Scattering characteristics of elastic waves by an elastic heterogeneity

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 "impedancetype" 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 INTRODUCTION Elastic wave scattering has become a topic of current interest in both general and exploration geophysics, because of its close relation with various kinds of heterogeneities in elastic media. The Earth has been found to be laterally inhomogeneous in every scale, and elastic wave scattering could be the most effective tool to examine these inhomogeneities. In general geophysics, seismic wave scattering by the heterogeneities near the mantle-core boundary of the Earth was proposed by Haddon and Cleary (1974) to interpret the precursors of PKIKP in seismograms (see Doornbos, 1976). The phase and amplitude fluctuation across a large seismic array (such as LASA or NORSAR) were used to estimate the parameters of velocity inhomogeneities under the arrays based on Chernov's theory of wave scattering in random media (Aki, 1973; Capon, 1974; Berteussen et al., 1975). Seismic coda waves from local earthquakes were attributed to backscattering of seismic waves (Aki, 1969; Aki and Chouet, 1975) and attempts have been made to infer the properties of local small-scale heterogeneities 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 I-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. from their studies (Aki, 1982; Sato, 1982b; Wu and Aki, 1984). Apparent attenuation caused by seismic wave scattering and its relation with the intrinsic absorption were also discussed (Aki, 1980a, 1980b, 1982;Sato, 1981, 1982a;Wu, 1980, 1982a,1982b; Richards and Menke, 1983) and the problem is still open. In many of the problems mentioned above the solution is obtained by a scalar wave scattering theory without guarantee of correctness. We need to develop full elastic wave treatments for these problems. In exploration geophysics, following the introduction of shear wave sources and three-component geophones, the need for applying elastic wave scattering to the complex object and structural exploration has become apparent and pressing. Especially in the case of vertical seismic profiling (VSP), where the source and receiver arrangements are favorable for receiving wide-angle reflected or scattered waves and the targets are often complicated, seismic wave scattering has vast possibilities of application. Some VSP experiments have been done to examine the explosion-formed fracture volume and hydrofractures (Turpening, 1984; Turpening and Blackway, 1984). Therefore, for both general and exploration geophysics, we Presented at the 53rd Annual International SEG Meeting, September 12, 1983, in Las Vegas. Manuscript received by the Editor January 31, 1984; revised manuscript received October 31, 1984. *Presently Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139; Institute of Geophysical and Geochemical Prospecting, Baiwanzhuang, Beijing, China. tDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute ofTechnology, Cambridge, MA02139. C 1985 Society of ExplorationGeophysicists. Allrights reserved. 582 D ow nl oa de d 06 /0 3/ 16 to 1 28 .1 14 .6 9. 17 . R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / Scauering by an Elastic Heterogeneity 583 RAYLEIGH SCATTERING OF ELASTIC WAVES