Time-shift imaging condition in seismic migration

Imaging condition in wave-equation imaging

A traditional imaging condition for shot-record migration, often referred-to as $U D^*$ imaging condition (Claerbout, 1985), consists of time cross-correlation at every image location between the source and receiver wavefields, followed by image extraction at zero time:

$$U \left ({ \bf m}, t \right ) = U_r \left ({ \bf m}, t \right )\ast U_s \left ({ \bf m}, t \right )\;,$$ (1)

$$R \left ({ \bf m}\right ) = U \left ({ \bf m},t=0 \right )\;,$$ (2)

where the symbol $\ast$ denotes cross-correlation in time. Here, ${ \bf m}= \left[ m_x,m_y,m_z \right]$ is a vector describing the locations of image points, $U_s({ \bf m},t)$ and $U_r({ \bf m},t)$ are source and receiver wavefields respectively, and $R({ \bf m})$ denotes a migrated image. A final image is obtained by summation over shots.