Shtivelman and Canning 1988 show how data can be continued through known near-surface velocity structures. McMechan and Chen 1990 implicitly include effects due to topography and near-surface velocity variation in prestack migration. In order for both of these methods to work, the near-surface velocity must be known. The more general situation I plan to addressed is one for which the near-surface velocity structure is unknown. In this situation I propose upward continuing the data to some planar datum above the topography with some fictitious extrapolation velocity. This should unravel the distortions caused by the irregular acquisition topography and allow standard imaging techniques to be applied to the data. Finding the correct extrapolation velocity which unravels the distortions may turn out to be an interesting optimization problem. One can envision an iterative velocity scheme with the objective of creating hyperbolic trajectories in the output space.
The proposed method is similar to wave-equation datuming applied to marine data to compensate for irregular water-bottom topography Berryhill (1979, 1984, 1986); Yilmaz and Lucas (1986). In the marine case, the data are downward continued to the water-bottom with the water velocity, and then upward continued with the water-bottom velocity. This removes the distortions due to the irregular water-bottom.
I claim that land data is like the marine data which has been continued to the water-bottom. I propose that the land data be extrapolated upward to some flat datum with an appropriate velocity. This extrapolation velocity should be close to the near-surface velocity. Once the data have been extrapolated to the flat datum, standard velocity analysis methods may be used to determine the near-surface velocity. Processing and imaging can be performed at the new datum or the data can be downward continued to some lower datum.
The proposed method can be thought of as an inverse problem where the upward continuation is the reverse of the downward continuation step performed in the marine layer replacement methods.