# Browsing by Subject "Inverse problems (Differential equations)"

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Item On a class of two-dimensional inverse problems: wavefield-based shape detection and localization and material profile reconstruction(2006) Na, Seong-Won; Kallivokas, Loukas F.Show more In this dissertation we discuss the numerical treatment of two classical inverse problems: firstly, we are interested in the shape detection and localization problem that arises when it is desirable to identify the location and shape of an unknown object embedded in a host medium using response measurements at remote stations. Secondly, we are concerned with the reconstruction of a medium’s material profile given, again, scant response data. For both problems we use acoustic (or equivalent) waves, to illuminate the interrogated object/medium; however, the mathematical/numerical treatment presented herein extends directly to other wave types. There is a wide, and ever widening, spectrum of possible applications that stand to benefit: of particular interest here are geotechnical applications that arise during site characterization efforts. To tackle both inverse problems we adopt the systematic framework of governing-equation-constrained optimization. Accordingly, misfit functionals are augmented with appropriate regularization terms, and with the weak imposition of the equations describing the physics of the wave interrogation. The governing equations may be either of the partial-differential or integral kind, subject only to user preference or problem bias. The framework is flexible enough to accommodate various misfit norms and regularization terms. We seek solutions that minimize the augmented functional by requiring that the first-order optimality conditions vanish at the optimum, thereby giving rise to Karush-Kuhn-Tucker-type systems. We then solve the associated state, adjoint, and control problems with a reduced-space approach. To alleviate the theoretical and numerical difficulties inherent to all inverse problems that are present here as well, we seek to narrow the solution feasibility space by adopting special schemes. In the shape detection and localization problem we adopt amplitude-based misfit functionals, and a frequencyand directionality-continuation scheme, somewhat akin to multigrid methods, that, thus far, have lend robustness to the inversion process. The mathematical details are based on integral equations, where, in addition, the control problem is cast in the elegant framework of total or material derivatives that allow computational speed-up when compared to finite-difference-based gradient schemes. Similarly, in the material profile reconstruction problem we adopt a time-dependent regularization scheme that exhibits superior performance to classical Tikhonov-type regularizations and is shown to be capable of recovering both sharp and smooth material distributions, while being relatively insensitive to the choice of initial guesses and regularization factors. These schemes constitute particular contributions of this work. We describe the mathematical framework and report numerical results. Specifically, with respect to the shape detection and localization problem we report on the two-dimensional case of sound-hard objects embedded in fullspace; with respect to the material profile reconstruction problem, we report results on the one-dimensional case of horizontally-layered systems, and on the two-dimensional case of finite or infinite-extent domains. We discuss the algorithmic performance in the presence of both noise-free and noisy data and provide recommendations for possible extensions of this work.Show more Item Stochastic inversion of pre-stack seismic data to improve forecasts of reservoir production(2003-08) Varela Londoño, Omar Javier; Torres-Verdín, Carlos; Lake, Larry W.Show more Reservoir characterization is a significant component of the commercial evaluation and production of hydrocarbon assets. Accurate reservoir characterization reduces uncertainty in both estimation of reserves and forecast of hydrocarbon production. It also provides optimal strategies for well placement and enhanced recovery processes. Despite continued progress, often the practice of reservoir characterization does not make quantitative and direct use of seismic amplitude measurements, especially pre-stack seismic data. This dissertation develops a novel algorithm for the estimation of elastic and petrophysical properties of complex hydrocarbon reservoirs. The algorithm quantitatively integrates 3D pre-stack seismic amplitude measurements, wireline logs, and geological information. A statistical link between petrophysical properties and elastic parameters is established through joint probability density functions that are adjusted to reflect a vertical resolution consistent with both well logs and seismic data. The estimation of inter-well petrophysical properties is performed with a global inversion technique that effectively extrapolates well-log data laterally away from wells while honoring the full gather of 3D pre-stack seismic data and prescribed global histograms. In addition, the inversion algorithm naturally lends itself to an efficient and robust numerical procedure to assess uncertainty of the constructed 3D spatial distributions of petrophysical and elastic properties. Validation and testing of the inversion algorithm is performed on realistic synthetic data sets. These studies indicate that pre-stack seismic data embody significantly more sensitivity than post-stack seismic data to detecting time-lapse reservoir changes and suggest that rock and fluid properties can be reliably estimated from pre-stack seismic data. Limitations to the quantitative use of seismic data arise in cases of thin reservoir units, low-porosity formations (porosity below 15%), low contrasts in fluid densities, and lack of correlation between petrophysical and elastic parameters. Numerical experiments with the novel algorithm show that petrophysical models constructed with the use of prestack seismic data are more accurate than those generated with standard geostatistical techniques provided that a good correlation exists between petrophysical and elastic parameters. Benefits of the developed algorithm for data integration include the reduction of uncertainty in the construction of rock property distributions such as porosity, fluid saturation, and shale volume. Property distributions constructed in this manner can be used to guide the reliable estimation of other important fluid-flow parameters, such as permeability and permeability anisotropy, that could have a substantial impact on dynamic reservoir behavior.Show more Item Velocity estimation from seismic data by nonlinear inversion and characterization of gas hydrate deposits offshore Oregon(2003) Wang, Chengshu; Tatham, R. H.; Sen, Mrinal K.Show more Seismic attributes such as traveltimes and reflection amplitude variation with offset contain information on the elastic parameters of subsurface rocks. The aim of generalized inversion of seismic data is to estimate values of the elastic parameters such as P-wave velocity, S-wave velocity and density for lithology discrimination and direct detection of hydrocarbons. My dissertation research comprises two parts: development of a method to improve the least-squares and the preconditioned conjugate gradient algorithm, and estimation of detailed velocity structure of gas hydrate-bearing sediments offshore Oregon from Ocean-bottom seismometers (OBS) and multi-channel streamer (MCS) data. I developed a new nonlinear inversion algorithm for estimating velocities from fully stacked reflection data with application to a field data set consisting of well logs viii from Ocean Drilling Program (ODP) Leg 170 and multi-channel seismic reflection (MCS) data offshore Costa Rica. Inversion of post-stack seismic data generally yields reflection coefficients or impedance as a function of two-way reflection time. In this experiment, fully stacked seismic data and density logs at selected locations along a 2-D seismic line are inverted to estimate seismic velocities. Mathematically, generalized inversion provides the best estimate of earth model parameters by minimizing the socalled cost (or misfit between observed and computed seismic data) function, which is a function of the data covariance matrix CD and the a priori model covariance matrix CM. Matrices CD and CM (generally approximated by scalars σd and σm) introduce stability to the process and robustness and thus have strong influence on the quality of the final inversion solution. Based on the least-squares and the preconditioned conjugate gradient algorithm, I have developed a 2-step procedure to solve this nonlinear inverse problem by first determining the two matrices CD and CM using the two-step procedure that involves mapping the sensitivity of model smoothness and data error to the parameters σd and σm .I found that there always exits an area in the σdσm plane in which the low values of the cost function lie, and hence a large 2-dimensional search space can be reduced to a significantly smaller search region. This led to the easy application of this method. The results from this experiment show that almost every identified reflector of seismic data is very well matched by final synthetic seismograms and the density from borehole log data, which confirms that my estimates of velocities are reliable. Combination of the inverted velocity and density profiles allows identification of major stratigraphic boundaries. The improved inversion method is extended to the inversion of pre-stack seismic data, which is applied to estimate seismic velocities of gas hydrate-bearing sediments, offshore Oregon. Gas hydrates are recognized as a target for major future energy reserves, are believed to be a potential source of an important greenhouse gas, and are considered to be a possible cause of submarine geo-hazard. A simple indicator of gas hydrate is a bottom-simulating reflector (BSR), which marks the transition between hydrate-bearing sediments with high Vp above free gas with low Vp. A 3-D streamer and ocean bottom seismometer (OBS) survey in the Hydrate Ridge, offshore Oregon was conducted to image structures controlling the migration of methane-rich fluid and free gas and to map the gas-hydrate distribution. Preliminary Vp and Vs profiles obtained from OBS data by interactive analysis are used as a starting model to estimate Vp from the streamer data. The results of my inversion and interpretation study in Hydrate Ridge are summarized below: Both 3-D streamer and OBS data show a strong BSR indicating the presence of gas hydrate above and free gas below. Interactive P- and S-wave velocity analysis of OBS data allows us to identify the presence of a “conversion surface” in the gas hydrate-bearing sediments. The conversion surface separates the overlying low P-wave velocity layer and underlying high P-wave velocity layer. Inverted velocity profiles show a low-velocity layer existing below the sea floor and above the normal gas hydrate, suggesting a new geological model of gas hydrates. Two types of hydrate fabrics, massive and porous hydrates, observed by deeptowed video survey, were identified in the P-wave velocity profiles. Three main layers of gas hydrate-sediments separated by the conversion surface and BSR are distinguished. Below the free gas is the normal sedimentary section. The profiles reflecting the physical properties of sediments, such as the Pwave velocity, acoustic impedance and Poisson’s ratio profiles, are able to map the distribution of gas hydrates and show very similar trends of lateral variation of the main layers. A series of faults in the accretionary complex under the ridge not only offer pathways for methane and fluid ascending from deeper layers but also control the distribution of the porous hydrates with low velocity below the seafloor. Hornbach et al. (2003) suggest their results using velocity analysis of seismic reflection data on the Blake Ridge is the first direct seismic detection of concentrated hydrate confirmed by velocity analysis. My results of direct inversion of seismic data extend these results to greater resolution of the entire seismic data set. Further, my results may be the first seismic indication of a visually observed porous hydrate zone.Show more