Hybrid Fast Sweeping Methods for Anisotropic Eikonal Equation in Two-Dimensional Tilted Transversely Isotropic Media
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Hybrid Fast Sweeping Methods for Anisotropic Eikonal Equation in Two-Dimensional Tilted Transversely Isotropic Media Guangnan Huang1 · Songting Luo2 Received: 12 June 2019 / Revised: 2 July 2020 / Accepted: 7 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords TTI eikonal equation · Hybrid fast sweeping method · Source singularity · Factorization approach · Super-convergence Mathematics Subject Classification 65N06 · 86-08
GH is partially supported by National Natural Science Foundation of China (41504095). SL is partially supported by NSF DMS 1418908 and 1719907.
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Songting Luo [email protected] Guangnan Huang [email protected]
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Department of Geophysics, East China University of Technology, Nanchang 330013, China
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Department of Mathematics, Iowa State University, Ames, IA 50011, USA 0123456789().: V,-vol
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1 Introduction Anisotropy has been widely observed and investigated by geophysicists in the crust, the upper mantle and the inner core of the Earth [40,60,62]. For example, the inner core is a complex asymmetric structure with strongly anisotropic property [18], and the anisotropic strength of rock in the upper mantle depends on the percentage of different minerals. The kinematic and dynamic features of the seismic wave are very different when it propagates in isotropic and anisotropic media. In isotropic media, there are only compressional wave and shear wave. The phase velocity is equal to the group velocity for compressional and shear waves. While in anisotropic media, which according to the orientational angle of the symmetric axis can be divided into
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