The computation of traveltimes

Initiated by Vidale (1988), several finite difference algorithms for calculating traveltimes have been developed (Van Trier and Symes, 1990; Podvin and Lecomte, 1991). These algorithms solve the eikonal equation as follows:

$$\left({\partial \tau_i \over \partial x }\right)^2+ \left({\partial \tau_i \over \partial z }\right)^2=m^2(x,z),$$ (7)

where subscript i indicates the source number. The initial condition is that the traveltime function $\tau=\tau_i(x,z)$is equal to zero at the source location. These finite difference algorithms can efficiently compute the traveltimes on an even grid of (x,z). The traveltime $\tau_i(r)$ can be found by evaluating function $\tau_i(x,z)$ at each receiver location.

Using the advantages of the existing algorithms, I have developed an adaptive finite difference scheme that is always stable and fully vectorizable (Zhang, 1991a). Further, I interpreted the finite difference method as local wavefront propagations through local ray tracing, and proposed a more accurate method that I call the local paraxial ray method (Zhang, 1991b).