First definition: 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 DMO operation the migration to zero-offset (MZO) concept has to be used. 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+DMO.

 

MZOdip
Figure 1 Geometry for a dipping reflector in a constant velocity medium. The zero-offset ray JR and the nonzero-offset ray bounce at the same reflection point R. The dipping angle is $\theta$.

Taking into account the geometry in Figure the MZO correction in constant velocity media is  

$$\left \{ \begin{array} {l} \Delta y_0 = \displaystyle{ {2h^2... ... {t_h^2-4h^2{{\cos^2\theta} \over v^2} } },\end{array} \right.$$ (1)

where h is the half offset, t₀ is twice the zero-offset travel time from the reflection point to the surface, v is the velocity of the medium and t_h is the constant-offset travel time.

The MZO correction for a given dip $\theta$ is  

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

To clarify the sign convention for the spatial coordinate in equation (2), we can notice that the y-axis is oriented to the left, and the angles are positive if the reflector dips toward the right and negative if the reflector dips toward the left. In Figure the angle $\theta$ is negative. This can be also derived from the equation

$${{dt_0} \over {dy_0}} = {{ 2 \sin \theta} \over v} = {{k_{y_0}} \over {\omega_0}}$$

where the sign of dy₀ determines the sign of the angle $\theta$.

The differentials of the new variables are  

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

The 2-D Fourier transform of the zero-offset field is  

$$P_0(\omega_0,k_{y_0},h)={\int_{t_0}dt_0\int_{y_0}dy_0 \: {e^{i (\omega_{0} t_0- k_{y_0} y_0) }}p_0(t_0,y_0,h)}.$$ (4)

Following Hale's (1984) technique, replace the variables t₀ and y₀ in equation (4) with their expression in equation (2). Fortunately, it is not necessary to express explicitly t_h=t_h(t₀,y₀) and y=y(t₀,y₀), though the respective dependencies are assumed.

After replacing the variables y₀ and t₀ in equation (4), the migration to zero-offset transformation becomes  

$$\begin{array} {lcl} P_0(\omega_0,k_{y_0},h) & = & \displayst... ...4h^2}\over v^2}+ h^2 {k_{y_0}^2 \over \omega_0^2}}}}\end{array}$$ (5)

Note that by replacing the quantity

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

in equation (5), we get the same phase as the one obtained by Hale and Zhang, but the Jacobian differs from Zhang's Jacobian by a factor  

$${J_M \over J_Z} = {t_h \over t_n} = {\sqrt {1 + {{4h^2} \over {v^2 t_n^2}} }},$$ (6)

which corresponds to the Jacobian of the NMO transformation.