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I have derived kinematic and dynamic equations for residual time migration
in the form of a continuous velocity continuation process. This
derivation explicitly decomposes prestack
velocity continuation into three parts corresponding to
zero-offset continuation, residual NMO, and residual DMO. These three
parts can be treated separately both for simplicity of theoretical
analysis and for practical purposes. It is important to note that in
the case of a three-dimensional migration, all three components of
velocity continuation have different dimensionality. Zero-offset
continuation is fully 3-D. It can be split into two 2-D continuations
in the in- and cross-line directions. Residual DMO is a
two-dimensional common-azimuth process. Residual NMO is a 1-D
single-trace procedure.
The dynamic properties of zero-offset velocity
continuation are precisely equivalent to the dynamic properties of
conventional post-stack migration methods such as Kirchhoff
migration. Moreover, the Kirchhoff migration operator coincides with
the integral solution of the velocity continuation differential
equation for continuation from the zero velocity plane.

This rigorous theory of velocity continuation can give us new insights
into the methods of prestack migration velocity analysis. However, its
practical applicability faces several important problems. One of them
concerns the comparative value of the residual DMO term. Another
problem is the choice of the implementation method for velocity
continuation. Both these problems require further research efforts.

** Next:** Acknowledgments
** Up:** Fomel: Velocity continuation
** Previous:** Dynamics of Residual DMO
Stanford Exploration Project

11/12/1997