Statistical methods for determining f₀ and its variance from single- and multi-station HVSR measurements

The horizontal-to-vertical spectral ratio (HVSR) of ambient noise measurement is a reliable technique to estimate a site’s resonance frequency (f₀). However, the random nature of ambient noise (both in time and space), in conjunction with variable environmental conditions and sensor coupling issues, can lead to uncertainty in f₀, [subscript HVSR] estimates, which can be classified as temporal, azimuthal and spatial uncertainty. Hence, it is important to report f₀, [subscript HVSR] in a statistical manner (e.g., as a mean or median value with standard deviation). First, a lognormal distribution to describe f₀, [subscript HVSR] is proposed, which is consistent with earthquake ground motion processing and allows for a seamless transition between HVSR statistics in terms of both frequency and its reciprocal, period. A new frequency-domain window-rejection algorithm is introduced to decrease temporal uncertainty. The algorithm is applied to 114 high-variance HVSR measurements and 77 low-variance HVSR measurements collected at two case study sites to demonstrate the effectiveness. The HVSR curve usually is computed as the ratio of the Fourier Amplitude Spectra (FAS) of a single representative horizontal component and the vertical component, which cannot account for azimuthal variability. Instead, the proposed statistical model and algorithm are applied to horizontal components at all azimuths separately. Passed f₀, [subscript HVSR] from all azimuths are combined in a total sample set to account for azimuthal variability. However, the procedure results in unequal numbers of accepted f₀, [subscript HVSR] for different azimuths and introduces bias to the final estimate. A weighting scheme is introduced to eliminate the bias. The application on 114 HVSR measurements shows the total sample set, coupled with the weighting scheme are able to capture the azimuthal variability. Finally, Voronoi Tessellation is used to eliminate sampling bias and obtain an unbiased spatial estimate of f₀, [subscript HVSR] from multiple HVSR measurements at different locations. Three case studies are shown to illustrate the application of the spatial estimate. The first case shows how effective the method in correcting sampling bias. The second case shows how to use the method for site classification. The last case shows how the spatial estimate can be used to compare the 2D/3D effect of two sites.