Velocity continuation and the anatomy of residual prestack time migration

Conclusions

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 those 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 gives us new insights into the methods of prestack migration velocity analysis. Extensions to the case of depth migration in a variable velocity background are developed by Liu and McMechan (1996) and Adler (2002). A practical application of velocity continuation to migration velocity analysis is demonstrated in the companion paper (Fomel, 2003b), where the general theory is used to design efficient and practical algorithms.

Velocity continuation and the anatomy of residual prestack time migration