An approach to upscaling for seismic waves in statistically isotropic heterogeneous elastic media

For complex heterogeneous models, smoothing is often required for imaging or forward modeling, e.g., ray tracing. Common smoothing methods average the slowness or squared slowness. However, these methods are unable to account for the difference between scattering caused by fluctuations of the different elastic moduli and density. Here, we derive a new smoothing method that properly accounts for all of the parameters of an isotropic elastic medium. We treat the geophysical problem of optimum smoothing of heterogeneous elastic media as a problem of moving-window upscaling of elastic media. In seismology, upscaling a volume of a heterogeneous medium means replacing it with a volume of a homogeneous medium. In the low-frequency limit, this replacement should leave the propagating seismic wave-field approximately unchanged. A rigorous approach to upscaling is given by homogenization theory. For randomly heterogeneous models, it is possible to reduce the problem of homogenization to calculating the coherent wavefield (mean field) in the low-frequency limit. After deriving analytical expressions for the coherent wave-field in weakly heterogeneous and statistically isotropic random media, we obtain a smoothing algorithm. We apply this algorithm to random media and to deterministic models. The smoothing algorithm is frequency dependent, i.e., for different dominant frequencies, different smooth versions of the same medium should be considered. Several numerical examples using finite differences demonstrate the advantages of our approach over common smoothing schemes. In addition, using a numerical eikonal equation solver we show that, in the case of complex heterogeneous media, appropriate initial smoothing is important for high-frequency modeling.