Sensitivity kernels for static and dynamic tomography of scattering and absorbing media with elastic waves: a probabilistic approach

Abstract

Summary Scattered seismic coda waves are frequently used to characterize small scale medium heterogeneities, intrinsic attenuation or temporal changes of wave velocity. Spatial variability of these properties raises questions about the spatial sensitivity of seismic coda waves. Especially the continuous monitoring of medium perturbations using ambient seismic noise led to a demand for approaches to image perturbations observed with coda waves. An efficient approach to localize spatial and temporal variations of medium properties is to invert the observations from different source-receiver combinations and different lapse times in the coda for the location of the perturbations. For such an inversion, it is key to calculate the coda-wave sensitivity kernels which describe the connection between observations and the perturbation. Most discussions of sensitivity kernels use the acoustic approximation in a spatially uniform medium and often assume wave propagation in the diffusion regime. We model 2-D multiple non-isotropic scattering in a random elastic medium with spatially variable heterogeneity and attenuation using the radiative transfer equations which we solve with the Monte-Carlo method. Recording of the specific energy density of the wavefield that contains the complete information about the energy density at a given position, time and propagation direction allows us to calculate sensitivity kernels according to rigorous theoretical derivations. The practical calculation of the kernels involves the solution of the adjoint radiative transport equations. We investigate sensitivity kernels that describe the relationships between changes of the model in P- and S-wave velocity, P- and S-wave attenuation, and the strength of fluctuation on the one hand and seismogram envelope, travel time changes and waveform decorrelation as observables on the other hand. These sensitivity kernels reflect the effect of the spatial variations of medium properties on the wavefield and constitute the first step in the development of a tomographic inversion approach for the distribution of small-scale heterogeneity based on scattered waves.

Implications of Laterally Varying Scattering Properties for Subsurface Monitoring With Coda Wave Sensitivity Kernels: Application to Volcanic and Fault Zone Setting

“…As previously put forward in theoretical studies (Margerin et al., 2016; Snieder et al., 2019; Zhang et al., 2021), there are two unique energy sources to be considered in the kernel computation for either uniform or non‐uniform cases: the actual (physical) source of energy as well as the detector. Following the terminology of the adjoint formalism, they are referred to as the source of forward intensity and backward intensity, respectively.…”

Adjoint envelope tomography for scattering and absorption using radiative transfer theory

“…The statistical properties that describe the small-heterogeneity have no direct expression in the wave equation which limits the applicability of FWI for spatial imaging of absorption and scattering. In our study, the forward problem is solved by modelling the multiple non-isotropic scattering in a random elastic medium based on the Radiative Transfer Equation using the Monte Carlo method (Zhang et al 2021). The spatial variability of scattering and absorption is described by the spatial distribution of fluctuation strength ε and intrinsic quality factors Q −1 P and Q −1 S in the random medium.…”

“…A similar parametrization has been used for example by Takeuchi (2016). These dependencies are discussed in detail in Zhang et al (2021) and can be expressed as…”

“…δε 2 (r ), δ Q −1 P (r ) and δ Q −1 S (r ) indicate the perturbation of ε 2 , 1/Q P and 1/Q S at the position r respectively. V d is the full space in dimension d. According to Zhang et al (2021), the scattering sensitivity kernel and the attenuation sensitivity kernels read:…”

“…We omit the derivation that is almost the same as shown in Tromp et al (2005) for the wave equation. Note that the reciprocity relationship of the elastic radiative transfer equations is (Zhang et al 2021):…”