MZO from prestack migration

The constant velocity prestack migration in offset-midpoint coordinates can be formulated as:  

$$p(t=0,k_y,h=0,z)= {\int d\omega \int d k_h \; e^{ik_z(\omega,k_y,k_h)z} p(\omega,k_y,k_h,z=0)}$$ (1)

where $p(\omega,k_y,k_h,z=0)$ is the 3-D Fourier transform of the field p(t,y,h,z=0) recorded at the surface, using Claerbout's (1985) sign convention:

$$p(\omega,k_y,k_h,z=0)= \int dt \; e^{i\omega t} \int dy e^{-ik_yy} \int dh e^{-ik_hh} p(t,y,h,z=0).$$

The constant $k_z(\omega,k_y,k_h)$ is defined in the dispersion relation as  

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

The constant velocity zero-offset migration in offset-midpoint coordinates can be formulated as:  

$$p(t=0,k_y,z)= {\int d\omega_0 \; e^{ik_z(\omega_0,k_y)z} p(\omega_0,k_y,z=0)}$$ (3)

where $p(\omega_0,k_y,z=0)$ is the 2-D Fourier transform of the field p(t,y,z=0). The constant $k_z(\omega_0,k_y)$ is defined in the dispersion relation as  

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

The goal of the ensuing derivation is to convert equation (1) into a form similar to equation (3). Using a change of variables and integrating over the variable k_h, equation (1) will be transformed into the form:

$$p(t=0,k_y,h=0,z)= {\int d\omega_0 \; e^{ik_z(\omega_0,k_y)z} p_0(\omega_0,k_y,z=0)},$$

where $p_0(\omega_0,k_y,z=0)$ represents the zero-offset data field. Following Hale's (1983) derivation, a new variable $\omega_0$ is introduced in order to isolate the zero-offset migration operator. The variable $\omega$ is expressed as  

$$\omega \equiv {\omega_0 { \left[ 1+{ {v^2 k_h^2 } \over { 4 \omega_0^2-v^2 k_y^2}} \right]}^{1 \over 2}},$$ (5)

where $\omega_0$ is considered variable and k_y, k_h are constant. Substituting $\omega$ in equation (2), the downward continuation operator $k_z(\omega,k_y,k_h)$ is transformed into  

$$k_z \equiv -{ {2 \omega_0} \over v} {\left[ 1 - {{v^2 k_y^2} \over {4 \omega_0^2}} \right]}^{1 \over 2}$$ (6)

which now has the same form as the operator in equation (4). In order to isolate the zero-offset migration operator, the variable $\omega_0$ is substituted for $\omega$ in equation (1) and the order of integration is changed between $\omega_0$ and k_h. The integration boundaries have to be observed carefully as they are modified after each change of variables and integration order. However, for the sake of simplicity, I will ignore in the following demonstration the integration limits, which are discussed in Appendix A.

For simplicity I define the variables

$$\begin{array} {lcl} v_h & = & {{v k_h} \over 2} \\ \\ v_y & = & {{v k_y} \over 2}.\end{array}$$

By substituting the variable $\omega$ with the expression in $\omega_0$ in equation (1) and changing the integration order between $\omega_0$ and k_h the prestack migration equation becomes  

$$\begin{array} {lcl} p(t=0,k_y,h=0,z) & = & \displaystyle{ {\... ...ga_0 \; e^{ik_z(\omega_0,k_y)z} p_0(\omega_0,k_y)}}.\end{array}$$ (7)

The new field $p^*(\omega_0,k_y,k_h,z=0)$ represents a remapping (interpolation) from $\omega$ to $\omega_0$ of the field $p(\omega,k_y,k_h,z=0)$.The meaning of this remapping will be discussed in the next section. The field $p_0(\omega_0,k_y)$ defined as  

$$p(\omega_0,k_y)=\int dk_h {\left[{{d \omega} \over {d \omega_0}}\right]} p^*(\omega_0,k_y,k_h,z=0)$$ (8)

with the Jacobian  

$$\begin{array} {lcl} J & = & \displaystyle{ {\left[{{d \omega... ..._h^2 v_y^2} \over { (\omega_0^2-v_y^2)^2}} \right ]}\end{array}$$ (9)

represents the zero-offset field.

The last equation in (7) represents zero-offset downward continuation and imaging as introduced by Gazdag (1978) or Stolt (1978). Equation (8) represents a way of obtaining the zero-offset section from constant-offset sections.

So far the operations needed to obtain the zero-offset stacked section from the constant-offset field are:

1.

Fourier transform the constant-offset field $p(t,y,h) \rightarrow p(\omega,k_y,k_h)$.

2.

Remap (interpolate) the $\omega$ axis into $\omega_0$.

3.

Multiply by the Jacobian.

4.

Integrate over k_h.

5.

Inverse Fourier transform $p_0(\omega_0,k_y) \rightarrow p_0(t_0,y)$.