Traveltime calculations for qP, qSV, and qSH waves in tilted transversely isotropic media using a fast sweeping method with a Newton iterative solver

Abstract

Efficient and accurate traveltime calculations of seismic waves have important applications in traveltime tomography for initial velocity model building, Kirchhoff depth migration, earthquake location, etc. Because anisotropy significantly affects the traveltimes of seismic waves, neglecting it would result in inaccurate imaging and inversion of the underground structure. Therefore, it is necessary to consider anisotropy in the traveltime calculations. The fast sweeping method (FSM) has important applications in computing the anisotropic first-arrival traveltime. The conventional method, which solves a transformed traveltime quartic equation combined with the FSM, is well suited for general anisotropic media and does not rely on approximations to Christoffel’s equation. However, this method has the following problems: (1) the form of the traveltime quartic equation is highly complex and exhibits multiple solutions, requiring additional effort to find solution intervals and determine which solution meets the criteria, and (2) Solving high-order polynomial equations may encounter numerical instability, especially when coefficients undergo minor variations or rounding errors occur for 3D problems. To address the stability and efficiency issues in anisotropic traveltime calculation, we analyze the characteristics of the constructed triangular-pyramid local solver and the quartic coupled slowness equation for the qP and qSV waves in tilted transversely isotropic (TTI) media. We observe that the decomposed slowness equation yields only one or no solution in the triangular-pyramid local solver and satisfies monotonicity conditions. Therefore, we develop the use of the Newton method to solve the factorized slowness equation efficiently, thus addressing stability and computational efficiency concerns inherent in the conventional approach. For the qSH wave, its slowness equation is quadratic and simple to solve. Our method provides an efficient and stable procedure for the traveltime calculations of qP, qSV, and qSH waves in 3D general TTI media. Numerical examples verify the efficiency and accuracy of our method.