Seismic Ray Tracing in Anisotropic Media - A Modified Newton Algorithm for Solving Nonlinear Systems

Abstract

Summary Seismic ray tracing with a path bending method leads to a nonlinear system that has much higher nonlinearity in anisotropic media than the counterpart in isotropic media. Any path perturbation causes changes in directional velocity which depends on not only the spatial position but also the local velocity direction in anisotropic media. To combat the high nonlinearity of the problem, Newton-type iterative algorithm is modified by enforcing Fermat’s minimum-time principle as a constraint for the solution update, instead of conventional error minimization in the nonlinear system. As the algebraic problem is integrated with the physical principle, the solution is robust for such a high-nonlinear problem as ray tracing in realistically complicated anisotropic media. This modified algorithm is applied to two ray-tracing schemes. The first scheme is newly derived raypath equations. The latter are approximate for anisotropic media, but the minimum-time constraint will ensure the solution steadily converges to the true solution. The second scheme based on the minimal variation principle is more efficient, as it solves a tridiagonal system in each iteration and does not need to compute the Jacobian and its inverse. Even in this second scheme Fermat’s minimum-time constraint is still employed for the solution update, as same as the other, to guarantee a robust convergence of the iterative solution in anisotropic media.