19c3c95c3c16c19c17c24c20c25c3c16c19c17c22c25c20c3c16c19c17c21c19c23c3c16c19c17c19c23c25c3c14c19c17c20c20c23c3c14c19c17c21c26c25c3c14c19c17c23c22c27c3c14c19c17c25c19c21c3c14c19c17c26c25c27c3c14c19c17c28c22c24 c19c17c23c26c19c3c95c3c16c19c17c23c26c21c3c16c19c17c22c21c19c3c16c19c17c20c25c26c3c16c19c17c19c20c21c3c14c19c17c20c23c23c3c14c19c17c22c19c21c3c14c19c17c23c25c20c3c14c19c17c25c21c20c3c14c19c17c26c27c22c3c14c19c17c28c23c25 c19c17c23c27c19c3c95c3c16c19c17c23c21c26c3c16c19c17c21c26c27c3c16c19c17c20c21c27c3c14c19c17c19c21c22c3c14c19c17c20c26c25c3c14c19c17c22c22c19c3c14c19c17c23c27c24c3c14c19c17c25c23c21c3c14c19c17c27c19c19c3c14c19c17c28c24c28 c19c17c23c28c19c3c95c3c16c19c17c22c27c20c3c16c19c17c21c22c24c3c16c19c17c19c27c28c3c14c19c17c19c24c28c3c14c19c17c21c19c28c3c14c19c17c22c24c28c3c14c19c17c24c20c20c3c14c19c17c25c25c23c3c14c19c17c27c20c28c3c14c19c17c28c26c23 c19c17c24c19c19c3c95c3c16c19c17c22c22c23c3c16c19c17c20c28c21c3c16c19c17c19c23c27c3c14c19c17c19c28c26c3c14c19c17c21c23c22c3c14c19c17c22c28c19c3c14c19c17c24c22c28c3c14c19c17c25c27c27c3c14c19c17c27c22c28c3c14c19c17c28c28c21 c19c17c24c20c19c3c95c3c16c19c17c21c27c26c3c16c19c17c20c23c26c3c16c19c17c19c19c26c3c14c19c17c20c22c24c3c14c19c17c21c26c27c3c14c19c17c23c21c21c3c14c19c17c24c25c27c3c14c19c17c26c20c23c3c14c19c17c27c25c21c3c14c20c17c19c20c20 c19c17c24c21c19c3c95c3c16c19c17c21c22c28c3c16c19c17c20c19c21c3c14c19c17c19c22c25c3c14c19c17c20c26c23c3c14c19c17c22c20c23c3c14c19c17c23c24c25c3c14c19c17c24c28c27c3c14c19c17c26c23c20c3c14c19c17c27c27c25c3c14c20c17c19c22c21 c19c17c24c22c19c3c95c3c16c19c17c20c28c19c3c16c19c17c19c24c25c3c14c19c17c19c26c27c3c14c19c17c21c20c23c3c14c19c17c22c24c21c3c14c19c17c23c28c19c3c14c19c17c25c21c28c3c14c19c17c26c26c19c3c14c19c17c28c20c21c3c14c20c17c19c24c24 c19c17c24c23c19c3c95c3c16c19c17c20c23c21c3c16c19c17c19c20c19c3c14c19c17c20c21c21c3c14c19c17c21c24c24c3c14c19c17c22c28c19c3c14c19c17c24c21c24c3c14c19c17c25c25c21c3c14c19c17c27c19c19c3c14c19c17c28c22c28c3c14c20c17c19c26c28 c19c17c24c24c19c3c95c3c16c19c17c19c28c22c3c14c19c17c19c22c25c3c14c19c17c20c25c25c3c14c19c17c21c28c26c3c14c19c17c23c21c28c3c14c19c17c24c25c20c3c14c19c17c25c28c25c3c14c19c17c27c22c20c3c14c19c17c28c25c26c3c14c20c17c20c19c23 c19c17c24c25c19c3c95c3c16c19c17c19c23c22c3c14c19c17c19c27c22c3c14c19c17c21c20c19c3c14c19c17c22c22c28c3c14c19c17c23c25c27c3c14c19c17c24c28c27c3c14c19c17c26c22c19c3c14c19c17c27c25c21c3c14c19c17c28c28c25c3c14c20c17c20c22c20 c19c17c24c26c19c3c95c3c14c19c17c19c19c25c3c14c19c17c20c22c19c3c14c19c17c21c24c24c3c14c19c17c22c27c20c3c14c19c17c24c19c27c3c14c19c17c25c22c25c3c14c19c17c26c25c24c3c14c19c17c27c28c24c3c14c20c17c19c21c25c3c14c20c17c20c24c27 c19c17c24c27c19c3c95c3c14c19c17c19c24c25c3c14c19c17c20c26c26c3c14c19c17c22c19c19c3c14c19c17c23c21c23c3c14c19c17c24c23c28c3c14c19c17c25c26c23c3c14c19c17c27c19c20c3c14c19c17c28c21c27c3c14c20c17c19c24c26c3c14c20c17c20c27c26 c19c17c24c28c19c3c95c3c14c19c17c20c19c24c3c14c19c17c21c21c24c3c14c19c17c22c23c25c3c14c19c17c23c25c26c3c14c19c17c24c28c19c3c14c19c17c26c20c22c3c14c19c17c27c22c26c3c14c19c17c28c25c22c3c14c20c17c19c27c28c3c14c20c17c21c20c25 c19c17c25c19c19c3c95c3c14c19c17c20c24c24c3c14c19c17c21c26c22c3c14c19c17c22c28c20c3c14c19c17c24c20c20c3c14c19c17c25c22c20c3c14c19c17c26c24c21c3c14c19c17c27c26c23c3c14c19c17c28c28c26c3c14c20c17c20c21c20c3c14c20c17c21c23c25 c19c17c25c20c19c3c95c3c14c19c17c21c19c24c3c14c19c17c22c21c20c3c14c19c17c23c22c26c3c14c19c17c24c24c23c3c14c19c17c25c26c22c3c14c19c17c26c28c21c3c14c19c17c28c20c21c3c14c20c17c19c22c22c3c14c20c17c20c24c23c3c14c20c17c21c26c26 c19c17c25c21c19c3c95c3c14c19c17c21c24c24c3c14c19c17c22c25c27c3c14c19c17c23c27c22c3c14c19c17c24c28c27c3c14c19c17c26c20c24c3c14c19c17c27c22c21c3c14c19c17c28c24c19c3c14c20c17c19c25c27c3c14c20c17c20c27c27c3c14c20c17c22c19c28 c19c17c25c22c19c3c95c3c14c19c17c22c19c24c3c14c19c17c23c20c25c3c14c19c17c24c21c28c3c14c19c17c25c23c21c3c14c19c17c26c24c26c3c14c19c17c27c26c21c3c14c19c17c28c27c27c3c14c20c17c20c19c24c3c14c20c17c21c21c22c3c14c20c17c22c23c20 c19c17c25c23c19c3c95c3c14c19c17c22c24c23c3c14c19c17c23c25c23c3c14c19c17c24c26c24c3c14c19c17c25c27c26c3c14c19c17c26c28c28c3c14c19c17c28c20c21c3c14c20c17c19c21c26c3c14c20c17c20c23c20c3c14c20c17c21c24c26c3c14c20c17c22c26c23 c19c17c25c24c19c3c95c3c14c19c17c23c19c23c3c14c19c17c24c20c21c3c14c19c17c25c21c20c3c14c19c17c26c22c20c3c14c19c17c27c23c21c3c14c19c17c28c24c22c3c14c20c17c19c25c24c3c14c20c17c20c26c28c3c14c20c17c21c28c21c3c14c20c17c23c19c26 c19c17c25c25c19c3c95c3c14c19c17c23c24c22c3c14c19c17c24c25c19c3c14c19c17c25c25c26c3c14c19c17c26c26c25c3c14c19c17c27c27c23c3c14c19c17c28c28c23c3c14c20c17c20c19c24c3c14c20c17c21c20c25c3c14c20c17c22c21c27c3c14c20c17c23c23c20 c19c17c25c26c19c3c95c3c14c19c17c24c19c22c3c14c19c17c25c19c27c3c14c19c17c26c20c23c3c14c19c17c27c21c19c3c14c19c17c28c21c26c3c14c20c17c19c22c24c3c14c20c17c20c23c23c3c14c20c17c21c24c23c3c14c20c17c22c25c23c3c14c20c17c23c26c24 c19c17c25c27c19c3c95c3c14c19c17c24c24c21c3c14c19c17c25c24c25c3c14c19c17c26c25c19c3c14c19c17c27c25c24c3c14c19c17c28c26c19c3c14c20c17c19c26c26c3c14c20c17c20c27c23c3c14c20c17c21c28c21c3c14c20c17c23c19c19c3c14c20c17c24c20c19 c19c17c25c28c19c3c95c3c14c19c17c25c19c20c3c14c19c17c26c19c22c3c14c19c17c27c19c25c3c14c19c17c28c19c28c3c14c20c17c19c20c22c3c14c20c17c20c20c27c3c14c20c17c21c21c22c3c14c20c17c22c22c19c3c14c20c17c23c22c26c3c14c20c17c24c23c23 c19c17c26c19c19c3c95c3c14c19c17c25c24c19c3c14c19c17c26c24c20c3c14c19c17c27c24c21c3c14c19c17c28c24c23c3c14c20c17c19c24c25c3c14c20c17c20c24c28c3c14c20c17c21c25c22c3c14c20c17c22c25c27c3c14c20c17c23c26c22c3c14c20c17c24c26c28 336 c55c68c69c79c72c3c39c23c17c21c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c26c20c19c3c95c3c14c19c17c25c28c28c3c14c19c17c26c28c27c3c14c19c17c27c28c27c3c14c19c17c28c28c27c3c14c20c17c19c28c28c3c14c20c17c21c19c20c3c14c20c17c22c19c22c3c14c20c17c23c19c25c3c14c20c17c24c20c19c3c14c20c17c25c20c24 c19c17c26c21c19c3c95c3c14c19c17c26c23c27c3c14c19c17c27c23c24c3c14c19c17c28c23c23c3c14c20c17c19c23c22c3c14c20c17c20c23c21c3c14c20c17c21c23c22c3c14c20c17c22c23c22c3c14c20c17c23c23c24c3c14c20c17c24c23c26c3c14c20c17c25c24c19 c19c17c26c22c19c3c95c3c14c19c17c26c28c25c3c14c19c17c27c28c22c3c14c19c17c28c28c19c3c14c20c17c19c27c26c3c14c20c17c20c27c24c3c14c20c17c21c27c23c3c14c20c17c22c27c23c3c14c20c17c23c27c23c3c14c20c17c24c27c24c3c3c3c16c16c3c3 c19c17c26c23c19c3c95c3c14c19c17c27c23c24c3c14c19c17c28c23c19c3c14c20c17c19c22c24c3c14c20c17c20c22c20c3c14c20c17c21c21c27c3c14c20c17c22c21c25c3c14c20c17c23c21c23c3c14c20c17c24c21c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c24c19c3c95c3c14c19c17c27c28c22c3c14c19c17c28c27c25c3c14c20c17c19c27c20c3c14c20c17c20c26c25c3c14c20c17c21c26c20c3c14c20c17c22c25c26c3c14c20c17c23c25c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c25c19c3c95c3c14c19c17c28c23c20c3c14c20c17c19c22c22c3c14c20c17c20c21c25c3c14c20c17c21c21c19c3c14c20c17c22c20c23c3c14c20c17c23c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c26c19c3c95c3c14c19c17c28c27c27c3c14c20c17c19c26c28c3c14c20c17c20c26c20c3c14c20c17c21c25c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c27c19c3c95c3c14c20c17c19c22c25c3c14c20c17c20c21c25c3c14c20c17c21c20c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c28c19c3c95c3c14c20c17c19c27c22c3c14c20c17c20c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c19c19c3c95c3c14c20c17c20c22c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 337 c55c68c69c79c72c3c39c23c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c22c20c3c87c82c3c19c17c23c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c19c17c21c25c28c3c3c19c17c21c26c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c25c19c3c95c3c3c19c17c21c25c24c3c3c19c17c21c26c19c3c3c19c17c21c26c24c3c3c19c17c21c27c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c26c19c3c95c3c3c19c17c21c25c21c3c3c19c17c21c25c26c3c3c19c17c21c26c21c3c3c19c17c21c26c26c3c3c19c17c21c27c22c3c3c19c17c21c27c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c27c19c3c95c3c3c19c17c21c25c20c3c3c19c17c21c25c25c3c3c19c17c21c26c20c3c3c19c17c21c26c25c3c3c19c17c21c27c19c3c3c19c17c21c27c24c3c3c19c17c21c28c19c3c3c19c17c21c28c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c28c19c3c95c3c3c19c17c21c25c20c3c3c19c17c21c25c24c3c3c19c17c21c26c19c3c3c19c17c21c26c24c3c3c19c17c21c26c28c3c3c19c17c21c27c23c3c3c19c17c21c27c28c3c3c19c17c21c28c23c3c3c19c17c21c28c27c3c3c19c17c22c19c22 c19c17c20c19c19c3c95c3c3c19c17c21c25c21c3c3c19c17c21c25c25c3c3c19c17c21c26c19c3c3c19c17c21c26c24c3c3c19c17c21c26c28c3c3c19c17c21c27c23c3c3c19c17c21c27c27c3c3c19c17c21c28c22c3c3c19c17c21c28c26c3c3c19c17c22c19c21 c19c17c20c20c19c3c95c3c3c19c17c21c25c22c3c3c19c17c21c25c26c3c3c19c17c21c26c20c3c3c19c17c21c26c24c3c3c19c17c21c27c19c3c3c19c17c21c27c23c3c3c19c17c21c27c27c3c3c19c17c21c28c22c3c3c19c17c21c28c26c3c3c19c17c22c19c20 c19c17c20c21c19c3c95c3c3c19c17c21c25c24c3c3c19c17c21c25c28c3c3c19c17c21c26c22c3c3c19c17c21c26c26c3c3c19c17c21c27c20c3c3c19c17c21c27c24c3c3c19c17c21c27c28c3c3c19c17c21c28c22c3c3c19c17c21c28c26c3c3c19c17c22c19c21 c19c17c20c22c19c3c95c3c3c19c17c21c25c27c3c3c19c17c21c26c21c3c3c19c17c21c26c24c3c3c19c17c21c26c28c3c3c19c17c21c27c22c3c3c19c17c21c27c26c3c3c19c17c21c28c20c3c3c19c17c21c28c23c3c3c19c17c21c28c27c3c3c19c17c22c19c22 c19c17c20c23c19c3c95c3c3c19c17c21c26c21c3c3c19c17c21c26c24c3c3c19c17c21c26c27c3c3c19c17c21c27c21c3c3c19c17c21c27c24c3c3c19c17c21c27c28c3c3c19c17c21c28c21c3c3c19c17c21c28c25c3c3c19c17c22c19c19c3c3c19c17c22c19c23 c19c17c20c24c19c3c95c3c3c19c17c21c26c25c3c3c19c17c21c26c28c3c3c19c17c21c27c21c3c3c19c17c21c27c24c3c3c19c17c21c27c27c3c3c19c17c21c28c20c3c3c19c17c21c28c24c3c3c19c17c21c28c27c3c3c19c17c22c19c21c3c3c19c17c22c19c25 c19c17c20c25c19c3c95c3c3c19c17c21c27c19c3c3c19c17c21c27c22c3c3c19c17c21c27c24c3c3c19c17c21c27c27c3c3c19c17c21c28c20c3c3c19c17c21c28c24c3c3c19c17c21c28c27c3c3c19c17c22c19c20c3c3c19c17c22c19c24c3c3c19c17c22c19c27 c19c17c20c26c19c3c95c3c3c19c17c21c27c24c3c3c19c17c21c27c26c3c3c19c17c21c28c19c3c3c19c17c21c28c21c3c3c19c17c21c28c24c3c3c19c17c21c28c27c3c3c19c17c22c19c20c3c3c19c17c22c19c23c3c3c19c17c22c19c26c3c3c19c17c22c20c20 c19c17c20c27c19c3c95c3c3c19c17c21c28c19c3c3c19c17c21c28c21c3c3c19c17c21c28c23c3c3c19c17c21c28c26c3c3c19c17c21c28c28c3c3c19c17c22c19c21c3c3c19c17c22c19c24c3c3c19c17c22c19c27c3c3c19c17c22c20c20c3c3c19c17c22c20c23 c19c17c20c28c19c3c95c3c3c19c17c21c28c24c3c3c19c17c21c28c26c3c3c19c17c21c28c28c3c3c19c17c22c19c20c3c3c19c17c22c19c23c3c3c19c17c22c19c25c3c3c19c17c22c19c28c3c3c19c17c22c20c21c3c3c19c17c22c20c23c3c3c19c17c22c20c26 c19c17c21c19c19c3c95c3c86c19c17c22c19c20c3c3c19c17c22c19c21c3c3c19c17c22c19c23c3c3c19c17c22c19c25c3c3c19c17c22c19c27c3c3c19c17c22c20c20c3c3c19c17c22c20c22c3c3c19c17c22c20c25c3c3c19c17c22c20c27c3c3c19c17c22c21c20 c19c17c21c20c19c3c95c3c86c19c17c22c19c26c3c3c19c17c22c19c27c3c3c19c17c22c20c19c3c3c19c17c22c20c20c3c3c19c17c22c20c22c3c3c19c17c22c20c24c3c3c19c17c22c20c27c3c3c19c17c22c21c19c3c3c19c17c22c21c21c3c3c19c17c22c21c24 c19c17c21c21c19c3c95c3c86c19c17c22c20c22c3c86c19c17c22c20c23c3c3c19c17c22c20c24c3c3c19c17c22c20c26c3c3c19c17c22c20c28c3c3c19c17c22c21c19c3c3c19c17c22c21c21c3c3c19c17c22c21c24c3c3c19c17c22c21c26c3c3c19c17c22c21c28 c19c17c21c22c19c3c95c3c86c19c17c22c20c28c3c86c19c17c22c21c19c3c3c19c17c22c21c20c3c3c19c17c22c21c21c3c3c19c17c22c21c23c3c3c19c17c22c21c25c3c3c19c17c22c21c26c3c3c19c17c22c21c28c3c3c19c17c22c22c20c3c3c19c17c22c22c23 c19c17c21c23c19c3c95c3c86c19c17c22c21c24c3c86c19c17c22c21c25c3c3c19c17c22c21c26c3c3c19c17c22c21c27c3c3c19c17c22c22c19c3c3c19c17c22c22c20c3c3c19c17c22c22c22c3c3c19c17c22c22c23c3c3c19c17c22c22c25c3c3c19c17c22c22c27 c19c17c21c24c19c3c95c3c86c19c17c22c22c21c3c86c19c17c22c22c21c3c86c19c17c22c22c22c3c3c19c17c22c22c23c3c3c19c17c22c22c24c3c3c19c17c22c22c26c3c3c19c17c22c22c27c3c3c19c17c22c23c19c3c3c19c17c22c23c20c3c3c19c17c22c23c22 c19c17c21c25c19c3c95c3c86c19c17c22c22c27c3c86c19c17c22c22c28c3c86c19c17c22c22c28c3c3c19c17c22c23c19c3c3c19c17c22c23c20c3c3c19c17c22c23c21c3c3c19c17c22c23c23c3c3c19c17c22c23c24c3c3c19c17c22c23c26c3c3c19c17c22c23c27 c19c17c21c26c19c3c95c3c86c19c17c22c23c24c3c86c19c17c22c23c24c3c86c19c17c22c23c25c3c3c19c17c22c23c25c3c3c19c17c22c23c26c3c3c19c17c22c23c27c3c3c19c17c22c23c28c3c3c19c17c22c24c19c3c3c19c17c22c24c21c3c3c19c17c22c24c22 c19c17c21c27c19c3c95c3c86c19c17c22c24c21c3c86c19c17c22c24c21c3c86c19c17c22c24c21c3c86c19c17c22c24c21c3c3c19c17c22c24c22c3c3c19c17c22c24c23c3c3c19c17c22c24c24c3c3c19c17c22c24c25c3c3c19c17c22c24c26c3c3c19c17c22c24c28 c19c17c21c28c19c3c95c3c86c19c17c22c24c28c3c86c19c17c22c24c27c3c86c19c17c22c24c27c3c86c19c17c22c24c28c3c3c19c17c22c24c28c3c3c19c17c22c25c19c3c3c19c17c22c25c20c3c3c19c17c22c25c21c3c3c19c17c22c25c22c3c3c19c17c22c25c23 c19c17c22c19c19c3c95c3c86c19c17c22c25c24c3c86c19c17c22c25c24c3c86c19c17c22c25c24c3c86c19c17c22c25c24c3c3c19c17c22c25c25c3c3c19c17c22c25c25c3c3c19c17c22c25c26c3c3c19c17c22c25c27c3c3c19c17c22c25c28c3c3c19c17c22c26c19 c19c17c22c20c19c3c95c3c86c19c17c22c26c21c3c86c19c17c22c26c21c3c86c19c17c22c26c21c3c86c19c17c22c26c21c3c3c19c17c22c26c21c3c3c19c17c22c26c21c3c3c19c17c22c26c22c3c3c19c17c22c26c23c3c3c19c17c22c26c23c3c3c19c17c22c26c24 c19c17c22c21c19c3c95c3c86c19c17c22c26c28c3c86c19c17c22c26c28c3c86c19c17c22c26c27c3c86c19c17c22c26c27c3c3c19c17c22c26c27c3c3c19c17c22c26c27c3c3c19c17c22c26c28c3c3c19c17c22c27c19c3c3c19c17c22c27c19c3c3c19c17c22c27c20 c19c17c22c22c19c3c95c3c86c19c17c22c27c25c3c86c19c17c22c27c25c3c86c19c17c22c27c24c3c86c19c17c22c27c24c3c86c19c17c22c27c24c3c3c19c17c22c27c24c3c3c19c17c22c27c24c3c3c19c17c22c27c25c3c3c19c17c22c27c25c3c3c19c17c22c27c26 c19c17c22c23c19c3c95c3c86c19c17c22c28c23c3c86c19c17c22c28c22c3c86c19c17c22c28c21c3c86c19c17c22c28c20c3c86c19c17c22c28c20c3c3c19c17c22c28c20c3c3c19c17c22c28c20c3c3c19c17c22c28c21c3c3c19c17c22c28c21c3c3c19c17c22c28c22 c19c17c22c24c19c3c95c3c86c19c17c23c19c20c3c86c19c17c22c28c28c3c86c19c17c22c28c28c3c86c19c17c22c28c27c3c86c19c17c22c28c27c3c3c19c17c22c28c27c3c3c19c17c22c28c27c3c3c19c17c22c28c27c3c3c19c17c22c28c27c3c3c19c17c22c28c28 c19c17c22c25c19c3c95c3c86c19c17c23c19c27c3c86c19c17c23c19c25c3c86c19c17c23c19c24c3c86c19c17c23c19c24c3c86c19c17c23c19c23c3c3c19c17c23c19c23c3c3c19c17c23c19c23c3c3c19c17c23c19c23c3c3c19c17c23c19c23c3c3c19c17c23c19c24 c19c17c22c26c19c3c95c3c86c19c17c23c20c24c3c86c19c17c23c20c22c3c86c19c17c23c20c21c3c86c19c17c23c20c20c3c86c19c17c23c20c20c3c3c19c17c23c20c20c3c3c19c17c23c20c19c3c3c19c17c23c20c19c3c3c19c17c23c20c19c3c3c19c17c23c20c20 c19c17c22c27c19c3c95c3c86c19c17c23c21c21c3c86c19c17c23c21c19c3c86c19c17c23c20c28c3c86c19c17c23c20c27c3c86c19c17c23c20c27c3c3c19c17c23c20c26c3c3c19c17c23c20c26c3c3c19c17c23c20c26c3c3c19c17c23c20c26c3c3c19c17c23c20c26 c19c17c22c28c19c3c95c3c86c19c17c23c21c28c3c86c19c17c23c21c26c3c86c19c17c23c21c25c3c86c19c17c23c21c24c3c86c19c17c23c21c23c3c3c19c17c23c21c23c3c3c19c17c23c21c22c3c3c19c17c23c21c22c3c3c19c17c23c21c22c3c3c19c17c23c21c22 c19c17c23c19c19c3c95c3c86c19c17c23c22c25c3c86c19c17c23c22c23c3c86c19c17c23c22c22c3c86c19c17c23c22c21c3c86c19c17c23c22c20c3c3c19c17c23c22c19c3c3c19c17c23c22c19c3c3c19c17c23c21c28c3c3c19c17c23c21c28c3c3c19c17c23c21c28 c19c17c23c20c19c3c95c3c86c19c17c23c23c22c3c86c19c17c23c23c20c3c86c19c17c23c23c19c3c86c19c17c23c22c27c3c86c19c17c23c22c26c3c3c19c17c23c22c26c3c3c19c17c23c22c25c3c3c19c17c23c22c25c3c3c19c17c23c22c24c3c3c19c17c23c22c24 c19c17c23c21c19c3c95c3c86c19c17c23c24c19c3c86c19c17c23c23c27c3c86c19c17c23c23c26c3c86c19c17c23c23c24c3c3c19c17c23c23c23c3c3c19c17c23c23c22c3c3c19c17c23c23c21c3c3c19c17c23c23c21c3c3c19c17c23c23c21c3c3c19c17c23c23c20 c19c17c23c22c19c3c95c3c86c19c17c23c24c26c3c86c19c17c23c24c24c3c86c19c17c23c24c22c3c86c19c17c23c24c21c3c3c19c17c23c24c20c3c3c19c17c23c24c19c3c3c19c17c23c23c28c3c3c19c17c23c23c27c3c3c19c17c23c23c27c3c3c19c17c23c23c27 c19c17c23c23c19c3c95c3c86c19c17c23c25c23c3c86c19c17c23c25c21c3c86c19c17c23c25c19c3c86c19c17c23c24c28c3c3c19c17c23c24c26c3c3c19c17c23c24c25c3c3c19c17c23c24c24c3c3c19c17c23c24c24c3c3c19c17c23c24c23c3c3c19c17c23c24c23 c19c17c23c24c19c3c95c3c86c19c17c23c26c20c3c86c19c17c23c25c28c3c86c19c17c23c25c26c3c86c19c17c23c25c24c3c3c19c17c23c25c23c3c3c19c17c23c25c22c3c3c19c17c23c25c21c3c3c19c17c23c25c20c3c3c19c17c23c25c20c3c3c19c17c23c25c19 c19c17c23c25c19c3c95c3c86c19c17c23c26c27c3c86c19c17c23c26c25c3c86c19c17c23c26c23c3c86c19c17c23c26c21c3c3c19c17c23c26c20c3c3c19c17c23c25c28c3c3c19c17c23c25c27c3c3c19c17c23c25c27c3c3c19c17c23c25c26c3c3c19c17c23c25c25 c19c17c23c26c19c3c95c3c86c19c17c23c27c24c3c86c19c17c23c27c22c3c86c19c17c23c27c20c3c86c19c17c23c26c28c3c3c19c17c23c26c26c3c3c19c17c23c26c25c3c3c19c17c23c26c24c3c3c19c17c23c26c23c3c3c19c17c23c26c22c3c3c19c17c23c26c22 c19c17c23c27c19c3c95c3c86c19c17c23c28c21c3c86c19c17c23c27c28c3c86c19c17c23c27c26c3c3c19c17c23c27c24c3c3c19c17c23c27c23c3c3c19c17c23c27c22c3c3c19c17c23c27c20c3c3c19c17c23c27c19c3c3c19c17c23c27c19c3c3c19c17c23c26c28 c19c17c23c28c19c3c95c3c86c19c17c23c28c28c3c86c19c17c23c28c25c3c86c19c17c23c28c23c3c3c19c17c23c28c21c3c3c19c17c23c28c19c3c3c19c17c23c27c28c3c3c19c17c23c27c27c3c3c19c17c23c27c26c3c3c19c17c23c27c25c3c3c19c17c23c27c24 c19c17c24c19c19c3c95c3c86c19c17c24c19c24c3c86c19c17c24c19c22c3c86c19c17c24c19c20c3c3c19c17c23c28c28c3c3c19c17c23c28c26c3c3c19c17c23c28c25c3c3c19c17c23c28c23c3c3c19c17c23c28c22c3c3c19c17c23c28c21c3c3c19c17c23c28c20 c19c17c24c20c19c3c95c3c86c19c17c24c20c21c3c86c19c17c24c20c19c3c86c19c17c24c19c26c3c3c19c17c24c19c24c3c3c19c17c24c19c23c3c3c19c17c24c19c21c3c3c19c17c24c19c20c3c3c19c17c24c19c19c3c3c19c17c23c28c28c3c3c19c17c23c28c27 c19c17c24c21c19c3c95c3c86c19c17c24c20c28c3c86c19c17c24c20c25c3c3c19c17c24c20c23c3c3c19c17c24c20c21c3c3c19c17c24c20c19c3c3c19c17c24c19c28c3c3c19c17c24c19c26c3c3c19c17c24c19c25c3c3c19c17c24c19c24c3c3c19c17c24c19c23 c19c17c24c22c19c3c95c3c86c19c17c24c21c25c3c86c19c17c24c21c22c3c3c19c17c24c21c20c3c3c19c17c24c20c28c3c3c19c17c24c20c26c3c3c19c17c24c20c24c3c3c19c17c24c20c23c3c3c19c17c24c20c21c3c3c19c17c24c20c20c3c3c19c17c24c20c19 c19c17c24c23c19c3c95c3c86c19c17c24c22c21c3c86c19c17c24c22c19c3c3c19c17c24c21c26c3c3c19c17c24c21c24c3c3c19c17c24c21c22c3c3c19c17c24c21c20c3c3c19c17c24c21c19c3c3c19c17c24c20c28c3c3c19c17c24c20c27c3c3c19c17c24c20c25 c19c17c24c24c19c3c95c3c86c19c17c24c22c28c3c3c19c17c24c22c25c3c3c19c17c24c22c23c3c3c19c17c24c22c21c3c3c19c17c24c22c19c3c3c19c17c24c21c27c3c3c19c17c24c21c25c3c3c19c17c24c21c24c3c3c19c17c24c21c23c3c3c19c17c24c21c22 c19c17c24c25c19c3c95c3c86c19c17c24c23c24c3c3c19c17c24c23c22c3c3c19c17c24c23c19c3c3c19c17c24c22c27c3c3c19c17c24c22c25c3c3c19c17c24c22c23c3c3c19c17c24c22c22c3c3c19c17c24c22c20c3c3c19c17c24c22c19c3c3c19c17c24c21c28 c19c17c24c26c19c3c95c3c3c19c17c24c24c21c3c3c19c17c24c23c28c3c3c19c17c24c23c26c3c3c19c17c24c23c24c3c3c19c17c24c23c22c3c3c19c17c24c23c20c3c3c19c17c24c22c28c3c3c19c17c24c22c27c3c3c19c17c24c22c25c3c3c19c17c24c22c24 c19c17c24c27c19c3c95c3c3c19c17c24c24c28c3c3c19c17c24c24c25c3c3c19c17c24c24c22c3c3c19c17c24c24c20c3c3c19c17c24c23c28c3c3c19c17c24c23c26c3c3c19c17c24c23c24c3c3c19c17c24c23c23c3c3c19c17c24c23c22c3c3c19c17c24c23c20 c19c17c24c28c19c3c95c3c3c19c17c24c25c24c3c3c19c17c24c25c21c3c3c19c17c24c25c19c3c3c19c17c24c24c26c3c3c19c17c24c24c24c3c3c19c17c24c24c22c3c3c19c17c24c24c21c3c3c19c17c24c24c19c3c3c19c17c24c23c28c3c3c19c17c24c23c27 c19c17c25c19c19c3c95c3c3c19c17c24c26c21c3c3c19c17c24c25c28c3c3c19c17c24c25c25c3c3c19c17c24c25c23c3c3c19c17c24c25c21c3c3c19c17c24c25c19c3c3c19c17c24c24c27c3c3c19c17c24c24c26c3c3c19c17c24c24c24c3c3c19c17c24c24c23 c19c17c25c20c19c3c95c3c3c19c17c24c26c27c3c3c19c17c24c26c24c3c3c19c17c24c26c22c3c3c19c17c24c26c19c3c3c19c17c24c25c27c3c3c19c17c24c25c25c3c3c19c17c24c25c23c3c3c19c17c24c25c22c3c3c19c17c24c25c20c3c3c19c17c24c25c19 c19c17c25c21c19c3c95c3c3c19c17c24c27c23c3c3c19c17c24c27c21c3c3c19c17c24c26c28c3c3c19c17c24c26c26c3c3c19c17c24c26c23c3c3c19c17c24c26c21c3c3c19c17c24c26c20c3c3c19c17c24c25c28c3c3c19c17c24c25c27c3c3c19c17c24c25c25 c19c17c25c22c19c3c95c3c3c19c17c24c28c20c3c3c19c17c24c27c27c3c3c19c17c24c27c24c3c3c19c17c24c27c22c3c3c19c17c24c27c20c3c3c19c17c24c26c28c3c3c19c17c24c26c26c3c3c19c17c24c26c24c3c3c19c17c24c26c23c3c3c19c17c24c26c21 c19c17c25c23c19c3c95c3c3c19c17c24c28c26c3c3c19c17c24c28c23c3c3c19c17c24c28c21c3c3c19c17c24c27c28c3c3c19c17c24c27c26c3c3c19c17c24c27c24c3c3c19c17c24c27c22c3c3c19c17c24c27c20c3c3c19c17c24c27c19c3c3c19c17c24c26c27 c19c17c25c24c19c3c95c3c3c19c17c25c19c22c3c3c19c17c25c19c20c3c3c19c17c24c28c27c3c3c19c17c24c28c25c3c3c19c17c24c28c22c3c3c19c17c24c28c20c3c3c19c17c24c27c28c3c3c19c17c24c27c27c3c3c19c17c24c27c25c3c3c19c17c24c27c24 c19c17c25c25c19c3c95c3c3c19c17c25c20c19c3c3c19c17c25c19c26c3c3c19c17c25c19c23c3c3c19c17c25c19c21c3c3c19c17c25c19c19c3c3c19c17c24c28c26c3c3c19c17c24c28c25c3c3c19c17c24c28c23c3c3c19c17c24c28c21c3c3c19c17c24c28c20 c19c17c25c26c19c3c95c3c3c19c17c25c20c25c3c3c19c17c25c20c22c3c3c19c17c25c20c19c3c3c19c17c25c19c27c3c3c19c17c25c19c25c3c3c19c17c25c19c23c3c3c19c17c25c19c21c3c3c19c17c25c19c19c3c3c19c17c24c28c27c3c3c19c17c24c28c26 c19c17c25c27c19c3c95c3c3c19c17c25c21c21c3c3c19c17c25c20c28c3c3c19c17c25c20c26c3c3c19c17c25c20c23c3c3c19c17c25c20c21c3c3c19c17c25c20c19c3c3c19c17c25c19c27c3c3c19c17c25c19c25c3c3c19c17c25c19c23c3c3c19c17c25c19c22 c19c17c25c28c19c3c95c3c3c19c17c25c21c27c3c3c19c17c25c21c25c3c3c19c17c25c21c22c3c3c19c17c25c21c19c3c3c19c17c25c20c27c3c3c19c17c25c20c25c3c3c19c17c25c20c23c3c3c19c17c25c20c21c3c3c19c17c25c20c19c3c3c19c17c25c19c28 c19c17c26c19c19c3c95c3c3c19c17c25c22c24c3c3c19c17c25c22c21c3c3c19c17c25c21c28c3c3c19c17c25c21c26c3c3c19c17c25c21c23c3c3c19c17c25c21c21c3c3c19c17c25c21c19c3c3c19c17c25c20c27c3c3c19c17c25c20c26c3c3c19c17c25c20c24 338 c55c68c69c79c72c3c39c23c17c22c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c26c20c19c3c95c3c3c19c17c25c23c20c3c3c19c17c25c22c27c3c3c19c17c25c22c24c3c3c19c17c25c22c22c3c3c19c17c25c22c19c3c3c19c17c25c21c27c3c3c19c17c25c21c25c3c3c19c17c25c21c23c3c3c19c17c25c21c22c3c3c19c17c25c21c20 c19c17c26c21c19c3c95c3c3c19c17c25c23c26c3c3c19c17c25c23c23c3c3c19c17c25c23c20c3c3c19c17c25c22c28c3c3c19c17c25c22c25c3c3c19c17c25c22c23c3c3c19c17c25c22c21c3c3c19c17c25c22c19c3c3c19c17c25c21c28c3c3c19c17c25c21c26 c19c17c26c22c19c3c95c3c3c19c17c25c24c22c3c3c19c17c25c24c19c3c3c19c17c25c23c26c3c3c19c17c25c23c24c3c3c19c17c25c23c22c3c3c19c17c25c23c19c3c3c19c17c25c22c27c3c3c19c17c25c22c25c3c3c19c17c25c22c24c3c3c3c16c16c3c3 c19c17c26c23c19c3c95c3c3c19c17c25c24c28c3c3c19c17c25c24c25c3c3c19c17c25c24c22c3c3c19c17c25c24c20c3c3c19c17c25c23c28c3c3c19c17c25c23c25c3c3c19c17c25c23c23c3c3c19c17c25c23c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c24c19c3c95c3c3c19c17c25c25c24c3c3c19c17c25c25c21c3c3c19c17c25c24c28c3c3c19c17c25c24c26c3c3c19c17c25c24c24c3c3c19c17c25c24c21c3c3c19c17c25c24c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c25c19c3c95c3c3c19c17c25c26c20c3c3c19c17c25c25c27c3c3c19c17c25c25c25c3c3c19c17c25c25c22c3c3c19c17c25c25c20c3c3c19c17c25c24c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c26c19c3c95c3c3c19c17c25c26c26c3c3c19c17c25c26c23c3c3c19c17c25c26c21c3c3c19c17c25c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c27c19c3c95c3c3c19c17c25c27c22c3c3c19c17c25c27c19c3c3c19c17c25c26c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c28c19c3c95c3c3c19c17c25c27c28c3c3c19c17c25c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c19c19c3c95c3c3c19c17c25c28c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 339 c55c68c69c79c72c3c39c23c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c22c20c3c87c82c3c19c17c23c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c23c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c24c19c3c95c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c25c19c3c95c3c3c19c17c20c26c23c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c26c19c3c95c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c27c19c3c95c3c3c19c17c20c25c27c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c25c3c3c19c17c20c27c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c19c28c19c3c95c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c25c3c3c19c17c20c27c28c3c3c19c17c20c28c21 c19c17c20c19c19c3c95c3c3c19c17c20c25c24c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c23c3c3c19c17c20c27c26c3c3c19c17c20c27c28 c19c17c20c20c19c3c95c3c3c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c24c3c3c19c17c20c27c26 c19c17c20c21c19c3c95c3c3c19c17c20c25c22c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c25 c19c17c20c22c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c20c23c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c20c24c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c20c25c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c20c26c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c20c27c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c20c28c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c24c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c19c19c3c95c3c86c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c20c19c3c95c3c86c19c17c20c25c20c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c21c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c22c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c23c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c3c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c24c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c25c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c26c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c27c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c21c28c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19 c19c17c22c19c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c19 c19c17c22c20c19c3c95c3c86c19c17c20c25c21c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c21c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c22c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c23c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c24c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c25c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20 c19c17c22c26c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c20 c19c17c22c27c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c23c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c22c28c19c3c95c3c86c19c17c20c25c20c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c19c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c20c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c86c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c21c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c22c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c23c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21 c19c17c23c24c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22 c19c17c23c25c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c23c26c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c86c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c23c27c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c23c28c19c3c95c3c86c19c17c20c25c19c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c19c19c3c95c3c86c19c17c20c24c28c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c20c19c3c95c3c86c19c17c20c24c28c3c86c19c17c20c25c22c3c86c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c21c19c3c95c3c86c19c17c20c24c28c3c86c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c22c19c3c95c3c86c19c17c20c24c28c3c86c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c23c19c3c95c3c86c19c17c20c24c28c3c86c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22 c19c17c24c24c19c3c95c3c86c19c17c20c24c28c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c23 c19c17c24c25c19c3c95c3c86c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c23 c19c17c24c26c19c3c95c3c3c19c17c20c24c28c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c24c27c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c24c28c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c25c19c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c25c20c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c25c21c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c25c22c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c21c3c3c19c17c20c27c23 c19c17c25c23c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c22c3c3c19c17c20c25c26c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c26c3c3c19c17c20c27c19c3c3c19c17c20c27c21c3c3c19c17c20c27c24 c19c17c25c24c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c21c3c3c19c17c20c27c24 c19c17c25c25c19c3c95c3c3c19c17c20c25c19c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c21c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c24 c19c17c25c26c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c24 c19c17c25c27c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c24c3c3c19c17c20c26c27c3c3c19c17c20c27c19c3c3c19c17c20c27c22c3c3c19c17c20c27c24 c19c17c25c28c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22c3c3c19c17c20c27c24 c19c17c26c19c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c23c3c3c19c17c20c25c26c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22c3c3c19c17c20c27c24 340 c55c68c69c79c72c3c39c23c17c23c3c70c82c81c87c76c81c88c72c71 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c57c3c3c3c95c3c3c19c17c22c20c19c3c3c19c17c22c21c19c3c3c19c17c22c22c19c3c3c19c17c22c23c19c3c3c19c17c22c24c19c3c3c19c17c22c25c19c3c3c19c17c22c26c19c3c3c19c17c22c27c19c3c3c19c17c22c28c19c3c3c19c17c23c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c26c20c19c3c95c3c3c19c17c20c25c20c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c19c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c27c3c3c19c17c20c27c20c3c3c19c17c20c27c22c3c3c19c17c20c27c25 c19c17c26c21c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c20c3c3c19c17c20c26c22c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c22c3c3c19c17c20c27c25 c19c17c26c22c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c19c17c20c27c23c3c3c3c16c16c3c3 c19c17c26c23c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c25c3c3c19c17c20c26c28c3c3c19c17c20c27c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c24c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c24c3c3c19c17c20c25c27c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c19c17c20c26c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c25c19c3c95c3c3c19c17c20c25c21c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c20c3c3c19c17c20c26c23c3c3c19c17c20c26c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c26c19c3c95c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c19c17c20c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c27c19c3c95c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c19c17c20c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c28c19c3c95c3c3c19c17c20c25c22c3c3c19c17c20c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c19c19c3c95c3c3c19c17c20c25c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c27c20c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 341 c55c68c69c79c72c3c39c24c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c23c20c3c87c82c3c19c17c24c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c23c20c19c3c3c19c17c23c21c19c3c3c19c17c23c22c19c3c3c19c17c23c23c19c3c3c19c17c23c24c19c3c3c19c17c23c25c19c3c3c19c17c23c26c19c3c3c19c17c23c27c19c3c3c19c17c23c28c19c3c3c19c17c24c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c19c19c3c95c3c3c20c20c17c25c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c20c19c3c95c3c3c20c19c17c24c28c3c3c20c19c17c28c27c3c3c20c20c17c22c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c21c19c3c95c3c3c28c17c25c27c20c3c3c20c19c17c19c25c3c3c20c19c17c23c23c3c3c20c19c17c27c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c22c19c3c95c3c3c27c17c27c25c26c3c3c28c17c21c23c22c3c3c28c17c25c20c22c3c3c28c17c28c26c27c3c3c20c19c17c22c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c23c19c3c95c3c3c27c17c20c23c21c3c3c27c17c24c19c25c3c3c27c17c27c25c26c3c3c28c17c21c21c22c3c3c28c17c24c26c23c3c3c28c17c28c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c24c19c3c95c3c3c26c17c23c28c23c3c3c26c17c27c23c26c3c3c27c17c20c28c26c3c3c27c17c24c23c21c3c3c27c17c27c27c23c3c3c28c17c21c21c22c3c3c28c17c24c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c25c19c3c95c3c3c25c17c28c20c24c3c3c26c17c21c24c25c3c3c26c17c24c28c23c3c3c26c17c28c21c28c3c3c27c17c21c25c21c3c3c27c17c24c28c20c3c3c27c17c28c20c26c3c3c28c17c21c22c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c26c19c3c95c3c3c25c17c22c28c26c3c3c25c17c26c21c24c3c3c26c17c19c24c21c3c3c26c17c22c26c26c3c3c26c17c25c28c28c3c3c27c17c19c20c27c3c3c27c17c22c22c24c3c3c27c17c25c23c28c3c3c27c17c28c25c19c3c3c3c16c16c3c3 c19c17c20c27c19c3c95c3c3c24c17c28c22c22c3c3c25c17c21c23c28c3c3c25c17c24c25c23c3c3c25c17c27c26c27c3c3c26c17c20c27c28c3c3c26c17c23c28c28c3c3c26c17c27c19c26c3c3c27c17c20c20c21c3c3c27c17c23c20c24c3c3c27c17c26c20c24 c19c17c20c28c19c3c95c3c3c24c17c24c20c27c3c3c24c17c27c21c21c3c3c25c17c20c21c24c3c3c25c17c23c21c26c3c3c25c17c26c21c27c3c3c26c17c19c21c27c3c3c26c17c22c21c25c3c3c26c17c25c21c21c3c3c26c17c28c20c25c3c3c27c17c21c19c28 c19c17c21c19c19c3c95c3c3c24c17c20c23c24c3c3c24c17c23c22c26c3c3c24c17c26c21c28c3c3c25c17c19c21c19c3c3c25c17c22c20c19c3c3c25c17c25c19c19c3c3c25c17c27c27c27c3c3c26c17c20c26c25c3c3c26c17c23c25c20c3c3c26c17c26c23c24 c19c17c21c20c19c3c95c3c3c23c17c27c20c19c3c3c24c17c19c28c20c3c3c24c17c22c26c20c3c3c24c17c25c24c21c3c3c24c17c28c22c21c3c3c25c17c21c20c20c3c3c25c17c23c28c19c3c3c25c17c26c25c27c3c3c26c17c19c23c24c3c3c26c17c22c21c20 c19c17c21c21c19c3c95c3c3c23c17c24c20c19c3c3c23c17c26c26c28c3c3c24c17c19c23c28c3c3c24c17c22c20c28c3c3c24c17c24c27c27c3c3c24c17c27c24c27c3c3c25c17c20c21c27c3c3c25c17c22c28c25c3c3c25c17c25c25c24c3c3c25c17c28c22c21 c19c17c21c22c19c3c95c3c3c23c17c21c23c19c3c3c23c17c23c28c27c3c3c23c17c26c24c26c3c3c24c17c19c20c26c3c3c24c17c21c26c26c3c3c24c17c24c22c26c3c3c24c17c26c28c26c3c3c25c17c19c24c26c3c3c25c17c22c20c25c3c3c25c17c24c26c24 c19c17c21c23c19c3c95c3c3c22c17c28c28c25c3c3c23c17c21c23c24c3c3c23c17c23c28c23c3c3c23c17c26c23c22c3c3c23c17c28c28c23c3c3c24c17c21c23c23c3c3c24c17c23c28c24c3c3c24c17c26c23c26c3c3c24c17c28c28c26c3c3c25c17c21c23c27 c19c17c21c24c19c3c95c3c3c22c17c26c26c26c3c3c23c17c19c20c24c3c3c23c17c21c24c24c3c3c23c17c23c28c24c3c3c23c17c26c22c25c3c3c23c17c28c26c27c3c3c24c17c21c21c19c3c3c24c17c23c25c22c3c3c24c17c26c19c24c3c3c24c17c28c23c27 c19c17c21c25c19c3c95c3c3c22c17c24c26c28c3c3c22c17c27c19c27c3c3c23c17c19c22c27c3c3c23c17c21c26c19c3c3c23c17c24c19c21c3c3c23c17c26c22c24c3c3c23c17c28c25c28c3c3c24c17c21c19c22c3c3c24c17c23c22c27c3c3c24c17c25c26c22 c19c17c21c26c19c3c95c3c3c22c17c23c19c19c3c3c22c17c25c21c19c3c3c22c17c27c23c21c3c3c23c17c19c25c24c3c3c23c17c21c27c28c3c3c23c17c24c20c23c3c3c23c17c26c22c28c3c3c23c17c28c25c25c3c3c24c17c20c28c21c3c3c24c17c23c21c19 c19c17c21c27c19c3c95c3c3c22c17c21c22c27c3c3c22c17c23c24c19c3c3c22c17c25c25c23c3c3c22c17c27c26c28c3c3c23c17c19c28c23c3c3c23c17c22c20c20c3c3c23c17c24c21c28c3c3c23c17c26c23c27c3c3c23c17c28c25c26c3c3c24c17c20c27c26 c19c17c21c28c19c3c95c3c3c22c17c19c28c20c3c3c22c17c21c28c25c3c3c22c17c24c19c21c3c3c22c17c26c19c28c3c3c22c17c28c20c26c3c3c23c17c20c21c25c3c3c23c17c22c22c26c3c3c23c17c24c23c27c3c3c23c17c26c25c19c3c3c23c17c28c26c22 c19c17c22c19c19c3c95c3c3c21c17c28c24c27c3c3c22c17c20c24c24c3c3c22c17c22c24c23c3c3c22c17c24c24c23c3c3c22c17c26c24c24c3c3c22c17c28c24c26c3c3c23c17c20c25c19c3c3c23c17c22c25c24c3c3c23c17c24c26c19c3c3c23c17c26c26c25 c19c17c22c20c19c3c95c3c3c21c17c27c22c27c3c3c22c17c19c21c27c3c3c22c17c21c21c19c3c3c22c17c23c20c21c3c3c22c17c25c19c26c3c3c22c17c27c19c21c3c3c22c17c28c28c28c3c3c23c17c20c28c25c3c3c23c17c22c28c24c3c3c23c17c24c28c23 c19c17c22c21c19c3c95c3c3c21c17c26c21c27c3c3c21c17c28c20c21c3c3c22c17c19c28c26c3c3c22c17c21c27c22c3c3c22c17c23c26c20c3c3c22c17c25c25c19c3c3c22c17c27c24c19c3c3c23c17c19c23c20c3c3c23c17c21c22c22c3c3c23c17c23c21c26 c19c17c22c22c19c3c95c3c3c21c17c25c21c27c3c3c21c17c27c19c25c3c3c21c17c28c27c24c3c3c22c17c20c25c24c3c3c22c17c22c23c26c3c3c22c17c24c22c19c3c3c22c17c26c20c23c3c3c22c17c27c28c28c3c3c23c17c19c27c24c3c3c23c17c21c26c21 c19c17c22c23c19c3c95c3c3c21c17c24c22c27c3c3c21c17c26c20c19c3c3c21c17c27c27c22c3c3c22c17c19c24c26c3c3c22c17c21c22c22c3c3c22c17c23c20c19c3c3c22c17c24c27c27c3c3c22c17c26c25c27c3c3c22c17c28c23c27c3c3c23c17c20c21c28 c19c17c22c24c19c3c95c3c3c21c17c23c24c24c3c3c21c17c25c21c21c3c3c21c17c26c27c28c3c3c21c17c28c24c27c3c3c22c17c20c21c28c3c3c22c17c22c19c19c3c3c22c17c23c26c22c3c3c22c17c25c23c26c3c3c22c17c27c21c21c3c3c22c17c28c28c27 c19c17c22c25c19c3c95c3c3c21c17c22c27c19c3c3c21c17c24c23c20c3c3c21c17c26c19c23c3c3c21c17c27c25c27c3c3c22c17c19c22c22c3c3c22c17c20c28c28c3c3c22c17c22c25c26c3c3c22c17c24c22c24c3c3c22c17c26c19c24c3c3c22c17c27c26c25 c19c17c22c26c19c3c95c3c3c21c17c22c20c21c3c3c21c17c23c25c27c3c3c21c17c25c21c25c3c3c21c17c26c27c24c3c3c21c17c28c23c24c3c3c22c17c20c19c25c3c3c22c17c21c25c28c3c3c22c17c23c22c21c3c3c22c17c24c28c26c3c3c22c17c26c25c22 c19c17c22c27c19c3c95c3c3c21c17c21c24c19c3c3c21c17c23c19c20c3c3c21c17c24c24c23c3c3c21c17c26c19c28c3c3c21c17c27c25c23c3c3c22c17c19c21c20c3c3c22c17c20c26c28c3c3c22c17c22c22c26c3c3c22c17c23c28c26c3c3c22c17c25c24c27 c19c17c22c28c19c3c95c3c3c21c17c20c28c22c3c3c21c17c22c23c19c3c3c21c17c23c27c28c3c3c21c17c25c22c28c3c3c21c17c26c28c19c3c3c21c17c28c23c21c3c3c22c17c19c28c24c3c3c22c17c21c24c19c3c3c22c17c23c19c24c3c3c22c17c24c25c21 c19c17c23c19c19c3c95c3c3c21c17c20c23c20c3c3c21c17c21c27c23c3c3c21c17c23c21c28c3c3c21c17c24c26c24c3c3c21c17c26c21c20c3c3c21c17c27c25c28c3c3c22c17c19c20c27c3c3c22c17c20c25c28c3c3c22c17c22c21c19c3c3c22c17c23c26c21 c19c17c23c20c19c3c95c3c3c21c17c19c28c23c3c3c21c17c21c22c23c3c3c21c17c22c26c23c3c3c21c17c24c20c25c3c3c21c17c25c24c28c3c3c21c17c27c19c22c3c3c21c17c28c23c27c3c3c22c17c19c28c23c3c3c22c17c21c23c20c3c3c22c17c22c27c28 c19c17c23c21c19c3c95c3c3c21c17c19c24c20c3c3c21c17c20c27c26c3c3c21c17c22c21c23c3c3c21c17c23c25c21c3c3c21c17c25c19c20c3c3c21c17c26c23c20c3c3c21c17c27c27c21c3c3c22c17c19c21c23c3c3c22c17c20c25c26c3c3c22c17c22c20c21 c19c17c23c22c19c3c95c3c3c21c17c19c20c22c3c3c21c17c20c23c24c3c3c21c17c21c26c27c3c3c21c17c23c20c21c3c3c21c17c24c23c27c3c3c21c17c25c27c23c3c3c21c17c27c21c21c3c3c21c17c28c25c19c3c3c22c17c20c19c19c3c3c22c17c21c23c19 c19c17c23c23c19c3c95c3c3c20c17c28c26c26c3c3c21c17c20c19c25c3c3c21c17c21c22c25c3c3c21c17c22c25c26c3c3c21c17c23c28c28c3c3c21c17c25c22c21c3c3c21c17c26c25c25c3c3c21c17c28c19c20c3c3c22c17c19c22c26c3c3c22c17c20c26c23 c19c17c23c24c19c3c95c3c3c20c17c28c23c24c3c3c21c17c19c26c20c3c3c21c17c20c28c26c3c3c21c17c22c21c24c3c3c21c17c23c24c23c3c3c21c17c24c27c22c3c3c21c17c26c20c23c3c3c21c17c27c23c25c3c3c21c17c28c26c27c3c3c22c17c20c20c21 c19c17c23c25c19c3c95c3c3c20c17c28c20c25c3c3c21c17c19c22c28c3c3c21c17c20c25c21c3c3c21c17c21c27c26c3c3c21c17c23c20c21c3c3c21c17c24c22c28c3c3c21c17c25c25c25c3c3c21c17c26c28c24c3c3c21c17c28c21c23c3c3c22c17c19c24c24 c19c17c23c26c19c3c95c3c3c20c17c27c28c19c3c3c21c17c19c20c19c3c3c21c17c20c22c19c3c3c21c17c21c24c21c3c3c21c17c22c26c23c3c3c21c17c23c28c27c3c3c21c17c25c21c21c3c3c21c17c26c23c27c3c3c21c17c27c26c23c3c3c22c17c19c19c20 c19c17c23c27c19c3c95c3c3c20c17c27c25c25c3c3c20c17c28c27c22c3c3c21c17c20c19c20c3c3c21c17c21c21c19c3c3c21c17c22c23c19c3c3c21c17c23c25c19c3c3c21c17c24c27c21c3c3c21c17c26c19c23c3c3c21c17c27c21c27c3c3c21c17c28c24c21 c19c17c23c28c19c3c95c3c3c20c17c27c23c24c3c3c20c17c28c24c28c3c3c21c17c19c26c24c3c3c21c17c20c28c20c3c3c21c17c22c19c27c3c3c21c17c23c21c24c3c3c21c17c24c23c23c3c3c21c17c25c25c23c3c3c21c17c26c27c23c3c3c21c17c28c19c25 c19c17c24c19c19c3c95c3c3c20c17c27c21c25c3c3c20c17c28c22c27c3c3c21c17c19c24c19c3c3c21c17c20c25c23c3c3c21c17c21c26c27c3c3c21c17c22c28c23c3c3c21c17c24c20c19c3c3c21c17c25c21c26c3c3c21c17c26c23c23c3c3c21c17c27c25c22 c19c17c24c20c19c3c95c3c3c20c17c27c19c28c3c3c20c17c28c20c27c3c3c21c17c19c21c28c3c3c21c17c20c23c19c3c3c21c17c21c24c20c3c3c21c17c22c25c23c3c3c21c17c23c26c27c3c3c21c17c24c28c21c3c3c21c17c26c19c26c3c3c21c17c27c21c22 c19c17c24c21c19c3c95c3c3c20c17c26c28c23c3c3c20c17c28c19c20c3c3c21c17c19c19c28c3c3c21c17c20c20c26c3c3c21c17c21c21c26c3c3c21c17c22c22c26c3c3c21c17c23c23c27c3c3c21c17c24c25c19c3c3c21c17c25c26c22c3c3c21c17c26c27c26 c19c17c24c22c19c3c95c3c3c20c17c26c27c20c3c3c20c17c27c27c24c3c3c20c17c28c28c20c3c3c21c17c19c28c26c3c3c21c17c21c19c23c3c3c21c17c22c20c21c3c3c21c17c23c21c20c3c3c21c17c24c22c20c3c3c21c17c25c23c20c3c3c21c17c26c24c21 c19c17c24c23c19c3c95c3c3c20c17c26c25c28c3c3c20c17c27c26c21c3c3c20c17c28c26c24c3c3c21c17c19c26c28c3c3c21c17c20c27c23c3c3c21c17c21c28c19c3c3c21c17c22c28c25c3c3c21c17c24c19c23c3c3c21c17c25c20c21c3c3c21c17c26c21c20 c19c17c24c24c19c3c95c3c3c20c17c26c24c28c3c3c20c17c27c24c28c3c3c20c17c28c25c20c3c3c21c17c19c25c22c3c3c21c17c20c25c25c3c3c21c17c21c25c28c3c3c21c17c22c26c23c3c3c21c17c23c26c28c3c3c21c17c24c27c24c3c3c21c17c25c28c20 c19c17c24c25c19c3c95c3c3c20c17c26c24c19c3c3c20c17c27c23c27c3c3c20c17c28c23c27c3c3c21c17c19c23c27c3c3c21c17c20c23c28c3c3c21c17c21c24c19c3c3c21c17c22c24c22c3c3c21c17c23c24c25c3c3c21c17c24c24c28c3c3c21c17c25c25c23 c19c17c24c26c19c3c95c3c3c20c17c26c23c21c3c3c20c17c27c22c28c3c3c20c17c28c22c25c3c3c21c17c19c22c24c3c3c21c17c20c22c23c3c3c21c17c21c22c22c3c3c21c17c22c22c22c3c3c21c17c23c22c24c3c3c21c17c24c22c25c3c3c21c17c25c22c28 c19c17c24c27c19c3c95c3c3c20c17c26c22c25c3c3c20c17c27c22c20c3c3c20c17c28c21c25c3c3c21c17c19c21c22c3c3c21c17c20c21c19c3c3c21c17c21c20c27c3c3c21c17c22c20c25c3c3c21c17c23c20c24c3c3c21c17c24c20c24c3c3c21c17c25c20c24 c19c17c24c28c19c3c95c3c3c20c17c26c22c20c3c3c20c17c27c21c23c3c3c20c17c28c20c27c3c3c21c17c19c20c21c3c3c21c17c20c19c27c3c3c21c17c21c19c22c3c3c21c17c22c19c19c3c3c21c17c22c28c26c3c3c21c17c23c28c24c3c3c21c17c24c28c23 c19c17c25c19c19c3c95c3c3c20c17c26c21c26c3c3c20c17c27c20c27c3c3c20c17c28c20c19c3c3c21c17c19c19c22c3c3c21c17c19c28c26c3c3c21c17c20c28c20c3c3c21c17c21c27c25c3c3c21c17c22c27c20c3c3c21c17c23c26c26c3c3c21c17c24c26c23 c19c17c25c20c19c3c95c3c3c20c17c26c21c23c3c3c20c17c27c20c22c3c3c20c17c28c19c23c3c3c20c17c28c28c24c3c3c21c17c19c27c26c3c3c21c17c20c26c28c3c3c21c17c21c26c21c3c3c21c17c22c25c25c3c3c21c17c23c25c20c3c3c21c17c24c24c25 c19c17c25c21c19c3c95c3c3c20c17c26c21c20c3c3c20c17c27c20c19c3c3c20c17c27c28c28c3c3c20c17c28c27c27c3c3c21c17c19c26c27c3c3c21c17c20c25c28c3c3c21c17c21c25c20c3c3c21c17c22c24c22c3c3c21c17c23c23c24c3c3c3c16c16c3c3 c19c17c25c22c19c3c95c3c3c20c17c26c21c19c3c3c20c17c27c19c26c3c3c20c17c27c28c23c3c3c20c17c28c27c21c3c3c21c17c19c26c20c3c3c21c17c20c25c19c3c3c21c17c21c24c19c3c3c21c17c22c23c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c23c19c3c95c3c3c20c17c26c21c19c3c3c20c17c27c19c24c3c3c20c17c27c28c20c3c3c20c17c28c26c26c3c3c21c17c19c25c23c3c3c21c17c20c24c21c3c3c21c17c21c23c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c24c19c3c95c3c3c20c17c26c21c19c3c3c20c17c27c19c23c3c3c20c17c27c27c27c3c3c20c17c28c26c22c3c3c21c17c19c24c28c3c3c21c17c20c23c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c25c19c3c95c3c3c20c17c26c21c20c3c3c20c17c27c19c22c3c3c20c17c27c27c25c3c3c20c17c28c26c19c3c3c21c17c19c24c23c3c3c21c17c20c22c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c26c19c3c95c3c3c20c17c26c21c21c3c3c20c17c27c19c23c3c3c20c17c27c27c24c3c3c20c17c28c25c27c3c3c21c17c19c24c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c27c19c3c95c3c3c20c17c26c21c24c3c3c20c17c27c19c24c3c3c20c17c27c27c24c3c3c20c17c28c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c28c19c3c95c3c3c20c17c26c21c27c3c3c20c17c27c19c25c3c3c20c17c27c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c19c19c3c95c3c3c20c17c26c22c20c3c3c20c17c27c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c20c19c3c95c3c3c20c17c26c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 342 c55c68c69c79c72c3c39c24c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c23c20c3c87c82c3c19c17c24c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c23c20c19c3c3c19c17c23c21c19c3c3c19c17c23c22c19c3c3c19c17c23c23c19c3c3c19c17c23c24c19c3c3c19c17c23c25c19c3c3c19c17c23c26c19c3c3c19c17c23c27c19c3c3c19c17c23c28c19c3c3c19c17c24c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c19c19c3c95c3c14c20c20c17c24c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c20c19c3c95c3c14c20c19c17c21c26c3c14c20c19c17c27c20c3c14c20c20c17c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c21c19c3c95c3c14c28c17c20c27c19c3c14c28c17c26c20c23c3c14c20c19c17c21c23c3c14c20c19c17c26c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c22c19c3c95c3c14c27c17c21c20c21c3c14c27c17c26c22c22c3c14c28c17c21c23c25c3c14c28c17c26c24c21c3c14c20c19c17c21c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c23c19c3c95c3c14c26c17c22c24c24c3c14c26c17c27c25c19c3c14c27c17c22c25c19c3c14c27c17c27c24c22c3c14c28c17c22c23c19c3c14c28c17c27c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c24c19c3c95c3c14c25c17c24c28c24c3c14c26c17c19c27c23c3c14c26c17c24c25c28c3c14c27c17c19c23c27c3c14c27c17c24c21c21c3c14c27c17c28c28c20c3c14c28c17c23c24c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c25c19c3c95c3c14c24c17c28c21c20c3c14c25c17c22c28c23c3c14c25c17c27c25c21c3c14c26c17c22c21c26c3c14c26c17c26c27c27c3c14c27c17c21c23c23c3c14c27c17c25c28c25c3c14c28c17c20c23c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c26c19c3c95c3c14c24c17c22c21c23c3c14c24c17c26c27c19c3c14c25c17c21c22c21c3c14c25c17c25c27c21c3c14c26c17c20c21c28c3c14c26c17c24c26c21c3c14c27c17c19c20c20c3c14c27c17c23c23c25c3c14c27c17c27c26c26c3c3c3c16c16c3c3 c19c17c20c27c19c3c95c3c14c23c17c26c28c24c3c14c24c17c21c22c23c3c14c24c17c25c26c19c3c14c25c17c20c19c24c3c14c25c17c24c22c26c3c14c25c17c28c25c25c3c14c26c17c22c28c22c3c14c26c17c27c20c25c3c14c27c17c21c22c25c3c14c27c17c25c24c21 c19c17c20c28c19c3c95c3c14c23c17c22c21c26c3c14c23c17c26c23c28c3c14c24c17c20c25c28c3c14c24c17c24c27c27c3c14c25c17c19c19c25c3c14c25c17c23c21c20c3c14c25c17c27c22c23c3c14c26c17c21c23c24c3c14c26c17c25c24c22c3c14c27c17c19c24c27 c19c17c21c19c19c3c95c3c14c22c17c28c20c22c3c14c23c17c22c20c27c3c14c23c17c26c21c22c3c14c24c17c20c21c25c3c14c24c17c24c21c28c3c14c24c17c28c22c20c3c14c25c17c22c22c19c3c14c25c17c26c21c28c3c14c26c17c20c21c24c3c14c26c17c24c20c27 c19c17c21c20c19c3c95c3c14c22c17c24c23c26c3c14c22c17c28c22c25c3c14c23c17c22c21c24c3c14c23c17c26c20c23c3c14c24c17c20c19c21c3c14c24c17c23c27c28c3c14c24c17c27c26c25c3c14c25c17c21c25c20c3c14c25c17c25c23c24c3c14c26c17c19c21c26 c19c17c21c21c19c3c95c3c14c22c17c21c21c23c3c14c22c17c24c28c26c3c14c22c17c28c26c20c3c14c23c17c22c23c24c3c14c23c17c26c20c28c3c14c24c17c19c28c22c3c14c24c17c23c25c25c3c14c24c17c27c22c28c3c14c25c17c21c20c20c3c14c25c17c24c27c20 c19c17c21c22c19c3c95c3c14c21c17c28c22c27c3c14c22c17c21c28c25c3c14c22c17c25c24c25c3c14c23c17c19c20c24c3c14c23c17c22c26c25c3c14c23c17c26c22c25c3c14c24c17c19c28c26c3c14c24c17c23c24c26c3c14c24c17c27c20c26c3c14c25c17c20c26c25 c19c17c21c23c19c3c95c3c14c21c17c25c27c25c3c14c22c17c19c22c19c3c14c22c17c22c26c24c3c14c22c17c26c21c20c3c14c23c17c19c25c28c3c14c23c17c23c20c25c3c14c23c17c26c25c23c3c14c24c17c20c20c21c3c14c24c17c23c25c19c3c14c24c17c27c19c27 c19c17c21c24c19c3c95c3c14c21c17c23c25c22c3c14c21c17c26c28c23c3c14c22c17c20c21c25c3c14c22c17c23c24c28c3c14c22c17c26c28c22c3c14c23c17c20c21c28c3c14c23c17c23c25c23c3c14c23c17c27c19c19c3c14c24c17c20c22c26c3c14c24c17c23c26c22 c19c17c21c25c19c3c95c3c14c21c17c21c25c26c3c14c21c17c24c27c24c3c14c21c17c28c19c23c3c14c22c17c21c21c24c3c14c22c17c24c23c26c3c14c22c17c27c26c19c3c14c23c17c20c28c23c3c14c23c17c24c20c28c3c14c23c17c27c23c23c3c14c24c17c20c26c19 c19c17c21c26c19c3c95c3c14c21c17c19c28c24c3c14c21c17c23c19c19c3c14c21c17c26c19c27c3c14c22c17c19c20c26c3c14c22c17c22c21c26c3c14c22c17c25c22c28c3c14c22c17c28c24c20c3c14c23c17c21c25c24c3c14c23c17c24c26c28c3c14c23c17c27c28c23 c19c17c21c27c19c3c95c3c14c20c17c28c23c22c3c14c21c17c21c22c26c3c14c21c17c24c22c22c3c14c21c17c27c22c20c3c14c22c17c20c22c19c3c14c22c17c23c22c20c3c14c22c17c26c22c22c3c14c23c17c19c22c25c3c14c23c17c22c23c19c3c14c23c17c25c23c23 c19c17c21c28c19c3c95c3c14c20c17c27c20c19c3c14c21c17c19c28c22c3c14c21c17c22c26c27c3c14c21c17c25c25c25c3c14c21c17c28c24c23c3c14c22c17c21c23c24c3c14c22c17c24c22c25c3c14c22c17c27c21c28c3c14c23c17c20c21c22c3c14c23c17c23c20c27 c19c17c22c19c19c3c95c3c14c20c17c25c28c22c3c14c20c17c28c25c25c3c14c21c17c21c23c21c3c14c21c17c24c20c28c3c14c21c17c26c28c26c3c14c22c17c19c26c27c3c14c22c17c22c25c19c3c14c22c17c25c23c22c3c14c22c17c28c21c26c3c14c23c17c21c20c21 c19c17c22c20c19c3c95c3c14c20c17c24c28c20c3c14c20c17c27c24c24c3c14c21c17c20c21c20c3c14c21c17c22c27c27c3c14c21c17c25c24c27c3c14c21c17c28c21c27c3c14c22c17c21c19c20c3c14c22c17c23c26c24c3c14c22c17c26c24c19c3c14c23c17c19c21c25 c19c17c22c21c19c3c95c3c14c20c17c24c19c22c3c14c20c17c26c24c27c3c14c21c17c19c20c23c3c14c21c17c21c26c22c3c14c21c17c24c22c22c3c14c21c17c26c28c24c3c14c22c17c19c24c28c3c14c22c17c22c21c23c3c14c22c17c24c28c19c3c14c22c17c27c24c27 c19c17c22c22c19c3c95c3c14c20c17c23c21c25c3c14c20c17c25c26c22c3c14c20c17c28c21c20c3c14c21c17c20c26c20c3c14c21c17c23c21c21c3c14c21c17c25c26c25c3c14c21c17c28c22c20c3c14c22c17c20c27c27c3c14c22c17c23c23c25c3c14c22c17c26c19c24 c19c17c22c23c19c3c95c3c14c20c17c22c25c19c3c14c20c17c24c28c28c3c14c20c17c27c22c28c3c14c21c17c19c27c20c3c14c21c17c22c21c23c3c14c21c17c24c26c19c3c14c21c17c27c20c26c3c14c22c17c19c25c24c3c14c22c17c22c20c24c3c14c22c17c24c25c26 c19c17c22c24c19c3c95c3c14c20c17c22c19c23c3c14c20c17c24c22c24c3c14c20c17c26c25c26c3c14c21c17c19c19c21c3c14c21c17c21c22c27c3c14c21c17c23c26c24c3c14c21c17c26c20c24c3c14c21c17c28c24c25c3c14c22c17c20c28c27c3c14c22c17c23c23c21 c19c17c22c25c19c3c95c3c14c20c17c21c24c25c3c14c20c17c23c27c19c3c14c20c17c26c19c24c3c14c20c17c28c22c21c3c14c21c17c20c25c20c3c14c21c17c22c28c21c3c14c21c17c25c21c23c3c14c21c17c27c24c27c3c14c22c17c19c28c22c3c14c22c17c22c21c28 c19c17c22c26c19c3c95c3c14c20c17c21c20c25c3c14c20c17c23c22c22c3c14c20c17c25c24c21c3c14c20c17c27c26c21c3c14c21c17c19c28c23c3c14c21c17c22c20c27c3c14c21c17c24c23c22c3c14c21c17c26c26c19c3c14c21c17c28c28c27c3c14c22c17c21c21c27 c19c17c22c27c19c3c95c3c14c20c17c20c27c22c3c14c20c17c22c28c23c3c14c20c17c25c19c25c3c14c20c17c27c21c19c3c14c21c17c19c22c24c3c14c21c17c21c24c21c3c14c21c17c23c26c20c3c14c21c17c25c28c20c3c14c21c17c28c20c22c3c14c22c17c20c22c25 c19c17c22c28c19c3c95c3c14c20c17c20c24c26c3c14c20c17c22c25c20c3c14c20c17c24c25c26c3c14c20c17c26c26c24c3c14c20c17c28c27c23c3c14c21c17c20c28c24c3c14c21c17c23c19c27c3c14c21c17c25c21c21c3c14c21c17c27c22c26c3c14c22c17c19c24c23 c19c17c23c19c19c3c95c3c14c20c17c20c22c25c3c14c20c17c22c22c23c3c14c20c17c24c22c24c3c14c20c17c26c22c26c3c14c20c17c28c23c19c3c14c21c17c20c23c24c3c14c21c17c22c24c21c3c14c21c17c24c25c19c3c14c21c17c26c26c19c3c14c21c17c28c27c20 c19c17c23c20c19c3c95c3c14c20c17c20c21c19c3c14c20c17c22c20c22c3c14c20c17c24c19c27c3c14c20c17c26c19c23c3c14c20c17c28c19c21c3c14c21c17c20c19c21c3c14c21c17c22c19c22c3c14c21c17c24c19c24c3c14c21c17c26c19c28c3c14c21c17c28c20c24 c19c17c23c21c19c3c95c3c14c20c17c20c19c28c3c14c20c17c21c28c26c3c14c20c17c23c27c26c3c14c20c17c25c26c27c3c14c20c17c27c26c19c3c14c21c17c19c25c24c3c14c21c17c21c25c19c3c14c21c17c23c24c26c3c14c21c17c25c24c25c3c14c21c17c27c24c25 c19c17c23c22c19c3c95c3c14c20c17c20c19c21c3c14c20c17c21c27c24c3c14c20c17c23c26c19c3c14c20c17c25c24c25c3c14c20c17c27c23c23c3c14c21c17c19c22c22c3c14c21c17c21c21c23c3c14c21c17c23c20c25c3c14c21c17c25c19c28c3c14c21c17c27c19c23 c19c17c23c23c19c3c95c3c14c20c17c19c28c28c3c14c20c17c21c26c27c3c14c20c17c23c24c27c3c14c20c17c25c22c28c3c14c20c17c27c21c21c3c14c21c17c19c19c25c3c14c21c17c20c28c21c3c14c21c17c22c26c28c3c14c21c17c24c25c27c3c14c21c17c26c24c27 c19c17c23c24c19c3c95c3c14c20c17c20c19c19c3c14c20c17c21c26c23c3c14c20c17c23c23c28c3c14c20c17c25c21c25c3c14c20c17c27c19c24c3c14c20c17c28c27c23c3c14c21c17c20c25c25c3c14c21c17c22c23c27c3c14c21c17c24c22c21c3c14c21c17c26c20c26 c19c17c23c25c19c3c95c3c14c20c17c20c19c22c3c14c20c17c21c26c22c3c14c20c17c23c23c24c3c14c20c17c25c20c26c3c14c20c17c26c28c20c3c14c20c17c28c25c26c3c14c21c17c20c23c22c3c14c21c17c22c21c20c3c14c21c17c24c19c20c3c14c21c17c25c27c20 c19c17c23c26c19c3c95c3c14c20c17c20c20c19c3c14c20c17c21c26c25c3c14c20c17c23c23c22c3c14c20c17c25c20c21c3c14c20c17c26c27c21c3c14c20c17c28c24c22c3c14c21c17c20c21c24c3c14c21c17c21c28c28c3c14c21c17c23c26c23c3c14c21c17c25c24c20 c19c17c23c27c19c3c95c3c14c20c17c20c21c19c3c14c20c17c21c27c21c3c14c20c17c23c23c24c3c14c20c17c25c19c28c3c14c20c17c26c26c24c3c14c20c17c28c23c22c3c14c21c17c20c20c20c3c14c21c17c21c27c20c3c14c21c17c23c24c21c3c14c21c17c25c21c23 c19c17c23c28c19c3c95c3c14c20c17c20c22c20c3c14c20c17c21c28c19c3c14c20c17c23c23c28c3c14c20c17c25c20c19c3c14c20c17c26c26c21c3c14c20c17c28c22c25c3c14c21c17c20c19c19c3c14c21c17c21c25c25c3c14c21c17c23c22c22c3c14c21c17c25c19c21 c19c17c24c19c19c3c95c3c14c20c17c20c23c24c3c14c20c17c22c19c19c3c14c20c17c23c24c25c3c14c20c17c25c20c23c3c14c20c17c26c26c21c3c14c20c17c28c22c21c3c14c21c17c19c28c22c3c14c21c17c21c24c24c3c14c21c17c23c20c27c3c14c21c17c24c27c22 c19c17c24c20c19c3c95c3c14c20c17c20c25c20c3c14c20c17c22c20c22c3c14c20c17c23c25c24c3c14c20c17c25c20c28c3c14c20c17c26c26c23c3c14c20c17c28c22c20c3c14c21c17c19c27c27c3c14c21c17c21c23c26c3c14c21c17c23c19c26c3c14c21c17c24c25c26 c19c17c24c21c19c3c95c3c14c20c17c20c26c28c3c14c20c17c22c21c26c3c14c20c17c23c26c26c3c14c20c17c25c21c27c3c14c20c17c26c26c28c3c14c20c17c28c22c21c3c14c21c17c19c27c25c3c14c21c17c21c23c20c3c14c21c17c22c28c27c3c14c21c17c24c24c24 c19c17c24c22c19c3c95c3c14c20c17c20c28c28c3c14c20c17c22c23c23c3c14c20c17c23c28c19c3c14c20c17c25c22c27c3c14c20c17c26c27c25c3c14c20c17c28c22c25c3c14c21c17c19c27c26c3c14c21c17c21c22c28c3c14c21c17c22c28c21c3c14c21c17c24c23c25 c19c17c24c23c19c3c95c3c14c20c17c21c21c19c3c14c20c17c22c25c21c3c14c20c17c24c19c24c3c14c20c17c25c24c19c3c14c20c17c26c28c24c3c14c20c17c28c23c21c3c14c21c17c19c28c19c3c14c21c17c21c22c28c3c14c21c17c22c27c27c3c14c21c17c24c22c28 c19c17c24c24c19c3c95c3c14c20c17c21c23c21c3c14c20c17c22c27c21c3c14c20c17c24c21c21c3c14c20c17c25c25c23c3c14c20c17c27c19c25c3c14c20c17c28c24c19c3c14c21c17c19c28c24c3c14c21c17c21c23c20c3c14c21c17c22c27c26c3c14c21c17c24c22c24 c19c17c24c25c19c3c95c3c14c20c17c21c25c25c3c14c20c17c23c19c22c3c14c20c17c24c23c20c3c14c20c17c25c26c28c3c14c20c17c27c20c28c3c14c20c17c28c25c19c3c14c21c17c20c19c21c3c14c21c17c21c23c24c3c14c21c17c22c27c28c3c14c21c17c24c22c22 c19c17c24c26c19c3c95c3c14c20c17c21c28c20c3c14c20c17c23c21c24c3c14c20c17c24c25c19c3c14c20c17c25c28c25c3c14c20c17c27c22c22c3c14c20c17c28c26c21c3c14c21c17c20c20c20c3c14c21c17c21c24c20c3c14c21c17c22c28c21c3c14c21c17c24c22c23 c19c17c24c27c19c3c95c3c14c20c17c22c20c26c3c14c20c17c23c23c28c3c14c20c17c24c27c20c3c14c20c17c26c20c24c3c14c20c17c27c23c28c3c14c20c17c28c27c24c3c14c21c17c20c21c20c3c14c21c17c21c24c28c3c14c21c17c22c28c26c3c14c21c17c24c22c25 c19c17c24c28c19c3c95c3c14c20c17c22c23c23c3c14c20c17c23c26c22c3c14c20c17c25c19c22c3c14c20c17c26c22c23c3c14c20c17c27c25c25c3c14c20c17c28c28c28c3c14c21c17c20c22c22c3c14c21c17c21c25c27c3c14c21c17c23c19c23c3c14c21c17c24c23c20 c19c17c25c19c19c3c95c3c14c20c17c22c26c21c3c14c20c17c23c28c28c3c14c20c17c25c21c26c3c14c20c17c26c24c24c3c14c20c17c27c27c24c3c14c21c17c19c20c24c3c14c21c17c20c23c26c3c14c21c17c21c26c28c3c14c21c17c23c20c21c3c14c21c17c24c23c26 c19c17c25c20c19c3c95c3c14c20c17c23c19c20c3c14c20c17c24c21c24c3c14c20c17c25c24c20c3c14c20c17c26c26c26c3c14c20c17c28c19c23c3c14c21c17c19c22c22c3c14c21c17c20c25c21c3c14c21c17c21c28c21c3c14c21c17c23c21c22c3c14c21c17c24c24c23 c19c17c25c21c19c3c95c3c14c20c17c23c22c19c3c14c20c17c24c24c22c3c14c20c17c25c26c25c3c14c20c17c27c19c19c3c14c20c17c28c21c24c3c14c21c17c19c24c20c3c14c21c17c20c26c27c3c14c21c17c22c19c24c3c14c21c17c23c22c23c3c3c3c16c16c3c3 c19c17c25c22c19c3c95c3c14c20c17c23c25c20c3c14c20c17c24c27c20c3c14c20c17c26c19c21c3c14c20c17c27c21c23c3c14c20c17c28c23c26c3c14c21c17c19c26c20c3c14c21c17c20c28c24c3c14c21c17c22c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c23c19c3c95c3c14c20c17c23c28c20c3c14c20c17c25c20c19c3c14c20c17c26c21c28c3c14c20c17c27c23c28c3c14c20c17c28c25c28c3c14c21c17c19c28c20c3c14c21c17c21c20c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c24c19c3c95c3c14c20c17c24c21c22c3c14c20c17c25c22c28c3c14c20c17c26c24c25c3c14c20c17c27c26c23c3c14c20c17c28c28c22c3c14c21c17c20c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c25c19c3c95c3c14c20c17c24c24c24c3c14c20c17c25c25c28c3c14c20c17c26c27c23c3c14c20c17c28c19c19c3c14c21c17c19c20c26c3c14c21c17c20c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c26c19c3c95c3c14c20c17c24c27c26c3c14c20c17c25c28c28c3c14c20c17c27c20c22c3c14c20c17c28c21c26c3c14c21c17c19c23c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c27c19c3c95c3c14c20c17c25c21c19c3c14c20c17c26c22c19c3c14c20c17c27c23c21c3c14c20c17c28c24c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c28c19c3c95c3c14c20c17c25c24c22c3c14c20c17c26c25c21c3c14c20c17c27c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c19c19c3c95c3c14c20c17c25c27c25c3c14c20c17c26c28c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c20c19c3c95c3c14c20c17c26c21c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 343 c55c68c69c79c72c3c39c24c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c23c20c3c87c82c3c19c17c24c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c23c20c19c3c3c19c17c23c21c19c3c3c19c17c23c22c19c3c3c19c17c23c23c19c3c3c19c17c23c24c19c3c3c19c17c23c25c19c3c3c19c17c23c26c19c3c3c19c17c23c27c19c3c3c19c17c23c28c19c3c3c19c17c24c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c19c19c3c95c3c3c19c17c22c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c20c19c3c95c3c3c19c17c22c19c25c3c3c19c17c22c20c19c3c3c19c17c22c20c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c21c19c3c95c3c3c19c17c22c19c25c3c3c19c17c22c20c19c3c3c19c17c22c20c23c3c3c19c17c22c20c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c22c19c3c95c3c3c19c17c22c19c26c3c3c19c17c22c20c20c3c3c19c17c22c20c24c3c3c19c17c22c20c28c3c3c19c17c22c21c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c23c19c3c95c3c3c19c17c22c19c27c3c3c19c17c22c20c21c3c3c19c17c22c20c25c3c3c19c17c22c21c19c3c3c19c17c22c21c23c3c3c19c17c22c21c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c24c19c3c95c3c3c19c17c22c19c28c3c3c19c17c22c20c22c3c3c19c17c22c20c26c3c3c19c17c22c21c20c3c3c19c17c22c21c24c3c3c19c17c22c21c28c3c3c19c17c22c22c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c25c19c3c95c3c3c19c17c22c20c21c3c3c19c17c22c20c24c3c3c19c17c22c20c28c3c3c19c17c22c21c21c3c3c19c17c22c21c25c3c3c19c17c22c22c19c3c3c19c17c22c22c23c3c3c19c17c22c22c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c26c19c3c95c3c3c19c17c22c20c23c3c3c19c17c22c20c26c3c3c19c17c22c21c20c3c3c19c17c22c21c23c3c3c19c17c22c21c27c3c3c19c17c22c22c21c3c3c19c17c22c22c24c3c3c19c17c22c22c28c3c3c19c17c22c23c22c3c3c3c16c16c3c3 c19c17c20c27c19c3c95c3c3c19c17c22c20c26c3c3c19c17c22c21c19c3c3c19c17c22c21c22c3c3c19c17c22c21c26c3c3c19c17c22c22c19c3c3c19c17c22c22c23c3c3c19c17c22c22c26c3c3c19c17c22c23c20c3c3c19c17c22c23c23c3c3c19c17c22c23c27 c19c17c20c28c19c3c95c3c3c19c17c22c21c19c3c3c19c17c22c21c22c3c3c19c17c22c21c25c3c3c19c17c22c21c28c3c3c19c17c22c22c22c3c3c19c17c22c22c25c3c3c19c17c22c22c28c3c3c19c17c22c23c22c3c3c19c17c22c23c25c3c3c19c17c22c24c19 c19c17c21c19c19c3c95c3c3c19c17c22c21c23c3c3c19c17c22c21c26c3c3c19c17c22c22c19c3c3c19c17c22c22c22c3c3c19c17c22c22c25c3c3c19c17c22c22c28c3c3c19c17c22c23c21c3c3c19c17c22c23c24c3c3c19c17c22c23c27c3c3c19c17c22c24c21 c19c17c21c20c19c3c95c3c3c19c17c22c21c27c3c3c19c17c22c22c19c3c3c19c17c22c22c22c3c3c19c17c22c22c25c3c3c19c17c22c22c28c3c3c19c17c22c23c21c3c3c19c17c22c23c24c3c3c19c17c22c23c27c3c3c19c17c22c24c20c3c3c19c17c22c24c23 c19c17c21c21c19c3c95c3c3c19c17c22c22c21c3c3c19c17c22c22c23c3c3c19c17c22c22c26c3c3c19c17c22c22c28c3c3c19c17c22c23c21c3c3c19c17c22c23c24c3c3c19c17c22c23c27c3c3c19c17c22c24c20c3c3c19c17c22c24c23c3c3c19c17c22c24c26 c19c17c21c22c19c3c95c3c3c19c17c22c22c25c3c3c19c17c22c22c27c3c3c19c17c22c23c20c3c3c19c17c22c23c22c3c3c19c17c22c23c25c3c3c19c17c22c23c28c3c3c19c17c22c24c20c3c3c19c17c22c24c23c3c3c19c17c22c24c26c3c3c19c17c22c25c19 c19c17c21c23c19c3c95c3c3c19c17c22c23c19c3c3c19c17c22c23c22c3c3c19c17c22c23c24c3c3c19c17c22c23c26c3c3c19c17c22c24c19c3c3c19c17c22c24c21c3c3c19c17c22c24c24c3c3c19c17c22c24c27c3c3c19c17c22c25c19c3c3c19c17c22c25c22 c19c17c21c24c19c3c95c3c3c19c17c22c23c24c3c3c19c17c22c23c26c3c3c19c17c22c23c28c3c3c19c17c22c24c21c3c3c19c17c22c24c23c3c3c19c17c22c24c25c3c3c19c17c22c24c28c3c3c19c17c22c25c20c3c3c19c17c22c25c23c3c3c19c17c22c25c26 c19c17c21c25c19c3c95c3c3c19c17c22c24c19c3c3c19c17c22c24c21c3c3c19c17c22c24c23c3c3c19c17c22c24c25c3c3c19c17c22c24c27c3c3c19c17c22c25c19c3c3c19c17c22c25c22c3c3c19c17c22c25c24c3c3c19c17c22c25c27c3c3c19c17c22c26c19 c19c17c21c26c19c3c95c3c3c19c17c22c24c24c3c3c19c17c22c24c26c3c3c19c17c22c24c28c3c3c19c17c22c25c20c3c3c19c17c22c25c22c3c3c19c17c22c25c24c3c3c19c17c22c25c26c3c3c19c17c22c25c28c3c3c19c17c22c26c21c3c3c19c17c22c26c23 c19c17c21c27c19c3c95c3c3c19c17c22c25c19c3c3c19c17c22c25c21c3c3c19c17c22c25c23c3c3c19c17c22c25c24c3c3c19c17c22c25c26c3c3c19c17c22c25c28c3c3c19c17c22c26c20c3c3c19c17c22c26c22c3c3c19c17c22c26c25c3c3c19c17c22c26c27 c19c17c21c28c19c3c95c3c3c19c17c22c25c25c3c3c19c17c22c25c26c3c3c19c17c22c25c28c3c3c19c17c22c26c19c3c3c19c17c22c26c21c3c3c19c17c22c26c23c3c3c19c17c22c26c25c3c3c19c17c22c26c27c3c3c19c17c22c27c19c3c3c19c17c22c27c21 c19c17c22c19c19c3c95c3c3c19c17c22c26c20c3c3c19c17c22c26c21c3c3c19c17c22c26c23c3c3c19c17c22c26c24c3c3c19c17c22c26c26c3c3c19c17c22c26c28c3c3c19c17c22c27c19c3c3c19c17c22c27c21c3c3c19c17c22c27c23c3c3c19c17c22c27c25 c19c17c22c20c19c3c95c3c3c19c17c22c26c25c3c3c19c17c22c26c27c3c3c19c17c22c26c28c3c3c19c17c22c27c19c3c3c19c17c22c27c21c3c3c19c17c22c27c23c3c3c19c17c22c27c24c3c3c19c17c22c27c26c3c3c19c17c22c27c28c3c3c19c17c22c28c20 c19c17c22c21c19c3c95c3c3c19c17c22c27c21c3c3c19c17c22c27c22c3c3c19c17c22c27c23c3c3c19c17c22c27c25c3c3c19c17c22c27c26c3c3c19c17c22c27c28c3c3c19c17c22c28c19c3c3c19c17c22c28c21c3c3c19c17c22c28c23c3c3c19c17c22c28c24 c19c17c22c22c19c3c95c3c3c19c17c22c27c27c3c3c19c17c22c27c28c3c3c19c17c22c28c19c3c3c19c17c22c28c20c3c3c19c17c22c28c21c3c3c19c17c22c28c23c3c3c19c17c22c28c24c3c3c19c17c22c28c26c3c3c19c17c22c28c27c3c3c19c17c23c19c19 c19c17c22c23c19c3c95c3c3c19c17c22c28c22c3c3c19c17c22c28c23c3c3c19c17c22c28c24c3c3c19c17c22c28c25c3c3c19c17c22c28c27c3c3c19c17c22c28c28c3c3c19c17c23c19c19c3c3c19c17c23c19c21c3c3c19c17c23c19c22c3c3c19c17c23c19c24 c19c17c22c24c19c3c95c3c3c19c17c22c28c28c3c3c19c17c23c19c19c3c3c19c17c23c19c20c3c3c19c17c23c19c21c3c3c19c17c23c19c22c3c3c19c17c23c19c23c3c3c19c17c23c19c24c3c3c19c17c23c19c26c3c3c19c17c23c19c27c3c3c19c17c23c20c19 c19c17c22c25c19c3c95c3c3c19c17c23c19c24c3c3c19c17c23c19c25c3c3c19c17c23c19c26c3c3c19c17c23c19c26c3c3c19c17c23c19c27c3c3c19c17c23c19c28c3c3c19c17c23c20c20c3c3c19c17c23c20c21c3c3c19c17c23c20c22c3c3c19c17c23c20c24 c19c17c22c26c19c3c95c3c3c19c17c23c20c20c3c3c19c17c23c20c21c3c3c19c17c23c20c21c3c3c19c17c23c20c22c3c3c19c17c23c20c23c3c3c19c17c23c20c24c3c3c19c17c23c20c25c3c3c19c17c23c20c26c3c3c19c17c23c20c27c3c3c19c17c23c21c19 c19c17c22c27c19c3c95c3c3c19c17c23c20c26c3c3c19c17c23c20c27c3c3c19c17c23c20c27c3c3c19c17c23c20c28c3c3c19c17c23c21c19c3c3c19c17c23c21c19c3c3c19c17c23c21c20c3c3c19c17c23c21c21c3c3c19c17c23c21c23c3c3c19c17c23c21c24 c19c17c22c28c19c3c95c3c3c19c17c23c21c22c3c3c19c17c23c21c22c3c3c19c17c23c21c23c3c3c19c17c23c21c23c3c3c19c17c23c21c24c3c3c19c17c23c21c25c3c3c19c17c23c21c26c3c3c19c17c23c21c27c3c3c19c17c23c21c28c3c3c19c17c23c22c19 c19c17c23c19c19c3c95c3c3c19c17c23c21c28c3c3c19c17c23c21c28c3c3c19c17c23c22c19c3c3c19c17c23c22c19c3c3c19c17c23c22c20c3c3c19c17c23c22c21c3c3c19c17c23c22c21c3c3c19c17c23c22c22c3c3c19c17c23c22c23c3c3c19c17c23c22c24 c19c17c23c20c19c3c95c3c3c19c17c23c22c24c3c3c19c17c23c22c24c3c3c19c17c23c22c25c3c3c19c17c23c22c25c3c3c19c17c23c22c26c3c3c19c17c23c22c26c3c3c19c17c23c22c27c3c3c19c17c23c22c28c3c3c19c17c23c23c19c3c3c19c17c23c23c19 c19c17c23c21c19c3c95c3c3c19c17c23c23c20c3c3c19c17c23c23c20c3c3c19c17c23c23c21c3c3c19c17c23c23c21c3c3c19c17c23c23c21c3c3c19c17c23c23c22c3c3c19c17c23c23c22c3c3c19c17c23c23c23c3c3c19c17c23c23c24c3c3c19c17c23c23c25 c19c17c23c22c19c3c95c3c3c19c17c23c23c26c3c3c19c17c23c23c26c3c3c19c17c23c23c27c3c3c19c17c23c23c27c3c3c19c17c23c23c27c3c3c19c17c23c23c28c3c3c19c17c23c23c28c3c3c19c17c23c24c19c3c3c19c17c23c24c19c3c3c19c17c23c24c20 c19c17c23c23c19c3c95c3c3c19c17c23c24c23c3c3c19c17c23c24c23c3c3c19c17c23c24c23c3c3c19c17c23c24c23c3c3c19c17c23c24c23c3c3c19c17c23c24c23c3c3c19c17c23c24c24c3c3c19c17c23c24c24c3c3c19c17c23c24c25c3c3c19c17c23c24c26 c19c17c23c24c19c3c95c3c3c19c17c23c25c19c3c3c19c17c23c25c19c3c3c19c17c23c25c19c3c3c19c17c23c25c19c3c3c19c17c23c25c19c3c3c19c17c23c25c19c3c3c19c17c23c25c20c3c3c19c17c23c25c20c3c3c19c17c23c25c21c3c3c19c17c23c25c21 c19c17c23c25c19c3c95c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c25c3c3c19c17c23c25c26c3c3c19c17c23c25c26c3c3c19c17c23c25c27 c19c17c23c26c19c3c95c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c21c3c3c19c17c23c26c22c3c3c19c17c23c26c22 c19c17c23c27c19c3c95c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c27c3c3c19c17c23c26c28 c19c17c23c28c19c3c95c3c3c19c17c23c27c24c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23c3c3c19c17c23c27c23 c19c17c24c19c19c3c95c3c3c19c17c23c28c20c3c3c19c17c23c28c19c3c3c19c17c23c28c19c3c3c19c17c23c28c19c3c3c19c17c23c27c28c3c3c19c17c23c27c28c3c3c19c17c23c27c28c3c3c19c17c23c28c19c3c3c19c17c23c28c19c3c3c19c17c23c28c19 c19c17c24c20c19c3c95c3c3c19c17c23c28c26c3c3c19c17c23c28c25c3c3c19c17c23c28c25c3c3c19c17c23c28c25c3c3c19c17c23c28c24c3c3c19c17c23c28c24c3c3c19c17c23c28c24c3c3c19c17c23c28c24c3c3c19c17c23c28c25c3c3c19c17c23c28c25 c19c17c24c21c19c3c95c3c3c19c17c24c19c22c3c3c19c17c24c19c22c3c3c19c17c24c19c21c3c3c19c17c24c19c21c3c3c19c17c24c19c20c3c3c19c17c24c19c20c3c3c19c17c24c19c20c3c3c19c17c24c19c20c3c3c19c17c24c19c20c3c3c19c17c24c19c20 c19c17c24c22c19c3c95c3c3c19c17c24c19c28c3c3c19c17c24c19c28c3c3c19c17c24c19c27c3c3c19c17c24c19c27c3c3c19c17c24c19c26c3c3c19c17c24c19c26c3c3c19c17c24c19c26c3c3c19c17c24c19c26c3c3c19c17c24c19c26c3c3c19c17c24c19c26 c19c17c24c23c19c3c95c3c3c19c17c24c20c25c3c3c19c17c24c20c24c3c3c19c17c24c20c23c3c3c19c17c24c20c23c3c3c19c17c24c20c22c3c3c19c17c24c20c22c3c3c19c17c24c20c22c3c3c19c17c24c20c22c3c3c19c17c24c20c22c3c3c19c17c24c20c22 c19c17c24c24c19c3c95c3c3c19c17c24c21c21c3c3c19c17c24c21c20c3c3c19c17c24c21c19c3c3c19c17c24c21c19c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c19c17c24c20c28 c19c17c24c25c19c3c95c3c3c19c17c24c21c27c3c3c19c17c24c21c26c3c3c19c17c24c21c25c3c3c19c17c24c21c25c3c3c19c17c24c21c24c3c3c19c17c24c21c24c3c3c19c17c24c21c24c3c3c19c17c24c21c23c3c3c19c17c24c21c23c3c3c19c17c24c21c23 c19c17c24c26c19c3c95c3c3c19c17c24c22c23c3c3c19c17c24c22c22c3c3c19c17c24c22c22c3c3c19c17c24c22c21c3c3c19c17c24c22c20c3c3c19c17c24c22c20c3c3c19c17c24c22c20c3c3c19c17c24c22c19c3c3c19c17c24c22c19c3c3c19c17c24c22c19 c19c17c24c27c19c3c95c3c3c19c17c24c23c19c3c3c19c17c24c22c28c3c3c19c17c24c22c28c3c3c19c17c24c22c27c3c3c19c17c24c22c26c3c3c19c17c24c22c26c3c3c19c17c24c22c25c3c3c19c17c24c22c25c3c3c19c17c24c22c25c3c3c19c17c24c22c25 c19c17c24c28c19c3c95c3c3c19c17c24c23c26c3c3c19c17c24c23c25c3c3c19c17c24c23c24c3c3c19c17c24c23c23c3c3c19c17c24c23c22c3c3c19c17c24c23c22c3c3c19c17c24c23c21c3c3c19c17c24c23c21c3c3c19c17c24c23c21c3c3c19c17c24c23c21 c19c17c25c19c19c3c95c3c3c19c17c24c24c22c3c3c19c17c24c24c21c3c3c19c17c24c24c20c3c3c19c17c24c24c19c3c3c19c17c24c23c28c3c3c19c17c24c23c28c3c3c19c17c24c23c27c3c3c19c17c24c23c27c3c3c19c17c24c23c27c3c3c19c17c24c23c26 c19c17c25c20c19c3c95c3c3c19c17c24c24c28c3c3c19c17c24c24c27c3c3c19c17c24c24c26c3c3c19c17c24c24c25c3c3c19c17c24c24c24c3c3c19c17c24c24c24c3c3c19c17c24c24c23c3c3c19c17c24c24c23c3c3c19c17c24c24c22c3c3c19c17c24c24c22 c19c17c25c21c19c3c95c3c3c19c17c24c25c24c3c3c19c17c24c25c23c3c3c19c17c24c25c22c3c3c19c17c24c25c21c3c3c19c17c24c25c20c3c3c19c17c24c25c20c3c3c19c17c24c25c19c3c3c19c17c24c25c19c3c3c19c17c24c24c28c3c3c3c16c16c3c3 c19c17c25c22c19c3c95c3c3c19c17c24c26c20c3c3c19c17c24c26c19c3c3c19c17c24c25c28c3c3c19c17c24c25c27c3c3c19c17c24c25c26c3c3c19c17c24c25c26c3c3c19c17c24c25c25c3c3c19c17c24c25c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c23c19c3c95c3c3c19c17c24c26c26c3c3c19c17c24c26c25c3c3c19c17c24c26c24c3c3c19c17c24c26c23c3c3c19c17c24c26c22c3c3c19c17c24c26c21c3c3c19c17c24c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c24c19c3c95c3c3c19c17c24c27c22c3c3c19c17c24c27c21c3c3c19c17c24c27c20c3c3c19c17c24c27c19c3c3c19c17c24c26c28c3c3c19c17c24c26c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c25c19c3c95c3c3c19c17c24c27c28c3c3c19c17c24c27c27c3c3c19c17c24c27c26c3c3c19c17c24c27c25c3c3c19c17c24c27c24c3c3c19c17c24c27c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c26c19c3c95c3c3c19c17c24c28c24c3c3c19c17c24c28c23c3c3c19c17c24c28c22c3c3c19c17c24c28c21c3c3c19c17c24c28c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c27c19c3c95c3c3c19c17c25c19c20c3c3c19c17c25c19c19c3c3c19c17c24c28c28c3c3c19c17c24c28c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c28c19c3c95c3c3c19c17c25c19c27c3c3c19c17c25c19c25c3c3c19c17c25c19c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c19c19c3c95c3c3c19c17c25c20c23c3c3c19c17c25c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c20c19c3c95c3c3c19c17c25c21c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 344 c55c68c69c79c72c3c39c24c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c23c20c3c87c82c3c19c17c24c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c23c20c19c3c3c19c17c23c21c19c3c3c19c17c23c22c19c3c3c19c17c23c23c19c3c3c19c17c23c24c19c3c3c19c17c23c25c19c3c3c19c17c23c26c19c3c3c19c17c23c27c19c3c3c19c17c23c28c19c3c3c19c17c24c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c19c28c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c19c19c3c95c3c3c19c17c20c28c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c20c19c3c95c3c3c19c17c20c28c19c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c21c19c3c95c3c3c19c17c20c27c27c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c22c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c23c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c24c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c25c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c21c19c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c26c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c3c16c16c3c3 c19c17c20c27c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21 c19c17c20c28c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20 c19c17c21c19c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c28c3c3c19c17c21c19c20 c19c17c21c20c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19 c19c17c21c21c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c27c3c3c19c17c21c19c19 c19c17c21c22c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c21c23c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c21c24c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c21c25c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27 c19c17c21c26c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27 c19c17c21c27c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c25c3c3c19c17c20c28c27 c19c17c21c28c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c25c3c3c19c17c20c28c27 c19c17c22c19c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27 c19c17c22c20c19c3c95c3c3c19c17c20c27c21c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27 c19c17c22c21c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27 c19c17c22c22c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c22c23c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c22c24c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c22c25c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c22c26c19c3c95c3c3c19c17c20c27c22c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c26c3c3c19c17c20c28c28 c19c17c22c27c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c20c28c28 c19c17c22c28c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c20c28c28 c19c17c23c19c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19 c19c17c23c20c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27c3c3c19c17c21c19c19 c19c17c23c21c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27c3c3c19c17c21c19c19 c19c17c23c22c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c19 c19c17c23c23c19c3c95c3c3c19c17c20c27c23c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c19 c19c17c23c24c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20 c19c17c23c25c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20 c19c17c23c26c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c20c28c28c3c3c19c17c21c19c20 c19c17c23c27c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c20 c19c17c23c28c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c20 c19c17c24c19c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21 c19c17c24c20c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21 c19c17c24c21c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c19c3c3c19c17c21c19c21 c19c17c24c22c19c3c95c3c3c19c17c20c27c24c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c19c17c21c19c21 c19c17c24c23c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c19c17c21c19c21 c19c17c24c24c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c19c17c21c19c22 c19c17c24c25c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c19c17c21c19c22 c19c17c24c26c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c19c17c21c19c22 c19c17c24c27c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c20c3c3c19c17c21c19c22 c19c17c24c28c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c22 c19c17c25c19c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c27c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23 c19c17c25c20c19c3c95c3c3c19c17c20c27c25c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c27c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23 c19c17c25c21c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c3c16c16c3c3 c19c17c25c22c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c19c17c21c19c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c23c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c19c17c20c28c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c24c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c25c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c20c3c3c19c17c20c28c22c3c3c19c17c20c28c24c3c3c19c17c20c28c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c26c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c27c28c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c19c17c20c28c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c27c19c3c95c3c3c19c17c20c27c26c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c19c17c20c28c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c28c19c3c95c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c19c17c20c28c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c19c19c3c95c3c3c19c17c20c27c27c3c3c19c17c20c28c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c20c19c3c95c3c3c19c17c20c27c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c26c21c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 345 c55c68c69c79c72c3c39c25c17c20c3c51c68c85c68c80c72c87c72c85c3c37c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c24c20c3c87c82c3c19c17c25c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c24c20c19c3c3c19c17c24c21c19c3c3c19c17c24c22c19c3c3c19c17c24c23c19c3c3c19c17c24c24c19c3c3c19c17c24c25c19c3c3c19c17c24c26c19c3c3c19c17c24c27c19c3c3c19c17c24c28c19c3c3c19c17c25c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c28c19c3c95c3c3c27c17c23c28c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c19c19c3c95c3c3c27c17c19c21c26c3c3c27c17c22c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c20c19c3c95c3c3c26c17c24c28c24c3c3c26c17c27c25c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c21c19c3c95c3c3c26c17c20c28c27c3c3c26c17c23c25c22c3c3c26c17c26c21c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c22c19c3c95c3c3c25c17c27c22c23c3c3c26c17c19c28c20c3c3c26c17c22c23c27c3c3c26c17c25c19c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c23c19c3c95c3c3c25c17c23c28c28c3c3c25c17c26c23c28c3c3c25c17c28c28c27c3c3c26c17c21c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c24c19c3c95c3c3c25c17c20c28c20c3c3c25c17c23c22c22c3c3c25c17c25c26c24c3c3c25c17c28c20c25c3c3c26c17c20c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c25c19c3c95c3c3c24c17c28c19c27c3c3c25c17c20c23c21c3c3c25c17c22c26c26c3c3c25c17c25c20c21c3c3c25c17c27c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c26c19c3c95c3c3c24c17c25c23c26c3c3c24c17c27c26c24c3c3c25c17c20c19c21c3c3c25c17c22c22c19c3c3c25c17c24c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c27c19c3c95c3c3c24c17c23c19c26c3c3c24c17c25c21c27c3c3c24c17c27c23c27c3c3c25c17c19c25c28c3c3c25c17c21c28c19c3c3c25c17c24c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c28c19c3c95c3c3c24c17c20c27c25c3c3c24c17c23c19c19c3c3c24c17c25c20c23c3c3c24c17c27c21c27c3c3c25c17c19c23c22c3c3c25c17c21c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c19c19c3c95c3c3c23c17c28c27c21c3c3c24c17c20c27c28c3c3c24c17c22c28c26c3c3c24c17c25c19c24c3c3c24c17c27c20c22c3c3c25c17c19c21c21c3c3c25c17c21c22c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c20c19c3c95c3c3c23c17c26c28c23c3c3c23c17c28c28c24c3c3c24c17c20c28c25c3c3c24c17c22c28c27c3c3c24c17c25c19c19c3c3c24c17c27c19c22c3c3c25c17c19c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c21c19c3c95c3c3c23c17c25c21c19c3c3c23c17c27c20c24c3c3c24c17c19c20c19c3c3c24c17c21c19c25c3c3c24c17c23c19c22c3c3c24c17c24c28c28c3c3c24c17c26c28c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c22c19c3c95c3c3c23c17c23c25c19c3c3c23c17c25c23c28c3c3c23c17c27c22c27c3c3c24c17c19c21c27c3c3c24c17c21c20c28c3c3c24c17c23c20c19c3c3c24c17c25c19c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c23c19c3c95c3c3c23c17c22c20c21c3c3c23c17c23c28c24c3c3c23c17c25c26c28c3c3c23c17c27c25c22c3c3c24c17c19c23c28c3c3c24c17c21c22c24c3c3c24c17c23c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c24c19c3c95c3c3c23c17c20c26c23c3c3c23c17c22c24c21c3c3c23c17c24c22c20c3c3c23c17c26c20c19c3c3c23c17c27c28c19c3c3c24c17c19c26c20c3c3c24c17c21c24c21c3c3c24c17c23c22c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c25c19c3c95c3c3c23c17c19c23c26c3c3c23c17c21c21c19c3c3c23c17c22c28c22c3c3c23c17c24c25c27c3c3c23c17c26c23c22c3c3c23c17c28c20c27c3c3c24c17c19c28c24c3c3c24c17c21c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c26c19c3c95c3c3c22c17c28c22c19c3c3c23c17c19c28c26c3c3c23c17c21c25c25c3c3c23c17c23c22c24c3c3c23c17c25c19c24c3c3c23c17c26c26c25c3c3c23c17c28c23c27c3c3c24c17c20c21c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c27c19c3c95c3c3c22c17c27c21c19c3c3c22c17c28c27c22c3c3c23c17c20c23c26c3c3c23c17c22c20c21c3c3c23c17c23c26c26c3c3c23c17c25c23c23c3c3c23c17c27c20c20c3c3c23c17c28c26c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c28c19c3c95c3c3c22c17c26c20c28c3c3c22c17c27c26c27c3c3c23c17c19c22c26c3c3c23c17c20c28c26c3c3c23c17c22c24c27c3c3c23c17c24c21c19c3c3c23c17c25c27c22c3c3c23c17c27c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c19c19c3c95c3c3c22c17c25c21c24c3c3c22c17c26c26c28c3c3c22c17c28c22c23c3c3c23c17c19c28c19c3c3c23c17c21c23c26c3c3c23c17c23c19c24c3c3c23c17c24c25c22c3c3c23c17c26c21c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c20c19c3c95c3c3c22c17c24c22c27c3c3c22c17c25c27c27c3c3c22c17c27c22c28c3c3c22c17c28c28c20c3c3c23c17c20c23c23c3c3c23c17c21c28c26c3c3c23c17c23c24c20c3c3c23c17c25c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c21c19c3c95c3c3c22c17c23c24c26c3c3c22c17c25c19c22c3c3c22c17c26c24c19c3c3c22c17c27c28c27c3c3c23c17c19c23c26c3c3c23c17c20c28c25c3c3c23c17c22c23c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c22c19c3c95c3c3c22c17c22c27c21c3c3c22c17c24c21c23c3c3c22c17c25c25c26c3c3c22c17c27c20c20c3c3c22c17c28c24c25c3c3c23c17c20c19c21c3c3c23c17c21c23c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c23c19c3c95c3c3c22c17c22c20c20c3c3c22c17c23c24c19c3c3c22c17c24c28c19c3c3c22c17c26c22c20c3c3c22c17c27c26c21c3c3c23c17c19c20c23c3c3c23c17c20c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c24c19c3c95c3c3c22c17c21c23c25c3c3c22c17c22c27c21c3c3c22c17c24c20c27c3c3c22c17c25c24c24c3c3c22c17c26c28c22c3c3c22c17c28c22c21c3c3c23c17c19c26c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c25c19c3c95c3c3c22c17c20c27c25c3c3c22c17c22c20c27c3c3c22c17c23c24c20c3c3c22c17c24c27c24c3c3c22c17c26c20c28c3c3c22c17c27c24c24c3c3c22c17c28c28c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c26c19c3c95c3c3c22c17c20c21c28c3c3c22c17c21c24c27c3c3c22c17c22c27c27c3c3c22c17c24c20c28c3c3c22c17c25c24c19c3c3c22c17c26c27c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c27c19c3c95c3c3c22c17c19c26c26c3c3c22c17c21c19c22c3c3c22c17c22c22c19c3c3c22c17c23c24c26c3c3c22c17c24c27c25c3c3c22c17c26c20c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c28c19c3c95c3c3c22c17c19c21c27c3c3c22c17c20c24c20c3c3c22c17c21c26c24c3c3c22c17c23c19c19c3c3c22c17c24c21c24c3c3c22c17c25c24c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c19c19c3c95c3c3c21c17c28c27c22c3c3c22c17c20c19c22c3c3c22c17c21c21c23c3c3c22c17c22c23c25c3c3c22c17c23c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c20c19c3c95c3c3c21c17c28c23c19c3c3c22c17c19c24c27c3c3c22c17c20c26c26c3c3c22c17c21c28c25c3c3c22c17c23c20c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c21c19c3c95c3c3c21c17c28c19c20c3c3c22c17c19c20c25c3c3c22c17c20c22c21c3c3c22c17c21c23c28c3c3c22c17c22c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c22c19c3c95c3c3c21c17c27c25c23c3c3c21c17c28c26c26c3c3c22c17c19c28c20c3c3c22c17c21c19c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c23c19c3c95c3c3c21c17c27c22c19c3c3c21c17c28c23c20c3c3c22c17c19c24c21c3c3c22c17c20c25c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c24c19c3c95c3c3c21c17c26c28c28c3c3c21c17c28c19c26c3c3c22c17c19c20c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c25c19c3c95c3c3c21c17c26c25c28c3c3c21c17c27c26c24c3c3c21c17c28c27c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c26c19c3c95c3c3c21c17c26c23c21c3c3c21c17c27c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c27c19c3c95c3c3c21c17c26c20c26c3c3c21c17c27c20c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c28c19c3c95c3c3c21c17c25c28c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 346 c55c68c69c79c72c3c39c25c17c21c3c51c68c85c68c80c72c87c72c85c3c38c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c24c20c3c87c82c3c19c17c25c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c24c20c19c3c3c19c17c24c21c19c3c3c19c17c24c22c19c3c3c19c17c24c23c19c3c3c19c17c24c24c19c3c3c19c17c24c25c19c3c3c19c17c24c26c19c3c3c19c17c24c27c19c3c3c19c17c24c28c19c3c3c19c17c25c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c28c19c3c95c3c14c27c17c23c25c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c19c19c3c95c3c14c26c17c28c20c19c3c14c27c17c21c28c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c20c19c3c95c3c14c26c17c23c19c27c3c14c26c17c26c27c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c21c19c3c95c3c14c25c17c28c24c19c3c14c26c17c22c20c27c3c14c26c17c25c27c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c22c19c3c95c3c14c25c17c24c22c23c3c14c25c17c27c28c20c3c14c26c17c21c23c26c3c14c26c17c25c19c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c23c19c3c95c3c14c25c17c20c24c24c3c14c25c17c24c19c20c3c14c25c17c27c23c26c3c14c26c17c20c28c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c24c19c3c95c3c14c24c17c27c20c19c3c14c25c17c20c23c25c3c14c25c17c23c27c20c3c14c25c17c27c20c25c3c14c26c17c20c23c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c25c19c3c95c3c14c24c17c23c28c24c3c14c24c17c27c21c20c3c14c25c17c20c23c26c3c14c25c17c23c26c21c3c14c25c17c26c28c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c26c19c3c95c3c14c24c17c21c20c19c3c14c24c17c24c21c24c3c14c24c17c27c23c20c3c14c25c17c20c24c25c3c14c25c17c23c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c27c19c3c95c3c14c23c17c28c24c19c3c14c24c17c21c24c25c3c14c24c17c24c25c21c3c14c24c17c27c25c27c3c14c25c17c20c26c23c3c14c25c17c23c27c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c28c19c3c95c3c14c23c17c26c20c23c3c14c24c17c19c20c19c3c14c24c17c22c19c26c3c14c24c17c25c19c23c3c14c24c17c28c19c20c3c14c25c17c20c28c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c19c19c3c95c3c14c23c17c23c28c28c3c14c23c17c26c27c25c3c14c24c17c19c26c23c3c14c24c17c22c25c21c3c14c24c17c25c24c20c3c14c24c17c28c23c19c3c14c25c17c21c21c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c20c19c3c95c3c14c23c17c22c19c23c3c14c23c17c24c27c21c3c14c23c17c27c25c20c3c14c24c17c20c23c20c3c14c24c17c23c21c20c3c14c24c17c26c19c21c3c14c24c17c28c27c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c21c19c3c95c3c14c23c17c20c21c25c3c14c23c17c22c28c25c3c14c23c17c25c25c26c3c14c23c17c28c22c28c3c14c24c17c21c20c20c3c14c24c17c23c27c23c3c14c24c17c26c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c22c19c3c95c3c14c22c17c28c25c24c3c14c23c17c21c21c26c3c14c23c17c23c28c19c3c14c23c17c26c24c23c3c14c24c17c19c20c27c3c14c24c17c21c27c22c3c14c24c17c24c23c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c23c19c3c95c3c14c22c17c27c21c19c3c14c23c17c19c26c22c3c14c23c17c22c21c27c3c14c23c17c24c27c23c3c14c23c17c27c23c20c3c14c24c17c19c28c28c3c14c24c17c22c24c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c24c19c3c95c3c14c22c17c25c27c26c3c14c22c17c28c22c23c3c14c23c17c20c27c20c3c14c23c17c23c22c19c3c14c23c17c25c26c28c3c14c23c17c28c22c19c3c14c24c17c20c27c20c3c14c24c17c23c22c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c25c19c3c95c3c14c22c17c24c25c26c3c14c22c17c27c19c26c3c14c23c17c19c23c26c3c14c23c17c21c27c28c3c14c23c17c24c22c20c3c14c23c17c26c26c24c3c14c24c17c19c20c28c3c14c24c17c21c25c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c26c19c3c95c3c14c22c17c23c24c28c3c14c22c17c25c28c20c3c14c22c17c28c21c24c3c14c23c17c20c25c19c3c14c23c17c22c28c25c3c14c23c17c25c22c22c3c14c23c17c27c26c20c3c14c24c17c20c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c27c19c3c95c3c14c22c17c22c25c20c3c14c22c17c24c27c26c3c14c22c17c27c20c23c3c14c23c17c19c23c21c3c14c23c17c21c26c21c3c14c23c17c24c19c21c3c14c23c17c26c22c23c3c14c23c17c28c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c28c19c3c95c3c14c22c17c21c26c22c3c14c22c17c23c28c21c3c14c22c17c26c20c22c3c14c22c17c28c22c24c3c14c23c17c20c24c28c3c14c23c17c22c27c22c3c14c23c17c25c19c27c3c14c23c17c27c22c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c19c19c3c95c3c14c22c17c20c28c22c3c14c22c17c23c19c26c3c14c22c17c25c21c21c3c14c22c17c27c22c27c3c14c23c17c19c24c24c3c14c23c17c21c26c23c3c14c23c17c23c28c22c3c14c23c17c26c20c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c20c19c3c95c3c14c22c17c20c21c20c3c14c22c17c22c21c28c3c14c22c17c24c22c28c3c14c22c17c26c23c28c3c14c22c17c28c25c20c3c14c23c17c20c26c23c3c14c23c17c22c27c27c3c14c23c17c25c19c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c21c19c3c95c3c14c22c17c19c24c26c3c14c22c17c21c25c19c3c14c22c17c23c25c23c3c14c22c17c25c25c28c3c14c22c17c27c26c24c3c14c23c17c19c27c21c3c14c23c17c21c28c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c22c19c3c95c3c14c22c17c19c19c19c3c14c22c17c20c28c26c3c14c22c17c22c28c25c3c14c22c17c24c28c25c3c14c22c17c26c28c26c3c14c22c17c28c28c28c3c14c23c17c21c19c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c23c19c3c95c3c14c21c17c28c23c28c3c14c22c17c20c23c20c3c14c22c17c22c22c24c3c14c22c17c24c22c19c3c14c22c17c26c21c25c3c14c22c17c28c21c22c3c14c23c17c20c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c24c19c3c95c3c14c21c17c28c19c22c3c14c22c17c19c28c20c3c14c22c17c21c27c19c3c14c22c17c23c26c19c3c14c22c17c25c25c20c3c14c22c17c27c24c23c3c14c23c17c19c23c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c25c19c3c95c3c14c21c17c27c25c22c3c14c22c17c19c23c25c3c14c22c17c21c22c20c3c14c22c17c23c20c25c3c14c22c17c25c19c22c3c14c22c17c26c28c20c3c14c22c17c28c27c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c26c19c3c95c3c14c21c17c27c21c27c3c14c22c17c19c19c26c3c14c22c17c20c27c26c3c14c22c17c22c25c27c3c14c22c17c24c24c19c3c14c22c17c26c22c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c27c19c3c95c3c14c21c17c26c28c27c3c14c21c17c28c26c21c3c14c22c17c20c23c27c3c14c22c17c22c21c24c3c14c22c17c24c19c22c3c14c22c17c25c27c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c28c19c3c95c3c14c21c17c26c26c20c3c14c21c17c28c23c21c3c14c22c17c20c20c23c3c14c22c17c21c27c25c3c14c22c17c23c25c19c3c14c22c17c25c22c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c19c19c3c95c3c14c21c17c26c23c27c3c14c21c17c28c20c24c3c14c22c17c19c27c22c3c14c22c17c21c24c21c3c14c22c17c23c21c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c20c19c3c95c3c14c21c17c26c21c28c3c14c21c17c27c28c22c3c14c22c17c19c24c26c3c14c22c17c21c21c21c3c14c22c17c22c27c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c21c19c3c95c3c14c21c17c26c20c23c3c14c21c17c27c26c22c3c14c22c17c19c22c23c3c14c22c17c20c28c25c3c14c22c17c22c24c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c22c19c3c95c3c14c21c17c26c19c20c3c14c21c17c27c24c26c3c14c22c17c19c20c24c3c14c22c17c20c26c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c23c19c3c95c3c14c21c17c25c28c20c3c14c21c17c27c23c23c3c14c21c17c28c28c27c3c14c22c17c20c24c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c24c19c3c95c3c14c21c17c25c27c23c3c14c21c17c27c22c23c3c14c21c17c28c27c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c25c19c3c95c3c14c21c17c25c26c28c3c14c21c17c27c21c25c3c14c21c17c28c26c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c26c19c3c95c3c14c21c17c25c26c26c3c14c21c17c27c21c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c27c19c3c95c3c14c21c17c25c26c26c3c14c21c17c27c20c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c28c19c3c95c3c14c21c17c25c26c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 347 c55c68c69c79c72c3c39c25c17c22c3c48c72c68c81c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c24c20c3c87c82c3c19c17c25c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c24c20c19c3c3c19c17c24c21c19c3c3c19c17c24c22c19c3c3c19c17c24c23c19c3c3c19c17c24c24c19c3c3c19c17c24c25c19c3c3c19c17c24c26c19c3c3c19c17c24c27c19c3c3c19c17c24c28c19c3c3c19c17c25c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c28c19c3c95c3c3c19c17c22c24c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c19c19c3c95c3c3c19c17c22c24c24c3c3c19c17c22c24c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c20c19c3c95c3c3c19c17c22c24c27c3c3c19c17c22c25c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c21c19c3c95c3c3c19c17c22c25c19c3c3c19c17c22c25c22c3c3c19c17c22c25c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c22c19c3c95c3c3c19c17c22c25c22c3c3c19c17c22c25c25c3c3c19c17c22c25c28c3c3c19c17c22c26c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c23c19c3c95c3c3c19c17c22c25c25c3c3c19c17c22c25c28c3c3c19c17c22c26c21c3c3c19c17c22c26c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c24c19c3c95c3c3c19c17c22c25c28c3c3c19c17c22c26c21c3c3c19c17c22c26c24c3c3c19c17c22c26c27c3c3c19c17c22c27c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c25c19c3c95c3c3c19c17c22c26c22c3c3c19c17c22c26c24c3c3c19c17c22c26c27c3c3c19c17c22c27c20c3c3c19c17c22c27c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c26c19c3c95c3c3c19c17c22c26c25c3c3c19c17c22c26c28c3c3c19c17c22c27c21c3c3c19c17c22c27c23c3c3c19c17c22c27c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c27c19c3c95c3c3c19c17c22c27c19c3c3c19c17c22c27c22c3c3c19c17c22c27c24c3c3c19c17c22c27c27c3c3c19c17c22c28c19c3c3c19c17c22c28c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c28c19c3c95c3c3c19c17c22c27c23c3c3c19c17c22c27c26c3c3c19c17c22c27c28c3c3c19c17c22c28c20c3c3c19c17c22c28c23c3c3c19c17c22c28c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c19c19c3c95c3c3c19c17c22c27c27c3c3c19c17c22c28c20c3c3c19c17c22c28c22c3c3c19c17c22c28c24c3c3c19c17c22c28c27c3c3c19c17c23c19c19c3c3c19c17c23c19c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c20c19c3c95c3c3c19c17c22c28c22c3c3c19c17c22c28c24c3c3c19c17c22c28c26c3c3c19c17c22c28c28c3c3c19c17c23c19c20c3c3c19c17c23c19c23c3c3c19c17c23c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c21c19c3c95c3c3c19c17c22c28c26c3c3c19c17c22c28c28c3c3c19c17c23c19c20c3c3c19c17c23c19c22c3c3c19c17c23c19c24c3c3c19c17c23c19c27c3c3c19c17c23c20c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c22c19c3c95c3c3c19c17c23c19c21c3c3c19c17c23c19c23c3c3c19c17c23c19c25c3c3c19c17c23c19c27c3c3c19c17c23c20c19c3c3c19c17c23c20c21c3c3c19c17c23c20c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c23c19c3c95c3c3c19c17c23c19c25c3c3c19c17c23c19c27c3c3c19c17c23c20c19c3c3c19c17c23c20c21c3c3c19c17c23c20c23c3c3c19c17c23c20c25c3c3c19c17c23c20c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c24c19c3c95c3c3c19c17c23c20c20c3c3c19c17c23c20c22c3c3c19c17c23c20c24c3c3c19c17c23c20c25c3c3c19c17c23c20c27c3c3c19c17c23c21c19c3c3c19c17c23c21c21c3c3c19c17c23c21c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c25c19c3c95c3c3c19c17c23c20c25c3c3c19c17c23c20c27c3c3c19c17c23c20c28c3c3c19c17c23c21c20c3c3c19c17c23c21c22c3c3c19c17c23c21c24c3c3c19c17c23c21c25c3c3c19c17c23c21c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c26c19c3c95c3c3c19c17c23c21c20c3c3c19c17c23c21c21c3c3c19c17c23c21c23c3c3c19c17c23c21c25c3c3c19c17c23c21c26c3c3c19c17c23c21c28c3c3c19c17c23c22c20c3c3c19c17c23c22c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c27c19c3c95c3c3c19c17c23c21c25c3c3c19c17c23c21c26c3c3c19c17c23c21c28c3c3c19c17c23c22c19c3c3c19c17c23c22c21c3c3c19c17c23c22c23c3c3c19c17c23c22c24c3c3c19c17c23c22c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c28c19c3c95c3c3c19c17c23c22c20c3c3c19c17c23c22c21c3c3c19c17c23c22c23c3c3c19c17c23c22c24c3c3c19c17c23c22c26c3c3c19c17c23c22c27c3c3c19c17c23c23c19c3c3c19c17c23c23c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c19c19c3c95c3c3c19c17c23c22c25c3c3c19c17c23c22c27c3c3c19c17c23c22c28c3c3c19c17c23c23c19c3c3c19c17c23c23c21c3c3c19c17c23c23c22c3c3c19c17c23c23c24c3c3c19c17c23c23c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c20c19c3c95c3c3c19c17c23c23c21c3c3c19c17c23c23c22c3c3c19c17c23c23c23c3c3c19c17c23c23c24c3c3c19c17c23c23c25c3c3c19c17c23c23c27c3c3c19c17c23c23c28c3c3c19c17c23c24c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c21c19c3c95c3c3c19c17c23c23c26c3c3c19c17c23c23c27c3c3c19c17c23c23c28c3c3c19c17c23c24c19c3c3c19c17c23c24c20c3c3c19c17c23c24c22c3c3c19c17c23c24c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c22c19c3c95c3c3c19c17c23c24c21c3c3c19c17c23c24c22c3c3c19c17c23c24c23c3c3c19c17c23c24c24c3c3c19c17c23c24c25c3c3c19c17c23c24c27c3c3c19c17c23c24c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c23c19c3c95c3c3c19c17c23c24c27c3c3c19c17c23c24c27c3c3c19c17c23c24c28c3c3c19c17c23c25c19c3c3c19c17c23c25c20c3c3c19c17c23c25c22c3c3c19c17c23c25c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c24c19c3c95c3c3c19c17c23c25c22c3c3c19c17c23c25c23c3c3c19c17c23c25c24c3c3c19c17c23c25c25c3c3c19c17c23c25c26c3c3c19c17c23c25c27c3c3c19c17c23c25c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c25c19c3c95c3c3c19c17c23c25c27c3c3c19c17c23c25c28c3c3c19c17c23c26c19c3c3c19c17c23c26c20c3c3c19c17c23c26c21c3c3c19c17c23c26c22c3c3c19c17c23c26c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c26c19c3c95c3c3c19c17c23c26c23c3c3c19c17c23c26c24c3c3c19c17c23c26c24c3c3c19c17c23c26c25c3c3c19c17c23c26c26c3c3c19c17c23c26c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c27c19c3c95c3c3c19c17c23c26c28c3c3c19c17c23c27c19c3c3c19c17c23c27c20c3c3c19c17c23c27c20c3c3c19c17c23c27c21c3c3c19c17c23c27c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c28c19c3c95c3c3c19c17c23c27c24c3c3c19c17c23c27c24c3c3c19c17c23c27c25c3c3c19c17c23c27c26c3c3c19c17c23c27c26c3c3c19c17c23c27c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c19c19c3c95c3c3c19c17c23c28c19c3c3c19c17c23c28c20c3c3c19c17c23c28c21c3c3c19c17c23c28c21c3c3c19c17c23c28c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c20c19c3c95c3c3c19c17c23c28c25c3c3c19c17c23c28c26c3c3c19c17c23c28c26c3c3c19c17c23c28c27c3c3c19c17c23c28c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c21c19c3c95c3c3c19c17c24c19c21c3c3c19c17c24c19c21c3c3c19c17c24c19c21c3c3c19c17c24c19c22c3c3c19c17c24c19c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c22c19c3c95c3c3c19c17c24c19c26c3c3c19c17c24c19c27c3c3c19c17c24c19c27c3c3c19c17c24c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c23c19c3c95c3c3c19c17c24c20c22c3c3c19c17c24c20c22c3c3c19c17c24c20c23c3c3c19c17c24c20c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c24c19c3c95c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c19c17c24c20c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c25c19c3c95c3c3c19c17c24c21c23c3c3c19c17c24c21c23c3c3c19c17c24c21c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c26c19c3c95c3c3c19c17c24c22c19c3c3c19c17c24c22c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c27c19c3c95c3c3c19c17c24c22c25c3c3c19c17c24c22c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c28c19c3c95c3c3c19c17c24c23c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 348 c55c68c69c79c72c3c39c25c17c23c3c47c16c86c70c68c79c72c3c82c73c3c47c16c74c68c80c80c68c3c71c76c86c87c85c76c69c88c87c76c82c81c3c68c86c3c73c88c81c70c87c76c82c81c3c82c73c3c87c75c72c3c80c72c71c76c68c81 c68c81c71c3c89c68c79c88c72c86c3c11c19c17c24c20c3c87c82c3c19c17c25c19c12c3c82c73c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c16c16c181c86c181c3c71c72c81c82c87c72c86c3c68c3c86c76c80c88c79c68c87c72c71c3c89c68c79c88c72c17 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c48c72c71c76c68c81c3c89c68c79c88c72c86 c3c3c95c3c3c3c95c3c57c68c79c88c72c86c3c73c82c85c3c76c81c87c72c85c16c87c72c85c70c76c79c72c3c85c68c81c74c72c3c16c16c16c16c16c16c16c33c3 c3c3c57c3c3c3c95c3c3c19c17c24c20c19c3c3c19c17c24c21c19c3c3c19c17c24c22c19c3c3c19c17c24c23c19c3c3c19c17c24c24c19c3c3c19c17c24c25c19c3c3c19c17c24c26c19c3c3c19c17c24c27c19c3c3c19c17c24c28c19c3c3c19c17c25c19c19 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 c19c17c20c27c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c20c28c19c3c95c3c3c19c17c21c19c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c19c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c20c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c21c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c22c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c23c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c24c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c25c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c26c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c27c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c21c28c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c19c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c20c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c21c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c22c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c23c19c3c95c3c3c19c17c21c19c19c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c24c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c25c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c26c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c27c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c22c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c22c28c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c22c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c19c19c3c95c3c3c19c17c21c19c20c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c19c17c21c20c20c3c3c19c17c21c20c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c20c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c19c17c21c20c20c3c3c19c17c21c20c22c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c21c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c22c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c23c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c19c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c24c19c3c95c3c3c19c17c21c19c21c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c19c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c25c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c19c17c21c20c21c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c26c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c27c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c19c28c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c23c28c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c19c17c21c20c20c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c19c19c3c95c3c3c19c17c21c19c22c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c20c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c27c3c3c19c17c21c20c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c21c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c19c17c21c20c19c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c22c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c23c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c26c3c3c19c17c21c19c28c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c24c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c25c19c3c95c3c3c19c17c21c19c23c3c3c19c17c21c19c25c3c3c19c17c21c19c27c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c26c19c3c95c3c3c19c17c21c19c24c3c3c19c17c21c19c25c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c27c19c3c95c3c3c19c17c21c19c24c3c3c19c17c21c19c26c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c24c28c19c3c95c3c3c19c17c21c19c24c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c19c17c25c19c19c3c95c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3c3c3c3c16c16c3c3 c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32c32 349 APPENDIX E Computer Programs The Perl programming language was used universally for this dissertation. Ports of several FORTRAN algorithms were made, and references are provided where necessary. The executable programs end in c17c83c79 extensions, and the modules end in c17c83c80 extensions. 350 lmoments.pl c6c4c18c88c86c85c18c69c76c81c18c83c72c85c79c3c16c90 c88c86c72c3c86c87c85c76c70c87c30 c6c3c51c72c85c79c3c83c85c82c74c85c68c80c3c87c82c3c70c82c80c83c88c87c72c3c47c16c80c82c80c72c81c87c86c3c88c86c76c81c74c3c88c81c69c76c68c86c72c71c15c3c83c79c82c87c87c76c81c74c16c83c82c86c76c87c76c82c81c15 c6c3c68c81c71c3c83c85c76c82c85c16c51c85c82c69c68c69c76c79c76c87c92c3c58c72c76c74c75c87c72c71c3c48c82c80c72c81c87c86c17c3c3c55c75c72c3c83c85c82c71c88c70c87c3c80c82c80c72c81c87c86c3c70c68c81c3c68c79c86c82 c6c3c69c72c3c70c82c80c83c88c87c72c71c17c3c3c36c79c74c82c85c76c87c75c80c86c3c71c72c85c76c89c72c71c3c73c85c82c80c3c81c88c80c72c85c82c88c86c3c86c82c88c85c70c72c86c3c69c92c3c45c17c53c17c48c3c43c82c86c78c76c81c74 c6c3c68c81c71c3c52c17c45c17c3c58c68c81c74c17c3c3c38c76c87c68c87c76c82c81c86c3c83c85c82c89c76c71c72c71c3c76c81c3c71c76c86c86c72c85c87c68c87c76c82c81c3c69c92c3c58c17c43c17c3c36c86c84c88c76c87c75c17 c6c3c36c56c55c43c50c53c29c3c58c76c79c79c76c68c80c3c43c17c3c36c86c84c88c76c87c75c15c3c36c88c74c88c86c87c3c21c19c19c21 c6c3c7c36c88c87c75c82c85c29c3c90c68c86c84c88c76c87c75c3c7 c6c3c7c39c68c87c72c29c3c21c19c19c21c18c20c20c18c19c25c3c20c23c29c22c19c29c22c22c3c7 c6c3c7c53c72c89c76c86c76c82c81c29c3c20c17c20c22c3c7 c88c86c72c3c70c82c81c86c87c68c81c87c3c86c84c85c87c51c44c3c32c33c3c86c70c68c79c68c85c3c86c84c85c87c11c22c17c20c23c20c24c28c21c25c12c30 c80c92c3c7c39c36c55c36c30c3c3c6c3c75c68c86c75c3c85c72c73c72c85c72c81c70c72c3c87c82c3c87c75c72c3c83c85c82c69c68c69c76c79c76c87c76c72c86c3c68c81c71c3c71c68c87c68 c80c92c3c7c49c50c39c36c55c36c3c32c3c182c16c16c183c30 c6c3c58c82c85c78c3c82c81c3c90c75c68c87c3c87c92c83c72c3c82c73c3c83c85c82c69c68c69c76c79c76c87c92c3c90c72c76c74c75c87c72c71c3c80c82c80c72c81c87 c6c3c70c82c80c83c88c87c68c87c76c82c81c86c3c68c85c72c3c74c82c76c81c74c3c87c82c3c69c72c3c83c72c85c73c82c85c80c72c71c3c68c81c71c3c83c85c82c70c72c86c86c3c82c87c75c72c85 c6c3c70c82c80c80c68c81c71c3c79c76c81c72c3c82c83c87c76c82c81c86c17 c88c86c72c3c42c72c87c82c83c87c29c29c47c82c81c74c30 c80c92c3c8c50c51c55c54c3c32c3c11c12c30c3c6c3c70c82c80c80c68c81c71c3c79c76c81c72c3c82c83c87c76c82c81c86 c80c92c3c35c82c83c87c76c82c81c86c3c32c3c84c90c3c11c3c83c83c3c91c73c3c88c69c3c83c80c3c71c68c87c68c3c71c68c87c68c79c76c80c76c87c86c3c41c71c76c89c20c19c19c3c75c72c79c83c3c12c30 c6c3c87c75c72c86c72c3c68c85c72c3c87c75c72c3c89c68c79c76c71c3c70c82c80c80c68c81c71c3c79c76c81c72c3c82c83c87c76c82c81c86 c9c42c72c87c50c83c87c76c82c81c86c11c63c8c50c51c55c54c15c3c35c82c83c87c76c82c81c86c12c30c3c6c3c83c68c85c86c72c3c87c75c72c3c70c82c80c80c68c81c71c3c79c76c81c72c3c82c83c87c76c82c81c86 c9c43c72c79c83c11c12c15c3c72c91c76c87c3c76c73c11c7c50c51c55c54c94c75c72c79c83c96c12c30 c80c92c3c7c87c92c83c72c3c32c3c11c7c50c51c55c54c94c83c83c96c12c3c34c3c180c51c51c181c3c29 c3c3c3c3c3c3c3c3c3c3c3c11c7c50c51c55c54c94c91c73c96c12c3c34c3c180c59c41c181c3c29 c3c3c3c3c3c3c3c3c3c3c3c11c7c50c51c55c54c94c88c69c96c12c3c34c3c180c56c37c181c3c29 c3c3c3c3c3c3c3c3c3c3c3c11c7c50c51c55c54c94c83c80c96c12c3c34c3c180c51c48c181c3c29c3c88c81c71c72c73c30 c71c76c72c3c180c48c82c80c72c81c87c3c70c82c80c83c88c87c68c87c76c82c81c3c87c92c83c72c3c76c86c3c81c82c87c3c71c72c73c76c81c72c71c3c69c92c3c180c17 c3c3c3c3c180c16c88c69c15c3c16c83c83c15c3c16c91c73c15c3c82c85c3c16c83c80c3c82c81c3c87c75c72c3c70c82c80c80c68c81c71c3c79c76c81c72c63c81c181c3c88c81c79c72c86c86c11c7c87c92c83c72c12c30 c6c3c40c81c71c3c82c73c3c70c82c80c80c68c81c71c3c79c76c81c72c3c82c83c87c76c82c81c3c75c68c81c71c79c76c81c74c17 c83c85c76c81c87c3c180c6c3c47c16c48c50c48c40c49c55c54c3c50c41c3c36c3c38c56c48c56c47c36c55c44c57c40c3c51c40c53c38c40c49c55c36c42c40c3c43c60c39c53c50c42c53c36c51c43c63c81c181c30c3 c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c40c81c87c72c85c3c86c83c68c70c72c3c71c72c79c76c80c76c87c72c71c3c70c88c80c88c79c68c87c76c89c72c3c83c85c82c69c68c69c76c79c76c87c92c3c68c81c71c3c71c68c87c68c3 c89c68c79c88c72c86c17c63c81c181c30 c76c73c11c7c87c92c83c72c3c72c84c3c182c59c41c183c12c3c94 c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c58c82c85c78c76c81c74c3c82c81c3c7c87c92c83c72c3c11c83c85c76c82c85c16c51c58c48c12c17c63c81c181c30 c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c50c81c72c3c83c68c76c85c3c83c72c85c3c79c76c81c72c17c63c81c181c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c180c6c3c41c3c3c3c3c59c63c81c181c30 c96 c72c79c86c72c3c94 c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c58c82c85c78c76c81c74c3c82c81c3c7c87c92c83c72c3c11c51c51c15c3c83c79c82c87c87c76c81c74c3c83c82c86c76c87c76c82c81c30c3c56c37c15c3c88c81c69c76c68c86c72c71c30c3c51c48c15c3 c83c85c82c71c88c70c87c3c80c82c80c72c81c87c12c17c63c81c181c30 c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c44c73c3c82c81c72c3c89c68c79c88c72c3c76c86c3c74c76c89c72c81c15c3c87c75c72c81c3c76c87c3c76c86c3c88c86c72c71c17c3c3c44c73c3c87c90c82c3c89c68c79c88c72c86c3c68c85c72c3 c74c76c89c72c81c15c3c87c75c72c81c3c87c75c72c63c81c181c15 351 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c180c6c3c86c72c70c82c81c71c3c82c81c72c3c76c86c3c88c86c72c71c17c3c3c55c75c76c86c3c73c72c68c87c88c85c72c3c83c72c85c80c76c87c86c3c86c90c76c87c70c75c76c81c74c3c69c72c87c90c72c72c81c3 c87c75c72c3c83c85c76c82c85c16c51c58c48c63c81c181c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c180c6c3c80c72c87c75c82c71c3c68c81c71c3c87c75c72c3c82c87c75c72c85c3c80c72c87c75c82c71c86c3c90c76c87c75c82c88c87c3c68c3c70c75c68c81c74c72c3c76c81c3c87c75c72c3c76c81c83c88c87c3 c86c87c85c72c68c80c17c63c81c181c30 c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c50c81c72c3c89c68c79c88c72c3c82c85c3c41c3c3c3c59c3c83c68c76c85c3c83c72c85c3c79c76c81c72c63c81c181c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c180c6c3c59c3c3c82c85c3c3c41c11c76c74c81c82c85c72c71c12c3c3c3c59c63c81c181c30 c96 c80c92c3c7c76c3c32c3c19c30 c80c92c3c7c41c66c70c82c88c81c87c66c69c72c79c82c90c66c19c24c30 c90c75c76c79c72c11c20c12c3c94 c3c3c7c66c3c32c3c31c54c55c39c44c49c33c30 c3c3c79c68c86c87c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c66c12c30 c3c3c81c72c91c87c3c76c73c11c18c62c68c16c93c36c16c61c64c95c65c6c18c12c30c3c6c3c84c88c76c72c87c79c92c3c86c78c76c83c3c68c81c92c3c81c82c81c81c88c80c72c85c76c70c3c79c76c81c72c86c3c82c85c3c79c72c68c71c76c81c74c3c6 c3c3c70c75c82c80c83c30 c3c3c79c68c86c87c3c76c73c11c7c66c3c72c84c3c180c180c12c30 c3c3c80c92c3c11c7c41c15c3c7c91c12c3c32c3c86c83c79c76c87c11c18c63c86c14c18c82c15c3c7c66c15c3c21c12c30 c3c3 c3c3c6c3c76c81c86c72c85c87c3c71c88c80c80c92c3c41c3c76c73c3c87c92c83c72c3c56c37c3c82c85c3c51c51c3c68c81c71c3c82c81c79c92c3c82c81c72c3c89c68c79c88c72c3c72c81c87c72c85c72c71c17 c3c3c6c3c68c3c83c85c76c82c85c76c3c41c3c76c86c3c81c82c87c3c81c72c72c71c72c71c3c73c82c85c3c81c82c81c16c83c51c58c48c3c70c82c80c83c88c87c68c87c76c82c81c86 c3c3c11c7c91c15c3c7c41c12c3c32c3c11c7c41c15c3c7c76c12c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c3c68c81c71c3c7c87c92c83c72c3c81c72c3c180c59c41c181c12c30c3 c3c3c7c41c66c70c82c88c81c87c66c69c72c79c82c90c66c19c24c14c14c3c76c73c11c7c41c3c31c3c19c17c24c12c30 c3c3c81c72c91c87c3c76c73c11c81c82c87c3c9c76c86c49c88c80c69c72c85c11c7c41c12c3c82c85c3c81c82c87c3c9c76c86c49c88c80c69c72c85c11c7c91c12c12c30 c3c3c7c41c3c18c32c3c20c19c19c3c76c73c11c7c50c51c55c54c94c182c41c71c76c89c20c19c19c183c96c12c30c3c6c3c87c75c72c3c83c85c82c69c68c69c76c79c76c87c76c72c86c3c70c68c81c3c76c81c3c68c86c3c83c72c85c70c72c81c87c68c74c72c86 c3c3c71c76c72c3c180c37c68c71c3c71c68c87c68c3c72c81c87c72c85c72c71c15c3c41c3c76c86c3c81c82c87c3c71c72c73c76c81c72c71c17c63c81c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c41c12c30 c3c3c71c76c72c3c180c37c68c71c3c71c68c87c68c3c72c81c87c72c85c72c71c15c3c59c3c76c73c3c81c82c87c3c71c72c73c76c81c72c71c17c63c81c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c12c30 c3c3 c3c3c7c39c36c55c36c16c33c94c7c41c96c3c32c3c7c91c30 c3c3c7c76c14c14c30 c96 c83c85c76c81c87c3c54c55c39c40c53c53c3c180c6c3c51c85c82c69c68c69c76c79c76c87c76c72c86c3c68c81c71c3c39c68c87c68c3c75c68c89c72c3c69c72c72c81c3c72c81c87c72c85c72c71c17c3c17c3c17c63c81c181c30 c76c73c11c7c87c92c83c72c3c81c72c3c182c51c48c183c12c3c94 c3c3c80c92c3c11c7c51c58c48c15c3c7c47c48c50c48c12c30 c3c3c80c92c3c7c80c72c71c76c68c81c3c32c3c88c81c71c72c73c30 c3c3c76c73c11c7c87c92c83c72c3c72c84c3c182c56c37c183c12c3c94 c3c3c3c3c3c80c92c3c35c71c68c87c68c30 c3c3c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c78c72c92c86c3c8c7c39c36c55c36c12c3c94 c3c3c3c3c3c3c3c83c88c86c75c11c35c71c68c87c68c15c3c7c39c36c55c36c16c33c94c7c41c96c12c30 c3c3c3c3c3c96 c3c3c3c3c3c7c47c48c50c48c3c32c3c9c47c76c81c72c68c85c48c82c80c72c81c87c86c11c62c35c71c68c87c68c64c15c3c20c12c30 c3c3c3c3c3c7c51c58c48c3c3c32c3c9c47c48c50c48c21c51c58c48c11c7c47c48c50c48c12c30 c3c3c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c78c72c92c3c11c78c72c92c86c3c8c7c47c48c50c48c12c3c94 c3c3c3c3c3c3c3c3c7c47c48c50c48c16c33c94c7c78c72c92c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c47c48c50c48c16c33c94c7c78c72c92c96c12c30 c3c3c3c3c3c96 c3c3c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c83c3c11c35c7c51c58c48c12c3c94 c3c3c3c3c3c3c3c3c7c83c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c83c12c30 c3c3c3c3c3c96 c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c9c48c72c71c76c68c81c11c35c71c68c87c68c12c30 c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c76c81c87c11c20c19c19c19c19c13c7c80c72c71c76c68c81c12c18c20c19c19c19c19c3c76c73c11c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c12c30 c3c3c96 c3c3c72c79c86c72c3c94 c3c3c3c3c76c73c11c7c87c92c83c72c3c72c84c3c182c59c41c183c12c3c94 352 c3c3c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c9c48c72c71c76c68c81c66c90c76c87c75c66c51c85c76c82c85c51c85c82c69c68c69c76c79c76c87c76c72c86c11c7c39c36c55c36c12c30 c3c3c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c76c81c87c11c20c19c19c19c19c13c7c80c72c71c76c68c81c12c18c20c19c19c19c19c3c76c73c11c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c12c30 c3c3c3c3c96 c3c3c3c3c72c79c86c72c3c94 c3c3c3c3c3c3c3c80c92c3c35c71c68c87c68c30 c3c3c3c3c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c78c72c92c86c3c8c7c39c36c55c36c12c3c94 c3c3c3c3c3c3c3c3c3c83c88c86c75c11c35c71c68c87c68c15c3c7c39c36c55c36c16c33c94c7c41c96c12c30 c3c3c3c3c3c3c3c96 c3c3c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c9c48c72c71c76c68c81c11c35c71c68c87c68c12c30 c3c3c3c3c3c3c3c7c80c72c71c76c68c81c3c32c3c76c81c87c11c20c19c19c19c19c13c7c80c72c71c76c68c81c12c18c20c19c19c19c19c3c76c73c11c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c12c30 c3c3c3c3c96 c3c3c3c3c7c51c58c48c3c3c32c3c9c70c82c80c83c88c87c72c51c58c48c11c7c87c92c83c72c15c7c39c36c55c36c12c30 c3c3c3c3c7c47c48c50c48c3c32c3c9c51c58c48c21c47c48c50c48c11c7c51c58c48c12c30 c3c3c96 c3c3c7c80c72c71c76c68c81c3c32c3c180c16c16c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c12c30 c3c3c83c85c76c81c87c3c180c6c3c47c16c80c82c80c72c81c87c3c38c82c80c83c88c87c68c87c76c82c81c63c81c181c30 c3c3c83c85c76c81c87c3c9c51c58c48c68c86c54c87c85c76c81c74c11c7c51c58c48c12c15c181c63c81c181c30 c3c3c83c85c76c81c87c3c9c47c48c50c48c68c86c54c87c85c76c81c74c11c7c47c48c50c48c12c15c181c63c81c181c30 c3c3c83c85c76c81c87c3c180c6c3c54c68c80c83c79c72c86c3c32c3c7c76c63c81c181c30 c3c3c83c85c76c81c87c3c180c6c3c48c72c68c81c3c3c3c47c16c86c70c68c79c72c3c3c3c47c16c38c57c3c3c3c3c47c16c86c78c72c90c3c3c3c47c16c78c88c85c87c82c86c76c86c3c3c3c55c68c88c24c3c3c3c48c72c71c76c68c81c63c81c181c30 c3c3c83c85c76c81c87c3c9c47c48c50c48c68c86c53c82c90c11c7c47c48c50c48c12c15c181c3c3c3c7c80c72c71c76c68c81c63c81c181c30 c96 c72c79c86c72c3c94 c3c3c3c80c92c3c35c71c68c87c68c30 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c78c72c92c86c3c8c7c39c36c55c36c12c3c94 c3c3c3c3c3c3c83c88c86c75c11c35c71c68c87c68c15c3c7c39c36c55c36c16c33c94c7c41c96c12c30 c3c3c3c96 c3c3c3c80c92c3c7c86c68c80c83c79c72c86c3c32c3c86c70c68c79c68c85c11c35c71c68c87c68c12c30 c3c3c3c80c92c3c7c80c76c81c3c3c3c3c3c32c3c7c71c68c87c68c62c19c64c30 c3c3c3c80c92c3c7c80c68c91c3c3c3c3c3c32c3c7c71c68c87c68c62c7c6c71c68c87c68c64c30 c3c3c3c80c92c3c7c80c72c71c76c68c81c3c3c32c3c9c48c72c71c76c68c81c11c35c71c68c87c68c12c30c3 c3c3c3c3c3c3c7c80c72c71c76c68c81c3c3c32c3c76c81c87c11c20c19c19c19c19c13c7c80c72c71c76c68c81c12c18c20c19c19c19c19c3c76c73c11c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c12c30 c3c3c3c80c92c3c7c75c68c85c80c72c68c81c3c32c3c9c43c68c85c80c82c81c76c70c48c72c68c81c11c35c71c68c87c68c12c30 c3c3c3c80c92c3c7c74c72c82c80c72c68c81c3c32c3c9c42c72c82c80c72c87c85c76c70c48c72c68c81c11c35c71c68c87c68c12c30 c3c3c3c80c92c3c11c7c80c72c68c81c15c7c86c87c71c72c89c15c7c86c78c72c90c15c7c78c88c85c87c12c3c32c3c9c51c85c82c71c88c70c87c48c82c80c72c81c87c86c11c35c71c68c87c68c12c30 c3c3c3c80c92c3c7c70c89c3c32c3c7c86c87c71c72c89c3c18c3c7c80c72c68c81c30 c3c3c3c7c80c72c71c76c68c81c3c3c32c3c11c71c72c73c76c81c72c71c3c7c80c72c71c76c68c81c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c80c72c71c76c68c81c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c75c68c85c80c72c68c81c3c32c3c11c71c72c73c76c81c72c71c3c7c75c68c85c80c72c68c81c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c75c68c85c80c72c68c81c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c74c72c82c80c72c68c81c3c32c3c11c71c72c73c76c81c72c71c3c7c74c72c82c80c72c68c81c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c74c72c82c80c72c68c81c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c80c72c68c81c3c3c3c3c32c3c11c71c72c73c76c81c72c71c3c7c80c72c68c81c3c3c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c80c72c68c81c3c3c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c86c87c71c72c89c3c3c3c32c3c11c71c72c73c76c81c72c71c3c7c86c87c71c72c89c3c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c86c87c71c72c89c3c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c86c78c72c90c3c3c3c3c32c3c11c71c72c73c76c81c72c71c3c7c86c78c72c90c3c3c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c86c78c72c90c3c3c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c78c88c85c87c3c3c3c3c32c3c11c71c72c73c76c81c72c71c3c7c78c88c85c87c3c3c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c78c88c85c87c3c3c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c7c70c89c3c3c3c3c3c3c32c3c11c71c72c73c76c81c72c71c3c7c70c89c3c3c3c3c3c12c3c34c3c86c83c85c76c81c87c73c11c180c8c19c17c25c73c181c15c7c70c89c3c3c3c3c3c12c3c29c3c182c88c81c71c72c73c183c30 c3c3c3c83c85c76c81c87c3c180c6c3c51c85c82c71c88c70c87c3c48c82c80c72c81c87c3c38c82c80c83c88c87c68c87c76c82c81c63c81c181c30 c3c3c3c83c85c76c81c87c3c180c6c3c54c68c80c83c79c72c86c3c32c3c7c86c68c80c83c79c72c86c63c81c181c30 c3c3c3c83c85c76c81c87c3c180c48c72c68c81c3c3c3c54c87c71c72c89c3c3c38c57c3c3c54c78c72c90c3c3c3c46c88c85c87c82c86c76c86c3c3c48c72c71c76c68c81c3c3c180c15 c3c3c3c3c3c3c3c3c3c180c42c72c82c80c72c87c85c76c70c48c72c68c81c3c3c43c68c85c80c82c81c76c70c48c72c68c81c63c81c181c30 c3c3c3c83c85c76c81c87c3c77c82c76c81c11c180c3c3c3c180c15c11c7c80c72c68c81c15c7c86c87c71c72c89c15c7c70c89c15c7c86c78c72c90c15c7c78c88c85c87c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c7c74c72c82c80c72c68c81c15c7c75c68c85c80c72c68c81c12c12c15c181c63c81c181c30 c96 c83c85c76c81c87c3c9c71c88c80c83c39c68c87c68c11c7c39c36c55c36c12c3c76c73c11c7c50c51c55c54c94c182c71c68c87c68c183c96c12c30 c83c85c76c81c87c3c9c71c88c80c83c39c68c87c68c47c76c80c76c87c86c11c7c39c36c55c36c12c3c76c73c11c7c50c51c55c54c94c182c71c68c87c68c79c76c80c76c87c86c183c96c12c30 c83c85c76c81c87c3c180c63c81c181c30 353 c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6 c6c3c54c56c37c53c50c56c55c44c49c40c54 c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6c6 c86c88c69c3c70c82c80c83c88c87c72c51c58c48c3c94 c3c3c3c80c92c3c11c7c87c92c83c72c15c3c7c71c68c87c68c12c3c32c3c35c66c30 c3c3c3c80c92c3c7c83c90c80c30 c3c3c3c83c85c76c81c87c3c180c6c3c51c58c48c86c3c82c73c3c87c92c83c72c3c7c87c92c83c72c3c68c85c72c3c86c87c68c85c87c72c71c3c180c30 c3c3c3c80c68c83c3c94c3c7c83c90c80c16c33c62c7c66c64c3c32c3c9c74c72c87c51c58c48c11c7c87c92c83c72c15c7c66c15c7c71c68c87c68c12c3c96c3c11c19c17c17c23c12c30 c3c3c3c83c85c76c81c87c3c180c68c81c71c3c70c82c80c83c79c72c87c72c71c17c63c81c181c30 c3c3c3c85c72c87c88c85c81c3c7c83c90c80c30 c96 c86c88c69c3c74c72c87c51c58c48c3c94 c3c3c3c80c92c3c11c7c90c75c76c70c75c12c3c32c3c86c75c76c73c87c30 c3c3c3c85c72c87c88c85c81c3c9c74c72c87c51c58c48c69c92c51c51c11c35c66c12c3c76c73c11c7c90c75c76c70c75c3c72c84c3c182c51c51c183c12c30 c3c3c3c85c72c87c88c85c81c3c9c74c72c87c51c58c48c69c92c59c41c11c35c66c12c3c76c73c11c7c90c75c76c70c75c3c72c84c3c182c59c41c183c12c30 c3c3c3c71c76c72c3c180c74c72c87c51c58c48c3c86c75c82c88c79c71c3c81c82c87c3c69c72c3c75c72c85c72c3c79c82c74c76c70c68c79c79c92c63c81c181c30 c96 c86c88c69c3c74c72c87c51c58c48c69c92c51c51c3c94 c3c3c3c80c92c3c11c7c82c85c71c72c85c15c3c7c71c68c87c68c12c3c32c3c35c66c30 c3c3c3c80c92c3c7c69c72c87c68c3c32c3c19c30 c3c3c3c80c92c3c35c41c86c3c3c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c78c72c92c86c3c8c7c71c68c87c68c30 c3c3c3c80c92c3c7c81c41c3c3c3c32c3c86c70c68c79c68c85c11c35c41c86c12c30 c3c3c3c80c92c3c7c76c3c3c3c3c32c3c20c30 c3c3c3 c3c3c3c85c72c87c88c85c81c3c7c49c50c39c36c55c36c3c76c73c11c7c81c41c3c32c32c3c19c3c82c85c3c7c82c85c71c72c85c14c20c3c33c3c7c81c41c12c30 c3c3c3 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c35c41c86c12c3c94 c3c3c3c3c3c3c7c69c72c87c68c3c14c32c3c7c71c68c87c68c16c33c94c7c41c96c3c13c3c11c11c7c76c16c19c17c22c24c12c18c7c81c41c12c13c13c7c82c85c71c72c85c30 c3c3c3c3c3c3c7c76c14c14c30 c3c3c3c96 c3c3c3 c3c3c3c85c72c87c88c85c81c3c9c66c71c72c70c76c80c68c79c86c11c7c69c72c87c68c18c7c81c41c12c30 c96 c86c88c69c3c74c72c87c51c58c48c69c92c59c41c3c94 c3c3c3c80c92c3c11c7c82c85c71c72c85c15c3c7c71c68c87c68c12c3c32c3c35c66c30 c3c3c3c80c92c3c7c69c72c87c68c3c32c3c19c30 c3c3c3c80c92c3c35c41c86c3c3c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c78c72c92c86c3c8c7c71c68c87c68c30 c3c3c3c80c92c3c7c81c41c3c3c3c32c3c86c70c68c79c68c85c11c35c41c86c12c30 c3c3c3c80c92c3c11c7c59c15c3c7c71c41c12c3c32c3c19c30 c3c3c3c80c92c3c7c41c30 c3c3c3c85c72c87c88c85c81c3c7c49c50c39c36c55c36c3c76c73c11c7c81c41c3c32c32c3c19c3c82c85c3c7c82c85c71c72c85c14c20c3c33c3c7c81c41c12c30 c3c3c3 c3c3c3c6c3c58c82c85c78c3c82c81c3c73c76c85c86c87c3c71c68c87c68c3c83c82c76c81c87 c3c3c3c7c41c3c3c3c3c3c32c3c3c3c7c41c86c62c19c64c30 c3c3c3c7c71c41c3c3c3c3c32c3c11c3c7c41c86c62c20c64c3c14c3c7c41c86c62c19c64c3c12c3c18c3c21c30 c3c3c3c6c7c71c41c3c3c3c3c32c3c11c3c7c41c86c62c20c64c3c12c3c18c3c21c30 c3c3c3c7c59c3c3c3c3c3c32c3c7c71c68c87c68c16c33c94c7c41c96c30 c3c3c3c7c69c72c87c68c3c14c32c3c7c59c3c13c3c7c41c13c13c7c82c85c71c72c85c3c13c3c7c71c41c30 354 c3c3c3 c3c3c3c6c3c58c82c85c78c3c82c81c3c80c76c71c71c79c72c3c71c68c87c68c3c83c82c76c81c87c86 c3c3c3c73c82c85c11c80c92c3c7c76c32c20c30c7c76c31c32c7c6c41c86c16c20c30c7c76c14c14c12c3c94 c3c3c3c3c3c3c7c41c3c3c3c3c3c32c3c3c3c7c41c86c62c7c76c64c30 c3c3c3c3c3c3c7c71c41c3c3c3c3c32c3c11c3c7c41c86c62c7c76c14c20c64c3c16c3c7c41c86c62c7c76c16c20c64c3c12c3c18c3c21c30 c3c3c3c3c3c3c7c59c3c3c3c3c3c32c3c7c71c68c87c68c16c33c94c7c41c96c30 c3c3c3c3c3c3c7c69c72c87c68c3c14c32c3c7c59c3c13c3c7c41c13c13c7c82c85c71c72c85c3c13c3c7c71c41c30 c3c3c3c96 c3c3c3 c3c3c3c6c3c58c82c85c78c3c82c81c3c79c68c86c87c3c71c68c87c68c3c83c82c76c81c87 c3c3c3c7c41c3c3c32c3c7c41c86c62c7c6c41c86c64c30 c3c3c3c7c71c41c3c32c3c20c3c16c3c11c7c41c86c62c7c6c41c86c64c3c14c3c7c41c86c62c7c6c41c86c16c20c64c12c3c18c3c21c30 c3c3c3c6c7c71c41c3c32c3c11c20c3c16c3c7c41c86c62c7c6c41c86c16c20c64c12c18c21c30 c3c3c3c7c59c3c3c32c3c7c71c68c87c68c16c33c94c7c41c96c30 c3c3c3c7c69c72c87c68c3c14c32c3c7c59c3c13c3c7c41c13c13c7c82c85c71c72c85c3c13c3c7c71c41c30 c3c3c3 c3c3c3c85c72c87c88c85c81c3c9c66c71c72c70c76c80c68c79c86c11c7c69c72c87c68c12c30 c96 c86c88c69c3c51c58c48c21c47c48c50c48c3c94 c3c3c3c80c92c3c11c7c83c90c80c12c3c32c3c35c66c30 c3c3c3c80c92c3c35c83c3c32c3c35c7c83c90c80c30 c3c3c3c80c92c3c8c79c80c82c80c30 c3c3c3 c3c3c3c7c79c80c82c80c94c47c20c96c3c32c3c7c79c80c82c80c94c47c21c96c3c32 c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c7c79c80c82c80c94c55c22c96c3c32 c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c79c80c82c80c94c55c24c96c3c32c3c7c49c50c39c36c55c36c30 c3c3c3 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c83c62c19c64c3c72c84c3c7c49c50c39c36c55c36c12c30 c3c3c3c7c79c80c82c80c94c47c20c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c83c62c19c64c12c30 c3c3c3 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c83c62c20c64c3c72c84c3c7c49c50c39c36c55c36c12c30 c3c3c3c7c79c80c82c80c94c47c21c96c3c32c3c3c21c13c7c83c62c20c64c3c16c3c3c3c3c3c7c83c62c19c64c30 c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c11c7c79c80c82c80c94c47c20c96c12c3c34c3c7c79c80c82c80c94c47c21c96c18c7c79c80c82c80c94c47c20c96c3c29c3c180c80c72c68c81c32c93c72c85c82c181c30 c3c3c3c7c79c80c82c80c94c47c21c96c3c32c3c7c49c50c39c36c55c36c15c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c81c82c87c3c7c79c80c82c80c94c47c21c96c12c30 c3c3c3 c3c3c3c6c3c44c73c3c47c16c86c70c68c79c72c3c76c86c3c79c72c86c86c3c87c75c68c81c3c93c72c85c82c15c3c90c72c3c75c68c89c72c3c83c82c82c85c79c92c3c70c75c82c82c86c72c81 c3c3c3c6c3c41c3c68c81c71c3c59c3c83c68c76c85c86c17c3c3c48c82c85c72c3c71c68c87c68c3c76c86c3c79c76c78c72c79c92c3c81c72c72c71c72c71c3c87c82c3c71c72c73c76c81c72c3 c3c3c3c6c3c87c75c72c3c71c76c86c87c85c76c69c88c87c76c82c81c17 c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c7c79c80c82c80c94c47c21c96c3c32c3c7c49c50c39c36c55c36c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c79c80c82c80c94c47c21c96c3c31c3c19c12c30 c3c3c3c7c79c80c82c80c94c47c21c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c79c80c82c80c94c47c21c96c12c30 c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c79c80c82c80c94c55c21c96c12c30 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c83c62c21c64c3c72c84c3c7c49c50c39c36c55c36c12c30 c3c3c3c7c79c80c82c80c94c47c22c96c3c32c3c3c25c13c7c83c62c21c64c3c16c3c3c3c25c13c7c83c62c20c64c3c14c3c3c3c3c7c83c62c19c64c30 c3c3c3c7c79c80c82c80c94c55c22c96c3c32c3c7c79c80c82c80c94c47c22c96c18c7c79c80c82c80c94c47c21c96c30 c3c3c3 c3c3c3c6c3c44c73c3c47c16c86c78c72c90c3c76c86c3c79c72c86c86c3c87c75c68c81c3c16c20c3c82c85c3c74c85c72c68c87c72c85c3c87c75c68c81c3c20c15c3c90c72c3c75c68c89c72c3c83c82c82c85c79c92c3c70c75c82c82c86c72c81 c3c3c3c6c3c41c3c68c81c71c3c59c3c83c68c76c85c86c17c3c3c48c82c85c72c3c71c68c87c68c3c76c86c3c79c76c78c72c79c92c3c81c72c72c71c72c71c3c87c82c3c71c72c73c76c81c72 c3c3c3c6c3c87c75c72c3c71c76c86c87c85c76c69c88c87c76c82c81c17 c3c3c3c7c79c80c82c80c94c55c22c96c3c32c3c7c79c80c82c80c94c47c22c96c3c32c3c7c49c50c39c36c55c36c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c79c80c82c80c94c55c22c96c3c31c3c16c20c3c82c85c3c7c79c80c82c80c94c55c22c96c3c33c3c20c12c30 c3c3c3c7c79c80c82c80c94c55c22c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c79c80c82c80c94c47c22c96c18c7c79c80c82c80c94c47c21c96c12c30 355 c3c3c3 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c83c62c22c64c3c72c84c3c7c49c50c39c36c55c36c12c30 c3c3c3c7c79c80c82c80c94c47c23c96c3c32c3c21c19c13c7c83c62c22c64c3c16c3c3c22c19c13c7c83c62c21c64c3c14c3c20c21c13c7c83c62c20c64c3c16c3c3c3c3c7c83c62c19c64c30 c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c79c80c82c80c94c47c23c96c18c7c79c80c82c80c94c47c21c96c30 c3c3c3 c3c3c3c6c3c44c73c3c47c16c78c88c85c87c82c86c76c86c3c76c86c3c79c72c86c86c3c87c75c68c81c3c19c17c21c24c13c11c24c13c55c68c88c22c13c13c21c3c16c3c20c12c3c82c85c3c74c85c72c68c87c72c85c3c87c75c68c81c3c20c15 c3c3c3c6c3c90c72c3c75c68c89c72c3c83c82c82c85c79c92c3c70c75c82c82c86c72c81c3c41c3c68c81c71c3c59c3c83c68c76c85c86c17c3c3c48c82c85c72c3c71c68c87c68c3c76c86c3c79c76c78c72c79c92c3c81c72c72c71c72c71c3c87c82 c3c3c3c6c3c71c72c73c76c81c72c3c87c75c72c3c71c76c86c87c85c76c69c88c87c76c82c81c17 c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c79c80c82c80c94c47c23c96c3c32c3c7c49c50c39c36c55c36c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c79c80c82c80c94c55c23c96c3c31c3c11c19c17c21c24c13c11c24c13c7c79c80c82c80c94c55c22c96c13c13c21c3c16c20c3c12c3c12c3c82c85 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c7c79c80c82c80c94c55c23c96c3c33c3c20c12c30 c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c79c80c82c80c94c55c23c96c12c30 c3c3c3 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c83c62c23c64c3c72c84c3c7c49c50c39c36c55c36c12c30 c3c3c3c7c79c80c82c80c94c47c24c96c3c32c3c26c19c13c7c83c62c23c64c3c16c3c20c23c19c13c7c83c62c22c64c3c14c3c28c19c13c7c83c62c21c64c3c16c3c21c19c13c7c83c62c20c64c3c14c3c7c83c62c19c64c30 c3c3c3 c3c3c3c6c3c55c68c88c24c3c70c68c81c3c81c82c87c3c69c72c3c74c85c72c68c87c72c85c3c87c75c68c81c3c20c15c3c76c87c3c75c68c86c3c68c3c79c82c90c72c85c3c79c76c80c76c87c15c3c69c88c87c3c43c82c86c78c76c81c74 c3c3c3c6c3c71c76c71c3c81c82c87c3c71c72c85c76c89c72c3c76c81c3c87c75c72c3c20c28c28c19c3c83c68c83c72c85c17 c3c3c3c7c79c80c82c80c94c55c24c96c3c32c3c7c79c80c82c80c94c47c24c96c18c7c79c80c82c80c94c47c21c96c30 c3c3c3c7c79c80c82c80c94c55c24c96c3c32c3c7c79c80c82c80c94c47c24c96c3c32c3c7c49c50c39c36c55c36c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c3c76c73c11c7c79c80c82c80c94c55c24c96c3c33c3c20c12c30 c3c3c3c7c79c80c82c80c94c55c24c96c3c32c3c9c66c71c72c70c76c80c68c79c86c11c7c79c80c82c80c94c55c24c96c12c30 c3c3c3c3 c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c30 c96 c86c88c69c3c47c48c50c48c21c51c58c48c3c94 c3c3c3c80c92c3c11c7c79c80c82c80c12c3c32c3c35c66c30 c3c3c3c80c92c3c35c83c3c32c3c11c7c49c50c39c36c55c36c15c3c7c49c50c39c36c55c36c15c3c7c49c50c39c36c55c36c15c3c7c49c50c39c36c55c36c15c3c7c49c50c39c36c55c36c15c12c30 c3c3c3c76c73c11c7c79c80c82c80c16c33c94c47c20c96c3c81c72c3c7c49c50c39c36c55c36c12c3c94 c3c3c3c3c3c3c7c83c62c19c64c3c32c3c3c3c3c3c3c3c3c7c79c80c82c80c16c33c94c47c20c96c30 c3c3c3c96 c3c3c3c76c73c11c7c79c80c82c80c16c33c94c47c21c96c3c81c72c3c7c49c50c39c36c55c36c12c3c94 c3c3c3c3c3c3c7c83c62c20c64c3c32c3c3c3c19c17c24c13c11c7c79c80c82c80c16c33c94c47c21c96c14c7c83c62c19c64c12c30 c3c3c3c96 c3c3c3c76c73c11c7c79c80c82c80c16c33c94c55c22c96c3c81c72c3c7c49c50c39c36c55c36c12c3c94 c3c3c3c3c3c3c7c83c62c21c64c3c32c3c11c20c18c25c12c13c11c7c79c80c82c80c16c33c94c47c21c96c13c7c79c80c82c80c16c33c94c55c22c96c14c25c13c7c83c62c20c64c16c7c83c62c19c64c12c30 c3c3c3c96 c3c3c3c76c73c11c7c79c80c82c80c16c33c94c55c23c96c3c81c72c3c7c49c50c39c36c55c36c12c3c94 c3c3c3c3c3c3c7c83c62c22c64c3c32c3c11c20c18c21c19c12c13c11c7c79c80c82c80c16c33c94c47c21c96c13c7c79c80c82c80c16c33c94c55c23c96c14c22c19c13c7c83c62c21c64c16c20c21c13c7c83c62c20c64c14c7c83c62c19c64c12c30 c3c3c3c96 c3c3c3c76c73c11c7c79c80c82c80c16c33c94c55c24c96c3c81c72c3c7c49c50c39c36c55c36c12c3c94 c3c3c3c3c3c3c7c83c62c23c64c3c32c3c11c20c18c26c19c12c13c11c7c79c80c82c80c16c33c94c47c21c96c13c7c79c80c82c80c16c33c94c55c24c96c14c20c23c19c13c7c83c62c22c64c3c16c3 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c28c19c13c7c83c62c21c64c14c21c19c13c7c83c62c20c64c16c7c83c62c19c64c12c30 c3c3c3c96 c3c3c3c85c72c87c88c85c81c3c63c35c83c30 c96 c86c88c69c3c47c76c81c72c68c85c48c82c80c72c81c87c86c3c94c3 c6c3c39c44c53c40c38c55c3c47c16c48c50c48c40c49c55c3c38c36c47c38c56c47c36c55c44c50c49c3 c6c3c68c71c82c83c87c72c71c3c73c85c82c80c3c52c17c45c17c3c58c68c81c74c15c3c39c76c85c72c70c87c3c86c68c80c83c79c72c3c72c86c87c76c80c68c87c82c85c86c3c82c73 c6c3c47c16c80c82c80c72c81c87c86c15c3c58c68c87c72c85c3c53c72c86c82c88c85c17c3c53c72c86c17c15c3c22c21c11c20c21c12c15c3c22c25c20c26c16c22c25c20c28 c6c3c68c81c71c3c80c82c71c76c73c76c72c71c3c73c82c85c3c55c24c3c89c76c68c3c72c80c68c76c79c3c73c85c82c80c3c39c85c17c3c58c68c81c74c3 c6c3c19c26c18c19c25c18c20c28c28c27c17c3c3c53c82c88c87c76c81c72c3c83c82c85c87c72c71c3c73c85c82c80c3c41c50c53c55c53c36c49c3c87c82c3c51c72c85c79c3c20c19c18c19c23c18c20c28c28c27 356 c6c3c51c68c86c86c3c71c68c87c68c3c68c85c85c68c92c3c76c81c87c82c3c85c82c88c87c76c81c72c3c68c81c71c3c85c82c88c87c76c81c72c3c85c72c87c88c85c81c86c3c87c75c72c3c47c16c80c82c80c72c81c87c3c68c85c85c68c92 c6c3c3c3c35c47c66c80c82c80c72c81c87c3c70c82c81c87c68c76c81c86c3c87c75c72c3c80c72c68c81c15c3c47c16c86c70c68c79c72c15c3c55c68c88c22c15c3c55c68c88c23c15c3c68c81c71c3c55c68c88c24 c3c3c80c92c3c11c3c7c71c68c87c68c85c72c73c15c3c7c81c72c72c71c86c82c85c87c12c3c32c3c35c66c30 c3c3c7c81c72c72c71c86c82c85c87c3c32c3c11c7c81c72c72c71c86c82c85c87c3c68c81c71c3c7c81c72c72c71c86c82c85c87c3c72c84c3c182c81c82c86c82c85c87c183c12c3c34c3c19c3c29c3c20c30 c3c3 c3c3c80c92c3c8c79c80c82c80c30 c3c3c3c3c3c7c79c80c82c80c94c47c20c96c3c32c3c7c79c80c82c80c94c47c21c96c3c32 c3c3c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c7c79c80c82c80c94c55c22c96c3c32 c3c3c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c79c80c82c80c94c55c24c96c3c32c3c7c49c50c39c36c55c36c30 c3c3 c3c3c80c92c3c11c3c35c47c66c80c82c80c72c81c87c86c12c3c32c3c11c12c30 c3c3c80c92c3c11c3c7c81c15c3c7c76c15c3c7c76c82c79c71c3c12c30 c3c3c80c92c3c11c3c7c47c20c15c7c47c21c15c7c47c22c15c7c47c23c15c7c47c24c3c12c30 c3c3c80c92c3c11c3c7c38c47c20c15c7c38c47c21c15c7c38c47c22c15c7c38c47c23c3c12c30 c3c3c80c92c3c11c3c7c38c53c20c15c7c38c53c21c15c7c38c53c22c15c7c38c53c23c3c12c30 c3c3c80c92c3c11c3c7c38c20c15c7c38c21c15c7c38c22c15c7c38c23c15c7c38c24c3c12c30 c3c3c80c92c11c35c59c12c3c32c3c35c7c71c68c87c68c85c72c73c30c3c3c3c3c6c3c71c72c85c72c73c72c85c72c81c70c72c3c87c75c72c3c83c68c86c86c72c71c3c86c70c68c79c68c85 c3c3c11c7c47c20c15c7c47c21c15c7c47c22c15c7c47c23c15c7c47c24c12c3c32c3c11c19c15c19c15c19c15c19c15c19c12c30c3c3c6c3c76c81c76c87c76c68c79c76c93c72c71c3c89c68c79c88c72c86c3c90c75c76c70c75c3c90c76c79c79c3c86c75c82c90 c3c3c11c7c38c20c15c7c38c21c15c7c38c22c15c7c38c23c15c7c38c24c12c3c32c3c11c19c15c19c15c19c15c19c15c19c12c30c3c3c6c3c68c81c3c88c81c76c81c76c87c76c68c79c76c93c72c71c3c89c68c79c88c72c3c90c68c85c81c76c81c74c3c3 c3c3c11c7c38c47c20c15c7c38c47c21c15c7c38c47c22c15c7c38c47c23c12c3c32c3c11c19c15c19c15c19c15c19c12c30 c3c3c11c7c38c53c20c15c7c38c53c21c15c7c38c53c22c15c7c38c53c23c12c3c32c3c11c19c15c19c15c19c15c19c12c30 c3c3c7c81c3c32c3c35c59c30c6c3c71c72c87c72c85c80c76c81c72c3c79c72c81c74c87c75c3c82c73c3c68c85c85c68c92 c3c3c76c73c11c81c82c87c3c7c81c12c3c94 c3c3c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c38c68c79c70c54c87c68c87c76c86c87c76c70c86c29c29c47c76c81c72c68c85c48c82c80c72c81c87c86c16c16c60c76c83c72c86c3c88c81c71c72c73c76c81c72c71c3c71c68c87c68c63c81c181c30 c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c30 c3c3c96 c3c3 c3c3c6c3c41c76c85c86c87c3c81c72c72c71c3c87c82c3c71c72c87c72c80c76c81c72c3c90c75c72c87c75c72c85c3c68c79c79c3c87c75c72c3c71c68c87c68c3c89c68c79c88c72c86c3c68c85c72c3c76c71c72c81c87c76c70c68c79 c3c3c80c92c3c7c73c76c85c86c87c89c68c79c3c32c3c7c59c62c19c64c30c3c3c6c3c76c81c76c87c76c68c79c76c93c72c3c87c75c72c3c73c76c85c86c87c3c89c68c79c88c72c3c76c81c3c68c85c85c68c92 c3c3c80c92c3c7c68c79c79c66c72c84c88c68c79c3c32c3c20c30c3c3c3c3c3c6c3c73c79c68c74c3c86c72c87c3c87c82c3c92c72c86c3c68c86c3c68c81c3c76c81c76c87c76c68c79c3c74c88c72c86c86 c3c3c73c82c85c72c68c70c75c3c80c92c3c7c89c68c79c3c11c35c59c12c3c94 c3c3c3c3c3c81c72c91c87c3c76c73c11c7c89c68c79c3c32c32c3c7c73c76c85c86c87c89c68c79c12c30c3c6c3c76c73c3c72c84c88c68c79c15c3c74c82c3c87c82c3c81c72c91c87c3c82c81c72 c3c3c3c3c3c7c68c79c79c66c72c84c88c68c79c3c32c3c19c30c3c3c3c3c3c3c3c3c3c3c3c3c3c6c3c68c87c3c79c72c68c86c87c3c82c81c72c3c76c86c3c71c76c73c73c72c85c72c81c87c15c3c82c78c3c87c82c3c70c82c81c87c76c81c88c72 c3c3c3c3c3c79c68c86c87c30 c3c3c96 c3c3c76c73c11c7c68c79c79c66c72c84c88c68c79c12c3c94c3c3c6c3c68c79c90c68c92c86c3c70c82c80c83c88c87c72c3c87c75c72c3c80c72c68c81c3 c3c3c3c3c3c3c3c85c72c87c88c85c81c3c94c3c16c80c72c68c81c3c32c33c3c9c48c72c68c81c11c7c71c68c87c68c85c72c73c12c15c3c16c79c66c86c70c68c79c72c3c32c33c3c19c15 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c16c87c68c88c22c3c32c33c3c88c81c71c72c73c15c3c16c87c68c88c23c3c32c33c3c88c81c71c72c73c15c3c16c87c68c88c24c3c32c33c3c88c81c71c72c73c3c96c30 c3c3c96 c3c3c3 c3c3c6c3c74c82c3c68c75c72c68c71c3c68c81c71c3c86c82c85c87c3c87c75c72c3c68c85c85c68c92c3 c3c3c35c59c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c11c35c59c12c3c76c73c11c7c81c72c72c71c86c82c85c87c12c30 c3 c3c3c6c3c56c86c72c3c87c75c72c3c50c16c86c87c68c87c76c86c87c76c70c86c3c86c88c69c85c82c88c87c76c81c72c3c87c82c3c70c68c79c70c88c79c68c87c72c71c3c90c75c68c87c72c89c72c85c3c47c16c80c82c80c72c81c87c86 c3c3c6c3c87c75c68c87c3c68c85c72c3c83c82c86c86c76c69c79c72c3c90c76c87c75c3c87c75c72c3c74c76c89c72c81c3c86c68c80c83c79c72c3c86c76c93c72c17 c3c3c6c3c55c75c76c86c3c86c72c70c87c76c82c81c3c81c72c72c71c86c3c73c88c85c87c75c72c85c3c71c72c89c72c79c82c83c80c72c81c87c16c16c20c18c21c28c18c20c28c28c28 c3c3c76c73c11c7c81c3c31c3c24c12c3c94 c3c3c3c3c3c80c92c3c7c80c72c68c81c3c32c3c9c48c72c68c81c11c7c71c68c87c68c85c72c73c12c30 c3c3c3c3c3c80c92c3c11c7c79c86c70c68c79c72c15c3c7c87c68c88c22c15c3c7c87c68c88c23c15c3c7c87c68c88c24c12c30 c3c3c3c3c3c80c92c3c7c82c86c87c68c87c86c3c32c3c9c74c72c81c72c85c68c87c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c3c63c35c59c15c3c182c81c82c86c82c85c87c183c12c30 c3c3c3c3c3c38c43c40c38c46c29c3c94 c3c3c3c3c3c3c3c80c92c3c7c82c30 c3c3c3c3c3c3c3c80c92c3c35c82c30 357 c3c3c3c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c20c12c30 c3c3c3c3c3c3c3c3c7c82c3c32c3c9c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c7c82c86c87c68c87c86c15c21c12c30c3c3c3c35c82c3c32c3c35c7c82c30 c3c3c3c3c3c3c3c3c7c79c86c70c68c79c72c3c32c3c19c17c24c13c11c7c82c62c20c64c3c16c3c7c82c62c19c64c12c30c3c3c3 c3c3c3c3c3c3c3c3 c3c3c3c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c21c12c30 c3c3c3c3c3c3c3c3c7c82c3c32c3c9c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c7c82c86c87c68c87c86c15c22c12c30c3c3c3c35c82c3c32c3c35c7c82c30 c3c3c3c3c3c3c3c3c7c87c68c88c22c3c32c3c11c20c18c22c12c13c11c7c82c62c21c64c3c16c3c21c13c7c82c62c20c64c3c14c3c7c82c62c19c64c12c30 c3c3c3c3c3c3c3c3c7c87c68c88c22c3c18c32c3c7c79c86c70c68c79c72c30 c3c3c3c3c3c3c3c3 c3c3c3c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c22c12c30 c3c3c3c3c3c3c3c3c7c82c3c32c3c9c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c7c82c86c87c68c87c86c15c23c12c30c3c3c3c35c82c3c32c3c35c7c82c30 c3c3c3c3c3c3c3c3c7c87c68c88c23c3c32c3c11c20c18c23c12c13c11c7c82c62c22c64c3c16c3c22c13c7c82c62c21c64c3c14c3c22c13c7c82c62c20c64c3c16c3c7c82c62c19c64c12c30 c3c3c3c3c3c3c3c3c7c87c68c88c23c3c18c32c3c7c79c86c70c68c79c72c30 c3c3c3c3c3c3c3c3 c3c3c3c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c23c12c30 c3c3c3c3c3c3c3c3c7c82c3c32c3c9c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c7c82c86c87c68c87c86c15c24c12c30c3c3c3c35c82c3c32c3c35c7c82c30 c3c3c3c3c3c3c3c3c7c87c68c88c24c3c32c3c11c20c18c24c12c13c11c7c82c62c23c64c3c16c3c23c13c7c82c62c22c64c3c14c3c25c13c7c82c62c21c64 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c16c3c23c13c7c82c62c20c64c3c14c3c3c3c7c82c62c19c64c3c12c30 c3c3c3c3c3c3c3c3c7c87c68c88c24c3c18c32c3c7c79c86c70c68c79c72c30 c3c3c3c3c3c96 c3c3c3c3c3c7c79c80c82c80c94c47c20c96c3c32c3c7c80c72c68c81c3c3c3c3c3c3c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c80c72c68c81c12c30 c3c3c3c3c3c7c79c80c82c80c94c47c21c96c3c32c3c7c79c86c70c68c79c72c3c3c3c3c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c79c86c70c68c79c72c12c30 c3c3c3c3c3c7c79c80c82c80c94c55c21c96c3c32c3c7c79c86c70c68c79c72c18c7c80c72c68c81c3c76c73c11c71c72c73c76c81c72c71c3c7c79c86c70c68c79c72c12c30 c3c3c3c3c3c7c79c80c82c80c94c55c22c96c3c32c3c7c87c68c88c22c3c3c3c3c3c3c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c87c68c88c22c12c30 c3c3c3c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c87c68c88c23c3c3c3c3c3c3c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c87c68c88c23c12c30 c3c3c3c3c3c7c79c80c82c80c94c55c24c96c3c32c3c7c87c68c88c24c3c3c3c3c3c3c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c87c68c88c24c12c30 c3c3c3c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c30 c3c3c96 c3c3c6c3c54c68c80c83c79c72c3c86c76c93c72c3c76c86c3c74c85c72c68c87c72c85c3c87c75c68c81c3c24c3c86c82c3c83c85c82c70c72c72c71c3c90c76c87c75c3c73c76c85c86c87c3c24c3c47c16c80c82c80c72c81c87c3 c70c68c79c70c88c79c68c87c76c82c81c3 c3c3c73c82c85c72c68c70c75c3c80c92c3c7c76c3c11c20c17c17c7c81c12c3c94 c3c3c3c3c7c38c47c20c3c32c3c7c76c16c20c30 c3c3c3c3c7c38c47c21c3c32c3c7c38c47c20c3c13c3c11c7c76c16c20c16c20c12c3c18c3c21c30 c3c3c3c3c7c38c47c22c3c32c3c7c38c47c21c3c13c3c11c7c76c16c20c16c21c12c3c18c3c22c30 c3c3c3c3c7c38c47c23c3c32c3c7c38c47c22c3c13c3c11c7c76c16c20c16c22c12c3c18c3c23c30 c3c3c3c3c7c38c53c20c3c32c3c7c81c16c7c76c30 c3c3c3c3c7c38c53c21c3c32c3c7c38c53c20c3c13c3c11c7c81c16c7c76c16c20c12c3c18c3c21c30 c3c3c3c3c7c38c53c22c3c32c3c7c38c53c21c3c13c3c11c7c81c16c7c76c16c21c12c3c18c3c22c30 c3c3c3c3c7c38c53c23c3c32c3c7c38c53c22c3c13c3c11c7c81c16c7c76c16c22c12c3c18c3c23c30c3c3c3c3c3 c3c3c3c3c7c47c20c3c14c32c3c7c59c62c7c76c16c20c64c30 c3c3c3c3c7c47c21c3c14c32c3c7c59c62c7c76c16c20c64c3c13c3c11c7c38c47c20c3c16c3c7c38c53c20c12c30 c3c3c3c3c7c47c22c3c14c32c3c7c59c62c7c76c16c20c64c3c13c3c11c7c38c47c21c3c16c3c21c13c7c38c47c20c13c7c38c53c20c3c14c3c7c38c53c21c12c30 c3c3c3c3c7c47c23c3c14c32c3c7c59c62c7c76c16c20c64c3c13c3c11c7c38c47c22c3c16c3c22c13c7c38c47c21c13c7c38c53c20c3c14c3c22c13c7c38c47c20c13c7c38c53c21c3c16c3c7c38c53c22c12c30 c3c3c3c3c7c47c24c3c14c32c3c7c59c62c7c76c16c20c64c3c13c3c11c7c38c47c23c3c16c3c23c13c7c38c47c22c13c7c38c53c20c3c14c3c25c13c7c38c47c21c13c7c38c53c21 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c16c3c23c13c7c38c47c20c13c7c38c53c22c3c14c3c7c38c53c23c12c30c3c3c3c3 c3c3c96 c3c3 c3c3c7c38c20c3c32c3c7c81c30 c3c3c7c38c21c3c32c3c7c38c20c3c13c3c11c7c81c16c20c12c3c18c3c21c30 c3c3c7c38c22c3c32c3c7c38c21c3c13c3c11c7c81c16c21c12c3c18c3c22c30 c3c3c7c38c23c3c32c3c7c38c22c3c13c3c11c7c81c16c22c12c3c18c3c23c30 c3c3c7c38c24c3c32c3c7c38c23c3c13c3c11c7c81c16c23c12c3c18c3c24c30 c3c3c7c47c20c3c32c3c7c47c20c3c18c3c7c38c20c30 358 c3c3c7c47c21c3c32c3c7c47c21c3c18c3c7c38c21c3c18c3c21c30 c3c3c7c47c22c3c32c3c7c47c22c3c18c3c7c38c22c3c18c3c22c30 c3c3c7c47c23c3c32c3c7c47c23c3c18c3c7c38c23c3c18c3c23c30 c3c3c7c47c24c3c32c3c7c47c24c3c18c3c7c38c24c3c18c3c24c30 c3c3 c3c3c7c79c80c82c80c94c47c20c96c3c32c3c7c47c20c30 c3c3c7c79c80c82c80c94c47c21c96c3c32c3c7c47c21c30 c3c3c7c79c80c82c80c94c55c21c96c3c32c3c7c47c21c18c7c47c20c30 c3c3c7c79c80c82c80c94c55c22c96c3c32c3c7c47c22c18c7c47c21c30 c3c3c7c79c80c82c80c94c55c23c96c3c32c3c7c47c23c18c7c47c21c30 c3c3c7c79c80c82c80c94c55c24c96c3c32c3c7c47c24c18c7c47c21c30 c3c3c85c72c87c88c85c81c3c63c8c79c80c82c80c30 c96c3c3c3c3c3c3 c6c3c38c82c80c83c88c87c72c3c80c72c68c81c3c82c73c3c68c81c3c68c85c85c68c92c3c85c72c73c72c85c72c81c70c72c17 c86c88c69c3c48c72c68c81c3c94 c3c3c3c80c92c3c11c7c68c85c85c68c92c85c72c73c12c3c32c3c35c66c30 c3c3c3c80c92c3c7c85c72c86c88c79c87c30 c3c3c3c80c68c83c3c94c3c7c85c72c86c88c79c87c3c14c32c3c7c66c3c96c3c11c3c35c7c68c85c85c68c92c85c72c73c3c12c30 c3c3c3c85c72c87c88c85c81c3c7c85c72c86c88c79c87c3c18c3c35c7c68c85c85c68c92c85c72c73c30 c96 c86c88c69c3c66c71c72c70c76c80c68c79c86c3c94c3c85c72c87c88c85c81c3c11c7c66c62c19c64c3c72c84c3c182c16c16c183c12c3c34c3c7c66c62c19c64c3c29c3c86c83c85c76c81c87c73c11c180c8c19c17c23c73c181c15c3c7c66c62c19c64c12c30c3c96 c86c88c69c3c47c48c50c48c68c86c53c82c90c3c94 c3c3c3c80c92c3c11c7c79c80c82c80c12c3c32c3c35c66c30 c3c3c3c80c92c3c8c79c3c32c3c8c7c79c80c82c80c30 c3c3c3c85c72c87c88c85c81c3c77c82c76c81c11c180c3c3c3c180c15c3c35c79c94c84c90c11c47c20c3c47c21c3c55c21c3c55c22c3c55c23c3c55c24c12c96c12c30 c96 c86c88c69c3c47c48c50c48c68c86c54c87c85c76c81c74c3c94 c3c3c3c80c92c3c11c7c79c80c82c80c12c3c32c3c35c66c30 c3c3c3c80c92c3c8c79c3c32c3c8c7c79c80c82c80c30 c3c3c3c80c92c3c7c86c87c71c72c89c3c32c3c11c7c79c94c47c21c96c3c72c84c3c7c49c50c39c36c55c36c12c3c34c3c7c79c94c47c21c96c3c29c3c9c66c71c72c70c76c80c68c79c86c11c7c79c94c47c21c96c13c86c84c85c87c51c44c12c30 c3c3c3c85c72c87c88c85c81c3c180c6c3c47c16c80c82c80c72c81c87c86c3c68c85c72c29c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c48c72c68c81c3c3c3c3c3c3c3c3c32c3c7c79c94c47c20c96c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c47c16c54c70c68c79c72c3c3c3c3c3c32c3c7c79c94c47c21c96c3c11c54c87c39c72c89c32c7c86c87c71c72c89c12c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c47c16c38c57c3c3c3c3c3c3c3c3c32c3c7c79c94c55c21c96c63c81c6c3c3c3c47c16c86c78c72c90c3c3c3c3c3c3c32c3c7c79c94c55c22c96c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c47c16c78c88c85c87c82c86c76c86c3c3c32c3c7c79c94c55c23c96c63c81c6c3c3c3c55c68c88c24c3c3c3c3c3c3c3c3c32c3c7c79c94c55c24c96c181c30c3c3 c96 c86c88c69c3c51c58c48c68c86c54c87c85c76c81c74c3c94 c3c3c3c80c92c3c11c7c83c90c80c12c3c32c3c35c66c30 c3c3c3c80c92c3c35c83c3c32c3c35c7c83c90c80c30 c3c3c3c85c72c87c88c85c81c3c180c6c3c51c85c82c68c69c76c79c76c87c92c3c58c72c76c74c75c87c72c71c3c48c82c80c72c81c87c86c3c68c85c72c29c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c37c72c87c68c19c3c3c32c3c7c83c62c19c64c63c81c6c3c3c3c37c72c87c68c20c3c3c32c3c7c83c62c20c64c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c37c72c87c68c21c3c3c32c3c7c83c62c21c64c63c81c6c3c3c3c37c72c87c68c22c3c3c32c3c7c83c62c22c64c63c81c181c17 c3c3c3c3c3c3c3c3c3c3c180c6c3c3c3c37c72c87c68c23c3c3c32c3c7c83c62c23c64c181c30 c96 c86c88c69c3c71c88c80c83c39c68c87c68c3c94 359 c3c3c3c80c92c3c11c7c71c68c87c68c12c3c32c3c35c66c30 c3c3c3c83c85c76c81c87c3c180c6c3c39c36c55c36c29c3c38c88c80c88c79c68c87c76c89c72c66c83c85c82c69c17c11c41c12c3c3c3c3c3c3c3c39c68c87c68c63c81c181c15 c3c3c3c3c3c3c3c3c3c180c6c3c39c36c55c36c29c3c49c82c81c72c91c70c72c72c71c68c81c70c72c66c83c85c82c69c17c3c3c3c3c3c3c3c89c68c79c88c72c63c81c181c30 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c78c72c92c86c3c8c7c71c68c87c68c12c3c94 c3c3c3c3c3c3c3c83c85c76c81c87c3c180c6c3c39c36c55c36c29c3c3c3c3c3c3c3c3c7c41c3c3c3c3c3c3c3c3c3c3c3c3c3c7c71c68c87c68c16c33c94c7c41c96c63c81c181c30 c3c3c3c96 c96 c86c88c69c3c71c88c80c83c39c68c87c68c47c76c80c76c87c86c3c94c3 c3c3c3c80c92c3c11c7c71c68c87c68c12c3c32c3c35c66c30 c3c3c3c80c92c3c11c7c73c80c76c81c15c3c7c73c80c68c91c15c3c7c91c80c76c81c15c3c7c91c80c68c91c12c30 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c41c3c11c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c78c72c92c86c3c8c7c71c68c87c68c12c3c94 c3c3c3c3c3c3c3c80c92c3c7c89c68c79c3c32c3c7c71c68c87c68c16c33c94c7c41c96c30 c3c3c3c3c3c3c3c7c73c80c76c81c3c32c3c11c81c82c87c3c71c72c73c76c81c72c71c3c7c73c80c76c81c3c82c85c3c3c3c7c41c3c31c3c7c73c80c76c81c12c3c34c3c3c3c7c41c3c29c3c7c73c80c76c81c30 c3c3c3c3c3c3c3c7c73c80c68c91c3c32c3c11c81c82c87c3c71c72c73c76c81c72c71c3c7c73c80c68c91c3c82c85c3c3c3c7c41c3c33c3c7c73c80c68c91c12c3c34c3c3c3c7c41c3c29c3c7c73c80c68c91c30 c3c3c3c3c3c3c3c7c91c80c76c81c3c32c3c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c80c76c81c3c82c85c3c7c89c68c79c3c31c3c7c91c80c76c81c12c3c34c3c7c89c68c79c3c29c3c7c91c80c76c81c30 c3c3c3c3c3c3c3c7c91c80c68c91c3c32c3c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c80c68c91c3c82c85c3c7c89c68c79c3c33c3c7c91c80c68c91c12c3c34c3c7c89c68c79c3c29c3c7c91c80c68c91c30 c3c3c3c96 c3c3c3c7c73c80c76c81c3c32c3c180c16c16c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c73c80c76c81c12c30 c3c3c3c7c73c80c68c91c3c32c3c180c16c16c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c73c80c68c91c12c30 c3c3c3c7c91c80c76c81c3c32c3c180c16c16c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c80c76c81c12c30 c3c3c3c7c91c80c68c91c3c32c3c180c16c16c181c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c91c80c68c91c12c30 c3c3c3c83c85c76c81c87c3c180c6c3c41c66c48c44c49c3c3c3c3c41c66c48c36c59c3c3c3c3c59c66c48c44c49c3c3c3c3c59c66c48c36c59c63c81c181c30 c3c3c3c83c85c76c81c87c3c180c6c3c7c73c80c76c81c3c3c3c3c7c73c80c68c91c3c3c3c3c7c91c80c76c81c3c3c3c3c7c91c80c68c91c63c81c181c30 c3c3c3c85c72c87c88c85c81c30 c96 c86c88c69c3c76c86c49c88c80c69c72c85c3c94 c3c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c66c62c19c64c3c12c3c94 c3c3c3c3c3c80c92c3c35c70c68c79c79c3c32c3c70c68c79c79c72c85c11c20c12c30 c3c3c3c3c3c80c68c83c3c94c3c7c70c68c79c79c62c7c66c64c3c32c3c180c180c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c70c68c79c79c62c7c66c64c12c3c96c3c11c19c17c17c7c6c70c68c79c79c12c30 c3c3c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c76c86c49c88c80c69c72c85c11c88c81c71c72c73c12c3c68c86c3c35c70c68c79c79c63c81c181c30 c3c3c3c3c3c85c72c87c88c85c81c3c19c30 c3c3c96 c3c3c7c66c62c19c64c3c32c97c3c18c65c63c86c13c62c14c16c64c34c63c71c14c63c17c34c63c71c13c63c86c13c7c18c82c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c95c95c3 c3c3c3c3c7c66c62c19c64c3c32c97c3c18c65c63c86c13c62c14c16c64c34c63c17c63c71c14c63c86c13c7c18c82c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c95c95 c3c3c3c3c3c3c7c66c62c19c64c3c32c97c3c18c65c63c86c13c62c14c16c64c34c63c71c14c63c17c34c63c71c13c62c72c40c64c62c14c16c64c34c63c71c14c63c86c13c7c18c82c3c95c95c3 c3c3c3c3c3c3c3c3c7c66c62c19c64c3c32c97c3c18c65c63c86c13c62c14c16c64c34c63c17c63c71c14c62c72c40c64c62c14c16c64c34c63c71c14c63c86c13c7c18c82c30 c96 c6c3c50c53c39c40c53c3c54c55c36c55c44c54c55c44c38c54 c6c3c74c72c81c72c85c68c87c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86 c6c3c38c68c79c70c88c79c68c87c72c3c87c75c72c3c50c16c86c87c68c87c76c86c87c76c70c86c17c3c3c42c76c89c72c81c3c68c3c71c68c87c68c3c68c85c85c68c92c3c87c75c68c87c3c70c68c81c3c69c72 c6c3c83c85c72c86c82c85c87c72c71c15c3c70c68c79c79c3c86c88c69c3c90c76c87c75c3c73c68c79c86c72c3c86c72c70c82c81c71c3c83c68c85c68c80c72c87c72c85c15c3c82c85c3c68c81c3c88c81c86c82c85c87c72c71 c6c3c68c85c85c68c92c15c3c70c68c79c79c3c86c88c69c3c90c76c87c75c3c87c85c88c72c3c86c72c70c82c81c71c3c83c68c85c68c80c72c87c72c85c15c3c87c75c76c86c3c86c88c69c85c82c88c87c76c81c72 c6c3c85c72c87c88c85c81c86c3c87c75c72c3c50c16c86c87c68c87c76c86c87c76c70c3c80c68c87c85c76c91c17c3c3c55c75c72c3c76c81c71c76c89c76c71c88c68c79c3c50c16c86c87c68c87c3c68c85c85c68c92c86 c6c3c68c85c72c3c85c72c87c85c76c72c89c72c71c3c73c85c82c80c3c87c75c72c3c50c16c86c87c68c87c76c86c87c76c70c3c80c68c87c85c76c91c3c88c86c76c81c74c3c85c72c87c85c76c72c89c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86 c6c3c7c85c72c73c3c32c3c85c72c87c85c76c72c89c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c74c72c81c72c85c68c87c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c11c63c35c71c68c87c68c12c15c20c12c30 c6c3c83c85c76c81c87c3c180c55c43c40c3c48c40c36c49c3c50c41c3c55c43c40c3c39c36c55c36c3c32c3c7c85c72c73c16c33c62c19c64c63c81c181c30 c6c3c3c3c3c55c75c72c3c50c16c86c87c68c87c76c86c87c76c70c86c3c68c85c72c3c87c75c72c3c72c91c83c72c70c87c68c87c76c82c81c86c3c82c73c3c87c75c72c3c88c86c88c68c79c3c82c85c71c72c85c3c86c87c68c87c76c86c87c76c70c86 c86c88c69c3c74c72c81c72c85c68c87c72c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c3c94 360 c3c3c3c80c92c3c11c7c71c68c87c68c85c72c73c15c3c7c81c72c72c71c86c82c85c87c12c3c32c3c11c3c86c75c76c73c87c15c3c86c75c76c73c87c12c30 c3c3c3c7c81c72c72c71c86c82c85c87c3c32c3c11c7c81c72c72c71c86c82c85c87c3c68c81c71c3c7c81c72c72c71c86c82c85c87c3c72c84c3c182c81c82c86c82c85c87c183c12c3c34c3c19c3c29c3c20c30 c3c3c3c80c92c3c35c59c3c32c3c35c7c71c68c87c68c85c72c73c30 c3c3c3c80c92c3c35c50c59c30c3c3 c3c3c3 c3c3c3c6c3c56c86c72c85c3c76c86c3c85c72c86c83c82c81c86c76c69c79c72c3c73c82c85c3c70c68c79c79c76c81c74c3c90c76c87c75c3c87c75c72c3c83c85c82c83c72c85 c3c3c3c6c3c86c82c85c87c76c81c74c3c86c90c76c87c70c75c17c3c3c55c75c72c3c71c68c87c68c3c68c85c85c68c92c3c80c88c86c87c3c69c72c3c86c82c85c87c72c71 c3c3c3c6c3c76c81c3c68c86c70c72c81c71c76c81c74c3c82c85c71c72c85c17 c3c3c3c35c59c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c35c59c3c76c73c11c7c81c72c72c71c86c82c85c87c12c30 c3c3c3 c3c3c3c80c92c3c7c81c3c32c3c86c70c68c79c68c85c11c35c59c12c30c3c6c3c86c76c93c72c3c82c73c3c71c68c87c68c3c3c3 c3c3c3c6c3c41c76c79c79c3c50c16c86c87c68c87c76c86c87c76c70c86c3c68c85c85c68c92c3c68c87c3c82c85c71c72c85c3c7c81c3c90c76c87c75c3c87c75c72c3c71c68c87c68 c3c3c3c7c50c59c62c7c81c64c3c32c3c62c3c35c59c3c64c30c3c3c6c3c70c85c72c68c87c72c3c81c72c90c3c68c81c82c81c92c80c82c88c86c3c68c85c85c68c92 c3c3c3 c3c3c3c6c3c3c85c3c3c3c3c3c3c3c3c71 c3c3c3c6c3c3c66c3c95c66c22c66c66c66c66c21c66c66c66c66c66c20c66c66c3c3c55c40c54c55c3c40c59c36c48c51c47c40 c3c3c3c6c3c3c19c3c95c3c20c3c3c3c24c18c22c3c3c3c3c22 c3c3c3c6c3c3c20c3c95c3c22c3c3c20c22c18c22 c3c3c3c6c3c3c21c3c95c3c24 c3c3c3c6c3c38c50c48c51c56c55c40c3c55c43c40c3c50c16c54c55c36c55c44c54c55c44c38c54c3c50c41c3c50c53c39c40c53c3c11c81c16c20c12c3c55c50c3c19 c3c3c3c6c3c11c86c72c72c3c72c84c88c68c87c76c82c81c3c21c17c22c3c76c81c3c46c68c76c74c75c15c3c58c17c39c17c15c3c68c81c71c3c39c85c76c86c70c82c79c79c15c3c48c17c41c17c15c3c20c28c27c26c15 c3c3c3c6c3c55c75c72c3c36c80c72c85c76c70c68c81c3c54c87c68c87c76c86c87c76c70c76c68c81c15c3c89c82c79c17c3c23c20c15c3c81c82c17c3c20c15c3c83c83c17c3c21c24c16c22c21c12 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c71c3c11c85c72c89c72c85c86c72c3c11c20c17c17c11c7c81c3c16c3c20c12c12c3c12c3c94 c3c3c3c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c85c3c11c19c17c17c11c7c71c3c16c3c20c12c12c3c94 c3c3c3c3c3c3c3c3c3c80c92c3c7c71c20c3c32c3c7c71c3c14c3c20c30 c3c3c3c3c3c3c3c3c3c80c92c3c7c59c20c3c32c3c7c50c59c62c7c71c20c64c62c7c85c64c30 c3c3c3c3c3c3c3c3c3c80c92c3c7c59c21c3c32c3c7c50c59c62c7c71c20c64c62c7c85c3c14c3c20c64c30 c3c3c3c3c3c3c3c3c3c7c50c59c62c7c71c64c62c7c85c64c3c32c3c11c3c11c11c7c71c16c7c85c12c3c13c3c7c59c20c12c3c14c3c11c11c7c85c14c20c12c3c13c3c7c59c21c12c3c12c3c18c3c11c7c71c14c20c12c30 c3c3c3c3c3c3c96 c3c3c3c96 c3c3c3c85c72c87c88c85c81c3c63c35c50c59c30c3c3c3 c96 c86c88c69c3c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c3c94 c3c3c3c80c92c3c11c3c7c50c59c85c72c73c15c3c7c82c85c71c72c85c15c3c7c79c82c90c88c83c12c3c32c3c35c66c30 c3c3c3c76c73c11c7c82c85c71c72c85c3c31c3c20c3c82c85c3c7c82c85c71c72c85c3c33c32c3c35c7c50c59c85c72c73c12c3c94 c3c3c3c3c3c3c6c83c85c76c81c87c3c54c55c39c40c53c53c3c180c74c72c87c50c85c71c72c85c54c87c68c87c76c86c87c76c70c86c29c3c53c72c87c85c76c72c89c68c79c3c82c85c71c72c85c3c182c7c82c85c71c72c85c183c3c180c17 c3c3c3c3c3c3c6c3c3c3c3c3c3c3c3c3c3c3c3c3c180c70c68c81c3c81c82c87c3c69c72c3c79c72c86c86c3c87c75c68c81c3c20c3c82c85c3c74c85c72c68c87c72c85c3c87c75c68c81c3c79c72c81c74c87c75c3c82c73c3c180c17 c3c3c3c3c3c3c6c3c3c3c3c3c3c3c3c3c3c3c3c3c180c87c75c72c3c50c16c86c87c68c87c76c86c87c76c70c86c3c180c17 c3c3c3c3c3c3c6c3c3c3c3c3c3c3c3c3c3c3c3c3c180c68c85c85c68c92c3c90c75c76c70c75c3c76c86c3c180c15c86c70c68c79c68c85c11c35c7c50c59c85c72c73c12c15c181c17c63c81c181c30 c3c3c3c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c30 c3c3c3c96 c3c3c3c72c79c86c76c73c11c7c79c82c90c88c83c12c3c94c3c3c6c3c89c68c79c88c72c86c3c86c75c82c88c79c71c3c69c72c3c72c76c87c75c72c85c3c182c79c82c90c72c85c183c3c82c85c3c182c88c83c83c72c85c183 c3c3c3c3c3c3c85c72c87c88c85c81c3c11c3c7c79c82c90c88c83c3c72c84c3c182c79c82c90c72c85c183c12c3c34c3c7c50c59c85c72c73c16c33c62c7c82c85c71c72c85c64c16c33c62c19c64c3c29 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c7c50c59c85c72c73c16c33c62c7c82c85c71c72c85c64c16c33c62c3c7c6c94c7c50c59c85c72c73c16c33c62c7c82c85c71c72c85c64c96c3c64c30 c3c3c3c96 c3c3c3c72c79c86c72c3c94c3c3c3c3c3c3c3c3c3 c3c3c3c3c3c3c85c72c87c88c85c81c3c7c50c59c85c72c73c16c33c62c7c82c85c71c72c85c64c30 c3c3c3c96 c96 c86c88c69c3c51c85c82c71c88c70c87c48c82c80c72c81c87c86c3c94 361 c3c3c80c92c3c11c35c59c12c3c32c3c35c66c30 c3c3c80c92c3c7c86c88c80c48c30 c3c3c80c92c3c7c81c3c32c3c35c59c30 c3c3c76c73c11c81c82c87c3c7c81c12c3c94 c3c3c3c3c3c83c85c76c81c87c3c54c55c39c40c53c53c3c180c51c85c82c71c88c70c87c48c82c80c72c81c87c86c16c16c60c76c83c72c86c3c88c81c71c72c73c76c81c72c71c3c71c68c87c68c63c81c181c30 c3c3c3c3c3c85c72c87c88c85c81c3c11c88c81c71c72c73c15c88c81c71c72c73c15c88c81c71c72c73c15c88c81c71c72c73c12c30 c3c3c96 c3c3c73c82c85c72c68c70c75c3c80c92c3c7c89c3c11c35c59c12c3c94c3c7c86c88c80c48c3c14c32c3c7c89c3c96c3 c3c3c80c92c3c7c80c72c68c81c3c32c3c7c86c88c80c48c18c7c81c30 c3c3 c3c3c80c92c3c11c7c86c88c80c54c55c39c15c3c7c86c88c80c54c46c58c15c3c7c86c88c80c46c56c53c12c3c32c3c11c19c15c3c19c15c3c19c12c30c3c3 c3c3c73c82c85c72c68c70c75c3c80c92c3c7c89c3c11c35c59c12c3c94 c3c3c3c3c7c86c88c80c54c55c39c3c14c32c3c11c7c89c16c7c80c72c68c81c12c13c13c21c30 c3c3c3c3c7c86c88c80c54c46c58c3c14c32c3c11c7c89c16c7c80c72c68c81c12c13c13c22c30 c3c3c3c3c7c86c88c80c46c56c53c3c14c32c3c11c7c89c16c7c80c72c68c81c12c13c13c23c30 c3c3c96 c3c3 c3c3c80c92c3c11c7c86c87c71c72c89c15c7c86c78c72c90c15c7c78c88c85c87c12c30 c3c3c38c43c40c38c46c29c3c94 c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c20c12c30 c3c3c3c3c3c7c86c87c71c72c89c3c32c3c11c7c86c88c80c54c55c39c18c11c7c81c16c20c12c12c13c13c19c17c24c30 c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c21c3c82c85c3c7c86c87c71c72c89c3c32c32c3c19c12c30c3c6c3c71c76c89c76c86c76c82c81c3c69c92c3c93c72c85c82c3c83c85c82c87c72c70c87c76c82c81 c3c3c3c3c3c7c86c78c72c90c3c32c3c7c81c13c11c7c86c88c80c54c46c58c12c18c11c11c7c81c16c20c12c13c11c7c81c16c21c12c13c7c86c87c71c72c89c13c13c22c12c30 c3c3c3c3c79c68c86c87c3c38c43c40c38c46c3c76c73c11c7c81c3c32c32c3c22c12c30 c3c3c3c3c3c7c78c88c85c87c3c32c3c11c11c7c81c13c11c7c81c14c20c12c12c18c11c11c7c81c16c20c12c13c11c7c81c16c21c12c13c11c7c81c16c22c12c12c12c13c11c7c86c88c80c46c56c53c18c11c7c86c87c71c72c89c13c13c23c12c12c30 c3c3c3c3c3c6c3c7c78c88c85c87c3c32c3c7c78c88c85c87c16c11c22c13c11c7c81c16c20c12c13c13c21c12c18c11c11c7c81c16c21c12c13c11c7c81c16c22c12c12c30c3c3c6c3c73c82c85c3c72c91c70c72c86c86c72c86c3c78c88c85c87c3 c3c3c96 c3c3c85c72c87c88c85c81c3c11c7c80c72c68c81c15c7c86c87c71c72c89c15c7c86c78c72c90c15c7c78c88c85c87c12c30 c96 c86c88c69c3c43c68c85c80c82c81c76c70c48c72c68c81c3c94 c3c3c3c80c92c3c35c71c68c87c68c3c32c3c35c66c30 c3c3c3c80c92c3c7c86c88c80c30 c3c3c3c80c92c3c7c70c82c88c81c87c3c32c3c86c70c68c79c68c85c11c35c71c68c87c68c12c30 c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c3c76c73c11c7c70c82c88c81c87c3c32c32c3c19c12c30c3c6c3c87c85c68c83c3c68c3c81c82c3c71c68c87c68c3c70c82c81c71c76c87c76c82c81 c3c3c3 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c89c68c79c3c11c35c71c68c87c68c12c3c94 c3c3c3c3c3c3c80c92c3c7c87c80c83c30 c3c3c3c3c3c3c72c89c68c79c3c94c3c7c87c80c83c3c32c3c20c18c7c89c68c79c3c96c30c3 c3c3c3c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c3c76c73c11c7c35c12c30c3c6c3c80c88c86c87c3c75c68c89c72c3c69c72c72c81c3c68c3c71c76c89c76c86c76c82c81c3c69c92c3c93c72c85c82c3c86c82c79c88c87c76c82c81 c3c3c3c3c3c3c7c86c88c80c3c14c32c3c7c87c80c83c30 c3c3c3c96 c3c3c3c7c86c88c80c3c18c32c3c7c70c82c88c81c87c30 c3c3c3c85c72c87c88c85c81c3c20c18c7c86c88c80c30 c96 c86c88c69c3c42c72c82c80c72c87c85c76c70c48c72c68c81c3c94 c3c3c3c80c92c3c35c71c68c87c68c3c32c3c35c66c30 c3c3c3c80c92c3c7c87c80c83c3c32c3c20c30 c3c3c3c80c68c83c3c94c3c7c87c80c83c3c13c32c3c7c66c3c96c3c35c71c68c87c68c30 c3c3c3c80c92c3c7c70c82c88c81c87c3c32c3c86c70c68c79c68c85c11c35c71c68c87c68c12c30 c3c3c3c6c3c87c85c68c83c3c73c82c85c3c85c82c82c87c86c3c82c73c3c81c72c74c68c87c76c89c72c3c81c88c80c69c72c85c86c3c68c81c71c3c73c82c85c3c81c82c3c71c68c87c68c3c68c87c3c68c79c79 c3c3c3c85c72c87c88c85c81c3c11c7c70c82c88c81c87c3c32c32c3c19c3c82c85c3c7c87c80c83c3c31c3c19c12c3c34c3c88c81c71c72c73c3c29c3c11c7c87c80c83c12c13c13c11c3c20c3c18c3c7c70c82c88c81c87c12c30 362 c96 c6c3c38c82c80c83c88c87c72c3c80c72c71c76c68c81c3c82c73c3c68c81c3c68c85c85c68c92 c86c88c69c3c48c72c71c76c68c81c3c94 c3c3c3c80c92c3c35c71c3c32c3c35c66c30 c3c3c3c3c3c3c35c71c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c35c71c30 c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c3c88c81c79c72c86c86c11c35c71c3c33c3c21c12c30c3c6c3c87c85c68c83c3c88c81c71c72c73c3c89c68c85c76c68c69c79c72c3c90c68c85c81c76c81c74 c3c3c3c85c72c87c88c85c81c3c11c35c71c3c8c3c21c12c3c34c3c7c71c62c35c71c18c21c64c3c29c3c11c7c71c62c35c71c18c21c3c16c3c20c64c14c7c71c62c35c71c18c21c64c12c18c21c30 c96 c6c3c38c82c80c83c88c87c72c3c80c72c71c76c68c81c3c82c73c3c68c81c3c68c85c85c68c92c3c90c76c87c75c3c83c85c76c82c85c3c83c85c82c69c68c69c76c79c76c87c76c72c86 c86c88c69c3c48c72c71c76c68c81c66c90c76c87c75c66c51c85c76c82c85c51c85c82c69c68c69c76c79c76c87c76c72c86c3c94 c3c3c3c80c92c3c7c71c68c87c68c3c32c3c86c75c76c73c87c30 c3c3c3c80c92c3c35c83c85c82c69c86c3c32c3c86c82c85c87c3c94c3c7c68c3c31c32c33c3c7c69c3c96c3c78c72c92c86c3c8c7c71c68c87c68c30 c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c3c88c81c79c72c86c86c11c35c83c85c82c69c86c12c30 c3c3c3c76c73c11c35c83c85c82c69c86c3c32c32c3c20c3c68c81c71c3c7c83c85c82c69c86c62c19c64c3c32c32c3c19c17c24c19c12c3c94 c3c3c3c3c3c3c85c72c87c88c85c81c3c7c71c68c87c68c16c33c94c7c83c85c82c69c86c62c19c64c96c30 c3c3c3c96 c3c3c3c80c92c3c11c7c83c79c82c15c3c7c83c75c76c12c3c32c3c11c88c81c71c72c73c15c88c81c71c72c73c12c30 c3c3c3c73c82c85c72c68c70c75c3c80c92c3c7c83c85c82c69c3c11c35c83c85c82c69c86c12c3c94 c3c3c3c3c3c3c85c72c87c88c85c81c3c7c71c68c87c68c16c33c94c7c83c85c82c69c96c3c76c73c11c7c83c85c82c69c3c32c32c3c19c17c24c19c12c30 c3c3c3c3c3c3c7c83c79c82c3c32c3c7c83c85c82c69c3c76c73c11c7c83c85c82c69c3c31c3c19c17c24c19c12c30 c3c3c3c3c3c3c7c83c75c76c3c32c3c7c83c85c82c69c15c3c79c68c86c87c3c76c73c11c7c83c85c82c69c3c33c3c19c17c24c19c12c30 c3c3c3c96 c3c3c3c90c68c85c81c3c180c6c6c6c6c6c6c6c6c6c6c6c6c3c7c83c79c82c3c68c81c71c3c7c83c75c76c63c81c181c30 c3c3c3c76c73c11c71c72c73c76c81c72c71c3c7c83c79c82c3c68c81c71c3c71c72c73c76c81c72c71c3c7c83c75c76c12c3c94 c3c3c3c3c3c3c80c92c3c11c7c79c82c15c3c7c75c76c12c3c32c3c11c7c71c68c87c68c16c33c94c7c83c79c82c96c15c3c7c71c68c87c68c16c33c94c7c83c75c76c96c12c30 c3c3c3c3c3c3c85c72c87c88c85c81c3c11c11c19c17c24c19c16c7c83c79c82c12c18c11c7c83c75c76c16c7c83c79c82c12c12c13c11c7c75c76c16c7c79c82c12c3c14c3c7c79c82c30 c3c3c3c96c3 c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c30 c96 c86c88c69c3c43c72c79c83c3c94 c3c3c80c92c3c11c7c36c56c55c43c50c53c15c3c7c37c56c44c47c39c39c36c55c40c15c3c7c57c40c53c54c44c50c49c12c3c32c3c9c74c72c87c38c57c54c86c87c68c80c83c86c11c12c30 c3c3c83c85c76c81c87c3c31c31c43c40c53c40 c49c36c48c40 c3c79c80c82c80c72c81c87c86c17c83c79c3c89c72c85c86c76c82c81c3c7c57c40c53c54c44c50c49c3c69c92c3c7c36c56c55c43c50c53c3c82c81c3c7c37c56c44c47c39c39c36c55c40 c39c40c54c38c53c44c51c55c44c50c49 c3c47c16c80c82c80c72c81c87c3c68c81c71c3c51c85c82c71c88c70c87c3c48c82c80c72c81c87c3c70c82c80c83c88c87c68c87c76c82c81c3c83c85c82c74c85c68c80 c3c11c86c72c72c3c81c82c87c72c86c3c69c72c79c82c90c12 c3c69c92c3c58c76c79c79c76c68c80c3c43c17c3c36c86c84c88c76c87c75c15c3c56c54c42c54c15c3c45c88c81c72c3c21c19c19c21 c3c3 c39c40c51c40c49c39c40c49c38c44c40c54 c3 c3c3 c56c54c36c42c40 c3c79c80c82c80c72c81c87c86c17c83c79c3c62c82c83c87c76c82c81c86c64 c3c3c3c3c3 c3c3c3c3c3c16c71c68c87c68c3c3c3c3c3c3c3c3c3c54c75c82c90c3c87c75c72c3c71c68c87c68c17 363 c3c3c3c3c3c16c71c68c87c68c79c76c80c76c87c86c3c3c3c54c75c82c90c3c87c75c72c3c79c76c80c76c87c86c3c82c73c3c87c75c72c3c71c68c87c68c17 c3c3c3c3c3c16c75c72c79c83c3c3c3c3c3c3c3c3c3c55c75c76c86c3c75c72c79c83c17 c3c3c3c3c3c16c88c69c3c3c3c3c3c3c3c3c3c3c3c47c16c80c82c80c72c81c87c86c3c69c92c3c88c81c69c76c68c86c72c71c3c72c86c87c76c80c68c87c82c85c86c17 c3c3c3c3c3c16c91c73c3c3c3c3c3c3c3c3c3c3c3c47c16c80c82c80c72c81c87c86c3c69c92c3c83c85c76c82c85c16c51c85c82c69c68c69c76c79c76c87c92c3c58c72c76c74c75c87c72c71c3c48c82c80c72c81c87c86c17 c3c3c3c3c3c16c83c83c3c3c3c3c3c3c3c3c3c3c3c47c16c80c82c80c72c81c87c86c3c69c92c3c83c79c82c87c87c76c81c74c3c83c82c86c76c87c76c82c81c17 c3c3c3c3c3c16c83c80c3c3c3c3c3c3c3c3c3c3c3c51c72c85c73c82c85c80c3c51c85c82c71c88c70c87c3c48c82c80c72c81c87c3c70c82c80c83c88c87c68c87c76c82c81c17 c3c3c3c3c3c16c41c71c76c89c20c19c19c3c3c3c3c3c3c55c75c72c3c81c82c81c72c91c70c72c72c71c68c81c70c72c3c83c85c82c69c68c69c76c79c76c87c92c3c86c87c85c72c68c80c3c76c86c3c76c81c3c83c72c85c70c72c81c87 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c81c82c87c3c73c85c68c70c87c76c82c81c68c79c3c83c85c72c70c72c81c87c3c11c83c85c82c69c68c69c76c79c76c87c92c12c3c86c82c3c71c76c89c76c71c72 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c3c87c75c72c3c89c68c79c88c72c86c3c69c92c3c20c19c19c17 c3c3c3c3c3 c3c3c3c3c3 c49c50c55c40c54 c3c3 c40c59c36c48c51c47c40c54 c58c36c53c49c44c49c42c54 c3 c43c40c53c40 c96 c86c88c69c3c74c72c87c38c57c54c86c87c68c80c83c86c3c94 c3c3c3c80c92c3c7c71c73c3c32c3c180c81c82c87c3c78c81c82c90c81c181c30 c3c3c3c80c92c3c11c7c68c88c87c75c15c3c7c71c68c87c72c15c3c7c89c72c85c12c3c32c3c11c7c71c73c15c3c7c71c73c15c3c7c71c73c12c30 c3c3c3c90c75c76c79c72c11c31c39c36c55c36c33c12c3c94 c3c3c3c3c3c3c79c68c86c87c3c76c73c11c18c70c89c86c11c63c86c14c12c34c72c81c71c18c76c82c12c30 c3c3c3c3c3c3c7c68c88c87c75c3c32c3c7c20c15c3c81c72c91c87c3c76c73c11c18c36c88c87c75c82c85c29c63c86c14c11c17c14c12c63c86c14c63c7c18c12c30 c3c3c3c3c3c3c7c71c68c87c72c3c32c3c7c20c15c3c81c72c91c87c3c76c73c11c18c39c68c87c72c29c63c86c14c11c17c14c12c63c86c14c63c7c18c12c30 c3c3c3c3c3c3c7c89c72c85c3c3c32c3c7c20c15c3c81c72c91c87c3c76c73c11c18c53c72c89c76c86c76c82c81c29c63c86c14c11c17c14c12c63c86c14c63c7c18c12c30 c3c3c3c3c3c3c79c68c86c87c3c76c73c11c7c68c88c87c75c3c81c72c3c7c71c73c3c68c81c71 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c7c71c68c87c72c3c81c72c3c7c71c73c3c68c81c71 c3c3c3c3c3c3c3c3c3c3c3c3c3c3c7c89c72c85c3c3c81c72c3c7c71c73c12c30 c3c3c3c96 c3c3c3c85c72c87c88c85c81c3c11c7c68c88c87c75c15c3c7c71c68c87c72c15c3c7c89c72c85c12c30 c96 c66c66c39c36c55c36c66c66 c6c3c7c36c88c87c75c82c85c29c3c90c68c86c84c88c76c87c75c3c7 c6c3c7c39c68c87c72c29c3c21c19c19c21c18c20c20c18c19c25c3c20c23c29c22c19c29c22c22c3c7 c6c3c7c53c72c89c76c86c76c82c81c29c3c20c17c20c22c3c7 c6c3c38c57c54c40c49c39 c20c30 364 useGammaFunctions.pl c6c4c18c88c86c85c18c69c76c81c18c83c72c85c79c3c16c90 c6c3c51c72c85c79c3c83c85c82c74c85c68c80c3c87c82c3c76c81c87c72c85c73c68c70c72c3c90c76c87c75c3c51c72c85c79c3c80c82c71c88c79c72c3c87c82c3c70c82c80c83c88c87c72c3c87c75c72c3 c6c3c42c68c80c80c68c3c68c81c71c3c44c81c70c82c80c83c79c72c87c72c3c42c68c80c80c68c3c73c88c81c70c68c87c76c82c81c86c17c3c3c55c75c72c3c68c79c74c82c85c76c87c75c80c86c3c76c81c3c87c75c72 c6c3c42c68c80c80c68c41c88c81c70c87c76c82c81c86c3c80c82c71c88c79c72c3c90c72c85c72c3c71c72c85c76c89c72c71c3c73c85c82c80c29 c6 c6c3c51c85c72c86c86c15c3c58c17c43c17c15c3c55c72c88c78c82c79c86c78c92c15c3c54c17c36c17c15c3c57c72c87c87c72c85c79c76c81c74c15c3c58c17c55c17c15c3c68c81c71 c6c3 c41c79c68c81c81c72c85c92c15c3c37c17c51c17c15c3c20c28c28c21c15c3c49c88c80c72c85c76c70c68c79c3c53c72c70c76c83c72c86c3c76c81c3c41c50c53c55c53c36c49c29c3c38c68c80c69c85c76c71c74c72 c6c3 c56c81c76c89c72c85c86c76c87c92c3c51c85c72c86c86c15c3c83c83c17c3c21c19c25c16c21c20c22c17 c6 c6c3c36c56c55c43c50c53c29c3c58c76c79c79c76c68c80c3c43c17c3c36c86c84c88c76c87c75c15c3c36c88c74c88c86c87c3c21c19c19c21 c6c3c7c36c88c87c75c82c85c29c3c90c68c86c84c88c76c87c75c3c7 c6c3c7c39c68c87c72c29c3c21c19c19c21c18c19c27c18c21c27c3c20c21c29c23c21c29c22c20c3c7 c6c3c7c53c72c89c76c86c76c82c81c29c3c20c17c22c3c7 c6c3c47c76c69c85c68c85c92c3c83c68c87c75c3c83c82c76c81c87c76c81c74c3c87c82c3c71c76c85c72c70c87c82c85c92c3c73c82c85c3c87c75c72c3c48c82c71c88c79c72c86c18c42c68c80c80c68c41c88c81c70c87c76c82c81c86c3c51c72c85c79 c6c3c80c82c71c88c79c72c3c87c75c68c87c3c76c86c3c183c88c86c72c71c183c3c76c81c3c87c75c72c3c81c72c91c87c3c86c87c68c87c72c80c72c81c87c17 c88c86c72c3c79c76c69c3c84c90c11c18c75c82c80c72c18c90c68c86c84c88c76c87c75c18c87c91c71c82c87c66c71c68c87c68c69c68c86c72c86c18c83c85c82c74c85c68c80c86c18c83c72c85c79c12c30 c6c3c76c80c83c82c85c87c3c87c90c82c3c73c88c81c70c87c76c82c81c86c3c73c85c82c80c3c87c75c72c3c80c82c71c88c79c72 c88c86c72c3c48c82c71c88c79c72c86c29c29c42c68c80c80c68c41c88c81c70c87c76c82c81c86c3c84c90c11c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c3c49c68c87c47c82c74c42c68c80c80c68c12c30 c6c3c53c72c80c76c81c71c3c87c75c72c3c88c86c72c85c3c82c81c3c75c82c90c3c87c75c72c3c83c85c82c74c85c68c80c3c90c82c85c78c86c17 c71c76c72c3c5c68c3c68c81c71c3c91c3c73c82c85c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c15c3c42c11c68c12c3c68c81c71c3c44c81c70c82c80c83c79c72c87c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c15c3c5c15 c3c3c3c3c5c51c11c68c15c91c12c3c68c85c72c3c85c72c74c88c76c85c72c71c3c82c81c3c70c82c80c80c68c81c71c3c79c76c81c72c63c81c5c3c88c81c79c72c86c86c11c35c36c53c42c57c3c32c32c3c21c12c30 c3c3c3c3 c11c7c68c15c3c7c91c12c3c32c3c35c36c53c42c57c30c3c6c3c74c72c87c3c87c75c72c3c87c90c82c3c70c82c80c80c68c81c71c3c79c76c81c72c3c68c85c74c88c80c72c81c87c86 c83c85c76c81c87c3c5c6c3c36c85c74c88c80c72c81c87c86c3c68c85c72c29c3c68c32c7c68c3c68c81c71c3c91c32c7c91c63c81c5c30c3c6c3c72c70c75c82c3c87c75c72c3c68c85c74c88c80c72c81c87c86 c80c92c3c7c42c3c32c3c72c91c83c11c9c49c68c87c47c82c74c42c68c80c80c68c11c7c68c12c12c30c3 c6c3c70c82c80c83c88c87c72c3c87c75c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81 c6c3c55c85c68c83c3c70c68c86c70c68c71c76c81c74c3c72c85c85c82c85c86c3c76c73c3c87c75c72c3c68c85c74c88c80c72c81c87c3c87c82c3c87c75c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c3c76c86c3c76c81c89c68c79c76c71c17 c71c76c72c3c5c39c44c40c39c29c3c42c68c80c80c68c11c7c68c12c3c76c86c3c81c82c87c3c71c72c73c76c81c72c71c63c81c5c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c42c12c30 c83c85c76c81c87c3c5c6c3c3c3c42c11c68c12c29c42c11c5c15c7c68c15c5c12c32c7c42c63c81c5c30c3c3c3c6c3c82c88c87c83c88c87c3c87c75c72c3c85c72c86c88c79c87c86 c80c92c3c7c51c3c32c3c9c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c11c7c68c15c7c91c12c30c3c3c3c6c3c70c82c80c83c88c87c72c3c87c75c72c3c44c81c70c82c80c83c79c72c87c72c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81 c6c3c55c85c68c83c3c70c68c86c70c68c71c76c81c74c3c72c85c85c82c85c86c3c76c73c3c87c75c72c3c68c85c74c88c80c72c81c87c86c3c68c85c72c3c76c81c89c68c79c76c71c17 c71c76c72c3c5c39c44c40c39c29c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c11c7c68c15c7c91c12c3c76c86c3c81c82c87c3c71c72c73c76c81c72c71c63c81c5c3c76c73c11c81c82c87c3c71c72c73c76c81c72c71c3c7c51c12c30 c83c85c76c81c87c3c5c6c3c51c11c68c15c91c12c29c51c11c5c15c7c68c15c5c15c5c15c7c91c15c5c12c32c7c51c63c81c5c30c3c3c6c3c82c88c87c83c88c87c3c87c75c72c3c85c72c86c88c79c87c86 c6c3c55c75c72c3c73c82c79c79c82c90c76c81c74c3c83c85c82c71c88c70c87c3c76c86c3c87c82c3c68c76c71c3c76c81c3c80c68c81c88c68c79c3c70c82c80c83c88c68c87c76c82c81c86c3c82c73c3c47c16c74c68c80c80c68c3c47c16c80c82c80c72c81c87c86 c83c85c76c81c87c3c5c6c3c42c11c68c12c13c51c11c68c15c91c12c3c32c3c5c15c7c51c13c7c42c15c5c63c81c5c30 c72c91c76c87c30 365 Modules::GammaFunctions.pm c83c68c70c78c68c74c72c3c48c82c71c88c79c72c86c29c29c42c68c80c80c68c41c88c81c70c87c76c82c81c86c30 c6c3c51c72c85c79c3c80c82c71c88c79c72c3c76c80c83c79c72c80c72c81c87c76c81c74c3c68c79c74c82c85c76c87c75c80c86c3c73c82c85c3c87c75c72c3c42c68c80c80c68c3c68c81c71c3c44c81c70c82c80c83c79c72c87c72 c6c3c42c68c80c80c68c3c73c88c81c70c87c76c82c81c86c3c71c72c85c76c89c72c71c3c73c85c82c80c29 c6c3c51c85c72c86c86c15c3c58c17c43c17c15c3c55c72c88c78c82c79c86c78c92c15c3c54c17c36c17c15c3c57c72c87c87c72c85c79c76c81c74c15c3c58c17c55c17c15c3c68c81c71c3c41c79c68c81c81c72c85c92c15c3c37c17c51c17c15c3c20c28c28c21c15 c6c3c3c3c3c49c88c80c72c85c76c70c68c79c3c53c72c70c76c83c72c86c3c76c81c3c41c50c53c55c53c36c49c29c3c38c68c80c69c85c76c71c74c72c3c56c81c76c89c72c85c86c76c87c92c3c51c85c72c86c86c15c3c83c83c17c3c21c19c25c16c21c20c22c17 c6 c6c3c36c56c55c43c50c53c29c3c58c76c79c79c76c68c80c3c43c17c3c36c86c84c88c76c87c75c15c3c36c88c74c88c86c87c3c21c19c19c21 c6c3c7c36c88c87c75c82c85c29c3c90c68c86c84c88c76c87c75c3c7 c6c3c7c39c68c87c72c29c3c21c19c19c21c18c19c27c18c21c27c3c20c21c29c23c21c29c22c20c3c7 c6c3c7c53c72c89c76c86c76c82c81c29c3c20c17c22c3c7 c88c86c72c3c86c87c85c76c70c87c30 c88c86c72c3c40c91c83c82c85c87c72c85c30 c88c86c72c3c89c68c85c86c3c84c90c11c35c38c50c41c3c35c44c54c36c3c35c40c59c51c50c53c55c66c50c46c12c30 c35c44c54c36c3c32c3c84c90c11c40c91c83c82c85c87c72c85c12c30 c35c40c59c51c50c53c55c66c50c46c3c32c3c84c90c11c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c3c49c68c87c47c82c74c42c68c80c80c68c12c30c3 c88c86c72c3c70c82c81c86c87c68c81c87c3c40c51c54c3c3c3c32c33c3c86c70c68c79c68c85c3c22c72c16c26c30 c88c86c72c3c70c82c81c86c87c68c81c87c3c44c55c48c36c59c3c32c33c3c86c70c68c79c68c85c3c20c19c19c30 c88c86c72c3c70c82c81c86c87c68c81c87c3c41c51c48c44c49c3c32c33c3c86c70c68c79c68c85c3c20c72c16c22c19c30 c88c86c72c3c70c82c81c86c87c68c81c87c3c54c55c51c3c3c3c32c33c3c86c70c68c79c68c85c3c21c17c24c19c25c25c21c27c21c26c23c25c22c20c19c19c19c24c30 c88c86c72c3c70c82c81c86c87c68c81c87c3c54c40c53c3c3c3c32c33c3c86c70c68c79c68c85c3c20c17c19c19c19c19c19c19c19c19c19c20c28c19c19c20c24c30 c35c38c50c41c3c32c3c11c3c26c25c17c20c27c19c19c28c20c26c21c28c23c26c20c23c25c3c3c3c3c3c3c15 c3c3c3c3c3c3c3c3c16c27c25c17c24c19c24c22c21c19c22c21c28c23c20c25c26c26c3c3c3c3c3c3c15 c3c3c3c3c3c3c3c3c3c21c23c17c19c20c23c19c28c27c21c23c19c27c22c19c28c20c3c3c3c3c3c3c15 c3c3c3c3c3c3c3c3c3c16c20c17c21c22c20c26c22c28c24c26c21c23c24c19c20c24c24c3c3c3c3c3c15 c3c3c3c3c3c3c3c3c3c3c19c17c19c19c20c21c19c27c25c24c19c28c26c22c27c25c25c20c26c28c3c3c15 c3c3c3c3c3c3c3c3c3c16c19c17c19c19c19c19c19c24c22c28c24c21c22c28c27c23c28c24c22c3c12c30 c86c88c69c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c3c94 c3c3c80c92c3c11c7c68c15c3c7c91c12c3c32c3c35c66c30 c3c3 c3c3c6c3c56c86c72c86c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c54c72c85c76c72c86c3c68c81c71c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c38c82c81c87c76c81c88c72c71c41c85c68c70c87c76c82c81 c3c3 c3c3c80c92c3c11c7c74c68c80c80c83c15c7c74c79c81c12c30 c3c3 c3c3c76c73c11c7c91c3c31c3c19c12c3c94 c3c3c3c3c3c90c68c85c81c3c5c37c68c71c3c86c72c70c82c81c71c3c68c85c74c88c80c72c81c87c3c76c81c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c29c3c91c11c7c91c12c3c31c3c19c63c81c5c30 c3c3c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c30 c3c3c96 c3c3c76c73c11c7c68c3c31c32c3c19c12c3c94 c3c3c3c3c3c90c68c85c81c3c5c37c68c71c3c73c76c85c86c87c3c68c85c74c88c80c72c81c87c3c76c81c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c29c3c68c11c7c68c12c3c31c32c3c19c63c81c5c30 c3c3c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c30 c3c3c96 c3c3c76c73c11c7c91c3c31c3c7c68c14c20c12c3c94 c3c3c3c3c3c11c7c74c68c80c80c83c15c3c7c74c79c81c12c3c32c3c9c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c54c72c85c76c72c86c11c7c68c15c7c91c12c30 366 c3c3c3c3c3c6c3c90c68c85c81c3c5c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c3c85c72c87c88c85c81c72c71c3c69c92c3c51c66c69c92c54c72c85c76c72c86c3c7c74c68c80c80c83c63c81c5c30 c3c3c3c3c3c85c72c87c88c85c81c3c11c90c68c81c87c68c85c85c68c92c12c3c34c3c11c7c74c68c80c80c83c15c7c74c79c81c12c3c29c3c11c7c74c68c80c80c83c12c30c3 c3c3c96 c3c3c72c79c86c72c3c94 c3c3c3c3c3c11c7c74c68c80c80c83c15c3c7c74c79c81c12c3c32c3c9c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c38c82c81c87c76c81c88c72c71c41c85c68c70c87c76c82c81c11c7c68c15c7c91c12c30 c3c3c3c3c3c7c74c68c80c80c83c3c32c3c20c3c16c3c7c74c68c80c80c83c30c3c6c3c87c68c78c72c3c70c82c80c83c79c76c80c72c81c87 c3c3c3c3c3c85c72c87c88c85c81c3c11c90c68c81c87c68c85c85c68c92c12c3c34c3c11c7c74c68c80c80c83c15c3c7c74c79c81c12c3c29c3c11c7c74c68c80c80c83c12c30c3 c3c3c96 c96 c86c88c69c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c54c72c85c76c72c86c3c94 c3c3c3c6c3c56c86c72c86c3c49c68c87c47c82c74c42c68c80c80c68 c3c3c3c6c3c53c72c87c88c85c81c86c3c87c75c72c3c76c81c70c82c80c83c79c72c87c72c3c74c68c80c80c68c3c73c88c81c70c87c76c82c81c3c51c11c68c15c91c12c3c72c89c68c79c88c68c87c72c71c3c69c92c3c76c87c86c3c86c72c85c76c72c86 c3c3c3c6c3c85c72c83c85c72c86c72c81c87c68c87c76c82c81c3c68c86c3c74c68c80c86c72c85c17c3c3c36c79c86c82c3c85c72c87c88c85c81c86c3c79c81c11c74c68c80c80c68c11c68c12c12c3c68c86c3c74c79c81c17 c3c3c3 c3c3c3c80c92c3c11c7c68c15c3c7c91c12c3c32c3c35c66c30 c3c3c3 c3c3c3c80c92c3c7c74c79c81c3c32c3c9c49c68c87c47c82c74c42c68c80c80c68c11c7c68c12c30 c3c3c3 c3c3c3c76c73c11c7c91c3c31c32c3c19c3c12c3c94 c3c3c3c3c3c3c90c68c85c81c3c5c91c3c31c32c3c19c3c76c81c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c54c72c85c76c72c86c63c81c5c30 c3c3c3c3c3c3c85c72c87c88c85c81c3c11c19c15c7c74c79c81c12c30 c3c3c3c96 c3c3 c3c3c3c80c92c3c7c68c83c3c3c32c3c7c68c30 c3c3c3c80c92c3c7c86c88c80c3c32c3c20c18c7c68c30 c3c3c3c80c92c3c7c71c72c79c3c32c3c7c86c88c80c30 c3c3c3 c3c3c3c80c92c3c7c70c79c72c68c81c66c72c91c76c87c3c32c3c19c30 c3c3c3c73c82c85c11c80c92c3c7c76c32c20c30c3c7c76c31c32c44c55c48c36c59c30c3c7c76c14c14c12c3c94 c3c3c3c3c3c3c7c68c83c14c14c30 c3c3c3c3c3c3c7c71c72c79c3c13c32c3c7c91c18c7c68c83c30 c3c3c3c3c3c3c7c86c88c80c3c14c32c3c7c71c72c79c30 c3c3c3c3c3c3c7c70c79c72c68c81c66c72c91c76c87c3c32c3c20c15c3c79c68c86c87c3c76c73c11c68c69c86c11c7c71c72c79c12c3c31c3c68c69c86c11c7c86c88c80c12c13c40c51c54c12c30 c3c3c3c96 c3c3c3c90c68c85c81c3c5c68c3c87c82c82c3c79c68c85c74c72c15c3c44c55c48c36c59c3c87c82c82c3c86c80c68c79c79c3c76c81c3c5c15 c3c3c3c3c3c3c3c3c5c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c54c72c85c76c72c86c63c81c5c3c76c73c11c81c82c87c3c7c70c79c72c68c81c66c72c91c76c87c12c30 c3c3c3 c3c3c3c80c92c3c7c74c68c80c86c72c85c3c32c3c7c86c88c80c3c13c3c72c91c83c11c16c7c91c3c14c3c7c68c3c13c3c79c82c74c11c7c91c12c3c16c3c7c74c79c81c12c30 c3c3c3c85c72c87c88c85c81c3c11c7c74c68c80c86c72c85c15c3c7c74c79c81c12c30 c96 c86c88c69c3c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c38c82c81c87c76c81c88c72c71c41c85c68c70c87c76c82c81c3c94 c3c3c3c6c3c56c86c72c86c3c49c68c87c47c82c74c42c68c80c80c68 c3c3c3c6c3c53c72c87c88c85c81c86c3c87c75c72c3c76c81c70c82c80c83c79c72c87c72c3c74c68c80c80c68c3c73c88c81c70c87c76c82c81c3c52c11c68c15c91c12c3c72c89c68c79c88c68c87c72c71c3c69c92c3c76c87c86c3 c3c3c3c6c3c70c82c81c87c76c81c88c72c71c3c73c85c68c70c87c76c82c81c3c85c72c83c85c72c86c72c81c87c68c87c76c82c81c17c3c3c53c72c87c88c85c81c86c3c79c81c62c74c68c80c80c68c11c68c12c64c3c87c82c82c17 c3c3c3 c3c3c3c80c92c3c11c7c68c15c3c7c91c12c3c32c3c35c66c30 c3c3c3c3c3c3 c3c3c3c80c92c3c7c74c79c81c3c32c3c9c49c68c87c47c82c74c42c68c80c80c68c11c7c68c12c30 c3c3c3c80c92c3c7c69c3c32c3c7c91c3c14c3c20c3c16c3c7c68c30 c3c3c3c80c92c3c7c70c3c32c3c20c18c41c51c48c44c49c30 c3c3c3c80c92c3c7c71c3c32c3c20c18c7c69c30 367 c3c3c3c80c92c3c7c75c3c32c3c7c71c30 c3c3c3c80c92c3c11c7c68c81c15c3c7c71c72c79c12c30 c3c3c3 c3c3c3c80c92c3c7c70c79c72c68c81c66c72c91c76c87c3c32c3c19c30 c3c3c3c73c82c85c11c80c92c3c7c76c32c20c30c7c76c31c32c44c55c48c36c59c30c7c76c14c14c12c3c94 c3c3c3c3c3c3c7c68c81c3c3c32c3c16c7c76c13c11c7c76c16c7c68c12c30 c3c3c3c3c3c3c7c69c3c3c14c32c3c21c30 c3c3c3c3c3c3c7c71c3c3c3c32c3c7c68c81c13c7c71c3c14c3c7c69c30 c3c3c3c3c3c3c7c71c3c3c3c32c3c41c51c48c44c49c3c76c73c11c68c69c86c11c7c71c12c3c31c3c41c51c48c44c49c12c30 c3c3c3c3c3c3c7c70c3c3c3c32c3c7c69c3c14c3c7c68c81c18c7c70c30 c3c3c3c3c3c3c7c70c3c3c3c32c3c41c51c48c44c49c3c76c73c11c68c69c86c11c7c70c12c3c31c3c41c51c48c44c49c12c30 c3c3c3c3c3c3c7c71c3c3c3c32c3c20c18c7c71c30 c3c3c3c3c3c3c7c71c72c79c3c32c3c7c71c13c7c70c30 c3c3c3c3c3c3c7c75c3c3c13c32c3c7c71c72c79c30 c3c3c3c3c3c3c7c70c79c72c68c81c66c72c91c76c87c3c32c3c20c15c3c79c68c86c87c3c76c73c11c68c69c86c11c7c71c72c79c3c16c3c20c12c3c31c3c40c51c54c12c30 c3c3c3c96 c3c3c3c90c68c85c81c3c5c68c3c87c82c82c3c79c68c85c74c72c15c3c44c55c48c36c59c3c87c82c82c3c86c80c68c79c79c3c76c81c3c5c15 c3c3c3c3c3c3c3c3c5c44c81c70c82c80c83c79c72c87c72c42c68c80c80c68c66c69c92c38c82c81c87c76c81c88c72c71c41c85c68c70c87c76c82c81c63c81c5c3c76c73c11c81c82c87c3c7c70c79c72c68c81c66c72c91c76c87c12c30 c3c3c3 c3c3c3c80c92c3c7c74c68c80c70c73c3c32c3c72c91c83c11c16c7c91c14c7c68c13c79c82c74c11c7c91c12c3c16c3c7c74c79c81c12c13c7c75c30 c3c3c3c85c72c87c88c85c81c3c11c7c74c68c80c70c73c15c3c7c74c79c81c12c30 c96 c86c88c69c3c49c68c87c47c82c74c42c68c80c80c68c3c94 c3c3c3c6c3c53c72c87c88c85c81c86c3c87c75c72c3c89c68c79c88c72c3c79c81c62c74c68c80c80c68c11c7c91c91c12c64c3c73c82c85c3c7c91c91c3c33c3c19 c3c3c3 c3c3c3c80c92c3c11c7c91c91c12c3c32c3c35c66c30 c3c3c3c85c72c87c88c85c81c3c88c81c71c72c73c3c76c73c11c7c91c91c3c31c32c3c19c12c30c3c6c3c71c76c89c76c86c76c82c81c3c69c92c3c93c72c85c82c3c83c85c82c87c72c70c87c76c82c81 c3c3c3c80c92c3c7c91c3c3c3c32c3c7c91c91c30 c3c3c3c80c92c3c7c92c3c3c3c32c3c7c91c30 c3c3c3c80c92c3c7c87c80c83c3c32c3c7c91c3c14c3c24c17c24c30c3c3c3 c3c3c3c3c3c3c7c87c80c83c3c32c3c11c7c91c3c14c3c19c17c24c12c13c79c82c74c11c7c87c80c83c12c16c7c87c80c83c30 c3c3c3c80c92c3c7c86c72c85c3c32c3c54c40c53c30 c3c3c3c73c82c85c11c80c92c3c7c77c32c19c30c7c77c31c32c24c30c7c77c14c14c12c3c94 c3c3c3c3c3c3c7c92c14c14c30 c3c3c3c3c3c3c7c86c72c85c3c14c32c3c7c38c50c41c62c7c77c64c18c7c92c30 c3c3c3c96 c3c3c3c85c72c87c88c85c81c3c7c87c80c83c3c14c3c79c82c74c11c54c55c51c13c7c86c72c85c3c18c3c7c91c12c30 c96 c20c30 368 APPENDIX F Supplemental Tables for Chapter 7. The tables contained in this appendix document the empirical hyetographs for the 0–12 hr, 12–24hr, and 24 hr and greater storm durations. These hyetographs are described in chapter 7. Tables F1, F3, and F5 document the actual percentiles of the empirical hyetographs; whereas tables F2, F4, and F6 document the L-moments of the empirical hyetographs. Tables F7, F8, and F9 document graphically smoothed values of the percentiles listed in tables F1, F3, and F5; the smoothed values are provided to increase the applicability of the percentiles in hydrologic design applications at the request of staff from the Texas Department of Transportation (Stolpa, personal commun., 2002) and consensus of the Texas Tech University, University of Houston, and Lamar University hyetograph research team. 369 Table F1. Percentile statistics for empirical hyetograph analysis for the 0–12 hr storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) ( ) (percent) (percent) (percent) (percent) (percent) 2.5 937 0.00 0.00 3.333 0.00 13.34 5.0 344 12.95 4.416 25.00 1.999 41.30 7.5 335 18.04 8.046 34.00 2.798 53.83 10.0 396 24.45 10.36 43.37 4.332 63.02 12.5 322 30.13 13.40 48.36 4.943 69.53 15.0 321 35.75 17.73 56.21 7.038 76.93 17.5 333 38.87 17.34 61.00 8.422 77.38 20.0 323 40.46 21.67 66.97 11.53 83.45 22.5 328 40.62 18.06 70.04 6.91 84.78 25.0 289 44.84 27.56 72.82 9.47 85.37 27.5 304 48.86 26.23 70.62 11.56 87.02 30.0 287 54.47 32.17 76.95 13.69 88.31 32.5 341 52.00 30.75 78.56 12.41 88.10 35.0 232 57.89 33.70 81.57 14.21 91.66 37.5 257 54.58 30.23 83.61 15.82 90.31 40.0 257 66.77 35.58 85.47 20.83 92.83 42.5 244 63.54 37.65 85.11 21.47 93.14 45.0 252 69.66 37.44 84.80 13.92 92.12 47.5 231 63.77 32.95 85.91 15.96 93.03 50.0 252 71.75 43.99 86.28 24.82 93.84 52.5 246 67.70 35.95 86.38 14.39 95.44 55.0 255 73.43 47.02 88.62 28.52 95.13 57.5 232 72.25 41.19 86.70 22.47 95.82 60.0 254 76.23 46.65 89.59 21.59 95.55 62.5 246 75.30 50.00 89.70 29.54 96.44 65.0 202 77.45 60.11 91.57 30.16 96.76 67.5 310 79.49 57.95 91.90 34.82 96.71 70.0 244 80.61 55.22 93.06 24.99 97.38 72.5 274 84.22 61.14 93.50 35.21 97.32 75.0 249 85.07 66.29 93.47 35.43 97.80 77.5 280 86.88 67.89 93.77 44.60 98.17 80.0 290 87.66 69.44 95.32 47.56 98.38 82.5 272 89.90 76.21 96.32 50.63 98.62 85.0 259 92.76 80.81 97.50 55.34 98.80 87.5 264 94.27 85.91 97.82 67.17 99.00 90.0 297 95.60 90.30 98.30 71.15 99.26 92.5 292 96.67 92.48 98.65 78.16 100.0 95.0 213 97.86 95.22 99.14 86.99 100.0 97.5 801 100.0 100.0 100.0 98.42 100.0 370 Table F2. L-moment statistics for empirical hyetograph analysis for the 0–12 hr storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Mean L-scale L-CV L-skew L-kurtosis (percent) ( ) (percent) (percent) ( ) ( ) ( ) ( ) 2.5 937 3.617 2.999 0.8292 0.6818 0.3782 0.1744 5.0 344 17.69 8.698 .4917 .2933 .1075 .07006 7.5 335 23.58 10.69 .4534 .2288 .08766 .04637 10.0 396 29.35 12.68 .4321 .2008 .07642 .03628 12.5 322 33.18 13.79 .4157 .1809 .07749 .02645 15.0 321 38.67 14.16 .3662 .1159 .07483 .02441 17.5 333 40.41 14.84 .3673 .08651 .01826 .01751 20.0 323 44.08 15.48 .3512 .07921 .01584 -.002722 22.5 328 44.18 16.59 .3755 .06167 -.009633 -.002197 25.0 289 47.88 15.68 .3274 .008481 .02000 -.006014 27.5 304 48.91 15.74 .3219 .007976 .01414 .004665 30.0 287 53.37 15.85 .2971 -.04174 .004482 -.008986 32.5 341 52.20 16.23 .3109 -.03955 .002352 -.01036 35.0 232 56.19 16.13 .2871 -.0634 .006389 -.007393 37.5 257 54.95 16.47 .2998 -.0423 -.01971 -.01620 40.0 257 60.40 15.95 .2640 -.1235 -.003574 -.006250 42.5 244 59.84 15.68 .2620 -.1003 .009002 -.01487 45.0 252 60.17 16.74 .2781 -.1670 -.004018 .02968 47.5 231 58.97 16.91 .2868 -.1200 -.02203 .009590 50.0 252 64.19 14.95 .2329 -.1799 .02796 .007438 52.5 246 61.33 16.69 .2721 -.1517 .004242 .01297 55.0 255 66.41 14.65 .2205 -.1758 .02404 -.008292 57.5 232 64.43 15.50 .2405 -.1847 .02611 .009597 60.0 254 66.96 15.27 .2280 -.2262 .03243 .009398 62.5 246 68.58 14.47 .2110 -.2233 .04760 -.006480 65.0 202 70.77 13.84 .1955 -.2711 .1110 -.03178 67.5 310 72.29 13.09 .1811 -.2591 .07725 -.02420 70.0 244 71.14 14.37 .2020 -.2840 .07629 -.009121 72.5 274 75.07 12.73 .1696 -.3194 .09683 -.01019 75.0 249 76.22 12.33 .1618 -.3283 .1173 -.02282 77.5 280 78.71 11.14 .1415 -.3394 .1165 -.01964 80.0 290 80.09 10.53 .1314 -.3382 .1182 -.03085 82.5 272 82.72 9.665 .1168 -.3939 .1718 -.06469 85.0 259 85.52 8.577 .1003 -.4495 .2176 -.1082 87.5 264 88.31 7.006 .07933 -.4827 .2770 -.1583 90.0 297 89.95 6.486 .07210 -.5550 .3471 -.2082 92.5 292 91.93 5.419 .05894 -.5692 .3614 -.2214 95.0 213 93.91 4.286 .04564 -.6395 .4548 -.3112 97.5 801 98.87 1.070 .01082 -.8943 .7668 -.6391 τ 5 371 Table F3. Percentile statistics for empirical hyetograph analysis for the 12–24 hr storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) ( ) (percent) (percent) (percent) (percent) (percent) 2.5 1251 1.505 0.000 7.965 0.000 18.61 5.0 632 16.94 6.065 28.21 2.274 48.43 7.5 624 26.87 9.896 42.88 4.612 58.75 10.0 599 34.64 13.41 52.06 6.849 69.96 12.5 556 38.81 18.53 58.18 8.219 78.73 15.0 529 45.37 23.36 61.99 10.54 82.27 17.5 517 49.58 25.19 63.42 9.546 81.84 20.0 464 52.42 25.82 69.45 9.247 84.20 22.5 449 56.07 30.70 71.06 12.44 86.03 25.0 440 56.14 25.67 74.01 13.44 86.85 27.5 430 59.84 31.60 80.50 13.26 87.49 30.0 374 64.81 35.70 83.74 14.66 89.79 32.5 388 60.55 32.87 87.36 15.20 92.29 35.0 343 63.34 35.76 90.91 12.76 95.42 37.5 350 60.56 31.71 87.87 13.15 95.75 40.0 349 59.81 36.87 86.78 15.26 96.10 42.5 320 62.18 37.43 89.07 14.76 95.98 45.0 307 68.11 42.68 89.43 19.64 96.59 47.5 337 65.30 41.65 88.89 18.04 96.70 50.0 329 67.22 40.27 86.87 19.47 96.39 52.5 352 68.06 41.78 90.58 21.49 97.19 55.0 311 66.67 42.58 87.10 19.29 95.73 57.5 313 67.17 46.01 90.87 25.14 97.22 60.0 301 74.07 49.98 92.61 26.26 97.28 62.5 294 76.94 54.90 93.30 32.00 98.04 65.0 297 80.52 59.27 93.59 34.42 98.26 67.5 306 81.33 57.63 93.82 36.80 98.62 70.0 278 79.85 57.62 93.59 36.97 97.93 72.5 301 81.65 58.37 95.07 39.248 98.74 75.0 286 83.41 60.43 94.94 42.75 99.02 77.5 326 85.53 66.34 94.68 46.01 98.71 80.0 316 86.88 68.84 95.88 49.18 98.67 82.5 308 88.35 71.49 96.82 49.44 99.20 85.0 337 87.84 72.34 96.46 43.85 98.77 87.5 373 90.34 78.62 97.35 50.92 99.47 90.0 388 92.19 82.80 98.11 62.71 99.53 92.5 350 93.04 87.48 98.17 65.05 99.45 95.0 378 95.59 90.20 98.70 65.70 99.77 97.5 897 100.0 97.67 100.0 91.05 100.0 372 Table F4. L-moment statistics for empirical hyetograph analysis for the 12–24 hr storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Mean L-scale L-CV L-skew L-kurtosis (percent) ( ) (percent) (percent) ( ) ( ) ( ) ( ) 2.5 1251 6.251 4.558 0.7291 0.5396 0.245367 0.136111 5.0 632 20.76 9.868 .4754 .2755 .138637 .088947 7.5 624 28.83 11.81 .4095 .1628 .071053 .059363 10.0 599 35.51 13.67 .3850 .1220 .043538 .037284 12.5 556 40.34 14.66 .3635 .09853 .056111 .016525 15.0 529 44.83 14.89 .3321 .04748 .05421 .01283 17.5 517 46.66 14.95 .3205 -.005984 .06574 .02673 20.0 464 49.70 15.12 .3041 -.05062 .06304 .03288 22.5 449 52.49 14.92 .2843 -.09168 .06198 .03205 25.0 440 51.97 15.87 .3054 -.07457 .003331 .03420 27.5 430 55.63 16.08 .2891 -.1311 -.002824 .01784 30.0 374 58.51 16.21 .2771 -.1583 -.007448 .01756 32.5 388 58.90 16.81 .2854 -.1247 -.02987 -.002221 35.0 343 60.62 17.19 .2836 -.1339 -.01799 -.01281 37.5 350 58.59 17.22 .2938 -.1000 -.01844 -.00867 40.0 349 59.08 16.68 .2823 -.08732 .002014 -.02434 42.5 320 60.31 16.71 .2771 -.1043 -.005200 -.01775 45.0 307 63.14 16.08 .2546 -.1411 .01482 -.02070 47.5 337 61.84 16.47 .2663 -.1233 .002887 -.01068 50.0 329 61.89 15.93 .2574 -.1161 .01174 -.01160 52.5 352 63.76 16.15 .2533 -.1318 -.01044 -.009891 55.0 311 62.11 15.79 .2542 -.1198 .006295 -.008991 57.5 313 65.54 14.88 .2271 -.1239 .009506 -.03097 60.0 301 69.02 14.64 .2122 -.2022 .02528 -.01438 62.5 294 70.96 13.66 .1925 -.1972 .03959 -.03362 65.0 297 72.85 13.31 .1827 -.2433 .06553 -.02381 67.5 306 73.75 13.08 .1773 -.2496 .06067 -.02114 70.0 278 72.95 13.12 .1799 -.2293 .05122 -.01645 72.5 301 74.15 12.99 .1751 -.2382 .05100 -.02954 75.0 286 76.26 12.18 .1597 -.2606 .07025 -.03829 77.5 326 78.44 11.04 .1407 -.2795 .08845 -.03992 80.0 316 79.58 10.92 .1373 -.3260 .1211 -.06497 82.5 308 80.83 10.73 .1328 -.3569 .1411 -.07603 85.0 337 79.97 11.14 .1393 -.3509 .1258 -.05265 87.5 373 83.42 9.765 .1171 -.4039 .1749 -.07394 90.0 388 86.47 8.260 .09553 -.4526 .2394 -.1554 92.5 350 87.69 7.686 .08765 -.5202 .3387 -.2197 95.0 378 89.42 7.173 .08021 -.5603 .3304 -.1753 97.5 897 96.58 2.929 .03033 -.7478 .5145 -.3540 τ 5 373 Table F5. Percentile statistics for empirical hyetograph analysis for the 24 hr and greater storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) ( ) (percent) (percent) (percent) (percent) (percent) 2.5 1526 3.390 0.000 11.63 0.000 25.68 5.0 719 13.64 6.061 30.58 2.439 48.26 7.5 662 20.11 8.587 41.01 4.062 59.20 10.0 647 23.11 11.120 50.164 5.994 68.39 12.5 594 27.20 13.14 56.58 5.861 79.13 15.0 589 31.58 16.74 55.18 7.645 80.18 17.5 546 40.12 19.44 59.85 9.328 80.45 20.0 480 43.18 21.19 63.97 8.5433 82.51 22.5 446 43.37 18.68 64.71 6.757 83.03 25.0 412 42.20 19.12 67.01 8.271 84.62 27.5 408 38.43 20.02 68.36 10.49 86.42 30.0 379 39.43 26.00 66.13 9.962 86.40 32.5 358 41.61 20.89 61.70 10.83 80.90 35.0 393 45.64 26.04 64.25 11.48 83.57 37.5 383 49.17 26.25 66.05 12.91 81.24 40.0 416 49.27 28.02 68.39 15.93 81.55 42.5 410 52.07 31.72 70.89 22.05 83.79 45.0 400 56.91 36.49 72.08 17.49 87.67 47.5 405 54.80 32.57 71.64 12.70 85.67 50.0 412 55.52 30.61 73.11 17.48 88.29 52.5 392 52.89 36.09 72.16 21.91 88.37 55.0 379 59.35 44.31 78.76 29.02 90.62 57.5 393 64.62 48.02 83.30 28.54 92.88 60.0 370 63.90 44.71 84.44 29.14 93.99 62.5 404 67.04 50.99 86.85 29.65 95.44 65.0 387 67.10 46.42 86.73 22.72 97.35 67.5 375 68.66 50.00 86.79 28.83 96.73 70.0 383 70.21 51.32 89.19 35.20 98.47 72.5 421 75.00 54.95 89.19 33.97 97.47 75.0 471 73.90 57.27 88.32 36.60 97.58 77.5 459 76.82 60.45 89.78 41.05 97.47 80.0 463 79.11 62.07 90.28 39.85 97.47 82.5 486 83.01 66.16 93.37 46.92 97.49 85.0 501 86.11 70.94 94.57 52.04 98.32 87.5 523 88.14 75.40 96.34 56.46 98.41 90.0 537 87.40 75.35 97.04 55.37 98.86 92.5 553 91.16 82.46 97.40 67.88 99.15 95.0 550 94.96 88.20 98.32 78.19 99.55 97.5 1238 99.38 96.22 100.0 89.63 100.0 374 Table F6. L-moment statistics for empirical hyetograph analysis for the 24 hr and greater storm duration and one or more inches of depth Center of percent of storm duration interval No. of samples Mean L-scale L-CV L-skew L-kurtosis (percent) ( ) (percent) (percent) ( ) ( ) ( ) ( ) 2.5 1526 8.790 6.028 0.6858 0.4929 0.2245 0.1339 5.0 719 21.03 10.50 .4994 .3178 .1242 .05618 7.5 662 26.73 12.11 .4529 .2445 .07722 .03255 10.0 647 31.27 13.81 .4417 .2345 .05807 .005247 12.5 594 35.51 15.25 .4294 .2033 .03884 -.01834 15.0 589 37.89 14.97 .3951 .1741 .06258 .003964 17.5 546 41.74 15.12 .3622 .09784 .05468 .01842 20.0 480 44.79 15.73 .3513 .05137 .03029 .01798 22.5 446 43.59 16.14 .3701 .06245 .02712 .02149 25.0 412 44.05 16.38 .3718 .06935 .008111 .01472 27.5 408 44.44 16.21 .3647 .08889 .01636 -.003169 30.0 379 45.47 15.36 .3378 .09027 .04793 -.009051 32.5 358 43.71 14.86 .3399 .08673 .05646 .03103 35.0 393 47.35 14.97 .3162 .04121 .05414 .02339 37.5 383 48.04 14.65 .3049 .009838 .04422 .04283 40.0 416 49.15 14.33 .2916 .008136 .03859 .02183 42.5 410 52.21 13.76 .2636 .02263 .03866 .004998 45.0 400 54.02 14.74 .2729 -.02930 .06430 .004514 47.5 405 52.32 15.35 .2933 -.04043 .05126 .02626 50.0 412 53.06 15.14 .2853 -.01078 .03271 .01334 52.5 392 53.83 14.01 .2603 .02288 .06559 -.01847 55.0 379 60.15 13.18 .2191 -.02562 .07155 -.04466 57.5 393 63.24 13.77 .2178 -.1220 .08023 -.03065 60.0 370 63.20 13.89 .2198 -.07004 .03157 -.04287 62.5 404 65.74 13.67 .2079 -.1138 .05475 -.04862 65.0 387 64.26 14.89 .2317 -.1205 .04477 -.02989 67.5 375 66.03 13.93 .2109 -.1321 .05598 -.03400 70.0 383 68.25 13.64 .1998 -.1233 .03936 -.05435 72.5 421 70.65 12.75 .1805 -.1597 .05603 -.03155 75.0 471 70.74 12.17 .1720 -.1452 .07805 -.04709 77.5 459 72.91 11.69 .1604 -.1863 .1014 -.06483 80.0 463 73.23 12.05 .1645 -.2278 .1099 -.05578 82.5 486 76.86 10.88 .1416 -.2556 .1058 -.05552 85.0 501 79.85 10.15 .1272 -.3067 .1296 -.06453 87.5 523 82.61 9.310 .1127 -.3432 .1411 -.08276 90.0 537 82.56 9.223 .1117 -.3073 .1090 -.06038 92.5 553 87.30 6.794 .07782 -.3380 .1523 -.09454 95.0 550 91.33 5.005 .05480 -.4113 .2077 -.1244 97.5 1238 96.60 2.580 .02671 -.5947 .3164 -.1937 τ 5 375 Table F7. Graphically smoothed percentile statistics for empirical hyetograph analysis for the 0–12 hr storm duration and one or more inches of depth Center of percent of storm duration interval Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) (percent) (percent) (percent) (percent) (percent) 2.5 3.00 2.00 6.00 1.00 9.00 5.0 11.00 4.416 25.00 1.999 41.30 7.5 18.04 8.046 34.00 2.798 53.83 10.0 24.45 10.36 43.37 4.332 63.02 12.5 30.13 13.40 48.36 4.943 69.53 15.0 35.75 15.00 56.21 7.038 76.93 17.5 38.87 17.34 61.00 8.00 80.00 20.0 40.46 20.00 67.50 8.00 83.45 22.5 42.00 22.00 70.04 8.50 84.78 25.0 44.84 24.00 72.82 9.47 85.37 27.5 48.86 26.50 74.00 11.56 87.02 30.0 51.50 30.00 76.95 13.00 88.31 32.5 54.00 30.75 78.56 14.00 88.10 35.0 56.50 32.00 81.57 14.21 89.00 37.5 59.50 33.00 83.61 15.82 90.31 40.0 62.00 34.00 84.50 16.50 91.00 42.5 63.54 36.00 85.11 17.50 91.50 45.0 66.00 36.50 85.00 18.00 92.12 47.5 68.00 37.50 85.91 19.50 93.03 50.0 70.00 39.50 86.28 20.00 93.84 52.5 71.00 40.50 86.38 21.00 95.00 55.0 72.50 42.00 87.00 22.00 95.13 57.5 73.50 44.00 88.00 22.47 95.82 60.0 75.00 46.65 89.59 25.00 95.55 62.5 76.50 50.00 89.70 27.50 96.44 65.0 77.45 53.00 91.57 30.16 96.76 67.5 79.49 56.00 91.90 32.00 96.71 70.0 81.50 58.00 93.06 33.50 97.38 72.5 83.50 61.14 93.50 35.21 97.32 75.0 85.07 65.00 93.47 38.50 97.80 77.5 86.88 67.89 93.77 43.50 98.17 80.0 87.66 72.00 95.32 47.56 98.38 82.5 89.90 76.21 96.32 50.63 98.62 85.0 92.76 80.81 97.50 55.34 98.80 87.5 94.27 85.91 97.82 64.00 99.00 90.0 95.60 90.30 98.30 71.15 99.26 92.5 96.67 92.48 98.65 78.16 100.0 95.0 97.86 95.22 99.14 86.99 100.0 97.5 99.40 98.90 99.90 98.42 100.0 376 Table F8. Graphically smoothed percentile statistics for empirical hyetograph analysis for the 12–24 hr storm duration and one or more inches of depth Center of percent of storm duration interval Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) (percent) (percent) (percent) (percent) (percent) 2.5 3.00 1.50 7.965 1.00 18.61 5.0 16.94 6.065 28.21 2.274 48.43 7.5 26.87 9.896 42.88 4.612 58.75 10.0 34.64 13.41 52.06 6.849 69.96 12.5 38.81 18.53 58.18 8.219 78.73 15.0 45.37 23.00 61.99 9.00 81.50 17.5 49.58 25.19 63.42 9.546 83.50 20.0 52.42 25.82 68.50 10.50 84.20 22.5 55.50 27.50 71.06 12.00 86.03 25.0 57.50 29.50 74.01 13.00 86.85 27.5 59.84 31.00 80.50 13.26 87.49 30.0 60.50 32.00 83.74 14.00 89.79 32.5 61.50 32.87 86.50 14.50 92.29 35.0 62.00 34.50 87.50 15.00 94.50 37.5 63.00 35.50 87.87 15.00 95.75 40.0 63.50 36.87 88.50 15.50 96.10 42.5 64.00 37.43 89.07 14.76 95.98 45.0 65.50 39.00 89.43 17.00 96.59 47.5 65.30 39.50 88.89 18.04 96.70 50.0 67.22 40.27 89.50 19.47 96.39 52.5 68.06 41.78 90.58 21.49 97.00 55.0 70.00 42.58 91.00 22.50 97.00 57.5 71.00 46.01 90.87 24.00 97.22 60.0 73.00 49.50 92.61 27.50 97.28 62.5 76.00 54.90 93.30 32.00 98.04 65.0 77.50 57.00 93.59 34.42 98.26 67.5 80.00 57.63 93.82 36.80 98.62 70.0 81.00 57.62 94.00 38.00 98.50 72.5 81.65 58.37 94.50 39.248 98.74 75.0 83.41 60.43 94.94 42.75 99.02 77.5 85.53 66.34 95.50 45.50 98.71 80.0 86.88 68.84 95.88 48.00 98.67 82.5 88.35 71.49 96.82 49.44 99.20 85.0 90.00 72.34 96.46 50.50 98.77 87.5 90.50 78.62 97.35 53.00 99.47 90.0 92.19 82.80 98.11 60.50 99.53 92.5 93.04 87.48 98.17 65.05 99.45 95.0 95.59 90.20 98.70 67.00 99.77 97.5 98.00 97.00 99.50 91.05 100.0 377 Table F9. Graphically smoothed percentile statistics for empirical hyetograph analysis for the 24 hr and greater storm duration and one or more inches of depth Center of percent of storm duration interval Median 50th percentile Lower quartile 25th percentile Upper quartile 75th percentile Lower decile 10th percentile Upper decile 90th percentile (percent) (percent) (percent) (percent) (percent) (percent) 2.5 5.00 2.50 11.63 0.50 25.68 5.0 13.64 6.061 30.58 2.439 48.26 7.5 20.11 8.587 41.01 4.062 59.20 10.0 24.00 11.120 50.164 5.994 68.39 12.5 27.20 13.14 54.00 7.00 78.00 15.0 31.58 16.74 57.00 7.645 80.18 17.5 35.50 18.00 59.85 8.50 81.50 20.0 37.50 19.00 63.97 8.5433 82.51 22.5 39.50 19.50 64.71 9.00 83.03 25.0 40.00 20.00 66.00 9.50 84.00 27.5 41.00 20.02 67.00 10.00 84.50 30.0 42.00 21.00 66.13 9.962 84.50 32.5 43.00 21.50 66.50 10.83 84.00 35.0 45.64 23.00 67.00 11.48 84.50 37.5 47.50 25.00 67.50 12.91 84.50 40.0 49.27 28.02 68.39 15.50 85.00 42.5 52.07 30.00 70.00 17.00 85.00 45.0 54.00 31.00 72.08 17.49 85.50 47.5 56.00 31.50 72.50 18.00 87.00 50.0 57.00 33.00 73.11 19.50 88.29 52.5 58.00 36.09 74.00 21.91 88.37 55.0 59.35 41.00 77.50 26.00 90.62 57.5 61.00 44.00 81.00 28.54 92.00 60.0 63.50 44.71 84.44 29.14 93.99 62.5 66.00 45.50 85.50 29.65 95.44 65.0 67.10 46.42 86.73 30.00 96.00 67.5 68.66 49.00 86.79 31.00 96.73 70.0 70.21 51.32 88.00 32.50 97.00 72.5 72.50 54.95 89.00 33.97 97.47 75.0 73.90 57.27 89.00 36.60 97.58 77.5 76.82 60.45 89.78 40.00 97.47 80.0 79.11 62.07 90.28 43.00 97.47 82.5 83.01 66.16 92.50 46.92 98.00 85.0 85.00 70.94 94.57 52.04 98.32 87.5 88.00 74.00 96.34 55.50 98.41 90.0 89.00 75.35 97.04 60.00 98.86 92.5 91.16 82.46 97.40 67.88 99.15 95.0 94.96 88.20 98.32 78.19 99.55 97.5 98.50 96.22 99.50 89.63 100.0 378 REFERENCES Abramowitz, M., and Stegun, I.A., 1964, Handbook of mathematical functions: National Bureau of Standards Applied Mathematics Series, vol. 55, reprinted in 1972, 1046 p. Aron, G., and Adl, I., 1992, Effects of storm patterns on runoff hydrographs: Water Resources Bulletin, American Water Resources Association, vol. 28, no. 3, pp. 569–575. Asquith, W.H., 1998, Depth-duration frequency of precipitation for Texas: U.S. Geological Survey Water-Resources Investigations Report 98–4044, 107 p. Asquith, W.H., 1999, Areal-reduction factors for the precipitation of the 1-day design storm in Texas: U.S. Geological Survey Water-Resources Investigations Report 99–4267, 81 p. Asquith, W.H., 2002, Effects of regulation on L-moments of annual peak streamflow in Texas: U.S. Geological Survey Water-Resources Investigations Report 01–4243, 66 p. Asquith, W.H., and Slade, R.M., 1997, Regional equations for estimation of peak- streamflow frequency for natural basins in Texas: U.S. Geological Survey Water- Resources Investigations Report 96–4307, 68 p. Balakrishnan, N., and Chan, P.S., 1995a, Maximum likelihood estimation for the log- gamma distribution under type-II censored samples and associated inference, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 22, CRC Press, Boca Raton, Fla., pp. 409–437. Balakrishnan, N., and Chan, P.S., 1995b, Maximum likelihood estimation for the three-parameter log-gamma distribution under type-II censored samples and associated inference, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 23, CRC Press, Boca Raton, Fla., pp. 439–453. Barnett, V., and Lewis, T., 1995, Outliers in statistical data, third edition: John Wiley and Sons Inc., New York, 584 p. Benjamin, J.R., and Cornell, C.A., 1970, Probability, statistics, and decision for civil engineers: McGraw-Hill, New York, 684 p. Bonta, J.V., and Rao, A.R., 1988a, Comparison of four design-storm hyetographs: Transactions American Society of Agricultural Engineers, vol. 31, no. 1, pp. 102–106. Bonta, J.V., and Rao, A.R., 1988b, Fitting equations to families of dimensionless cumulative hyetographs: Transactions American Society of Agricultural Engineers, vol. 31, no. 3, pp. 756–760. 379 Bonta, J.V., and Rao, A.R., 1989, Regionalization of storm hyetographs: Water Resources Bulletin, vol. 25, no. 1, pp. 211–217. Cheng, K.S., Hueter, I., Hsu, E.C., and Yeh, H.C., 2001, A scale-invariant Gauss- Markov model for design storm hyetographs: Journal of the American Water Resources Association, American Water Resources Association, vol. 28, no. 3, pp. 723–735. Chow, V.T., Maidment, D.R., Mays, L.W., 1988, Applied hydrology: McGraw-Hill Publishing Company, New York, 572 p. Chukwuma, G.O., and Schwab, G.O., 1983, Procedure for developing design hyetographs for small watersheds: Transactions American Society of Agricultural Engineers, vol. 26, no. 5, pp. 1386–1389. Croley II, T.E., 1980, Gamma synthetic hydrographs: Journal of Hydrology, vol. 47, pp. 41–52. Cunnane, C., 1989, Statistical distributions for flood frequency analysis: World Meteorological Organization Operational Hydrology Report No. 33, variously paged. David, H.A., 1981, Order statistics: John Wiley and Sons Inc., New York, 360 p. Davis, J.C., 1986, Statistics and data analysis in Geology: John Wiley and Sons Inc., New York, 646 p. Dingman, S.L., 2002, Physical Hydrology: Prentice Hall, New Jersey, 2nd edition, 646 p. Drufuca, G., and Rogers, R.R., 1978, Statistics of rainfall over paths from 1 to 50 kilometers: Atmospheric Environment, vol. 12, pp. 2333–2342. Edson, C.G., 1951, Parameters for relating unit hydrographs to watershed characteristics: American Geophysical Union Transactions, vol. 32, 591–596. Evans M., Hastings, N.A.J., and Peacock, J. B., 2000, Statistical distributions, 3rd ed.: John Wiley and Sons Inc., 221 p. Fill, H.D., and Stedinger, J.R., 1995, L-moment and probability plot correlation coefficient goodness-of-fit tests for the Gumbel distribution and impact of autocorrelation: Water Resources Research, vol. 31, no. 1, pp. 225–229. Frederick, R.H., Meyers, V.A., and Auciello, E.P., 1977, Five- to 60-minute precipitation frequency for the eastern and central United States: National Oceanic and Atmospheric Administration Technical Memorandum NWS HYDRO–35, 36 p. 380 French, R.H., 1983, Development of design hyetographs for southern Nevada: Water Resources Center, Desert Research Institute, University of Nevada, variously paged. Gilchrist, W.G., 2000, Statistical modeling with quantile functions: Chapman and Hall CRC Press, Boca Raton, Fla., 320 p. Gilchrist, W.G., 2002, E-mail written communication: W.G.Gilchrist@shu.ac.uk, Sheffield Hallam University, Sheffield, United Kingdom. Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments—definition and relation to parameters of several distributions expressible in inverse form: Water Resources Research, vol. 15, pp. 1029–1054. Haan, C.T., Barfield, B.J., and Hayes, J.C., 1994, Design hydrology and sedimentology for small catchments: Academic Press, San Diego, 558 p. Haktanir, T., and Bozduman, A., 1995, A study on sensitivity of the probability- weighted moments method on the choice of the plotting position formula: Journal of Hydrology, vol. 168, pp. 265–281. Hansen, W.R, 1991, Suggestions to authors of the reports of the United States Geological Survey, 7th edition: U.S. Government Printing Office, 289 p. Hald, A., 1998, A history of mathematical statistics from 1750 to 1930: John Wiley and Sons Inc., 795 p. Helsel, D.R., 1989, written communication, Statistics—New series of technical briefing papers: U.S. Geological Survey, Branch of Systems Analysis Technical Memorandum 89–01, June 19, 13 p. Helsel, D.R., and Hirsch, R.M., 1992, Statistical methods in water resources—Studies in Environmental Science 49: Elsevier, Amsterdam, 529 p. Herrmann, G., 2002, E-mail written communication: Texas Department of Transportation, San Angelo, Texas. Hershfield, D.M., 1962, Rainfall frequency atlas of the United States for durations from 30 minutes to 24 hours and return periods from 1 to 100 years: Washington, D.C., U.S. Weather Bureau Technical Paper 40, 61 p. Hollander, M., and Wolfe, D.A., 1973, Nonparametric statistical methods: John Wiley and Sons Inc., 503 p. Hosking, J.R.M., 1986, The theory of probability weighted moments, Research Report RC12210, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York. 381 Hosking, J.R.M., 1989, Some theoretical results concerning L-moments, Research Report RC14492, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York. Hosking, J.R.M., 1990, L-moments: Analysis and estimation of distributions using linear combination of order statistics: Journal Royal Statistical Society, B, vol. 52, no. 1, pp. 105–124. Hosking, J.R.M., 1992, Moments or L moments? An example comparing two measures of distributional shape: The American Statistician., vol 46, no. 3, pp. 186–189. Hosking, J.R.M., 1994, The four-parameter Kappa distribution: IBM Journal of Research and Development, vol. 38, no. 3, pp. 251–258. Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560. Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments, version 3: Research Report RC20525, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York. Hosking, J.R.M., 2002, E-mail written communication: hosking@watson.ibm.com, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York. Hosking, J.R.M., and Wallis, J.R., 1987, Parameter and quantile estimation for the generalized Pareto distribution: Technometrics, vol. 29, pp. 339–349. Hosking, J.R.M., and Wallis, J.R., 1993a, Some statistics useful in regional frequency analysis: Water Resources Research, vol. 29, no. 2, pp. 271–281. Hosking, J.R.M., and Wallis, J.R., 1993b, Some statistics useful in regional frequency analysis, Research Report RC-17096, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York, 23 p. Hosking, J.R.M., and Wallis, J.R., 1995, A comparison of unbiased and plotting- position estimators of L moments: Water Resources Research, vol. 31, no. 8, pp. 2019–2025. Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press, 224 p. Hosking, J.R.M., Wallis, J.R., and Wood, E.F., 1985, Estimation of the generalized extreme-value distribution by the method of probability-weighted moments: Technometrics, vol. 27, pp. 251–261. 382 Huff, F.A., 1967, Time distributions in heavy storms: Water Resources Research, no. 4, pp. 1007–1019. Huff, F.A., 1990, Time distributions of heavy rainstorms in Illinois: Illinois State Water Survey Circular 173, Champaign, 18 p. Jin, C.X., 1992, A deterministic gamma-type geomorphologic instantaneous unit hydrograph based on path types: Water Resources Research, vol. 28, no. 2, pp. 479–486. Jones, D.A., 1985, Applied Hydrology Informal Note 103: Institute of Hydrology, Wallingford, United Kingdom. Kaigh, W.D., and Driscoll, M.F., 1987, Numerical and graphical data summary using O-statistics: The American Statistician, vol. 41, no. 1, pp. 25–32. Karian, Z.A., Dudewicz, E.J., 2000, Fitting statistical distribution—the generalized Lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, Fla., 438 p. Keifer, C.J., and Chu, H.H., 1957, Synthetic storm pattern for drainage design: Journal Hydraulics Division, American Society of Civil Engineers, vol. 83, HY4, pp. 1–25. Kirby, W., 1974, Algebraic boundedness of sample statistics: Water Resources Research, vol. 10, pp. 220–222. Kite, G.W., 1988, Frequency and risk analyses in hydrology: Water Resources Publications, Littleton, Colorado, 257 p. Kroll, C.N., 2002, E-mail written communication: cnkroll@esf.edu, SUNY College of Environmental Science and Forestry, Syracuse, New York. Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979a, Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles: Water Resources Research, vol. 15, no. 5, pp. 1055–1064. Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979b, Estimation of parameters and quantiles of Wakeby distributions: Water Resources Research, vol. 15, no. 5, pp. 1362–1379. Lawless, J.F., 1980, Inference in the generalized gamma and log-gamma distribution: Technometrics, vol. 22, pp. 67–82. Mahler, B.J., 2002, personal communication, U.S. Geological Survey, Austin, Texas. Nash, J. E.,1959, Systematic determination of unit hydrograph parameters: Journal Geophysical Research 64, pp. 111–115. 383 Norbury, J.R., and White, J.K., 1975, Intensity-time profiles of high-intensity rainfall: The Meteorological Magazine, vol. 104, no. 1237, pp. 221–227. Panda, M.N., and Lake, L.W., 1994, Estimation of single-phase permeability from parameters of particle-size distribution: American Association of Petroleum Geologists Bulletin, vol. 78, no. 7, pp. 1028–1039. Pani, E.A., and Haragan, D.R., 1981, A comparison of Texas and Illinois temporal rainfall distributions: Fourth Conference on Hydrometeorology, American Meteorological Society, pp. 76–80. Parrett, C.P., 1998, Characteristics of extreme storms in Montana and methods for constructing synthetic storm hyetographs: U.S. Geological Survey Water- Resources Investigations Report 98–4100, 55 p. Pilgrim, D.H., and Cordery, I., 1975, Rainfall temporal patterns for design floods: Journal of the Hydraulics Division, American Society of Civil Engineers, vol. 101, HY1, pp. 81–95. Pilgrim, D.H., and Cordery, I., 1993, Flood runoff, in Handbook of Applied Hydrology, chapter 9, editor-in-chief D. A. Maidment: McGraw-Hill, New York. Pinkham, R.S., 1961, On the distribution of first significant digits: Annals of Mathematical Statistics, vol. 32, pp. 1223–1230. Prentice, R.L., 1974, A log-gamma model and its maximum likelihood estimation: Biometrika, vol. 61, pp. 539–544. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992, Numerical recipes in FORTRAN: Cambridge University Press, 963 p. Preul, H.C., and Papadakis, C.N., 1973, Development of design storm hyetographs for Cincinnati, Ohio: Water Resources Bulletin, vol. 9, no. 2, pp. 291–300. Ross, S., 1994, A first course in probability, 4th ed.: MacMillan College Publishing Company, New York, 473 p. Schaefer, M.G., 1989, Characteristics of extreme precipitation events in Washington State: Washington State Department of Ecology Report 89-51, variously paged. Schaefer, M.G., 1993, Dam safety guidelines technical note 3—Design storm construction: Washington State Department of Ecology Report 92-55G, variously paged. Serfling, R.J., 1980, Approximation theorems of mathematical statistics: John Wiley and Sons Inc., New York. Serfling, R.J., 2002, E-mail written communication: serfling@utdallas.edu, University of Texas at Dallas. 384 Shen, H.W., and Julien, P.Y., 1993, Erosion and sediment transport, in Handbook of Applied Hydrology, chapter 12, editor-in-chief D.A. Maidment: McGraw-Hill, New York. Soil Conservation Service, 1973, A method for estimating volume and rate of runoff in small watersheds: SCS-TP-149, U.S. Department of Agriculture, Soil Conservation Service, Washington, D.C. Stedinger, J.R., Vogel, R.M., and Foufoula-Georgiou, E., 1992, Frequency analysis of extreme events, in Handbook of Hydrology, chapter 18, editor-in-chief D. A. Maidment: McGraw-Hill, New York. Stolpa, D., 2002, E-mail written communication: Texas Department of Transportation, Austin, Texas. Texas Department of Water Resources, 1980, HIPLEX 1980 operations plan, Big Spring, Texas, LP-125: Austin, Texas, 47 pp. Thompson, D.B., 2001, personal communication: Department of Civil Engineering, Texas Tech University, Lubbock, Texas. Thompson, D.B., 2002, personal communication: Department of Civil Engineering, Texas Tech University, Lubbock, Texas. Tocher, K.D., 1954, The application of automatic computers to sampling experiments: Journal Royal Statistical Society, B, vol. 16, pp. 39–61. Troutman, B., 2002, E-mail written communication: troutman@usgs.gov, U.S. Geological Survey, Denver, Colorado. Veneziano, D., and Villani, P., 1999, Best linear unbiased design hyetograph: Water Resources Research, vol. 35, no. 9, pp. 2725–2738. Vogel, R.M., and Fennessey, N.M., 1993, L moment diagrams should replace product moment diagrams: Water Resources Research, vol. 29, no. 6, pp. 1745–1752. Vogel, R.M., 2002, E-mail written communication: rvogel@emerald.tufts.edu, Tufts University, Medford, Massachusetts. Wallis, J.R., 2002, E-mail written communication: james.wallis@yale.edu, Yale University, New Haven, Connecticut. Wallis, J.R., Matalas, N.C., and Slack, J.R., 1974, Just a moment!: Water Resources Research, vol. 10, pp. 211–219. Wang, Q.J., 1996a, Using partial probability weighted moments to fit the extreme value distributions to censored samples: Water Resources Research, vol. 32, no. 6, pp. 1767–1771. 385 Wang, Q.J., 1996b, Direct sample estimators of L-moments: Water Resources Research, vol. 32, no. 12., pp. 3617–3619. Wilks, D.S., 1995, Statistical methods in the atmospheric sciences: Academic Press, San Diego, 467 p. Yen, B.C., and Chow, V.T., 1980, Design hyetographs for small drainage structures: Journal of the Hydraulics Division, American Society of Civil Engineers, vol. 106, HY6, pp. 1055–1076. Yue, S., Ouarda, T.B.M.J., Bobée, B., Legendre, P., and Bruneau, P., 2002, Approach for describing statistical properties of flood hydrograph: Journal of Hydrologic Engineering, vol. 7, no. 2, pp. 147–153. Zafirakou-Koulouris, A., Vogel, R.M., Craig, S.M., and Habermeier, J., 1998, L-moment diagrams for censored observations: Water Resources Research, vol. 34, no. 5, pp. 1241–1249. 386 VITA William Harold Asquith was born in Dallas, Texas on October 4, 1969, the son of Anne Louise Asquith and Dr. George Benjamin Asquith. After completing his work at Canyon High School, Canyon, Texas, in 1988, he entered the University of Texas at Austin. During the summers of 1988 and 1989, he attended West Texas State University in Canyon. He received the degree of Bachelor of Science in Civil Engineering from University of Texas at Austin in December 1992, and after entering the Graduate School of the University of Texas at Austin in January 1993 he received the degree of Master of Science in Civil Engineering in December 1994. During the years between 1992 and the present he has been employed as a hydrologist at the United States Geological Survey in Austin. He has over 20 publications on surface water and precipitation hydrology in Texas. In January 1998 he again entered the Graduate School of the University of Texas at Austin in pursuit of a Ph.D. in Geosciences. Permanent Address: 8708 Little Laura Drive, Austin, Texas 78757 This dissertation was typed by the author using Adobe Framemaker 7.0.