Simulation of multiply scattered elastic waves with 3D wave-equation and radiative transfer equation for displacements and their gradients

In this study, we investigate the behavior of seismic waves in a highly scattering medium using numerical simulations of the full wavefield, based on Spectral Element Method solutions of the wave equation. The simulated 3D elastic medium was designed with Von Kármán-correlated heterogeneity, providing a realistic representation of the complexities present in natural seismic environments. We analyzed three distinct cases, each characterized by increasing levels of heterogeneity fluctuation, with standard deviations ranging from 10% to 25%, both at depth and at the free surface, allowing us to compare the behavior of seismic waves under varying conditions. We validated the consistency between theoretical predictions of the scattering theory and simulations of the elastic wave equation using Spectral Elements Methods. We compared the results in terms of the mean free path, long-term energy evolution, and the partitioning of energy between compressional and shear modes. For the latter, the validation was supported by numerical simulations of the elastic Radiative Transfer Equation. We show that under specific conditions, existing Spectral Elements simulation codes can effectively replicate wave propagation in a highly scattering medium. This implies that a greater part of the waveform, namely the late envelops, could be employed in inversion processes, thus opening up new possibilities in the realm of inversion studies. Furthermore, we investigated the energy envelope of the displacement wavefield and its gradient to demonstrate how the analysis of the rotational field can provide additional insights into the source mechanism, the propagation direction and the polarization of seismic waves.