3D Eikonal Solvers, Part I: First-Arrival Traveltimes

The article is concerned with the development and comparison of three di erent algorithms for the computation of rst-arrival traveltimes: the fast marching method (FMM), the group marching method (GMM), and a secondorder nite di erence eikonal solver. GMM is introduced as a variant of FMM. It proceeds the solution by advancing a selected group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. The second-order eikonal solver studied in the article is an expanding-box, essentially non-oscillatory (ENO) scheme for which the stability is enforced by the introduction of a down 'n' out (DNO) marching and a post sweeping (PS) iteration. Techniques such as the maximum angle condition, the average normal velocity, and cache-based implementation are introduced for the algorithms to improve the numerical accuracy and eÆciency. The algorithms are implemented for solving the eikonal equation in 3D isotropic media and their performances are compared. GMM is numerically veri ed to be faster than FMM. However, the second-order algorithm turns out to be superior to these rst-order level set methods in both accuracy and eÆciency; the incorporation of average normal velocity improves accuracy, in particular, dramatically for the second-order scheme.