next up previous [pdf]

Next: Angle transformation in wave-equation Up: Imaging condition in wave-equation Previous: Time-shift imaging condition

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:

$\displaystyle U \left ({ \bf m},{ \bf h}, t \right )$ $\textstyle =$ $\displaystyle U_r \left ({ \bf m}+{ \bf h}, t+{ \tau}\right )\ast
U_s \left ({ \bf m}-{ \bf h}, t-{ \tau}\right )\;,$ (9)
$\displaystyle R \left ({ \bf m},{ \bf h},{ \tau}\right )$ $\textstyle =$ $\displaystyle U \left ({ \bf m},{ \bf h},{ \tau},t=0 \right )\;.$ (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.