Boundary value ray tracing in a heterogeneous medium: a simple and versatile algorithm

Abstract

Traveltime calculations in 3‐D velocity models have become more commonplace during the past decade or so. Many schemes have been developed to deal with the initial value problem, which consists of tracing rays from a known source position and trajectory usually towards some distant surface. Less attention has been given to the more difficult problem of boundary value ray tracing in 3‐D. In this case, source and receiver positions are known and one, or more, minimum time paths are sought between fixed endpoints. A new technique for boundary value ray tracing is proposed. The scheme uses a common numerical integration technique for solving the initial value problem and iteratively updates the take‐off angles until the ray passes through the receiver. This type of ‘shooting’ technique is made efficient by using expressions describing the geometrical spreading of the wavefront to determine the relationship between the ray position at any time and the take‐off angles from the source. The use of numerical integration allows the method to be compatible with a wide variety of structures. These include models with velocity varying smoothly as a function of position and those with arbitrarily orientated surfaces of discontinuity. An examination of traveltime accuracy is given as well as a discussion of efficiency for a few classes of velocity model. To improve upon the first guess pair of take‐off angles, a small‐scale non‐linear inverse problem must be solved. The difference between the receiver position and the arrival point of a ray, on a plane through the receiver, describe a mis‐match surface as a function of the two take‐off angles of the ray. The shape of this surface can possess local minima and multiple ‘global’ minima even for relatively simple 1‐D velocity models. Its study provides some insight into the non‐linearities of a small‐scale geophysical inverse problem. Show more