Traveltimes from a given source position to a regular grid of subsurface positions can be effectively calculated by finite difference solution of the ray theoretic eikonal equation (Vidale, 1988, Van Trier and Symes, 1991), or by raytracing techniques such as the paraxial method (Beydoun and Keho, 1987). Since the traveltime field is associated with the phase of the WKBJ Green's function, it can be useful in integral solution applications such as Kirchhoff prestack depth migration (Keho and Beydoun, 1988, and Van Trier, 1991) and tomographic inversion (Luo and Schuster, 1990).
In such applications, the traveltime field, or its Frechet derivative, is required at each subsurface location, due to effective sources at each receiver and shot position. In 2-D, this rapidly becomes a computational burden, and in 3-D a seemingly insurmountable obstacle. Usually, in areas of significant lateral migration velocity variation, traveltime fields are required at every source and receiver position. For surface seismic data, this requirement ideally means at every surface CMP location along a line, at a 10-30 m spacing. For a depth migration along a 2-D 10 km line, this means evaluating several hundred traveltime tables, each of which may consist of 105-106 mesh points.
In a ``slow'' CPU environment, the traveltime fields may be precalculated prior to migration or inversion, stored on disk, and then accessed remotely during the migration or inversion procedure. Or, in a very fast CPU environment, each traveltime field may be regenerated redundantly ``on the fly'' as needed. The precompute strategy relieves some of the computational effort, but can be extremely taxing in terms of available disk storage and I/O bandwidth bottlenecks. The ``on the fly'' strategy pushes the limits of present-day supercomputing capabilities for realistically large 2-D prestack depth migration problems, and is probably out of reach for 3-D prestack depth migration. As an example, Cabrera et al. (1992) presented one of the few state-of-the-art 3-D prestack depth migrations publicly discussed, and used a massively parallel supercomputer (Intel iPSC/860) to precompute traveltimes at every 20 CMPs in a quasi v(z) background velocity model. The 3-D prestack depth migration was done as a second step, by retrieving the necessary traveltime values from the lookup tables stored on disk.
An alternate strategy to the two outlined above, would be to precompute a very sparse amount of traveltime fields over the survey area, thus reducing both computational effort and disk storage, and then subsequently interpolating the desired traveltimes from the given sparse set. I have in mind a goal of something on the order of 500 m spacings, even in complex laterally variable media. By reason of physical continuity in the (first arrival) traveltime field, there must be some mathematical basis for extrapolating and/or interpolating such traveltime fields from adjacent source positions.
With this in mind, I derive both an extrapolation equation and a system of interpolation equations for the traveltime field, based on the ray theoretic eikonal. In the first section of this paper, I briefly review the eikonal equation. Next, I derive an extrapolation equation to find the traveltime field due to a source, given the traveltime field of an adjacent source position. This is then extended to a system of interpolation equations, given several adjacent traveltime fields. I discuss how the interpolation system can be solved by standard least-squares or SVD techniques. Finally, I test the theory and the ensuing algorithm on velocity models that are: constant, constant gradient, and complex 2-D (Marmousi). The interpolation is found to be highly accurate for smooth (but strong) velocity gradient fields, and somewhat less accurate for the complex Marmousi velocity model. However, the results are encouraging and this work is still only in its infancy.