Second definition: MZO extracted from prestack migration

Hale (1984) starts with the prestack migration equation in common-midpoint and offset coordinates and tries to isolate four major processing steps. The four steps are

1.

DMO,

2.

NMO,

3.

Stacking,

4.

Zero-offset migration.

By considering steps 1, 2, and 3 as a single process (MZO), I isolate only two steps:

• Migration to zero-offset.
• Zero-offset migration.

After extracting the zero-offset migration from the prestack migration equation, we are left with the migration to zero-offset operator.

For a constant velocity, downward continue the field p(t,h,y,z=0) recorded at the surface, to a depth $z=\Delta z$  

$$p(\omega,k_h,k_y,z=\Delta z)= { e^{ik_z(\omega,k_h,k_y)\Delta z}p(\omega,k_h,k_y,z=0)}$$ (7)

for all values of $\omega,k_h,k_y$. The prestack migration imaging step consists of summing for all the values of $\omega$ and k_h after downward continuation. To image for t=0 and h=0:  

$$p(t=0,h=0,k_y,z)= {\int d\omega \int d k_h \; e^{ik_z(\omega,k_h,k_y)z} p(\omega,k_h,k_y,z=0)}$$ (8)

where k_z is given by  

$${k_z(\omega,k_h,k_y)} \equiv { -{\omega \over v} \left\{ \le... ...2 \over {4\omega^2}} (k_y-k_h)^2\right]^{1 \over 2} \right\} }.$$ (9)

Following Hale's (1984) derivation, introduce a new variable $\omega_0$ in order to isolate the zero-offset migration operator. The substitution is based on the equation  

$$\omega \equiv {\omega_0 { \left[ 1+{ {v^2 k_h^2 } \over { 4 \omega_0^2-v^2 k_y^2}} \right]}^{1 \over 2}}.$$ (10)

Substituting $\omega$ in equation (9) the downward continuation operator $k_z(\omega,k_h,k_y)$ is transformed into  

$$k_z \equiv -{ {2 \omega_0} \over v} {\left[ 1 - {{v^2 k_y^2} \over {4 \omega_0^2}} \right]}^{1 \over 2}$$ (11)

The downward continuation operator now has the same form as the one for zero-offset. In order to isolate the zero-offset migration operator, the variable $\omega_0$ is substituted for $\omega$ and then the order of integration is changed between $\omega_0$ and k_h. The integration boundaries have to be observed carefully.