Analytic Solution to the Moment Tensor—Anisotropy Inverse Problem

Abstract

We solve an idealized version of the moment tensor—anisotropy inverse problem. The medium in which faulting occurs has general anisotropy described by 21 parameters. The data are a set of moment tensors for all possible configurations of a unit fault; that is, all combinations of strike, dip and rake (a very idealized assumption, for tectonic constraints may limit available configurations). We first ask whether the anisotropic parameters can be inferred uniquely from the moment tensors and arrive at the answer, “almost”. The data constrain only 20 linear combinations of the 21 parameters; the isotropic Lamé parameter $\lambda$ cannot be determined. We provide an analytic solution algorithm, valid for both weak and strong anisotropy, that uses the data and a prior value of $\lambda$. We then ask whether measurements of the trace of the moment tensor and its smallest eigenvalue (proxies for its explosive and compensated linear vector dipole components, respectively) can determine uniquely the anisotropy. Here our analysis is limited to the weak anisotropy case. We find that these data are “almost” sufficient. They constrain only 19 linear combinations of the 21 parameters; the isotropic Lamé parameters $\lambda$ and $\mu$ cannot be determined. We provide a semi-analytic algorithm for solving the inverse problem, given prior values of $\lambda$ and $\mu$. Numerical tests demonstrate that both solution methods work and can be used to assess the sensitivity of the solution to erroneous prior information. However, because the methods require data for all fault configurations, their application to the sparse data available for most seismic source regions is limited; they complement but do not supplant existing inversion techniques.