The traveltimes of first arriving seismic rays through most velocity structures can be computed rapidly on a three-dimensional numerical grid by finite-difference extrapolation. Head waves are properly treated and shadow zones are filled by the appropriate diffractions. Differences of less than 0.11 percent are found between the results of this technique and ray tracing for a complex but smooth model. This scheme has proven useful for earthquake location and shows promise as an inexpensive, well-behaved substitute for ray tracing in forward-modeling and Kirchhoff inversion applications.

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