MZO by Fourier transform

Both Hale and Zhang separate the NMO correction from the DMO correction. However for variable velocity media this can not be done, and in order to generalize the $NMO \cdot DMO$ operation we have to use the migration to zero-offset (MZO) concept. Migration to zero-offset is the process that transforms a nonzero-offset (constant offset) section into a zero-offset section. In constant velocity media $MZO = NMO \cdot DMO$.The Fourier transform expression is simply obtained by using again equation (20) without separating the NMO step. We have  

$$\left \{ \begin{array} {l} y_0 = \displaystyle{y- {{2h^2 \si... ...r v^2}+4h^2{{\sin^2\theta} \over v^2} }} }.\end{array} \right.$$ (25)

The differentials of the new variables are  

$$\left \{ \begin{array} {l} dy_0 = \displaystyle{dy} \\ \\ dt... ..._0}^2 \over \omega_0^2})^{3 \over 2}}} dt_h \end{array} \right.$$ (26)

After replacing the variables y₀ and t₀ in equation (21) we obtain  

$$P_0(\omega_0,k_{y_0},h)={\int_{t_h}dt_h\int_{y}dy {{t_h(t_h^... ...}\over v^2}+ h^2 {k_{y_0}^2 \over \omega_0^2}}- k_{y_0} y)}} }.$$ (27)

Note that if we replace the quantity

$$t_h^2 - {{4h^2} \over v^2} = t_n^2$$

in equation (27) we get the same phase as the one obtained by Hale and Zhang, but the Jacobian differs from Zhang's Jacobian by a factor  

$${J_M \over J_Z} = {t_h \over t_n} = {\sqrt {1 + {{4h^2} \over {v^2 t_n^2}} }}$$ (28)

which corresponds to the Jacobian of the NMO transformation.