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

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. Since 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 fast sweeping method (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 meet the criteria. (2) Numerical instability: 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 analyzed 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 observed that the decomposed slowness equation yields only one or no solution in the triangular-pyramid local solver and satisfies monotonicity conditions. Therefore, we proposed to use the Newton method to efficiently solve the factorized slowness equation, thus addressing stability and computational efficiency concerns inherent in conventional approach. For the qSH wave, its slowness equation is quadratic and simple to solve. The proposed method provides an efficient and stable procedure for the traveltime calculations of qP, qSV, and qSH waves in 3D general TTI media. Numerical examples have verified the efficiency and accuracy of the proposed method.