The conventional approach to migration is to derive analytically the positions and the amplitudes of the operator coefficients before the application. Much effort has gone into calculating the correct amplitude response of migration operators, and a considerable cost is involved in applying them. DMO and prestack migration amplitude problems also appear to be difficult problems if amplitude characteristics are required, but the kinematics are fairly straightforward.
In this paper, we try to ignore much of the physics of the problem by attempting to start with a simple geometrical description of the positions of an operator, that is, the kinematic part of the problem, and then deriving the amplitude and phase information by applying a 2-D whitening operator to this kinematic-only part of the operator. This 2-D whitening operator may be described as a unitary operator that maps data from x-t space to x-z space along with the geometrical transformation. The 2-D whitening operator we use is based on the forms for 2-D prediction-error filters discussed by Claerbout 1992, page 198.
This approach might allow simpler and more efficient application of migration and DMO operators. If a short operator can be used, and if that operator allows the amplitude and phase information of the migration operator to be specified, migration and DMO processes might be implemented by a two-step process of applying the 2-D prediction-error filter operator to the input data, then applying a simple kinematic operation to the data. The 2-D prediction-error filter operator may be applied before or after the kinematic operations.
In this paper, we show results of applying this technique to three cases. The first case is a purely kinematic hyperbola, one with uniform amplitude. The second case is a hyperbola corrected with simple spherical divergence and cosine amplitude corrections. The third case is a hyperbola with the amplitude corrections and a half-derivative filter.
Although the prediction-error filter whitens the spectrum of the kinematic-only and the amplitude-corrected hyperbola, the calculated operator shows noise being built up in the evanescent zone. The amplitude-corrected hyperbola with the half-derivative filter is significantly damaged by the filter, while we would expect little change if the process had been successful.