All three of the formulations presented can accommodate lateral velocity variation. The chief drawback of the recursive methods is that they require a regular computational grid. The grid must be sampled with a sufficient vertical density so that the topography can be adequately discretized; otherwise, artifacts are introduced. The synthetic results show that Kirchhoff datuming produces the best output. But more importantly, it is the most efficient method for application to prestack data, and it can be easily applied to 3-D data. Furthermore, the Kirchhoff implementation handles the exact topographic elevations, not some discretized representation. The Kirchhoff method is not confined to a regular grid, so for the irregular geometries of land data acquired in rugged terrain and for 3-D implementation considerations, the datuming algorithm based on the Kirchhoff integral is the operator of choice.
In the remainder of this dissertation, I will demonstrate how the Kirchhoff datuming operator can be applied to improve imaging in two of the most challenging environments facing geophysicists: overthrust areas with substantial surface topography and areas of complex velocity structure.