Scattering characteristics of elastic waves by an elastic heterogeneity

Abstract

Elastic wave scattering by a general elastic heterogeneity having slightly different density and elastic constants from the surrounding medium is formulated using the equivalent source method and Born approximation. In the low‐frequency range (Rayleigh scattering) the scattered field by an arbitrary heterogeneity having an arbitrary variation of density and elastic constants can be equated to a radiation field from a point source composed of a unidirectional force proportional to the density contrast between the heterogeneity and the medium, and a force moment tensor proportional to the contrasts of elastic constant. It is also shown that the scattered field can be decomposed into an “impedance‐type” field, which has a main lobe in the backscattering direction and no scattering in the exact forward direction, and a “velocity type” scattered field, which has a main lobe in the forward scattering direction and no scattering in the exact backward direction. For Mie scattering we show that the scattered far field is a product of two factors: (1) elastic Rayleigh scattering of a unit volume, and (2) a scalar wave scattering factor for the parameter variation function of the heterogeneity which we call “volume factor.” For the latter we derive the analytic expressions for a uniform sphere and for a Gaussian heterogeneity. We show the relations between volume factors and the 3-D Fourier transform (or 1-D Fourier transform in the case of spherical symmetry) of the parameter variations of the heterogeneity. The scattering spatial pattern varies depending upon various combinations of density and elastic‐constant perturbations. Some examples of scattering pattern are given to show the general characteristics of the elastic wave scattering.