Seismic Ray Tracing in Anisotropic Media (2014)

Seismic ray tracing with a path bending method leads to a nonlinear system that has much stronger nonlinearity in anisotropic media than the counterpart in isotropic media. Any path perturbation causes changes to directional velocities, which depend not only upon the spatial position but also upon the local velocity direction in anisotropic media. 

To combat the high nonlinearity of the problem, Professor Wang proposed to modify the Newton-type iterative algorithm 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 incorporated with the physical principle, the proposed algorithm stabilises the solution for a highly nonlinear problem such as ray tracing in realistically complicated anisotropic media.

In anisotropic media, the ray paths are not perpendicular to the wavefronts (dotted black curves). As the group velocity vector is parallel to the energy flux, an exact expression for the group velocity vector in terms of the group angle is difficult to obtain and is too cumbersome for practical use. Practical implementation of a bending method often leads to a system of nonlinear equations. This nonlinear system in anisotropic media has much stronger nonlinearity than the counterpart in isotropic media. 

This article proposed a modified Newton algorithm for solving highly nonlinear systems, and presented two ray-tracing schemes. The first scheme involves newly derived ray path equations, which are approximate for anisotropic media but the minimum-time constraint will ensure that the solution steadily converges to the true solution. The second scheme is based on the minimal variation principle. It is more efficient than the first one as it solves a tridiagonal system and does not need to compute the Jacobian and its inverse in each iteration. 

Even in this second scheme Fermat’s minimum-time constraint is employed for the solution update, so as to guarantee a robust convergence of the iterative solution in anisotropic media.

This article is published in the GEOPHYSICS (2014, vol. 79, no. 1), doi: 10.1190/GEO2013-0110.1.