Properties of the zeroth-, first-, and higher-order approximations of attributes of elastic waves in weakly anisotropic media

摘要

Use of the perturbation theory in the study of attributes of elastic waves propagating in weakly anisotropic media leads to approximate but transparent and simple formulas, which have many applications in forward and inverse wave modeling. We present and study such formulas. We show that all studied attributes depend on elements of a matrix linearly dependent on parameters of a medium. We study this dependence with the goal to understand which parameters of the medium, and in which combinations, affect individual wave attributes. Alternative auxiliar vector bases, in which the matrix can be specified, are proposed and studied. The vector bases offer alternative specifications of polarization vectors of qS waves. One of the important observations is that the higher-order (n ≥ 2) perturbation formulas for qS waves are obtained separately for qS1 and qS2 waves. We also study effects of the use of the perturbation theory on the accuracy of the determination of the acoustical axes in weakly anisotropic media. We show that longitudinal directions in the first-order approximation are identical with actual ones. In singular directions, however, the first-order formulas provide directions, which may deviate from the exact ones, or they may even indicate false singular directions. Again, the above-mentioned matrix depending linearly on the parameters of the medium plays a central role in this study.