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
*k*_{x} and *k*_{h}. 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.

5/3/2005