A fast sweeping algorithm for high-order accurate solution of the anisotropic eikonal equation

High-frequency asymptotic methods, based on solving the eikonal equation, are widely used in many seismic applications including Kirchhoff migration and traveltime tomography. Finite-difference methods to solve the eikonal equation are computationally more efficient and attractive than ray tracing. But, finite-difference solution of the eikonal equation for a point-source suffers from inaccuracies due to singularity at the source location. Since the curvature of wavefront is large in the source neighborhood, the truncation error of the finite-difference approximation is also significant, leading to inaccuracies in the solution. Compared to several proposed approaches to tackle source-singularity, factorization of the unknown traveltime is computationally efficient and simpler to implement. Recently, a factorization algorithm has been proposed to obtain clean first-order accuracy for tilted transversely isotropic (TTI) media. However, high-order accuracy of traveltimes is needed for quantities that require computation of traveltime derivatives, such as takeoff angle and amplitude. I propose an iterative fast sweeping algorithm to obtain high-order accuracy using factorization and a high-order finite-difference stencil. Numerical test shows improvements in accuracy of the TTI eikonal solution. Once the source-singularity problem is tackled using factorization, high-order accurate solutions can be constructed easily. The method can be easily extended to media with lower anisotropic symmetries.