A Bayesian Approach to Microtremor Array Methods for Estimating Shallow S Wave Velocity Structures: Identifying Structural Singularities

Abstract

This paper proposes applying Bayesian inference to the problem of estimating shallow S wave velocity structures-in particular, those containing a structural singularity (a low-velocity layer sandwiched between high-velocity layers or a high-velocity layer between low-velocity layers)-by using Rayleigh-wave phase velocities obtained in a microtremor survey. The method proposed here uses the empirical Bayesian approach, whereby field measurement data are used to build the prior distribution. An optimal velocity structure model is selected, from among multiple candidate models with different layer thickness parameter values, on the basis of a Bayes factor. The use of the proposed method allows the layer thickness parameter values in a one-dimensional velocity structure model to be determined objectively for explaining an observed Rayleigh-wave phase velocity dispersion curve. It also stabilizes the inversion process by the use of the prior distribution and strengthens constraints on the unknown parameter, or S wave velocity in each layer, by using multiple Rayleigh-wave modes. These characteristics of the proposed method allow flexible and stable inverse analysis of highly variable shallow subsurface structures. The present study conducts numerical experiments to show that our method allows the assumed structural singularities are reproduced. Then field measurement data from three sites are used to show that the method has allowed shallow structural singularities (depths 5-30 m) to be identified appropriately.Research on model selection in microtremor array methods has only started in recent years. Giulio et al. (2012) applied a quantity called the corrected Akaike information criterion (c-AIC;Hurvich & Tsai, 1989;Sugiura, 1978) to selecting a velocity structure model. When identifying parameter values that maximized the likelihood, however, they set limits on their range of parameter search and did not allow velocity reversal layers in some cases. That is not an appropriate way to use the c-AIC, because there is no way to count the number of "free" parameters, which is needed to calculate the c-AIC, under those limitations. Molnar et al. (2010), by contrast, avoided the problem by setting limits on their range of search and using those limits as the prior distribution for Bayesian inference. Dettmer et al. (2012) adopted a Bayesian CHO AND IWATA 527

Limits and Benefits of the Spatial Autocorrelation Microtremor Array Method Due to the Incoherent Noise, With Special Reference to the Analysis of Long Wavelength Ranges

“…By the use of the sets of the two parameters (radius and NSR) given in Section 4, we can obtain a relation between the array radius and the NSR at the ULF as shown in Figure 10. The figure shows that NSRs generally decrease with the decrease of the array radii, supporting our experiences: we have had very small values of NSR (e.g., ) when we use very small arrays with radii about 1 m (e.g., Cho, 2020; Cho & Iwata, 2019; Cho et al., 2018, 2013). In such cases, we can treat long wavelength ranges reaching several tens of an array radius or more by the SPAC method.…”

“…The readers can refer to Tada et al (2007) and Cho (2018) for the technical details, including practical problems, of the compensation of the incoherent noise. Also, note that the value of NSR,  , can be used to determine the ULW as demonstrated in our previous papers (Cho & Iwata, 2019;Cho et al, 2013).…”

Shallow Microtremor Array Survey Using Miniature and Small Arrays: Strategy for Efficient and Feasible Dense Survey

“…In addition, a miniature array was arranged without a center point. The center point of a miniature array is used to evaluate the SNR, based on which the analysis limits are evaluated and noise compensation is performed (e.g., Cho & Iwata, 2019). Therefore, the center point is not necessary if only the standard SPAC method is applied.…”

“…This means that an inverted velocity structure model can be easily assumed from the corresponding dispersion curve, so we do not discuss the inversion results here. Readers interested in case‐specific techniques can refer to our previous studies (Cho, 2023; Cho & Iwata, 2019; Cho et al., 2021) and refer to Supporting Information for the individual analysis results presented in this study.…”

Shallow Subsurface Structure in the Hualien Basin and Relevance to the Damage Pattern and Fault Rupture during the 2018 Hualien Earthquake

“…Therefore, we constructed an initial model empirically [Ballard Jr, 1964]. The initial models (number of layers and Vs) is updated by an empirical Bayesian approach [Cho and Iwata, 2019] to better explain the phase velocity dispersion curve. It enables flexible modeling of shallowto-deep structure by automatically determining the number of layers based on the Bayes…”