Integration limits

From now on for simplicity I will use the variables

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

In equation (8), the values of the constant k_z have to be real. Imaginary values of k_z do not satisfy the downward continuation ordinary differential equation:  

$${{\partial^2 P} \over {\partial z^2}}= -k_z^2 P$$ (12)

and have to be excluded. Real values of k_z from equation (9) require both of the following conditions to be satisfied:

$$\left \{ \begin{array} {lcl} \mid \omega \mid & \geq & \mid ... ... \mid \omega \mid & \geq & \mid v_y-v_h \mid\end{array}\right .$$

which can be reduced to the condition  

$$\begin{array} {lcl} \mid \omega \mid & \geq & \mid v_y \mid + \mid v_h \mid\end{array}$$ (13)

Therefore, equation (8) should be written  

$$\begin{array} {lcl} p(t=0,h=0,k_y,z) & = & \displaystyle{ {\... ...a e^{ik_z(\omega,k_h,k_y)z} p(\omega,k_h,k_y,z=0)}}\end{array}$$ (14)

For each value of v_y the integration in $\omega$ is done in two regions: under the value $\omega=-(\mid v_h \mid + \mid v_y \mid )$ and over the value $\omega=(\mid v_h \mid + \mid v_y \mid )$ as displayed in Figure .

 

khkyomega
Figure 2 Regions of integration.

Before substituting the new variable $\omega_0$ in the prestack migration equation let us analyze the limits of integration for the new variable $\omega_0$.From Appendix A, the expression of $\omega_0(\omega,k_h,k_y)$ is

$$\omega_0^2 = {1 \over 2} \left [{ \omega^2+v_y^2-v_h^2 + {\sqrt {(\omega^2+v_y^2-v_h^2)^2-4 \omega^2 v_y^2}}} \right ]$$ (15)

and the boundary values for $\omega_0$ corresponding to the limits for $\omega$ in equation (13):

$$\begin{array} {lcl} \mid \omega_0 \mid & \geq & \sqrt{v_y^2+ \mid v_y v_h \mid}\end{array}.$$