Factored singularities and high-order Lax–Friedrichs sweeping schemes for point-source traveltimes and amplitudes

In the high frequency regime, the geometrical-optics approximation for the Helmholtz equation with a point source results in an Eikonal equation for traveltime and a transport equation for amplitude. Because the point-source traveltime field has an upwind singularity at the source point, all formally high-order finite-difference Eikonal solvers exhibit first-order convergence and relatively large errors. In this paper, we propose to first factor out the singularities of traveltimes, takeoff angles, and amplitudes, and then we design high-order Lax–Friedrichs sweeping schemes for point-source traveltimes, takeoff angles, and amplitudes. Numerical examples are presented to demonstrate the performance of our new method.