The migrated time section: an industry kludge

As geology becomes increasingly dramatic, reflection data gets more anomalous. The first thing noticeable is that the stacking velocity becomes unreasonable. In practice the available computer processes--based on inappropriate assumptions--will be tried anyway.

A stacking velocity will be chosen and a stack formed. How should the migration be done? Most basic migration programs omit the lens term. Although it is easy to include the lens term, the term is sensitive to lateral variation in velocity. Since estimates of lateral variation in velocity always have questionable reliability, use of a migration program with a lens term is usually limited to knowledgeable interpreters. The lens term is usually omitted from the basic migration utility program. Let us see what this means.

The migration equation is valid in some ``local plane wave'' sense, i.e.  

$$k_z ( y,z) \ \ =\ \ {\omega \over v(y,z)} \ \ \sqrt{ \ 1 \ ... ...\left( \ { v (y,z) \ k_y ( y,z ) \over \omega }\ \right)^2 \ }$$ (20)

A migrated time section is defined by transforming the depth variable z in (20) to a travel-time depth $\tau$. 

$$k_{\tau} ( y , \tau ) \ \ =\ \ \omega \ \sqrt{ 1 \ -\ \left( {v(y, \tau ) \ k_y ( y , \tau ) \over \omega }\ \right)^2 \ }$$ (21)

The implementation of equation (21) requires no lens terms, so no large sensitivity to lateral velocity variation is expected. Unfortunately, there is a pitfall. The (y,z) coordinate system is an orthogonal coordinate system, but the $( y, \tau )$ system is not orthogonal [unless v(y) =const]. So equation (20), which says that $\cos \theta = \sqrt { 1 - \sin^2 \theta }$,is not correctly interpreted by (21). A hyperbola would migrate to its top when it should be migrating toward the low-velocity side.

In summary: In a production environment a great deal of data gets processed before anyone has a clear idea of how much lateral velocity variation is present. So the lens term is omitted. The results are OK if the lens term happens to commute with the diffraction term. The terms do commute when the lateral velocity variation is slow enough. Otherwise, you should reprocess with the lens term. The reprocessing will be sensitive to errors in velocity. Be careful!