Instead, our last methodology attempts to include all propagation
directions by choosing portions of the wavenumber spectrum of the
receiver wavefield that are appropriately limited for each component
of the source wavefield (or vice-versa). While the goal of the
imaging condition is a cross-correlation of the two wavefields
followed by extraction of the zero time-lag as shown in equation (4),
the process in effect multiplies across the space axis. Further, to calculate
offset, a spatial cross-correlation without summation is employed. The
Fourier dual of these two implicit operations are convolution and
multiplication respectively. Thus, by transforming our wavefields into
the wavenumber domain, the imaging condition takes the
form of the convolution
Inspecting the above, we notice that the counting subscript j is actually the wavenumber offset axis with with the exponential inverse Fourier transforming the axis during summation to give a single offset panel. Thus, by not summing this dimension, we build the Fourier transform of the offset axis.
Using this formulation, and the fact that kr - ks = kh, we can bandlimit the image space by only allowing offset wavenumber combinations where kr - ks is less than the prescribed bandlimit. Thus, while calculating kx wavenumbers for the image space, only a limited and varying band from the offset axis is considered. In this manner, we can limit reflectors to different offset spectra depending on their structural dip. Figure (6) shows the result of this implementation. Several features are prominent. First, the thick, fast layer at (3000m, 2500m) contains dipping energy that is not in the impulse response. This is a dip ringing due to implementing a hard cutoff in the Fourier domain when selecting wavenumbers for imaging. This noise should cancel during summation of many shots.
Unfortunately, the convolutional imaging condition has Fourier domain periodicity problems that are well avoided by operating in the space domain. Further, there did not seem to be significant improvement over illumination angle as compared to the two previous methods proposed. Finally, given that our effective range of sub-surface offset is actually quite limited, the huge cost differential, O(nx nh) vs. O(nx2), makes the decision for us.