Time-shift imaging condition in seismic migration

Space-shift and time-shift imaging condition

To be even more general, we can formulate an imaging condition involving both space-shift and time-shift, followed by image extraction at zero time:

$$U \left ({ \bf m},{ \bf h}, t \right ) = U_r \left ({ \bf m}+{ \bf h}, t+{ \tau}\right )\ast U_s \left ({ \bf m}-{ \bf h}, t-{ \tau}\right )\;,$$ (9)

$$R \left ({ \bf m},{ \bf h},{ \tau}\right ) = U \left ({ \bf m},{ \bf h},{ \tau},t=0 \right )\;.$$ (10)

However, the cost involved in this transformation is large, so this general form does not have immediate practical value. Imaging conditions described by equations  (3)-(4) and (6)-(7) are special cases of equations (9)-(10) for ${ \bf h}=0$ and ${ \tau}=0$, respectively.