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 S wave, of Rayleigh 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 (ΔC44=0) when the cracks are aligned perpendicular to the x axis. We present the P, SV and SH radiation patterns for the two scatterers and discuss the implications for observation of fractured volumes. The most important result is that the zero perturbation ΔC44 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. ¿American Geophysical Union 1991