An adaptive finite‐difference method for traveltimes and amplitudes

The point‐source traveltime field has an upwind singularity at the source point. Consequently, all formally high‐order, finite‐difference eikonal solvers exhibit first‐order convergence and relatively large errors. Adaptive upwind finite‐difference methods based on high‐order Weighted Essentially NonOscillatory (WENO) Runge‐Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on a posteriori error estimation. It achieves prescribed accuracy at a far lower cost than does the fixed‐grid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as take‐off angles and geometric spreading factors.