Dislocations in inhomogeneous media via a moduli perturbation approach: General formulation and two‐dimensional solutions

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Abstract
Quasi‐static elastic dislocations in a homogeneous elastic half‐space are commonly used to model earthquake faulting processes. Recent studies of the 1989 Kalapana, Hawaii, and Loma Prieta, California, earthquakes suggest that spatial variations in elastic properties are necessary to reconcile geodetic and seismic results (Arnadottir et al., 1991; Eberhart‐Phillips and Stuart, 1992). In this paper, we use a moduli perturbation approach to investigate the effect of lateral and vertical variations in elastic properties on the elastic fields produced by dislocations. The method is simple, efficient, and in some cases leads to closed form solutions. The zero‐order solution is simply the solution for a homogeneous body. The first‐order correction for elastic heterogeneity is given by a volume integral involving the spatial variations in moduli, the displacements due to a dislocation in a homogeneous half‐space, and the half‐space Green's function. The same representation can be also used to obtain higher‐order solutions. If there are only piecewise constant variations in shear modulus, the volume integral can be reduced to a surface integral (or line integral in two‐dimensions). Comparisons with the analytical solutions for a screw dislocation in a layered medium suggest that the perturbation solutions are valid for nearly an order of magnitude variation in modulus. It is shown that a simple two‐dimensional model with both vertical and lateral variations in the elastic properties may explain a large part of the discrepancy between seismic and geodetically inferred fault depths for the 1989 Kalapana, Hawaii, earthquake.

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