When subsurface offset is introduced to the space-domain imaging condition, it is not a strict multiplication across the space axis. The lagged multiplication of the up- and down-going wavefields exist somewhere between simple multiplication and cross-correlation. By summing over the offset axis we are generating, we would be performing a rigorous correlation in space. Maintaining this axis invalidates the conventional relationships of operations in dual spaces that, in this case, results in symmetric (though not perfectly), imaging conditions in both the space and Fourier-domains.
In the theory below, we develop the imaging condition in terms of kx and kh. We then present synthetic migrations with the Fourier-domain imaging condition to show its equivalence with the space-domain imaging condition. There are, however, several key differences, associated with aliased replications, between the two results that can be seen by viewing the results in the x-h plane at the depth of an imaging point. These have important ramifications for the use of this form of the imaging condition at shallow depths. Finally, by inspecting the form of the equation, we can see how the implementation of anti-aliasing criteria can be appropriately applied post-migration. This happy fact is beneficial because the Fourier-domain imaging condition is much more expensive to calculate than its space-domain equivalent.