A Fast Solution for the Eikonal Equation Based on Quadratic Function in Weakly Tilted Transversely Isotropic Media

Calculating the first-arrival traveltimes of quasi-compressional (qP) waves has important applications in geophysics. In practice, geophysical problems often involve extensive calculations of traveltimes, especially when dealing with a large number of source-receiver pairs in complex geological structures. This necessitates a high level of computational efficiency. However, it is challenging to solve the anisotropic eikonal equation due to its complex nonlinearity. Conventional approaches to solving this equation in tilted transversely isotropic (TTI) media involve the use of various iterative algorithms combined with a fast sweeping method (FSM). When tackling large-scale geophysical problems, optimizing efficiency and minimizing the number of iterations is a crucial challenge. To address this problem, we developed a fitting algorithm to solve the anisotropic eikonal equation. We analyze the corresponding slowness surface equations of the qP wave in a local solver and observe a small difference between the slowness function and its quadratic fitting function. Therefore, we propose using the quadratic function to approximate the slowness equation based on three known traveltime points in the local solver. Subsequently, we obtain two roots from the fitting quadratic equation as approximate solutions. After obtaining the traveltime solutions, we check the traveltime causality to preserve the solution satisfying this condition. Three numerical tests are used to further illustrate the validity and computational efficiency of the proposed algorithm in computing traveltimes for 2-D/3-D weakly TTI models, which has been improved by more than three times compared with the conventional method.