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 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.