3-D prestack depth migration
We remind ourselves of the two prominent circumstances that require migration of seismic data before stack and in three dimensions:
- The compelling reason for doing time migration is dipping events. The compelling reason for doing time migration before stack is to account for conflicting dips with different stacking velocities. And the compelling reason for doing time migration in three dimensions is to account for the 3-D behavior of fault-plane reflections and reflections within fault blocks that give rise to the problem of conflicting dips. In a strict theoretical sense, in the presence of conflicting dips with different stacking velocities, you need to image the subsurface by migration of seismic data in time, before stack and in three dimensions.
- The compelling reason for doing depth migration is lateral velocity variations. The compelling reason for doing depth migration before stack is to account for nonhyperbolic moveout caused by lateral velocity variations. And the compelling reason for doing depth migration in three dimensions is to account for the 3-D behavior of complex overburden structures that give rise to lateral velocity variations. In a strict theoretical sense, in the presence of lateral velocity variations, you need to image the subsurface by migration of seismic data in depth, before stack and in three dimensions.
Figures 8.5-1 and 8.5-2 show subsurface images along the same inline traverse obtained from 2-D and 3-D, post- and prestack, and time and depth migrations. Examine the quality of imaging of the fault planes and the reflectors within the faulted zone in each of the sections. The fault-plane reflections and the gently dipping reflections within the fault blocks give rise to the problem of conflicting dips, and the lateral velocity variations across the fault blocks themselves give rise to the problem of nonhyperbolic moveout. These two problems also manifest themselves in three dimensions. Therefore, to get the best possible image, as predicated by the data set in Figures 8.5-1 and 8.5-2, one should do the migration before stack, in depth, and in three dimensions.
What then is an appropriate algorithm for 3-D prestack depth migration — Kirchhoff-summation, finite-difference, or frequency-wavenumber? The algorithm of choice must meet the following requirements:
- For it to qualify as a depth migration algorithm, first and foremost, the algorithm of our choice must be able to image steeply dipping reflectors in the presence of lateral velocity variations.
- 3-D prestack seismic data invariably suffer from irregular spatial sampling. Therefore, the algorithm of choice for prestack migration must cope with irregularly sampled data.
- Just as we use prestack time migration to estimate rms velocities (migration velocity analysis and 3-D prestack time migration), we often wish to use prestack depth migration to estimate interval velocities. When used as a velocity estimation tool, we do not have to generate image gathers from prestack depth migration at all CMP locations along a 2-D line or at all bin locations over a 3-D survey area. Instead, it often is sufficient to generate image gathers at sparse intervals along the line, or along selected lines or even on a sparse grid over the 3-D survey area.
Whatever the type of algorithm, requirement (a) cannot be waived for depth migration. Requirement (b) to handle irregular spatial sampling may be fulfilled by azimuth-moveout (AMO) correction  or inversion to common offset (ICO) . Once data are spatially regularized so that the resulting prestack data have uniform fold of coverage and have been corrected for source-receiver azimuth, then, in principle, any of the three categories of migration algorithms — Kirchhoff-summation, finite-difference, or frequency-wavenumber, can be used to perform 3-D prestack depth migration. While it has the advantage of producing an image in depth along a set of line traverses without having to produce an image volume in depth, Kirchhoff summation technique lacks the rigor to handle amplitudes that the frequency-wavenumber techniques can provide. Finally, the finite-difference and frequency-wavenumber migration methods are global methods; as such, they are not suitable to meet requirement (c). The better treatment of amplitudes by the frquency-wavenumber algorithms compared to the Kirchhoff summation technique, however, has greatly increased their use in practice .
- Biondi et al., 1998, Biondi, B., Fomel, S., and Chemingui, N., 1998, Azimuth moveout for 3-D prestack imaging: Geophysics, 63, 574–588.
- Chemingui and Biondi, 1999, Chemingui, N. and Biondi, B., 1999, Data regularization to inversion to common offset (ICO): 69th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1398–1401.
- Biondi and Palacharla, 1996, Biondi, B. and Palacharla, G., 1996, 3-D prestack migration of common-azimuth data: Geophysics, 61, 1822–1832.
- Kirchhoff summation
- Calculation of traveltimes
- The eikonal equation
- Fermat’s principle
- Summation strategies
- Migration aperture
- Operator antialiasing
- 3-D common-offset depth migration