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Time-shift imaging condition

Another possible imaging condition, advocated in this paper, involves shifting of the source and receiver wavefields in time, as opposed to space, followed by image extraction at zero time:
$\displaystyle U \left ({ \bf m},t,{ \tau}\right )$ $\textstyle =$ $\displaystyle U_r \left ({ \bf m},t+{ \tau}\right )\ast
U_s \left ({ \bf m},t-{ \tau}\right )\;,$ (6)
$\displaystyle R \left ({ \bf m},{ \tau}\right )$ $\textstyle =$ $\displaystyle U \left ({ \bf m},{ \tau},t=0 \right )\;.$ (7)

Here, ${ \tau}$ is a scalar describing the time-shift between the source and receiver wavefields prior to imaging. This imaging condition can be implemented in the Fourier domain using the expression
R \left ({ \bf m},{ \tau}\right )= \sum_\omega
U_r \left ...
...U_s^* \left ({ \bf m},\omega \right )e^{2i\omega { \tau}} \;,
\end{displaymath} (8)

which simply involves a phase-shift applied to the wavefields prior to summation over frequency $\omega $ for imaging at zero time.