A 3D VTI factored eikonal solver using six-tetrahedron pyramidal stencil

Abstract

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.