Mathieu Perton

Two perspectives on equipartition in diffuse elastic fields in three dimensions.

The elastodynamic Green function can be retrieved from the cross correlations of the motions of a diffuse field. To extract the exact Green function, perfect diffuseness of the illuminating field is required. However, the diffuseness of a field relies on the equipartition of energy, which is usually described in terms of the distribution of wave intensity in direction and polarization. In a full three dimensional (3D) elastic space, the transverse and longitudinal waves have energy densities in fixed proportions. On the other hand, there is an alternative point of view that associates equal energies with the independent modes of vibration. These two approaches are equivalent and describe at least two ways in which equipartition occurs. The authors gather theoretical results for diffuse elastic fields in a 3D full-space and extend them to the half-space problem. In that case, the energies undergo conspicuous fluctuations as a function of depth within about one Rayleigh wavelength. The authors derive diffuse energy densities from both approaches and find they are equal. The results derived here are benchmarks, where perfect diffuseness of the illuminating field was assumed. Some practical implications for the normalization of correlations for Green function retrieval arise and they have some bearing for medium imaging.

A computer code for forward calculation and inversion of the H/V spectral ratio under the diffuse field assumption

During a quarter of a century, the main characteristics of the horizontal-to-vertical spectral ratio of ambient noise HVSRN have been extensively used for site effect assessment. In spite of the uncertainties about the optimum theoretical model to describe these observations, over the last decade several schemes for inversion of the full HVSRN curve for near surface surveying have been developed.In this work, a computer code for forward calculation of H/V spectra based on the diffuse field assumption (DFA) is presented and tested. It takes advantage of the recently stated connection between the HVSRN and the elastodynamic Greens function which arises from the ambient noise interferometry theory.The algorithm allows for (1) a natural calculation of the Greens functions imaginary parts by using suitable contour integrals in the complex wavenumber plane, and (2) separate calculation of the contributions of Rayleigh, Love, P-SV and SH waves as well. The stability of the algorithm at high frequencies is preserved by means of an adaptation of the Wangs orthonormalization method to the calculation of dispersion curves, surface-waves medium responses and contributions of body waves.This code has been combined with a variety of inversion methods to make up a powerful tool for passive seismic surveying. H/V spectral ratios of ambient seismic noise are modeled by using full wavefield.The theoretical framework is consistent with ambient noise interferometry.The method provides separate calculations of different wave modes.The software is also suitable for coda waves and any other diffuse-like wavefields.It supports joint inversion of seismic velocity models from H/V and dispersion curves.

Green's function calculation from equipartition theorem.

A method is presented to calculate the elastodynamic Greens functions by using the equipartition principle. The imaginary parts are calculated as the average cross correlations of the displacement fields generated by the incidence of body and surface waves with amplitudes weighted by partition factors. The real part is retrieved using the Hilbert transform. The calculation of the partition factors is discussed for several geometrical configurations in two dimensional space: the full-space, a basin in a half-space and for layered media. For the last case, it results in a fast computation of the full Greens functions. Additionally, if the contribution of only selected states is desired, as for instance the surface wave part, the computation is even faster. Its use for full waveform inversion may then be advantageous.

Lateral heterogeneity imaged by small‐aperture ScS retrieval from the ambient seismic field

Interpreting core-related body wave travel-times is challenging for seismologists because Earths heterogeneities are averaged over thousands of kilometers and the sparsity of earthquake measurements makes these heterogeneities difficult to localize. We show how the ambient seismic wave field can be used to overcome these limitations. We present a regional-scale analysis of core-mantle boundary reflections (ScS) under Mexico. We show that body wave arrivals (i.e., P and ScS) are retrieved from higher-order cross-correlations (C3), a technique that provides a more uniform and controlled source distribution using the scattered waves of the coda of classical ambient field cross-correlations (C1). Then, we extract ScS travel times along a dense linear array in Mexico and find that lithospheric lateral heterogeneity, such as the subducting Cocos slab beneath Mexico, may have a strong impact on ScS travel times. In parallel, we show that lateral heterogeneity such as a possible ultra low velocity zone (ULVZ) near the core mantle boundary might also affect, although to a lesser extent, the travel time anomalies. Our results and interpretation are supported through numerical simulations that accounts for slab and ULVZ properties.

Shallow VS Imaging of the Groningen Area from Joint Inversion of Multimode Surface Waves and H/V Spectral Ratios

The Groningen gas field in the northern Netherlands is subject to production-induced earthquakes and has quickly become one of the seismologically best-instrumented areas on Earth. Accurate quantification of seismic hazard from potential future earthquakes requires accurate shallow velocity structure for ground-motion prediction. Toward this end, we present a shear-wave velocity model developed through the joint inversion of multimode Loveand Rayleigh-wave dispersion curves (DCs) and H/V spectral ratio (HVSR) measurements. We obtain local DCs from azimuthally averaged frequency–time analysis of the cross correlation of the ambient seismic field (ASF) between pairs of stations. HVSR is measured at each station from the directional energy density, that is, the autocorrelation of the ASF for all components. We simultaneously fit these observables at each station of the dense Loppersum array to infer a 1D velocity model from the surface to a depth of ∼900 m. In the frequency range considered (∼1–7 Hz), Rayleigh-wave DCs show high modal complexity, which makes clear identification of the modes challenging and leads us to downweight their contribution to the result. Fundamentaland higher-mode Love-wave dispersion is much clearer. We find good agreement between our model and independently derived models of shallow structure, which validates our approach and supports the value of HVSR analysis as a tool to map subsurface properties. Electronic Supplement: Frequency–time diagrams, theoretical kx , omega diagrams, example joint inversion for site 235587, and example of horizontal-to-vertical (H/V) spectral ratio (HVSR) at station site 235587.

Surface‐Wave Retrieval from Generalized Diffuse Fields in 2D Synthetic Models of Alluvial Valleys

Abstract Among other methods, passive imaging technique is widely applied to obtain surface‐wave velocities. This technique implies that the average cross correlations between diffuse wavefields recorded at two observers is proportional to the imaginary part of the Green’s function. For this purpose, most applications rely on both seismic ambient noise and the coda of earthquakes. Instead, we use a generalized diffuse field (GDF), defined as the waves produced by a multiplicity of distant seismic sources. These wavefields undergo multiple scatterings along their way and at the local surface geology. In this communication, we use GDF to extract the locally generated surface waves in a 2D alluvial valley model for both inplane and antiplane cases from the retrieved Green’s function. For the inplane case, an equipartitioned cocktail of plane P , SV , and Rayleigh waves is used, whereas for the antiplane case, the incidence is a set of plane SH waves. In addition to isotropic illumination, we explore the partial illumination from one side of the valley. In both cases, we obtain dispersion curves for the Rayleigh and Love waves’ group velocities from the retrieved Green’s functions and found good agreement with the exact result for the fundamental modes of both Love and Rayleigh waves in an infinite horizontal layer. This theoretical validation is a proof of concept within an ongoing project whose goal is to improve the characterization of Mexico City subsoil throughout tomography maps of surface‐wave velocities using a collection of historical strong earthquakes recorded by the Mexico City Accelerometric Network.

Diffuse seismic waves and site effects

The Greens function can be retrieved from averaging cross correlations of recorded motions within a diffuse field. When autocorrelation is performed the energy density at a point is proportional to the trace of the imaginary part of the Greens function tensor at the source itself. This is because the singularity of the Greens function is restricted to the real part. Thus, the Greens function availability may allow us to establish the theoretical energy density of a seismic diffuse field generated by a background equipartitioned excitation. We compute the imaginary part of the Greens functions for some cases and consider that it represents the spectral signature of the site. It is the pseudo-reflection response. The relative importance of the peaks of this energy spectrum, ruling out nonlinear effects, may have some bearing on the seismic site response for future earthquakes.