Preliminary investigations on computational methods for solving the two-point seismic ray-tracing problem in a heterogeneous and isotropic medium

Abstract

Two-point seismic ray tracing is an important problem in seismology. In this report, we study this problem by first deriving the differential equations governing seismic wave propagation between two end points in a 3-dimensional heterogeneous and isotropic medium. We then formulate three numerical schemes to solve the two-point ray-tracing problem. In the first method, introduced originally by R. L. Wesson, the second order ray equations are reduced by the central difference approximation to a set of simultaneous non-linear equations, which are then solved by Newton's method. In the second method, the second order ray equations are reduced to a set of first order equations which are solved by using an adaptive finite difference method introduced by M. Lentini and V. Pereyra. The third method, due originally to L. Euler uses a sum to approximate the integral for the travel time which are solved directly for the minimum time path. This formulation may be shown to be equivalent to Wesson's method. We programed Wesson's method for a 3-dimensional inhomogeneous velocity medium, and also applied Lentini and Pereyra's method for a 2-dimensional linear velocity model. Our programs are tested for constant velocity, and linear velocity in one, and two, space variable models. Both programs give satisfactory answers in comparison with known analytic solutions, and with respect to each other.