An iterative factored topography-dependent eikonal solver for anisotropic media

Accurate and efficient eikonal solvers for heterogeneous media play an important role in many areas of seismology, such as seismic tomography, migration, and earthquake localization. Incorporating seismic anisotropy and complex topography remain a computational challenge for finite-difference eikonal solvers. In recent years, the topography-dependent eikonal equation (TDEE) has been proposed as an effective way to calculate seismic travel-times for isotropic and anisotropic media with irregular topography. However, the Lax-Friedrichs sweeping method used in previous studies to approximate the viscosity solution of TDEE for anisotropic media is more dissipative and needs a much higher number of iterations to converge. In addition, the TDEE solution for the initial point source has an upwind source singularity, which makes all TDEE solvers, even the high-order ones, exhibit polluted convergence and relatively large errors that propagate from the point source to the entire computational domain. To solve these problems, we have formulated the factored topography-dependent anisotropic eikonal (FTDAE) equation in tilted transversely isotropic (TTI) media using the factorization principle. Then, the resulting quartic equation can be numerically solved by using a fixed-point iteration technique based on the simpler elliptical anisotropic eikonal (EAE) equation with a high-order source term due to anelliptical anisotropy introduced by TTI media. At each iteration, the unknown traveltime in the EAE equation is factored into two functions: One of the functions is specified analytically to capture the source singularity, such that the unknown factor is differentiable in the source neighborhood and could be solved by the fast sweeping method. Numerical examples indicate that our FTDAE solver can treat source singularity successfully and achieve high accuracy after just a few iterations, independently of the mesh size, which could provide a more efficient and robust tool for traveltime calculation in the presence of seismic anisotropy and complex surfaces.