Introduction to dip-moveout correction and prestack migration
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  first introduced the wave extrapolation equations for nonzero-offset data. Sherwood  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  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  recast the theory for PSPM in a form similar to Kirchhoff migration. Ottolini  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 ,    that is, mapping a far-offset section to a near-offset section, thereby collapsing all offsets to zero offset. Hale ,  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  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 , Hale and Artley  and Artley and Hale  extended the DMO theory to accommodate vertical velocity variations. French  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 , Cabrera and Levy , Granser  and Zhou  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  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  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  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.
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- 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
- Topics in Dip-Moveout Correction and Prestack Time Migration