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 solut
ion of the paraxial quasi-P eikonal equation, an evolution Hamilton-Jacobi
equation in depth. gives the first-arrival travel time along downward propa
gating rays. We devise nonlinear numerical algorithms to compute the paraxi
al Hamiltonian for quasi-P wave propagation ill general anisotropic media.
A second-order essentially nonoscillatory (ENO) Runge-Kutta scheme solves t
his paraxial eikonal equation with a point source as an initial condition i
n O(N) floating point operations, where N is the number of grid points, Num
erical experiments using 2-D transversely isotropic models with inclined sy
mmetry axes demonstrate the accuracy of the algorithms. (C) 2001 Academic P
ress.