# Introduction to dip-moveout correction and prestack migration

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Dip-moveout (DMO) correction is applied to normal-moveout corrected prestack data to preserve conflicting dips with different stacking velocities during stacking. As a result, DMO correction yields an improved stacked section that is a closer representation of a zero-offset section compared to conventional CMP stack based on normal-moveout correction, only. This then enables us to apply the theory of zero-offset migration discussed in migration to stacked data with greater confidence.

We remind ourselves from velocity analysis and statics corrections that stacking velocities are dip-dependent (equation 3-8). When a flat event is intersected by a dipping event, we can only choose a stacking velocity in favor of one of these events, not both. Therefore, conventional CMP stacking does not equally preserve events with conflicting dips with different stacking velocities. This is not the case for a zero-offset section, for it contains all events, regardless of dip. Thus, in the presence of conflicting dips, a stacked section is not identical to a zero-offset section.

Since a CMP-stacked section strictly is not equivalent to a zero-offset section, we expect that migration after stack would not produce a crisp image in the presence of conflicting dips with different stacking velocities. To circumvent the problem of conflicting dips, it can be suggested that migration should be done before rather than after stack.

A practical alternative to migration before stack is to correct for the dip effect on moveout velocities implied by Levin’s equation (3-8) prior to stacking. It can be suggested that prestack data first can be moveout-corrected using flat-event velocities. This *normal-moveout correction* (NMO) then is followed by a *dip-moveout correction* (DMO) to account for the dip effect on moveout. Stacking of NMO- and DMO-corrected CMP gathers yields a section that is a closer approximation to a zero-offset section than a conventional CMP-stacked section based on NMO correction only.

Conflicting dips with different stacking velocities often are encountered in two geological situations — reflections from steeply dipping fault planes conflicting with reflections associated with gently dipping strata, and diffractions and reflections off the flank of salt domes conflicting with, again, reflections associated with gently dipping strata.

The conflicting dips problem has been investigated extensively. Doherty ^{[1]} first introduced the wave extrapolation equations for nonzero-offset data. Sherwood ^{[2]} devised a method for mapping nonzero-offset data to zero offset in the presence of conflicting dips with different stacking velocities. Then, Yilmaz and Claerbout ^{[3]} suggested a prestack partial migration (PSPM) technique for solving the problem of conflicting dips. Specifically, they developed a wave theory to account for the difference between migration before stack and the result of conventional processing that includes moveout correction, CMP stacking and migration after stack (Section D.1). They recognized the fact that DMO correction actually is a partial migration process applied to moveout-corrected common-offset data. Nevertheless, the PSPM theory had one important drawback. Although valid for the layered earth velocity model, it was based on a small-offset approximation. Deregowski and Rocca ^{[4]} recast the theory for PSPM in a form similar to Kirchhoff migration. Ottolini ^{[5]} developed the PSPM equations in Snell-midpoint coordinates, the domain of constant-ray-parameter sections (Section F.2). The technique is theoretically accurate for layered medium as well as for all offsets and dips. This method was followed by another unique approach that involves offset continuation ^{[6]}, ^{[7]} ^{[8]} ^{[9]} that is, mapping a far-offset section to a near-offset section, thereby collapsing all offsets to zero offset. Hale ^{[10]}, ^{[11]} formulated a DMO method in the *f − k* domain. This method is exact for constant velocity, it can handle all dips and offsets, and is accurate as long as the vertical velocity gradient is moderate. Jacubowicz ^{[12]} developed a conceptually appealing technique that involves first dip decomposition of the input data and application of a DMO operator to each component, individually. Hale ^{[10]}, Hale and Artley ^{[13]} and Artley and Hale ^{[14]} extended the DMO theory to accommodate vertical velocity variations. French ^{[15]} developed a partial migration technique that tries to account for variations in source-receiver azimuths, a form particularly suitable for 3-D applications. Biondi and Ronen ^{[16]}, Cabrera and Levy ^{[17]}, Granser ^{[18]} and Zhou ^{[19]} designed DMO operators applied to shot profiles. All of these techniques are confined to vertical velocity variations for which time migration is appropriate. However, DMO correction cannot solve stack imperfections caused by lateral velocity variations. Principles of and techniques for DMO correction are described in principles of dip-moveout correction, and practical aspects of DMO correction are covered in dip-moveout correction in practice.

While the practical solution to the problem of conflicting dips with different stacking velocities is DMO correction combined with poststack time migration, the rigorous solution is prestack time migration. A theory for imaging nonzero-offset data based on the double-square-root equation is provided in Section D.1. Recall from migration principles that poststack migration, in principle, is based on summation of amplitudes along a zero-offset hyperbolic traveltime curve and placing the summed amplitude to the apex of the hyperbola. Similarly, prestack time migration, in principle, involves summation of amplitudes along the nonzero-offset traveltime curve in midpoint-offset coordinates and placing the summed amplitude at the apex of the surface. The nonzero-offset traveltime equation can be derived by stationary-phase approximation to the double-square-root equation (Section D.2). As with the zero-offset case, the velocity field dictates the curvature of the nonzero-offset summation paths. Each common-offset section can be imaged separately; the results then can be superimposed to produce the migrated section. In practice, however, processing sequence for prestack time migration often incorporates DMO correction and repicking velocities after migration. Fowler ^{[20]} developed a velocity-independent prestack imaging technique that incorporates a step to correct for dip-dependency of stacking velocities applied to constant-velocity stacked data. Gardner ^{[21]} combined DMO correction with a velocity-independent prestack imaging technique based on constant-time slices of prestack data in midpoint-offset coordinates. Bancroft and Geiger ^{[22]} developed a prestack imaging technique also based on constant-time slices of prestack data, but in shot-receiver coordinates. Practical aspects of prestack time migration are discussed in prestack time migration, while techniques for migration velocity analysis are described in migration velocity analysis.

## References

- ↑ Doherty (1975), Doherty, S. M., 1975, Structure-independent velocity estimation: Ph. D. thesis, Stanford University.
- ↑ Sherwood et al. (1978), Sherwood, J.W.C., Schultz, P.S., and Judson, D.R., 1978, Equalizing the stacking velocities of dipping events via Devilish: Presented at the 48th Ann. Internat. Soc. Expl. Geophys. Mtg.
- ↑ Yilmaz and Claerbout (1980), Yilmaz, O. and Claerbout, J.F., 1980, Prestack partial migration: Geophysics, 45, 1753–1777.
- ↑ Deregowski and Rocca (1981), Deregowski, S. M. and Rocca, F., 1981, Geometrical optics and wave theory for constant-offset sections in layered media: Geophys. Prosp., 29, 374–406.
- ↑ Ottolini (1982), Ottolini, R., 1982, Migration of seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University.
- ↑ Bolondi et al., 1982, Bolondi, G., Loinger, E. and Rocca, F., 1982, Offset continuation of seismic sections: Geophys. Prosp., 30, 813–828.
- ↑ 1984, Bolondi, G., Loinger, E. and Rocca, F., 1984, Offset continuation in theory and practice: Geophys. Prosp., 32, 1045–1073.
- ↑ Salvador and Savelli, 1982, Salvador, L. and Savelli, S., 1982, Offset continuation for seismic stacking: Geophys. Prosp., 30, 829–849.
- ↑ Bolondi and Rocca, 1985, Bolondi, G. and Rocca, F., 1985, Normal moveout correction, offset continuation and prestack partial migration compared as prestack processes: in Fitch, A. A., Ed., Developments in geophysical exploration methods, 6, Elsevier Applied science Publ., Ltd.
- ↑
^{10.0}^{10.1}Hale (1983), Hale, D., 1983, Dip moveout by Fourier transform: Ph.D. thesis, Stanford University. - ↑ Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
- ↑ Jacubowicz (1990), Jacubowicz, H., 1990, A simple efficient method of dip-moveout correction: Geophys. Prosp., 38, 221–245.
- ↑ Hale and Artley (1992), Hale, D. and Artley, C., 1992, Squeezing dip-moveout for depth-variable velocity: Geophysics, 58, 257–264.
- ↑ Artley and Hale (1994), Artley, C. and Hale, D., 1994, Dip-moveout processing for depth-variable velocity: Geophysics, 59, 610–622.
- ↑ French et al. (1984), French, W.S., Perkins, W.T., and Zoll, R.M., 1984, Partial migration via true CRP stacking: 54th Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 799–802.
- ↑ Biondi and Ronen (1987), Biondi, B. and Ronen, J., 1987, Dip moveout in shot profiles: Geophysics, 52, 1473–1482.
- ↑ Cabrera and Levy (1989), Cabrera, J. and Levy, S., 1989, Shot dip-moveout with logarithmic transformations: Geophysics, 54, 1038–1041.
- ↑ Granser (1994), Granser, H., 1994, Shot gather DMO in the double-log domain: Geophysics, 49, 1305–1307.
- ↑ Zhou et al. (1996), Zhou, B, Mason, I. M., and Greenalgh, S.A., 1996, Accurate and efficient shot-gather dip-moveout processing in the log-stretch domain: Geophys. Prosp., 43, 963–978.
- ↑ Fowler (1984), Fowler, P., 1984, Velocity-independent imaging of seismic reflectors: 54th Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 383–385.
- ↑ Gardner et al. (1986), Gardner, G. F. H., Wang, S. Y., Pan, N. D., and Zhang, Z., 1986, Dip moveout and prestack imaging: Expanded Abstracts, 75–84, 18th Ann. Offshore Tech. Conf.
- ↑ Bancroft and Geiger (1994), Bancroft, J. C. and Geiger, H. D., 1994, Equivalent-offset CRP gathers: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 672–675.

## See also

- Salt-flank reflections
- Fault-plane reflections
- DMO and stacking velocities
- Turning-wave reflections
- Principles of dip-moveout correction
- Dip-moveout correction in practice
- Prestack time migration
- Migration velocity analysis
- Exercises
- Topics in Dip-Moveout Correction and Prestack Time Migration