Common-azimuth Stolt residual migration

Common-azimuth data represent subsets of 3D prestack data that have been recorded or transformed to a common azimuth, which corresponds to zero cross-line offsets (${\bf h}_y=0$). Stolt migration for common-azimuth data involves the use of the following dispersion relation (9):  

$${\kzz}_x= \frac{1}{2}\sqrt{\frac{\ww^2}{v^2}- \lp {k_m}_x-{k... ... \frac{1}{2}\sqrt{\frac{\ww^2}{v^2}- \lp {k_m}_x+{k_h}_x\rp^2},$$ (40)

where the depth wavenumber for the common-azimuth dataset ($\kzz$) is written as  

$$\kzz = \sqrt{ {\kzz}_x^2 - {k_m}_y^2}.$$ (41)

We can rewrite dispersion.true and disp.true for a given reference velocity (v₀) and obtain the corresponding depth wavenumbers ${{\kzz}_x}_0$ and ${\kzz}_0$.Mathematically, the goal of common-azimuth Stolt residual migrations is also to obtain $\kzz$ from ${\kzz}_0$. Again, we can achieve this by eliminating the frequency $\omega$ from the expressions for ${\kzz}_0$ and $\kzz$, which leads to the 3D common-azimuth residual migration equations:  

$$\left\{ \bea{l} { \bea{r} {\kzz}_x= \frac{1}{2}\sqrt{\rho^2... ...a } \\ \kzz = \sqrt{ {\kzz}_x^2 - {k_m}_y^2} \;, \eea \right .$$ (42)

where, by definition, $\rho=\frac{v_0}{v}$.If we make the change of variables

$$\mu_c^2=\frac{ \lb {{\kzz}_0}^2 + {k_m}_y^2 + {k_h}_x^2 \rb ... ...}_0}^2 + {k_m}_y^2 + {k_m}_x^2 \rb } {{{\kzz}_0}^2+{k_m}_y^2}.$$ (43)

we obtain a simplified version of myresmig-ca:

$$\left\{ \bea{l} {\kzz}_x= \frac{1}{2}\sqrt{ \rho^2 \mu_c^2 -... ...p^2} \\ \kzz = \sqrt{ {\kzz}_x^2 - {k_m}_y^2} \;, \eea \right .$$ (44)

For 2D data, where k_my=0, $\kzz \equiv {\kzz}_x$ and ${\kzz}_0\equiv {{\kzz}_x}_0$,myresmig-ca reduce to the 2D prestack myresmig-2d-pr and post-stack myresmig-2d-po forms.