The seismic traveltimes and amplitudes associated with the WKB Green's functions can be computed by using two different approaches. One approach uses the dynamic ray tracing method to find the traveltimes and amplitudes along rays (Cervený, 1987), and then interpolates them onto a regular grid with the paraxial ray approximation (Beydoun and Keho, 1987). The other approach uses a finite difference method to extrapolate wavefronts so that the traveltimes and amplitudes are computed directly on a regular grid (Vidale, 1988; Vidale and Houston, 1990; Van Trier and Symes, 1991; Podvin and Lecomte, 1991; Zhang, 1991). The first approach is usually called the ray tracing method and the second the finite difference method.
The ray tracing method computes accurate traveltimes and amplitudes along rays. However, for complicated velocity media, the interpolation process may be theoretically invalid and computationally expensive. The finite difference method can efficiently and accurately compute the traveltimes in the presence of extremely severe, arbitrarily shaped velocity contrasts, but the amplitude calculation is limited to smoothly varying velocity structures. The finite difference method also requires that the grid sizes to be sufficiently small.
In this note, I propose a new method that combines the two approaches. Like the finite difference method, the new method does the local computation on a regular grid and the computation proceeds step by step. On the other hand, the local computation within a grid cell involves local dynamic ray tracing followed by local paraxial wavefront interpolation. Therefore, I call the new method the local paraxial ray method. This method turns into a ray tracing method if the whole velocity model is within one grid cell. And it becomes a finite difference method if the grid sizes are sufficiently so small that the assumptions of straight rays and plane wavefronts are locally valid.
This note describes the principles of the local paraxial ray method and discusses several issues related to its implementation.