Minimum traveltime calculation in 3-D graph theory

Abstract

Traveltime calculation is a crucial part of seismic migration schemes, especially prestack migration. There are many different ways to compute traveltimes. These methods can be divided into three categories: (1) Ray tracing (Julian and Gubbins, 1977; Červený et al., 1977). These treat the problem as a initial value problem by shooting rays from the source to the receivers. Or they can also treat the problem as a two‐point boundary value problem. An initial raypath is bent using perturbation theory until Fermat’s principle is satisfied. Nichols (1994) also computed traveltimes with the amplitude information attached to it in two dimensions. (2) Finite‐difference methods (Reshel and Kosloff, 1986; Vidale, 1988; van Trier and Symes, 1991). These solve the eikonal equation directly by using different numerical schemes such as the Runge‐Kutta method, wavefront expansion, or upwind finite difference. (3) Graph theory (Moser, 1991; Fisher and Lees, 1993; Meng et al. 1994). This method recasts the traveltime problem into a shortest path search over a network, which is constructed from the velocity model. This method is guaranteed to find a stable minimum traveltime with any velocity model.