We present hybrid fast sweeping methods for computing first-arrival traveltime of the qP, qSV and qSH waves in two-dimensional tilted transversely isotropic media, based on solving the anisotropic eikonal equation. A factorization approach is applied to resolve the source singularity near the point source, which leads to a factored anisotropic eikonal equation whose solutions can be computed with high accuracy. The proposed methods solve the factored equation in a neighborhood of the point source with the size of the neighborhood independent of the mesh, and solve the original equation outside the neighborhood. The methods enjoy all the appealing features, such as efficiency, accuracy and convergence, of the usual fast sweeping method. Furthermore, the “super-convergence” property of the first-order fast sweeping method, i.e., both its numerical solution and gradient are first-order accurate, allows us to design a second-order fast sweeping method based on a linear discontinuous Galerkin formulation. As a post-processing procedure of the first-order method, the second-order method reduces the local degrees of freedom from three to one in the linear discontinuous Galerkin formulation, which implies a simple local updating formula, hence an efficient second-order scheme. Numerical experiments are presented to demonstrate the proposed methods.

我们提出了基于二维各向异性方程的混合快速扫描方法，用于计算二维倾斜横向各向同性介质中qP，qSV和qSH波的初到传播时间。应用分解方法来解决点源附近的源奇点问题，这导致了一个分解的各向异性本征方程，其解可以被高精度地计算。所提出的方法求解点源邻域中的分解方程，其邻域的大小与网格无关，并求解邻域外的原始方程。该方法具有常规快速扫描方法的所有吸引人的特征，例如效率，准确性和收敛性。此外，一阶快速扫描方法的“超收敛”特性 它的数值解和梯度都是一阶精确的，这使我们能够基于线性不连续Galerkin公式设计二阶快速扫描方法。作为一阶方法的后处理过程，二阶方法将线性不连续Galerkin公式中的局部自由度从3减小到一个，这意味着简单的局部更新公式，因此是一种有效的二阶方案。数值实验表明了所提出的方法。二阶方法在线性不连续Galerkin公式中将局部自由度从3减小到1，这意味着一个简单的局部更新公式，因此是一种有效的二阶方案。数值实验表明了所提出的方法。二阶方法在线性不连续Galerkin公式中将局部自由度从3减小到1，这意味着一个简单的局部更新公式，因此是一种有效的二阶方案。数值实验表明了所提出的方法。