Development and numerical tests of a Bayesian approach to inferring shallow velocity structures using microtremor arrays

Abstract

Abstract. We propose an empirical Bayesian approach to inferring shallow (depth ranges from a few to several tens of metres) S-wave velocity structures using microtremor arrays and execute numerical tests to assess the feasibility of this approach. In our approach, the estimate of the S-wave structure (posterior) is derived from an empirical S-wave structure model (prior) and phase velocities of Rayleigh waves obtained with microtremor arrays. In other words, we aim to find a model that is close to the empirical model and is able to explain phase velocities with a 1D surface-wave theory. The inversion is stabilised by the constraints from the prior model so that model parameterisation with many thin layers can be adopted. The velocity structure is individually estimated for each of two cases (assumptions): the case where we assume fundamental-mode dominance and the case where we take into account the higher modes. Optimal values of the model parameters (e.g. a thickness parameter) are found, based on Akaike's Bayesian Information Criterion (ABIC), and the choice of the better assumption of the surface-wave theory is also based on ABIC. Numerical tests, where synthetic data is generated from a horizontally stratified two-layer model, indicate that the relative weight between a prior model and the observed data is appropriately adjusted by ABIC. It is revealed that a value of the thickness parameter required to reproduce the given two-layer model is successfully found by ABIC. We also suggest that we can make a plausible choice of the assumption of the surface-wave theory with ABIC, unless observation error is extremely large.

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

“…By setting small values of a and b , the layer thicknesses can be made so small that the model is fine enough to represent any structural singularity in shallower portions. We first proposed the use of equation for modeling the layer thicknesses in Cho and Iwata (), where we fixed a at 0.2 and studied several different values of b (2, 5, and 20). Broader ranges of both a and b will be studied in this paper to show how the model selection process performs differently when the proposed method is used and when a similar but slightly different technique, namely, a maximum likelihood method using an information criterion, is employed.…”

“…Equation has a representation that focuses on σ 2 , λ 2 , a , and b in particular, but the equation can in fact be used to compare all parameters that characterize the prior distribution and the likelihood function (e.g., Kass & Raftery, ). For example, Cho and Iwata () used the ABIC to compare the validity of the assumption that the wavefield of vertical‐motion microtremors is dominated by fundamental‐mode Rayleigh waves alone and the assumption that multiple modes of Rayleigh waves coexist.…”

“…Three different horizontally layered structure models (Figure ; Tables ), where a bottom layer with V S = 300 m/s is reached at a depth of 20 m, were used to calculate theoretical Rayleigh‐wave phase velocities for each one of them. Of the three models, Model A is a simple two‐layered structure that was also used by Cho and Iwata (). Models B and C each represent a singular structure with a low‐velocity layer and with a high‐velocity layer, respectively.…”

“…The layer thickness parameter a was fixed at a low value of 0.2 in this section, which was the value used in Cho and Iwata (), whereas b was assigned values from 1 to 25 at an interval of 1. The hyperparameter λ 2 was assigned seven values of 10 −4 , 10 −3 , 10 −2 , 10 −1 , 10 0 , 10 1 and 10 2 .…”

“…We have already conducted preliminary numerical experiments to find out how our approach works (Cho & Iwata, ). The present article is novel in focusing on whether the proposed method allows structural singularities to be identified and then carrying out detailed numerical experiments and real data analysis; the structural singularity means a low‐velocity layer sandwiched between high‐velocity layers or a high‐velocity layer between low‐velocity layers.…”