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.