Time-shift imaging condition for converted waves

Conventional imaging condition

A conventional imaging condition for shot-record migration, also known as $U \overline{D}$ 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. Mathematically, we can represent this process by the relations

$${ \bf u}\left({ \bf m}, t \right) = { \bf u}_s\left({ \bf m}, t \right) \ast { \bf u}_r\left({ \bf m}, t \right) \;,$$ (1)

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

Here, ${ \bf m}= \left[ m_x,m_y,m_z \right]$ is a vector describing the locations of image points, ${ \bf u}_s({ \bf m},t)$ and ${ \bf u}_r({ \bf m},t)$ are source and receiver wavefields respectively, and ${ \bf R}({ \bf m})$ denotes the migrated image, proportional to reflectivity at every location in space. The symbol $\ast$ denotes cross-correlation in time.

A typical implementation of this imaging condition is in the Fourier domain, where the image is produced using the expression

$${ \bf R}\left({ \bf m}\right) = \sum_\omega { \bf U}_s\lef... ...right) \overline{{ \bf U}_r\left({ \bf m},\omega \right)} \;,$$ (3)

where summation over frequency $\omega$ corresponds to imaging at zero time. The over-line represents a complex conjugate applied on the receiver wavefield ${ \bf U}_r$ in the Fourier domain.

Time-shift imaging condition for converted waves