Starting from the ray-theoretic eikonal equation, I derived an equation
to extrapolate source traveltimes, and a system of least squares equations
to interpolate Green's function traveltime fields. The equations are cast
in terms of traveltime gradient, and must be subsequently integrated to
traveltime. I solved the least squares system by a combination of Cramer's
Rule and SVD. The interpolation results were very impressive for a strong
constant gradient velocity model, with relative traveltime errors generally
much less than 1%. The results for the complex 2-D Marmousi velocity model
were less impressive, with unstable interpolation results in localized regions
of 5-20% relative traveltime error. These results suggest that this
interpolation method might be very good for velocity models with strong
vertical and lateral velocity gradients, provided those gradients are
*smooth*. In this case, I expect the interpolation to be at least an
order of magnitude faster in CPU time than direct forward modeling of the
eikonal equation, and require at least an order of magnitude less disk
or memory storage of precomputed traveltime tables. For velocity models
with strong and rapid spatial contrasts, the interpolation may cause
unacceptable traveltime errors due to the inherent violation of the
eikonal ray validity conditions.

I am currently working to make this algorithm robust and practical. If successful, this method could greatly reduce disk/memory storage and CPU requirements for traveltime intensive applications such as 3-D prestack depth migration and 3-D traveltime tomography.

11/17/1997