3-D common-offset depth migration
In 3-D prestack time migration we defined a strategy for 3-D prestack time migration based on the application of 3-D zero-offset migration theory to 3-D common-offset data. Specifically, following NMO and 3-D DMO correction, prestack data will have been corrected for dip and source-receiver azimuth effects. As a result, the common-offset volumes of data are decoupled representations of a 3-D zero-offset wavefield; thus, each common-offset volume can be migrated independently using a 3-D zero-offset time migration algorithm. For the subsequent processing sequence, refer to 3-D prestack time migration.
Decoupling of common-offset data volumes, in a strict theoretical sense, is only possible for events with hyperbolic moveout. For events with nonhyperbolic moveout that calls for depth migration, each of the resulting common-offset volumes following the applications of NMO and 3-D DMO corrections would not be a representation of a 3-D zero-offset wavefield. As such, and again in theory, a 3-D common-offset volume of data could not be depth-migrated.
Despite this theoretical mandate, it is surprisingly pleasant to see that indeed, in practice, the common-offset strategy for 3-D prestack time migration sometimes can also be applicable to 3-D prestack depth migration. While, following NMO and 3-D DMO correction, each of the common-offset volumes is time-migrated using a 3-D rms velocity field, depth migration of the common-offset volumes is done using a 3-D velocity-depth model.
There are two reasons why you may choose to apply the common-offset strategy to 3-D prestack depth migration:
- You may wish to use a 3-D zero-offset depth migration algorithm other than the Kirchhoff summation, and thus avoid the troublesome task of computing 3-D nonzero-offset traveltimes needed for the latter.
- For line output or for selected image gathers Kirch-hoff summation is the appropriate algorithm. For volume output from 3-D prestack depth migration other algorithms can be more efficiently applied to common-offset data.
Figure 8.5-13 shows selected inline sections derived from 3-D prestack depth migration using the Kirchhoff summation and the common-offset strategy. For the latter, a 3-D zero-offset phase-shift-plus-correction (PSPC)  was used to migrate all 32 volumes of 3-D DMO-corrected common-offset data; thus a full volume of image in depth was created. Comparison of the sections from the two approaches indicate that the imaging quality is largely comparable.
The common-offset strategy for depth migration will not work strictly for a case of complex overburden structure that gives rise to complex, nonhyperbolic moveout. Shown in Figure 8.5-14a is the image obtained from prestack depth migration based on Kirchhoff summation with a finite-difference solution to the eikonal equation to compute the traveltimes. The synthetic data  are associated with a salt sill model (Section K.2). The objective is to image the subsalt region accurately. Note that the common-offset strategy for prestack depth migration with or without DMO correction (Figure 8.5-14b,c) included in the workflow fails to produce an accurate image of the subsalt region with a level of accuracy that is comparable to the result from the Kirchhoff summation (Figure 8.5-14a).
As a final note, efficient and accurate implementation of 3-D prestack depth migration is a topic of active and intensive research in the industry. A choice of strategies for 3-D prestack depth migration is summarized below:
- You may wish to first apply azimuth-moveout (AMO) correction  to regularize the prestack data, then use a frequency-wavenumber algorithm  to perform the 3-D prestack depth migration. The method involves the design and application of a wavefield extrapolation operator on the whole of the 3-D prestack data without decoupling the common-offset volumes.
- Alternatively, you may wish to perform 3-D prestack depth migration using the Kirchhoff summation technique discussed in this section. The Kirchhoff summation implicitly handles geometry irregularities associated with the prestack data.
- A third strategy, which also is described in the latter part of this section, is based on migration of the decoupled 3-D common-offset volumes of data that have been corrected for geometry irregularities and reduced to zero offset by way of 3-D DMO correction. Again, the preferred choice for the algorithm to perform the 3-D common-offset depth migration is one based on wave extrapolation.
A general guideline for the choice of strategy for 3-D prestack depth migration is given below:
- If you are dealing with a low-relief structure and moderate lateral velocity variations, use 3-D prestack depth migration for conducting image-gather analysis for model verification and update, but not necessarily for imaging in depth. For the purpose of model verification and update, you need to work with only a sparse grid of image gathers; and the Kirchhoff summation is the suitable algorithm to produce such a set of image gathers without creating a whole image volume. As for imaging in depth, you may be content with 3-D poststack depth migration; and a finite-difference (Sections G.1 and G.2) or frequency-wavenumber algorithm (3-D poststack migration) that uses wave extrapolation is the suitable algorithm.
- If you are dealing with a complex structure and moderate-to-strong lateral velocity variations, use a 3-D prestack depth migration algorithm based on decoupling of common-offset volumes of data as demonstrated in this section.
- Finally, if you are dealing with a complex overburden structure, use of a 3-D prestack depth migration algorithm without decoupling the common-offset data is imperative.
- Kosloff and Kessler, 1987, Kosloff, D. and Kessler, D., 1987, Accurate depth migration by a generalized phase-shift method: Geophysics, 52, 1074–1084.
- O’Brien and Gray, 1996, O’Brien, M. and Gray, S., 1996, Can we image beneath salt?: The Leading Edge, 17–22.
- Biondi et al., 1998, Biondi, B., Fomel, S., and Chemingui, N., 1998, Azimuth moveout for 3-D prestack imaging: Geophysics, 63, 574–588.
- Biondi and Palacharla, 1996, Biondi, B. and Palacharla, G., 1996, 3-D prestack migration of common-azimuth data: Geophysics, 61, 1822–1832.
- 3-D prestack depth migration
- Kirchhoff summation
- Calculation of traveltimes
- The eikonal equation
- Fermat’s principle
- Summation strategies
- Migration aperture
- Operator antialiasing