Subsampling in $\omega$-$\mathbf{k}$ Space

The previous subsections showed that the velocity part of the computation can be precomputed and therefore the cost of the algorithm becomes essentially quadratic in the spatial dimensions. For prestack 3D migration that is still too expensive. Notice however, that the algorithm doesn't have any significant approximation, since the velocities can be binned as finely as required by their intrinsic accuracy without significantly affecting the cost.

The question is whether we can reduce the computation time significantly by introducing reasonable approximations in the computation of each trace of the wavefield. It should be clear from Equation (6) that the cost of the algorithm comes from having to consider every single trace of the wavefield in the computation of every wavefield trace. We could, for instance, compute only every other wavefield trace in each of the axis of the $\omega$-$\mathbf{k}$ space. For 3D prestack migration that alone would reduce the computation cost to one sixteenth. The extrapolated wavefield would then be interpolated at each depth step. Or, we can compute only say one in four traces in the cmp inline wavenumber axis and every trace in the cross-line offset wavenumber axis. This may be better since the cmp inline wavenumber axis is likely to be over-sampled whereas the xline offset wavenumber axis is not. Similarly, we may only consider traces of the wavefield in a given neighborhood for the computation of a given trace of the wavefield. If, for example, for the computation of each wavefield trace we use only the half traces closest to the trace being computed along each axis, again, for 3D prestack migration, that would imply a reduction of computation to only one sixteenth of the total computation. If we combine the two forms of computation savings we end up with an algorithm that may begin to be competitive with the mixed-domain algorithms, but that is simpler and more accurate in handling arbitrary lateral velocity variations.

Subsampling in the $\omega$-$\mathbf{k}$ domain is akin to reducing the lateral extent of the wavefield in the $\omega$-$\mathbf{x}$ domain. Whether this is acceptable and to what degree in each of the spatial axis is an unresolved issue at this point in our research. On physical grounds we can argue that the wavefield expands as it propagates so perhaps the approximation is valid at small depths but deteriorates at larger depths. Nothing prevents the subsampling to be a function of depth, making it an interesting issue to investigate further. Limiting the number of wavefield components that are actually used to the computation of another component may be acceptable in most cases since the wavefield is expected to be coherent in the $\omega$-$\mathbf{k}$ domain. However, in specific, important cases, the wavefield may be irregular in the presence of sharp velocity discontinuities. In those cases it is not clear to what extent the approximation deteriorates.