Topography‐dependent eikonal solver in tilted transverse isotropic media with anelliptical factorization

Abstract Incorporating anisotropy and complex topography is necessary to perform traveltime tomography in complex land environments while being a computational challenge when traveltimes are computed with finite‐difference eikonal solvers. Previous studies have taken this challenge by computing traveltimes in transverse isotropic media involving complex topography with a finite‐difference eikonal equation solver on a curvilinear grid. In this approach, the source singularity, which is a major issue in eikonal solvers, is managed with the elliptical multiplicative factorization method, where the total traveltime field is decomposed into an elliptical base traveltime map, which has a known analytical expression and an unknown perturbation field. However, the group velocity curve can deviate significantly from an ellipse in anellipitically anisotropic media. In this case, the elliptical base traveltime field differs significantly from the anelliptical counterpart, leading to potentially suboptimal traveltime solutions, even though it helps to mitigate the detrimental effects of the source singularity. To overcome this issue, we develop a more accurate topography‐dependent eikonal solver in transverse isotropic media that relies on anelliptical factorization. To achieve this, we first define the coordinate transform from the Cartesian to the curvilinear coordinate system, which provides the necessary framework to implement the topography‐dependent transverse isotropic finite‐difference eikonal solver with arbitrary source and receiver positioning. Then, we develop a semi‐analytical method for the computation of the topography‐dependent anelliptical base traveltime field. Finally, we efficiently solve the resulting quadratic elliptical equation using the fast sweeping method and a quartic anelliptical source term through fixed‐point iteration. We assess the computational efficiency, stability and accuracy of the new eikonal solver against the solver based on elliptical factorization using several transverse isotropic numerical examples. We conclude that this new solver provides a versatile and accurate forward engine for traveltime tomography in complex geological environments such as foothills and thrust belts. It can also be used in marine environments involving complex bathymetry when tomography is applied to redatumed data on the sea bottom.