theory

Common-azimuth data represent subsets of 3-D datasets that have been recorded or transformed to zero cross-line offsets (h_y=0). Stolt constant velocity migration for common-azimuth data involves the use of the following dispersion relation Biondi and Palacharla (1995):  

$${k_z}_x= \frac{1}{2}\sqrt{\left(\frac{\omega}{\v} \right)^2-... ...t(\frac{\omega}{\v} \right)^2- \left({k_m}_x+{k_h}_x\right)^2},$$ (1)

where the depth wavenumber of the common-azimuth dataset (k_z) is  

$$k_z= \sqrt{ {k_z}_x^2 - {k_m}_y^2},$$ (2)

and the midpoint and offset wavenumbers are defined as ${\bf k_m}={k_m}_x\i + {k_m}_y\j$ and ${\bf k_h}={k_h}_x\i + {k_h}_y\j$.

We can write equations (2) and (1) for a given reference velocity ($\v_0$) as  

$${{k_z}_x}_0= \frac{1}{2}\sqrt{\left(\frac{\omega}{\v_0} \rig... ...\frac{\omega}{\v_0} \right)^2- \left({k_m}_x+{k_h}_x\right)^2},$$ (3)

and  

$${k_z}_0= \sqrt{ {{k_z}_x}_0^2 - {k_m}_y^2}.$$ (4)

The goal of common-azimuth Stolt residual migrations is to obtain k_z (equations 2 and 1) from k_z0 (equations 4 and 3).

If we express the frequency $\omega$ from equation (3) as

$$\omega^2 = \v_0^2 \frac {\left[{k_z}_0^2+{k_m}_y^2+{k_h}_x^2... ...ft[{k_z}_0^2+{k_m}_y^2+{k_m}_x^2 \right]} {{k_z}_0^2+{k_m}_y^2}$$

and introduce it in equation (1), we obtain the common-azimuth residual migration equations  

$$\left\{ \begin{array} {l} { \begin{array} {r} {k_z}_x= \fra... ...ray}} \\ k_z= \sqrt{ {k_z}_x^2 - {k_m}_y^2}\end{array}\right .$$ (5)

Note that for 2-D prestack data, equations (5) reduce to the 2-D prestack form Sava (1999):  

 $$\begin{array} {r} k_z= \frac{1}{2}\sqrt{\frac{\v_0^2}{\v^2... ...ht]} {{k_z}_0^2}- \left({k_m}_x+{k_h}_x\right)^2} \end{array},$$

which, furthermore, reduces to the well-known Stolt 2-D post-stack residual migration equation Stolt (1996) :  

 $$k_z= \sqrt{\frac{\v_0^2}{\v^2} { \left[{k_z}_0^2+{k_m}_x^2 \right]}- {k_m}_x^2}.$$

As for the 2-D prestack data, the common-azimuth residual migration is velocity independent; that is, we need not make any assumption about the actual values of the velocities for the reference and improved migration, but only about their ratio. In this way, we can take an image and residually migrate it without knowing what velocity model has been used to image it in the first place.