Three-dimensional travel-time calculation based on Fermat's principle

We present a new travel-time calculation method based on Fermat's principle. In the method, travel times are recursively calculated on horizontal planes of increasing depth. For typical configurations of exploration geophysics, the distance between the planes is on the order of 100 m. The travel times on the first plane are calculated by connecting straight ray segments from the source point to the grid points on the plane and integrating the slowness along each segment. The times on the other planes are calculated by finding the minimum of the combination of the times on the plane above plus the additional time along segments connecting grid points on the two planes. The travel-time calculation method is designed for calculating either the first arrival times, or the time of the shortest travel path arrival. The method is extended to handle vertically transverse isotropic (VTI) media by an approach which increases the computing time only slightly. The algorithm is tested against synthetic examples for isotropic and VTI wave propagation.