A Study on the Selection of Seismic Anisotropic Parameterizations in Linearized Inversion Schemes Used in Seismic Tomography

In recent years, researchers have focused on seismic anisotropy and developed related seismic tomography methods. One common method is inverting P-wave travel times, which provides unique constraints on the Earth's interior elastic properties. To characterize anisotropy, 21 independent parameters need to be constrained. Assuming Earth has hexagonal symmetry simplifies the problem, reducing free parameters. In this case, P-wave velocity variations are described by mean slowness, anisotropy strength, and orientation. Three representations have been proposed for these anisotropic parameters for the purpose of tomography imaging: spherical, vectoral and the ABC (a modified vectoral parameterization). Although each is equivalent in describing hexagonal anisotropy, they are nonlinear, making it unclear which is most suitable for iterative inversion schemes in seismic tomography. Therefore, we designed numerical experiments to study how different parameterizations affect the performance of linearized inversion schemes in seismic tomography, providing suggestions for parameterization selection. In this paper, we study the stability and convergence characteristics of all three parameterizations in P-wave travel time linearized inversion, and evaluate the number of iterations and errors of each parameterization in solving the problem. For the linearized inversion scheme, we explore three common methods for minimizing the objective function: gradient descent, Newton, and Levenberg-Marquardt, and select the most suitable solver based on experiments. Our numerical experiments start with a simplified case of constraining two anisotropic parameters, then extend to a full anisotropic problem with four constrained parameters. Finally, we summarize the stability and performance of various anisotropic models and initial conditions under each parameterization. For the 2-D anisotropy numerical experiments, the results show that the ABC parameterization has excellent stability and performance, with the Levenberg-Marquardt method being the best solver method; for the full anisotropy numerical experiments, the combination of ABC and spherical parameterizations becomes the best parameterization scheme choice when using the Levenberg-Marquardt method with further constrained solver parameters.