Homogenization of Elastic Wave Equation using Renormalization Group Theory

Seismic waves traveling through the Earth interact with heterogeneities of all scales along their path. Generally, we are interested in travel time associated with coarse-scale structures; however, fine-scale structures also influence the amplitude and travel time for all phases. The distribution of fine-scale heterogeneities not only affects travel times but also impacts how we observe subsurface properties via these travel times. For instance, fine-scale isotropic heterogeneities in the medium, that may not be resolved in seismic imaging can induce extrinsic anisotropy on a coarser scale. Further, simulating seismic wavefield in a medium with all scales of heterogeneities requires a huge amount of computation, posing a significant challenge. To address these challenges, we propose an upscaling technique based on the Renormalization Group theory for the 2D elastic wave equation. It is helpful to understand how seismic waves propagate through a medium containing fine-scale structures and how their response is manifested on the seismogram. This approach aims to generate more cost-effective wave simulations by reducing the required number of grid points and providing effective properties at a coarser scale while preserving wavefield accuracy. To validate our approach, we tested different models for different levels of upscaling. The waveforms and wavefields for both original and coarse-scale models matched well, demonstrating that the Renormalization Group theory-based upscaled medium effectively represents the fine-scale medium over a surface-seismic frequency band.