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. ¿ American Geophysical Union 1989