Time-shift imaging condition in seismic migration

Time-shift imaging condition

Using the same definitions as the ones introduced in the preceding subsection, we can re-write equation (18) as

$$\vert{ \bf p}_{ \bf m}\vert^2 = 4 s^2 \cos^2 \theta \;,$$ (22)

from which we can derive an expression for angle-transformation after time-shift prestack imaging:

$$\cos \theta = \frac{\vert{ \bf p}_{ \bf m}\vert}{2s} \;.$$ (23)

Relation (23) can be interpreted using ray parameter vectors at image locations (Figure 2).

img3
Figure 2. Interpretation of angle-decomposition based on equation (23) for time-shift gathers.

Angle-domain common-image gathers can be obtained by transforming prestack migrated images using equation (23):

$$R \left ({ \bf m}, { \tau}\right )\Longrightarrow R \left ({ \bf m}, \theta \right )\;.$$ (24)

Equation (23) can be written as

$$\cos^2 \theta = \frac{\vert\nabla_{{ \bf m}} 2 { \tau}\vert... ...^2 + { \tau}_y^2 + { \tau}_z^2}{s^2 \left (x,y,z\right )} \;,$$ (25)

where ${ \tau}_x,{ \tau}_y,{ \tau}_z$ are partial derivatives of ${ \tau}$ relative to $x,y,z$. We can rewrite equation (25) as

$$\cos^2 \theta = \frac{{ \tau}_z^2}{s^2 \left (x,y,z \right )} \left (1+z_x^2+z_y^2 \right )\;,$$ (26)

where $z_x,z_y$ denote partial derivative of coordinate $z$ relative to coordinates $x$ and $y$, respectively. Equation (26) describes an algorithm in two steps for angle-decomposition after time-shift imaging: compute $\cos \theta$ through a slant-stack in $z-{ \tau}$ panels (find a change in ${ \tau}$ with respect to $z$), then apply a correction using the migration slowness $s$ and a function of the structural dips $\sqrt{1+z_x^2+z_y^2}$.

Time-shift imaging condition in seismic migration