Seismic ray tracing in anisotropic media: A modified Newton algorithm for solving highly nonlinear systems

Abstract

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 propagation direction in anisotropic media. To combat the high nonlinearity of the problem, the 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 incorporated with the physical principle, it is able to stabilize the solution for such a highly nonlinear problem as ray tracing in realistically complicated anisotropic media. With this modified algorithm, two ray-tracing schemes are presented. The first scheme involves newly derived raypath 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.

Complex Patterns of Past and Ongoing Crustal Deformations in Southern California Revealed by Seismic Azimuthal Anisotropy

“…In this study, we attempt to constrain crustal azimuthal anisotropy in southern California using a massive number of local P ‐wave arrival times accumulated in the past 40 years. This is technically challenging since ray tracing in anisotropic media is rather complex, especially in the highly heterogeneous crust (e.g., Wang, 2014). Here, we solve this problem by using a new adjoint‐state traveltime tomography technique that requires no ray tracing (Tong, 2021).…”

Adjoint‐State Traveltime Tomography for Azimuthally Anisotropic Media and Insight Into the Crustal Structure of Central California Near Parkfield

“…But conventional ray‐tracing methods including the shooting method and bending method may fail to converge to the true path even in mildly heterogeneous media (Rawlinson et al., 2008). Furthermore, due to the increasing complexity of ray tracing in anisotropic media (Wang, 2014; Gou et al., 2018), it is a common practice to use seismic ray paths computed in isotropic media to approximate the corresponding ones in weakly anisotropic media (Tong et al., 2021). Obviously, this kind of approximation affects the accuracy of the computed traveltimes and the accuracy of the derivatives of the objective function with respect to anisotropic model parameters, and consequently distorts the final tomographic results to some extent.…”

Selecting velocity models using Bayesian Information Criterion

“…Such an analysis is complicated even for relatively simple cases, such as horizontally layered media. Ray theory, which is invoked in this work, provides mathematical tools that simplify the analysis (e.g., Keller (1978); Červený (1985); Shearer and Chapman (1988); Slawinski and Webster (1999); Wang (2014); Slawinski et al (2003Slawinski et al ( , 2004).…”