Finite‐difference solutions of the 3‐D eikonal equation in spherical coordinates

Abstract

3D prestack Kirchhoff depth migration requires computation of traveltimes from many surface locations to large grids of subsurface points. Finite-difference solutions of the eikonal equation provide computationally efficient methods for generating these traveltime tables. For these computations, spherical coordinates have many advantages over Cartesian coordinates, although stability must be monitored carefully. Spherical coordinates become singular along the vertical axis, potentially causing computational instability. Also, stability and efficiency are adversely affected by the uneven spatial sampling induced by regular sampling in angular coordinates. Here I present a scheme for letting the sampling in the latitudinal coordinate depend on radial distance r, and the sampling in the longitudinal coordinate depend on both r and This sampling scheme enhances stability and reduces computational cost.