High-Order Accurate Method for Solving the Anisotropic Eikonal Equation

Abstract

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 take off angle and amplitude. I propose an iterative fast sweeping algorithm to obtain high order accuracy using factorization followed by Weighted Essentially Non-oscillatory (WENO) approximation of the derivatives. Although this method yields highly accurate traveltimes but it also results in increased computational load. Therefore, I propose a parallel fast sweeping algorithm to compute fast and accurate solution of the anisotropic eikonal equation. High accuracy is achieved first by using factorization followed by the WENO approximation of derivatives, whereas computational speed up is obtained by sweeping the computational domain in parallel. With a large number of CPUs, significant reduction in computational cost can be achieved for large 3D models. Numerical test shows improvements in accuracy of the TTI eikonal solution.