AI 中文总结
该研究针对VTI介质，开发了结合乘积因式化与六四面体金字塔模板的三维快速扫掠程函方程求解器，可抑制源相关误差并提升走时计算精度。
AI 中文摘要
程函方程的精确走时计算对于层析成像、偏移等地震应用至关重要。快速扫掠法（FSM）因具备无条件稳定性与计算效率而被广泛应用。我们开发了一种适用于垂直横向各向同性（VTI）介质的三维快速扫掠求解器，该方法结合了乘积因式化与六四面体金字塔模板。因式化可消除点源奇异性，而模板则能提升局部精度。在未因式化的公式中，六四面体有限差分格式仅需求解二次方程；但因式化后，扰动因子的局部更新变为四次方程，且斜模板面的更新系统远比非斜模板面复杂。为解决这些难题，我们采用带稳健根选择策略的Ferrari方法求解更新方程，并推导了所有斜面构型的完整更新公式。对于水平约束面，通过利用VTI哈密顿量的结构，特征约束简化为线性关系，因此局部系统仍可简化为四次方程；对于水平-垂直混合约束面，我们设计了一种基于二分法的迭代求解器。数值算例表明，所提方法可有效抑制与源相关的误差，提升走时精度。
英文摘要
Accurate traveltime computation for the eikonal equation is essential in seismic applications such as tomography and migration. The fast sweeping method (FSM) is widely used because of its unconditional stability and computational efficiency. We develop a 3D fast sweeping solver for vertical transverse isotropic (VTI) media that combines multiplicative factorization with a six-tetrahedron pyramidal stencil. The factorization removes the point-source singularity, while the stencil improves local accuracy. In the unfactored formulation, the six-tetrahedron finite-difference scheme requires solving only quadratic equations. After factorization, however, the local update for the perturbation factor becomes quartic, and the update systems on oblique stencil faces are substantially more complicated than those on non-oblique faces. To resolve these difficulties, we solve the update equation using Ferrari's method with a robust root-selection strategy and derive complete update formulas for all oblique-face configurations. For horizontally constrained faces, the characteristic constraint reduces to a linear relation by exploiting the structure of the VTI Hamiltonian, so the local system still reduces to a quartic equation. For mixed horizontal-vertical constrained faces, we design a bisection-based iterative solver. Numerical examples show that the proposed method effectively suppresses source-related errors and improves traveltime accuracy.
