Modeling and inversion of the microtremor H/V spectral ratio: physical basis behind the diffuse field approach

Abstract

Microtremor H/V spectral ratio (MHVSR) has gained popularity to assess the dominant frequency of soil sites. It requires measurement of ground motion due to seismic ambient noise at a site and a relatively simple processing. Theory asserts that the ensemble average of the autocorrelation of motion components belonging to a diffuse field at a given receiver gives the directional energy densities (DEDs) which are proportional to the imaginary parts of the Green's function components when both source and receiver are the same point and the directions of force and response coincide. Therefore, the MHVSR can be modeled as the square root of 2 × ImG 11 /ImG 33 , where ImG 11 and ImG 33 are the imaginary parts of Green's functions at the load point for the horizontal (sub-index 1) and vertical (subindex 3) components, respectively. This connection has physical implications that emerge from the duality DED force and allows understanding the behavior of the MHVSR. For a given model, the imaginary parts of the Green's functions are integrals along a radial wavenumber. To deal with these integrals, we have used either the popular discrete wavenumber method or the Cauchy's residue theorem at the poles that account for surface waves normal modes giving the contributions due to Rayleigh and Love waves. For the retrieval of the velocity structure, one can minimize the weighted differences between observations and calculated values using the strategy of an inversion scheme. In this research, we used simulated annealing but other optimization techniques can be used as well. This last approach allows computing separately the contributions of different wave types. An example is presented for the mouth of Andarax River at Almería, Spain.

Site Effect Assessment in Ulaanbaatar, Mongolia through Inversion Analysis of Microtremor H/V Spectral Ratios

“…According to the previous inversion studies of MHVR [10,15], joint inversion of MHVR and surface wave dispersion curve is more effective than a single inversion of MHVR to avoid the non-uniqueness of V S and thickness during the V S profile estimation. Therefore, the joint inversion of the MHVRs with dispersion curve obtained in the OD's microtremor array observations [3] are applied at Sites 1, 2 and 3.…”

“…In this theory, MHVR can be numerically computed as the imaginary parts of Green's functions, and the theoretical H/V ratio depends on underground layer thickness and elastic properties of soil. Hence the theory allowed for inverting V S profiles from MHVR [13][14][15][16]. One of the authors also examined the applicability of the DFA for the MHVR-based V S profiling and presented the validity of the technique for the site characterizations [17].…”

The inversion of spectral ratio H/V in a layered system using the diffuse field assumption (DFA)

“…Other experiments based on simple models with high impedance contrasts (e.g. Sánchez-Sesma et al, 2011a;Salinas et al, 2014;Sánchez-Sesma, 2016) allow to say that the main effect of anelastic attenuation is the decreasing of the main peak amplitude as the quality factor decreases, whereas the effects are almost negligible at higher frequencies, at which the Green's functions are dominated by surface waves. The imaginary parts of the Green function, for a given frequency, can be expressed as integrals in the horizontal (radial) wavenumber domain and can be evaluated numerically (see Sánchez-Sesma et al 2011a).…”

“…As shown in García-Jerez et al (2013;2016) the expressions of the imaginary part of the Green's function, for the special case when source and receiver coincide at the top of the layered media, can be written in a very compact form. It is clear that the contributions from surface and body waves can be isolated:…”

“…One possible approach is to apply Bouchon's (2007) Discrete Wavenumber Method, instead the focus of this research is to isolate the contributions of the various waves. To this end, the expressions obtained by García-Jerez et al (2013;2016) are used. The kernels in the ( , ) domain, where = horizontal (radial) component of the wave vector, are extended to the complex -plane.…”