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Migration velocity analysis is a routine part of prestack time
migration applications. It serves both as a tool for velocity
estimation Deregowski (1990) and as a tool for optimal stacking
of migrated seismic sections and modeling zero-offset data for depth
migration Kim et al. (1997). In the most common form, migration
velocity analysis amounts to residual moveout correction on CRP
(common reflection point) gathers. However, in the case of dipping
reflectors, this correction does not provide optimal focusing of
reflection energy, since it does not account for lateral movement of
reflectors caused by the change in migration velocity. In other words,
different points on a stacking hyperbola in a CRP gather can
correspond to different reflection points at the actual reflector. The
situation is similar to that of the conventional NMO velocity
analysis, where the reflection point dispersal problem is usually
overcome with the help of DMO Deregowski (1986); Hale (1991). An
analogous correction is required for optimal focusing in the
post-migration domain. In this paper, I propose and test velocity
continuation as a method of migration velocity analysis. The method
enhances the conventional residual moveout correction by taking into
account lateral movements of migrated reflection events.
Velocity continuation is an artificial process of transforming time
migrated images according to the changes in migration velocity. This
process has wave-like properties, which have been described in my
earlier papers Fomel (1994, 1996, 1997).
Hubral et al. (1996) and Schleicher et al. (1997) use the term
*image waves* to introduce a similar concept. Velocity
continuation extends the theory of residual and cascaded migrations
Larner and Beasley (1987); Rothman et al. (1985). In practice, the
continuation process can be modeled by finite-difference or spectral
methods Fomel and Claerbout (1997); Fomel (1998).

Applying velocity continuation to migration velocity analysis involves
the following steps:

- 1.
- prestack common-offset (and common-azimuth) migration - to
generate the initial data for continuation,
- 2.
- velocity continuation with stacking across different offsets -
to transform the offset data dimension into the velocity dimension,
- 3.
- picking the optimal velocity and slicing through the migrated
data volume - to generate an optimally focused image.

In this paper, I demonstrate all three steps, using a simple
two-dimensional dataset. For the implementation of velocity
continuation, I chose the Fourier spectral method. The method has its
limitations Fomel (1998), but looks optimal in terms of
the accuracy versus efficiency trade-off. It is important to note that
although the velocity continuation result could be achieved in
principle by using prestack residual migration in Kirchhoff
Etgen (1990) or Stolt Stolt (1996) formulation,
the first is evidently inferior in efficiency, and the second is not
convenient for velocity analysis across different offsets, because it
mixes them in the Fourier domain Sava (1999).

** Next:** Putting together prestack velocity
** Up:** Fomel: Velocity continuation
** Previous:** Fomel: Velocity continuation
Stanford Exploration Project

4/20/1999