Wiggins (1984) and Reshef (1991) implicitly include effects due to topography in their migration algorithms. Wiggins uses a Kirchhoff formulation to directly incorporate topography in prestack migration from a nonflat surface. Reshef's phase-shift migration method is also used to directly migrate data. He performs downward extrapolation from a flat datum and adds data to the extrapolated wavefield each time the topographic surface is intersected. Both these methods allow direct migration of data recorded on a nonflat topographic surface. Beasley and Lynn (1992) introduced an elegant and simple algorithm to correct for the error caused by the static time shift based on the ``zero-velocity layer'' concept. Not only is the static shift required before the migration, but this technique cannot be applied to the computationally attractive phase-shift algorithms, because it includes the nonphysical characteristic of zero velocity. In order for all of the above methods to work, the velocity must be known. The more general and problematic situation is one for which the velocity structure is unknown. In this situation, I upward continue the data to some planar datum above the topography with a replacement velocity. This unravels the distortions caused by the rugged acquisition topography and allows standard velocity estimation and imaging techniques to be applied to the data.
Yilmaz and Lucas (1986) and Berryhill (1986) demonstrate how to compensate for the raypath bending effects of a rugged ocean bottom with severe velocity contrast. They call the technique prestack layer replacement. The method uses wave-equation datuming twice; first to downward continue the wavefield from an initial surface to a datum, using a known velocity, and second to upward continue the wavefield from the datum to the initial surface, using a different velocity.
Schneider et al. (1995) apply the same layer-replacement idea to replace the low-velocity layer for an overthrust data set. They use refraction analysis to estimate velocity in the weathering layer. They use wave-equation datuming to downward continue the data to the base of the layer with the estimated velocity, and then upward continue with a replacement velocity. Their results show some improvement over refraction statics, but there is error associated with the refraction velocity and layer determination. Their final results show some ungeological features that were determined by the refraction analysis. In particular, their refraction model shows an unintuitive thickening of the low-velocity layer over a topographic high. This thickening often occurs because tunneling or diving waves are misidentified as refracted arrivals (Bevc, 1991). In portions of their final stack that correspond to topographic highs, static shift produces a better result than wave-equation datuming.