# DMO correction and common-offset migration

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

As stated earlier, a desired workflow for prestack time migration incorporates a step for updating the initial velocity field that was used to migrate the data. Specifically, we want to create common-reflection-point (CRP) gathers from prestack time migration and use them to perform conventional velocity analysis based on the hyperbolic moveout assumption. Finally, we can apply normal-moveout correction to the CRP gathers using the updated velocity field and stack them along to obtain the prestack time-migrated section.

In this section, we shall follow a most popular sequence for prestack time migration based on a practical variation of the constant-offset summation technique described above [1].

1. Perform velocity analysis at sparse intervals and pick just a few velocity functions with minimal dip effects.
2. Apply NMO correction using these flat-event velocities.
3. Sort data to common-offset sections and apply DMO correction.
4. Migrate each of the common-offset sections using your favorite zero-offset migration algorithm (migration) and the velocity field based on the flat-event velocity picks from step (a).
5. Sort the migrated common-offset data back to CMP gathers, and apply inverse NMO correction using the flat-event velocities from step (a).
6. Perform velocity analysis at frequent intervals as needed to derive an optimum stacking velocity field.
7. Apply NMO correction using the optimum stacking velocity field.
8. Stack the data and perform inverse migration (equivalent to 2-D zero-offset wavefield modeling) using the velocity field from step (d).
9. Finally, remigrate the result from step (h) using the updated velocity field from step (f).

This sequence differs from the constant-offset summation procedure for prestack time migration (Figure 5.3-1e) in one respect, only. In case of the latter, step (e) precedes step (d). Although this is more plausable, in practice, the two procedures yield comparable results.

Consider a zero-offset section that contains a set of pulses at 500-ms intervals placed on the center trace as shown in Figure 5.3-2a (left column). Now, assume that this same section represents a common-offset section with 1000-m offset. Apply DMO correction to this section to obtain the section in Figure (5.3-2b, left column). Repeat the same exercise by labeling the input section with pulses (Figure 5.3-2a, left column) as a common-offset section with 2000-m offset and apply DMO correction to get the section in Figure 5.3-2c (left column). Repeat this exercise once more with an offset value of 3000 m to get the section in Figure 5.3-2d (left column). The DMO-corrected common-offset sections shown on the left column of Figure 5.3-2 represent the DMO impulse responses for zero-offset, 1000-m, 2000-m and 3000-m offset cases.

Following the sequence described above, now, treat each of the DMO-corrected sections on the left column in Figure 5.3-2 as zero-offset sections and migrate them using a zero-offset constant-velocity algorithm. Results are shown on the right column of Figure 5.3-2. Migration of the section in Figure 5.3-2a (left column) yields a set of concentric circles (right column). This migrated section represents the zero-offset migration impulse response, and its circular trajectories are defined by equation (34a). Migrations of the sections in Figure 5.3-2b,c,d (left column) yield a set of elliptical trajectories (right column). These migrated sections represent the nonzero-offset (prestack) migration impulse responses, and their elliptical trajectories are defined by equation (33).

 ${\displaystyle {\frac {y^{2}}{(vt/2)^{2}}}+{\frac {z^{2}}{(vt/2)^{2}-h^{2}}}=1,}$ (33)

 ${\displaystyle {\frac {y^{2}}{(vt/2)^{2}}}+{\frac {z^{2}}{(vt/2)^{2}}}=1,}$ (34a)

Once again, consider the section in Figure 5.3-2a (left column) as a common-offset section with assigned offsets values of 0, 1000, 2000, and 3000 m. If we were to have migrated each of these common-offset sections directly using a prestack migration algorithm in lieu of first applying DMO correction then migrating using a zero-offset migration algorithm, we would have obtained the same results as those shown in Figure 5.3-2 (right column). Figure 5.3-3 shows the DMO impulse responses on the left column of Figure 5.3-2 superimposed on the migration impulse responses on the right column of the same figure. For the zero-offset case (Figure 5.3-3a), the DMO impulse response has zero aperture, while for the nonzero-offset case the lateral excursion of the DMO impulse response increases with offset and at shallow times. But, the lateral excursion of the DMO impulse response always is much less than that of the migration impulse response.

This experiment demonstrates that zero-offset migration of DMO-corrected common-offset data is equivalent to prestack migration of nonzero-offset data. Of course, this equivalence is largely valid only within the bounds of velocity variations judged to be acceptable for time migration. In this section, we shall apply the sequence outlined above to two common cases of conflicting dips with different stacking velocities — salt flanks and fault planes.

## References

1. Notfors and Godfrey, 1987, Marcoux, M. O., Godfrey, R. J., and Notfors, C. D., 1987, Migration for optimum velocity evaluation and stacking: Presented at the 49th Ann. Mtg. European Asn. Expl. Geophys.
2. Fowler, 1997, Fowler, P., 1997, A comparative overview of prestack time migration methods: 67th Ann. Int. Soc. Explor. Geophys. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1571–1574.