A heterogeneous mixture of isotropic elements may appear homogeneous and anisotropic when the scale of its fabric is smaller than the seismic wavelengths that measure it (Backus, 1962). These fabrics can result from thin layering or from oriented microcracks or fractures. In many situations the anisotropy is quite complex, resulting, for example, from fractures in previously anisotropic media. Since these anisotropies combine, the effect of each must be separated to permit the study of either the fractures or the preexisting anisotropy.

A finely layered medium or a system of parallel fractures in an otherwise homogeneous elastic background renders the medium anisotropic for long wavelengths. The anisotropy increases in complexity as the number of different systems incorporated into the medium increases. Using the group calculus formulation for layered media developed by Schoenberg and Muir, the effects of the individual constituents can be separated arithmetically after the properties of each constituent are transformed. When orthorhombic behavior results from a set of parallel fractures perpendicular to closely spaced bedding planes, the contribution of the fractures to the elastic stiffness matrix can be removed. The average fracture compliances can be evaluated by requiring the background to be transversely isotropic. The remaining stiffness matrix gives the elastic properties of the transversely isotropic background as if the background had no vertical fractures.

Summary In a system of plane parallel fractures, it is often assumed that resistance to slip on the fractures is independent of the direction of that slip. The fracture system then is transversely isotropic in its elastic properties and can be characterized by just two numbers: the compliances normal and tangential to the fractures, for example. But any relief on the fracture surfaces will destroy the symmetry and demand additional elastic constants. In general, a system of plane parallel fractures has a 3 times 3 matrix Z of fracture compliances which contributes to the overall 6times6 compliance matrix S of the fractured solid. For transversely isotropic fractures Z has only two independent elements, but for general fractures all that is known is that Z must be symmetric. This implies that six parameters are needed to describe a fracture system with triclinic behaviour. We find, however, that there is always a coordinate rotation which sets a symmetric pair of the off‐diagonal terms of Z to zero, so Z has, in fact, only five independent elements. The zeros of the rotated Z show that particle displacements tangential to the fractures and parallel to the new coordinate axes are decoupled from each other. Despite this decoupling, the medium is still fully triclinic because displacement normal to the fractures still couples with all the other displacements.

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