Integration order

By taking equation (14) and changing the integration variable from $\omega$ to $\omega_0$. In the same time the new variable $k_z(\omega,k_h,k_y)$ becomes function of only two variables in $k_z(\omega_0,k_y)$,in the form given by equation (11).  

$$\begin{array} {lcl} p(t=0,h=0,k_y,z) & = & \displaystyle{ {\... ...e^{ik_z(\omega_0,k_y)z} p^*(\omega_0,k_h,k_y,z=0)}}.\end{array}$$ (16)

Special attention should be given to the new field $p^*(\omega_0,k_h,k_y,z=0)$. As the variable $\omega$ is replaced with $\omega_0$, each value of $\omega_0$ has to be associated with the appropriate value of the field

$$p(\omega,k_h,k_y,z=0)= p(\omega={\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}}} ,k_h,k_y,z=0).$$

The new field $p^*(\omega_0,k_h,k_y,z=0)$ represents a remapping (or interpolation) of the original field $p(\omega,k_h,k_y,z=0)$.Each value in the new field $p^*(\omega_0,k_h,k_y,z=0)$ with coordinates $(\omega_0,k_h,k_y)$ corresponds to the value in the field $p(\omega,k_h,k_y,z=0)$ with coordinates $(\omega={\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}}},k_h,k_y)$. A more detailed analysis of this mapping is done in a companion paper in this report: Popovici (1993).

From equation (10) the Jacobian is  

$$J ={\left[{{d \omega} \over {d \omega_0}}\right]}= (1+{{v_h^... ...t [ 1 - {{v_h^2 v_y^2} \over { (\omega_0^2-v_y^2)^2}} \right ].$$ (17)

In the next step we change the integration order  

$$\begin{array} {lcl} p(t=0,h=0,k_y=0,z) & = & \displaystyle{ ... ...\omega_0 e^{ik_z(\omega_0,k_y)z} p_0(\omega_0,k_y)}}\end{array}$$ (18)

where $p_0(\omega_0,k_y)$ is defined as  

$$p_0(\omega_0,k_y)={\int_{{2(-\omega_0^2+v_y^2)} \over {v\mid... ...^2)} \over {v\mid v_y \mid }} dk_h J p^*(\omega_0,k_h,k_y,z=0)}$$ (19)

Equation (18) represents zero-offset downward continuation and imaging as introduced by Gazdag (1978) or Stolt (1978). Equation (19) represents a way of obtaining zero-offset section from constant-offset sections.

The operations needed to obtain the zero-offset stacked section from the constant-offset field described in equation (19) are:

1.

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

2.

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

3.

Multiply by the Jacobian.

4.

Integrate over k_h.

5.

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