JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. B12, PAGES 19,905-19,924, NOVEMBER 10, 1991 Elastic Wave Scattering by Anisotropic Obstacles: Application to Fractured Volumes RICHARD L. GIBSON, JR. ANDARI BEN-MENAHEM 1 Earth Resources Laboratory, Department of Earth, Atmospheric, and PlanetarySciences Massachusetts Institute of Technology, Cambridge The Born approximation is frequently applied to isotropic media to determine wave fields scattered from heterogeneous regions. We extend this theory to the general case of an anisotropic obstacle embedded within an anisotropic matrix and show that a perturbation to any of the 21 independent elastic constants acts as a secondary moment tensor source which radiates energy as it is encountered by the incident wave. We consider the case of an anisotropic obstacle in an isotropic background medium in more detail, since the well-known Green's tensor for isotropic, homogeneous media and the Born approximation allow an expression of the radiation patterns, P and $ wave, of l•ayleigh scattering due to a perturbation to any elastic constants. The wave fields scattered from an anisotropic obstacle differ significantly from the isotropic case in dependence on both the direction of observation and the direction of the incident wave. In order to provide a concrete example of these results, we examine the case of a small fractured volume. If the fractures within the volume are randomly oriented, the resulting material is isotropic, but if the cracks are aligned, the material is anisotropic with five independent elastic constants. In the latter case, the value of C44, analogous to the isotropic rigidity, is unchanged from the background value (5C44 -- 0) when the cracks are aligned perpendicular to the x axis. We present the P, $V and $H radiation patterns for the two scatterers and discuss the implications for observation of fractured volumes. The most important result is that the the zero perturbation 5C44 in the aligned crack case causes the scattered displacement field to vanish completely for incident shear waves polarized parallel to the crack plane. Determination of this direction for observations of a fractured zone would allow important insights into the nature of the fracturing. INTRODUCTION Scattering of elasticwaves by variations of the elastic prop- erties of the Earth is a fundamental problem with significant applications in several areas. As early as1896, the effects of lo- calized variations in bulk modulus and density on propagation of the acoustic waves were considered with what was essen- tially the first Bornapproximation [Rayleigh, 1945]. Morere- cently, Miles[1960] outlined the extension of the Bornapprox- imationto a fully elastic scattering obstacle.Chernov [1960] applied the same approach,under the name of the method of small perturbations, to the acousticcase in order to ex- aminepropagation in randommedia. The application of the Born approximationessentially consists of the insertion of a perturbation solution into the equation of motion, and the perturbations to the elastic properties of the medium then appear as source terms for waves which propagatein the un- perturbed, background medium. This method only accounts for single scattering, but nonetheless can be very useful. In seismology, the scattering due to random variations of velocityis often considered as an explanation for features of elasticwave propagation which are difficult to account for as deterministic body or surface wavepropagation.Aki [1969] suggested surface wave single scattering by heterogeneities concentrated along the Earth's surface asa source for the coda energyfollowing determinisitic arrivals of local earthquakes tNow at the Weizmann Institute of Science, l•ehovot, Israel. Copyright 1991 by the American Geophysical Union. Paper number 91JB01668. 0148-0227 / 91/91JB-01668505.00 and attempted to infer typical scatterersize and density. Sub- sequent work continued to explorethis hypothesis, consider- ing the effects of body wavescattering as well [e.g.,Aki and Chouet, 1975]. The application of the Born approximation to a single scattering model for P wavecoda was described by Hudson [1977], outlining some suggestions for application to statistical distributions of fluctuations of the elastic proper- ties of a given medium and the resultingstochastic variations of the wavefield. Subsequently, Hudson and Heritage[1981] thoroughlyanalyzedthe applicationof the Born approxima- tion to the elastodynamicscattering in an effort to estimate the applicabilityof the method to various scattering problems in earthquake seismology. Gao et al. [1983a, 1983b] presented some simple models for the effectsof multiple scattering on observed coda signals, while Wu andAki [1985a] applied a sin- glescattering model, including the Born approximation, for a volume distribution of scatterers and inferred mean amplitude deviationsof seismograms for given distributionsof scatter- ers. The influence of scattering on the observedattenuation of various waves is also important[e.g., Aki and Chouet, 1975; Dainty and ToksSz, 1977;Dainty, 1981;Aki, 1980]. A recent general model for the attenuation of elastic waves within the Earth includesa significant contribution from scatteringdue to inhomogeneity in the uppermost crust [ToksSz et al., 1988]. In addition to these studies of the effects of scattering on forward modeling of wave propagation, both stochastic and deterministic, scattering theory has important implications for various inversion algorithms. The applicationof the Born approximation allows a linearization of the inversionproce- dure which is often utilized in very generalinversion schemes in exploration seismology [e.g., Beydoun and Mendes, 1989; Cohen et al., 1986; Boyse and Keller, 1986; Claytonand Stolt, 19,905