Stabilization of theS-to-PEnergy Ratio of a Diffuse Wavefield from Reflections at Free Surfaces

Abstract

ABSTRACTSeismic interferometry relates the Green’s function between two observation sites to the cross correlation of recordings at these sites. In the formulation of seismic interferometry to retrieve the exact Green’s function, an assumption that the energy density ratio of S-to-P waves is 2(α/β)3 is used. Thus, understanding how and under what conditions a wavefield satisfies this energy partition ratio is important for these studies. Previous studies revealed that multiple scattering by random inhomogeneities of the medium equilibrates the S- to P-wave energy densities, and the ratio is 2(α/β)3. However, the effect of the free surface on the energy partition ratio is not fully understood. This study analyzes the effect of the free surface to the energy density ratio in a diffuse wavefield. We employ the definition of a diffuse wavefield by a random superposition of plane waves. We consider a wavefield where plane waves travel inside a homogeneous solid with a free surface and repeat P-to-S and S-to-P conversions at the free surface. Under this condition, it is known that the energy density ratio of S-to-P waves tends to a specific value. However, the value proposed in a previous study slightly deviated from 2(α/β)3, and it was argued that the deviation was due to a numerical error. We prove that the ratio exactly equals 2(α/β)3. Our results show that repeated reflections at the free surface as well as scattering in random media equilibrate the energy ratio of S-to-P waves into 2(α/β)3 regardless of the energy partition of the source.