Time-shift imaging condition in seismic migration

Space-shift imaging condition

Defining ${ \bf k}_{ \bf m}$ and ${ \bf k}_{ \bf h}$ as location and offset wavenumber vectors, and assuming $\vert{ \bf p}_{ \bf s}\vert=\vert{ \bf p}_{ \bf r}\vert=s$, where $s \left ({ \bf m}\right )$ is the slowness at image locations, we can replace $\vert{ \bf p}_{ \bf m}\vert= \vert{ \bf k}_{ \bf m}\vert/\omega$ and $\vert{ \bf p}_{ \bf h}\vert= \vert{ \bf k}_{ \bf h}\vert/\omega$ in equations (15)-(16):

$$\vert{ \bf k}_{ \bf h}\vert^2 = 2(\omega s)^2 (1 - \cos 2\theta ) \;,$$ (17)

$$\vert{ \bf k}_{ \bf m}\vert^2 = 2(\omega s)^2 (1 + \cos 2\theta ) \;.$$ (18)

Using the trigonometric identity

$$\cos(2\theta ) = \frac{1-\tan^2 \theta } {1+\tan^2 \theta } \;,$$ (19)

we can eliminate from equations (17)-(18) the dependence on frequency and slowness, and obtain an angle decomposition formulation after imaging by expressing $\tan \theta$ as a function of position and offset wavenumbers $({ \bf k}_{ \bf m},{ \bf k}_{ \bf h})$:

$$\tan \theta = \frac{\vert{ \bf k}_{ \bf h}\vert}{\vert{ \bf k}_{ \bf m}\vert} \;.$$ (20)

We can construct angle-domain common-image gathers by transforming prestack migrated images using equation (20)

$$R \left ({ \bf m}, { \bf h}\right )\Longrightarrow R \left ({ \bf m}, \theta \right )\;.$$ (21)

In 2D, this transformation is equivalent with a slant-stack on migrated offset gathers. For 3D, this transformation is described in more detail by Fomel (2004) or Sava and Fomel (2005).

Time-shift imaging condition in seismic migration