Multi-stencils fast marching method for factored eikonal equations with quadratic anisotropy

In this study, we propose a multi-stencils fast marching method to solve the factored eikonal equations for media with quadratic anisotropy, also known as Riemannian anisotropy. Compared with the conventional fast marching method, the new method has an improved accuracy in tracing moving wavefronts via computing the solution at each grid point along several stencils. To deal with the source singularity, we solve for the viscosity solution iteratively by factorize the unknown traveltime function into a function that captures the singularity and a multiplicative factor which is differentiable at the source location. By leveraging the strengths of multi-stencils and factorisation, the newly developed algorithm is highly accurate in solving anisotropic eikonal equations. Through a series of numerical examples, we explicitly show that the algorithm (1) has an obvious first (second) order convergence rate with the first (second) order finite-difference scheme, and (2) is unconditionally stable as long as the initial time table around the source is specified. Finally, we design an eikonal-based anisotropic ray tracing technique. We test the anisotropic ray tracing technique on both radial and azimuthal anisotropy cases and the numerical results demonstrate that the usage of multi-stencils scheme increase the accuracy of the anisotropic ray paths especially for azimuthal anisotropy case. This high-accurate eikonal solver and the associated ray tracing technique can be used in high-resolution seismic anisotropic tomography and other researches in the future.