Time-shift imaging condition in seismic migration

Time-shift imaging in Kirchhoff migration

The imaging condition described in the preceding section has an equivalent formulation in Kirchhoff imaging. Traditional construction of common-image gathers using Kirchhoff migration is represented by the expression

$$R \left ({ \bf m},{ \widehat{\bf h}}\right )= \sum_{{ \wideh... ...m},{ \widehat{\bf m}}+{ \widehat{\bf h}}\right ) \right] \;,$$ (36)

where $\widehat{U} \left ({ \widehat{\bf m}},{ \widehat{\bf h}},t\right )$ is the recorded wavefield at the surface as a function of surface midpoint ${ \widehat{\bf m}}$ and offset ${ \widehat{\bf h}}$ (Figure 6). $t_s$ and $t_r$ stand for traveltimes from sources and receivers at coordinates ${ \widehat{\bf m}}-{ \widehat{\bf h}}$ and ${ \widehat{\bf m}}+{ \widehat{\bf h}}$ to points in the subsurface at coordinates ${ \bf m}$. For simplicity, the amplitude and phase correction term $A \left ({ \bf m},{ \widehat{\bf m}},{ \widehat{\bf h}}\right )\frac{\partial}{\partial t}$ is omitted in equation (36).

kir
Figure 6. Notations for Kirchhoff imaging. S is a source and R is a receiver.

The time-shift imaging condition can be implemented in Kirchhoff imaging using a modification of equation (36) that is equivalent to equations (6) and (8):

$$R \left ({ \bf m},{ \tau}\right )=\sum_{{ \widehat{\bf m}}} ... ...at{\bf m}}+{ \widehat{\bf h}}\right )+ 2 { \tau} \right] \;.$$ (37)

Images obtained by Kirchhoff migration as discussed in equation (37) differ from image constructed with equation (36). Relation (37) involves a double summation over surface midpoint ${ \widehat{\bf m}}$ and offset ${ \widehat{\bf h}}$ to produce an image at location ${ \bf m}$. Therefore, the entire input data contributes potentially to every image location. This is advantageous because migrating using relation (36) different offsets ${ \widehat{\bf h}}$ independently may lead to imaging artifacts as discussed by Stolk and Symes (2004). After Kirchhoff migration using relation (37), images can be converted to the angle domain using equation (23).

Time-shift imaging condition in seismic migration