Some methods of seismic tomography, migration and velocity-inversion require tracing rays for given pairs of points. This is much more difficult than tracing rays for given initial conditions. Langan et al. (1985) solve this problem by using an iterative method. In each iteration, the initial conditions are updated according to the difference between the given destination and the actual destination of the last ray tracing. Obviously, this method is inefficient. Van Trier (1988) presents a ray tracing method that is vectorizable. He uses this method to trace many rays at once, and then interpolates the result to approximate two-point ray tracing. For complicated velocity models, this method may fail to trace rays to shadow zones.
Recently, several finite-difference schemes are developed to calculate the traveltimes for an arbitrary velocity model. Among them, Vidale's method (Vidale, 1988) and Van Trier's method (Van Trier, 1989) draw most attention. Because these methods give the traveltime field on a regular grid, one can plot the wavefronts by contouring this field. It is known that rays and wavefronts are orthogonal. Can one calculate another field whose contours define the trajectories of rays? The answer is yes.
From the orthogonal relation between the gradient directions of rays and wavefronts, I derived an partial differential equation. The solution of this equation is a function that has constant values along each ray. Therefore, two-point ray tracing can be efficiently done by tracing the contour lines of the function. This method handles rays traveling in one direction and corresponding to first arrivals. These properties may be disadvantages of the method in some applications. However, for others, such as seismic tomography and time-to-depth conversion of migrated sections, they are exactly what are needed.
In this paper, I first derive the orthogonal equation in the general case. Then I describe, in detail, how two-point ray tracing is done for point-source and line-source. Finally, I show two examples of the applications of the method.