Novel techniques for ray tracing in inhomogeneous media

Abstract

Tracing rays through inhomogeneous media generally involves numerically integrating a differential equation. Accuracy doubling,1 which takes advantage of the extremal property of a ray (namely, Fermat's principle) in a novel way, provides a framework for deriving new higher-order integration schemes from existing integration schemes. These accuracy doubled integration schemes will, in general, be of twice the order as the scheme from which they are derived. This technique has been applied to a commonly used fourth-order Runge-Kutta scheme introduced by Sharma et al.2 In the context of inhomogeneous ray tracing, however, the form of the differential equation makes it possible to derive a more efficient fourth-order Runge-Kutta scheme on which to apply the techniques of accuracy doubling. The resulting scheme is eighth order, but it requires fewer computations to achieve a given accuracy than the eighth-order scheme derived from Sharma's method. It so happens that, to derive these accuracy doubled schemes, the underlying Runge-Kutta scheme must be modified to provide differential ray information about the base ray being traced. A method of determining this differential ray information is presented, and this procedure has uses outside the context of accuracy doubling.