The first-arrival quasi-P wave traveltime field in an anisotropic elastic solid solves a first-order nonlinear partial differential equation, the qP eikonal equation. The difficulty in solving this eikonal equation by a finite-difference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of the traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty. Furthermore, the solution of the paraxial qP eikonal equation, an evolution equation in depth, gives the first-arrival traveltime along downward propagating rays. A second-order upwind finite-difference scheme solves this paraxial eikonal equation in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2-D and 3-D transversely isotropic models demonstrate the accuracy of the scheme.

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