Several interpolations are required to implement the imaging condition as a function of wave number. First, the input wavefields, and then the calculated model space, must be twice finer sampled in wavenumber. Completely inadequate results are obtained for complex earth models if both interpolation steps are not honored. These steps increase the memory and computational demands of the method to unacceptable levels. Further, an equal number of offset-wavenumbers must be calculated to avoid aliasing as opposed to O(10) for a space-domain implementation where one is reasonably confident in the accuracy of the velocity model.
Analyzing the form of the imaging condition allows us to make important conclusions about how to mitigate migration aliasing problems inherent with shot-profile migrations when the source and receiver sampling is unequal. Most importantly, anti-aliasing strategies can be implemented in the image domain after migration without needing to resort to the very expensive Fourier-domain imaging condition.
While the development of a Fourier-domain imaging condition for shot-profile migration has been presented, the periodicity of the process introduces unwanted artifacts into the image result. The form of the equation, however, provides rigor and understanding as to how to design anti-aliasing filters for data sets that do not have equal sampling of the source and receiver data axes. Further, these bandlimits can be applied on smaller post-migration volumes, possibly even during the course of converting subsurface offset to angle in the Fourier-domain at little to no additional cost.