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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:

prestack common-offset (and common-azimuth) migration - to generate the initial data for continuation,
velocity continuation with stacking across different offsets - to transform the offset data dimension into the velocity dimension,
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).
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Next: Putting together prestack velocity Up: Fomel: Velocity continuation Previous: Fomel: Velocity continuation
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