Paraxial Eikonal Solvers for Anisotropic Quasi-P Travel Times

The first-arrival quasi-P wave travel-time field in an anisotropic elastic solid solves a first-order nonlinear partial differential equation, the qP eikonal equation, which is a stationary Hamilton–Jacobi equation. The solution of the paraxial quasi-P eikonal equation, an evolution Hamilton–Jacobi equation in depth, gives the first-arrival travel time along downward propagating rays. We devise nonlinear numerical algorithms to compute the paraxial Hamiltonian for quasi-P wave propagation in general anisotropic media. A second-order essentially nonoscillatory (ENO) Runge–Kutta scheme solves this paraxial eikonal equation with a point source as an initial condition in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2-D transversely isotropic models with inclined symmetry axes demonstrate the accuracy of the algorithms.