A Fast Sweeping Method for the Eikonal Equation in 3-D TTI Media Based on a Semi-Analytical Solver

Summary Accurate traveltime computation is fundamental to high-accuracy 3-D seismic imaging and inversion. In anisotropic media, finite-difference schemes and conventional iterative fast sweeping methods (FSM) for the eikonal equation often suffer from numerical instability or convergence difficulties when the monotonicity of the slowness surface breaks down. Thus, we propose a traveltime computation method for 3-D tilted transversely isotropic (TTI) media that embeds a semi-analytical solver into the FSM framework. The proposed semi-analytical solver employs a lower triangular–diagonal–lower triangular transpose (LDLT) decomposition together with a resolvent cubic equation to robustly factorize the local quartic traveltime equation. Combined with a Newton-Raphson-based coefficient refinement strategy and a group-velocity-based causality check, the method directly and accurately identifies the physical root corresponding to the quasi-P (qP) wave. Numerical experiments show that the semi-analytical solver has better numerical stability than existing quartic solvers. For weakly anisotropic models, the proposed method achieves an accuracy comparable to that of Newton-based local solvers. Its main advantage lies in improved robustness in strongly anisotropic media or more complicated local quartic behavior, where admissible-root selection becomes more challenging.