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
10^{5}-10^{6} 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.

11/17/1997