Introduction

 The purpose of seismic migration is accurate focusing of surface measurements at subsurface reflection points. Straight-ray limitations degrade the ability of time-migration techniques to produce accurate subsurface images. Relaxing constant velocity constraints by allowing velocities to vary as functions of midpoint and time does not resolve the issue. Rays are still forbidden to bend and, as a result, data are not properly focused.

Modern depth-imaging techniques allow rays to bend and are understood well enough to have become a standard part of data processing. They are rapidly leading to sophisticated methods for velocity estimation and mapping. One might conclude that further work on time based algorithms is unnecessary but depth migrations do have an Achilles heel. They require an acceptable initial velocity model prior to application. Without good initial velocity estimates, iterative depth migration can lead to diasasterous results. Although estimation of prior velocity information is the subject of considerable discussion and research, approaches can be formulated to postpone velocity estimation until the data have been fully migrated.

The basic idea is to reverse the conventional processing sequence:

• Velocity Analysis (VA),
• Normal Moveout (NMO) Correction,
• Dip Moveout (DMO),
• Stack, (S) and
• Post-Stack Migration (PsSM)

to something like:

• Prestack Migration (PrSM),
• VA,
• Normal Moveout (NMO) Correction, and
• S.

In the latter approach, velocities are chosen after migration and so, in principle, are independent of dip. They are also estimated along image rays and, therefore, pertain more to true migrated positions than methods that do not incorporate migration in their formulation. A stacking-velocity field derived from migrated data provides an improved model for initial depth-imaging requirements.

There are several candidate techniques for reversing the processing sequence. Prestack-time-migration of normal-moveout-corrected common-offset sections followed by inverse NMO can be used to perform residual velocity analysis and in a sense provides a crude approach to the second processing sequence described above. Forel and Gardner 1988 develop a two-step dip-moveout (DMO) and prestack imaging (PSI) approach for forming common scatterpoint gathers. They claim that their method is completely velocity independent, and fully reverses the traditional processing sequence. Berryhill's 1996 non-imaging-shot-record migration is similar to Forel and Gardner's two-step approach and in fact M. Popovici 1994 claims that the two methods are equivalent. Moreover, Popovici 1994 shows that Berryhill's method is much faster computationally. This is an important issue for subsequent remarks and should be kept in mind as the paper progresses. The method described in Ferber et al. (1996); Ferber (1994) provides an additional approach to reversing the processing sequence, but is also similar in spirit to that of Forel and Gardner. Since it concentrates more on 3D issues it is of significant interest for that alone. Bancroft et al. 1994, 1995 introduced Equivalent-offset migration (EOM) suggesting that it is several orders of magnitude faster than more traditional Kirchhoff approaches. They claim Bancroft and Geiger (1994); Bancroft et al. (1995); Bancroft (1997), that a major benefit of their method is the production of high fold, large offset common scatterpoint gathers that improve the focusing of velocity semblance resulting in improved velocity estimates. They suggest that rough initial velocity estimates are the only requirement for achieving this result. Bancroft 1997 further claims that the relationship between EOM and DMO followed by PSI (DMO-PSI) is superficial.

In view of the similarities, this paper contrasts the Forel and Gardner 1988 method with Bancroft and Geiger (1994); Bancroft et al. (1995); Bancroft (1997) by first casting each within the same framework and notation. For the readers convenience, velocity-independent dip-moveout, prestack imaging, and equivalent-offset migration are each explained in separate sections using the same theoretical foundation. It is then clear that equivalent-offset migration only partially reverses the standard processing steps. The EOM schematic is VA, PrSM, NMO, and S. Moreover, EOM is asymptotically equivalent to the PSI step in DMO-PSI. On the other hand, dip-moveout followed by prestack imaging is a full reversal of the standard processing sequence. It is completely velocity independent. Both PSI and EOM perform the imaging step on constant time-slices by diffracting energy over appropriately defined curves. Not surprisingly, DMO can be implemented in exactly the same manner. The PSI curves are much simplier than those for EOM and so are significantly easier to implement. Although the details will not be discussed here, DMO-PSI can be formulated as a Fourier domain process and so has $O(n \log n)$ conputational complexity.

While all the results in this paper focus on 2D relationships, extension to 3D is straightforward.