REFERENCES

• Claerbout, J. C., 1985, Imaging the Earth's Interior: Blackwell Scientific Publications.

• Deregowski, S. M., and Rocca, F., 1981, Geometrical optics and wave theory of constant-offset sections in layered media: Geophysical Prospecting, 29, 374-406.

• Gazdag, J., 1978, Wave equation migration with the phase shift method: Geophysics, 43, 1342-1351.

• Hale, I. D., 1983, Dip-moveout by Fourier transform: Ph.D. thesis, Stanford University.

• Hale, I. D., 1988, Dip-moveout processing: Course notes from SEG continuing education course, SEG.

• Popovici, A. M., 1990, Prestack partial migration analysis: SEP-65, 17-28.

• Popovici, A. M., 1992, Dip-Moveout processing: A Tutorial. DMO basics and DMO by Fourier transform: SEP-75, 407-425.

• Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23-48.

• Zhang, L., 1988, A new Jacobian for dip moveout: SEP-59, 201-208.

APPENDIX A

In equation (1) the values of the constant k_z, given by equation (2), have to be real. Imaginary values of k_z do not satisfy the downward continuation ordinary differential equation

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

and have to be excluded. Real values of k_z from equation (2) require the conditions:

$$\begin{array} {lcl} \mid \omega \mid & \geq & \mid v_y+v_h \mid \\ \\ \mid \omega \mid & \geq & \mid v_y-v_h \mid\end{array}$$

which can be reduced to the condition  

$$\begin{array} {lcl} \mid \omega \mid & \geq & \mid v_y \mid + \mid v_h \mid\end{array}.$$ (16)

After the change of variable from $\omega$ to $\omega_0$ in equation (5)

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

we want to express $\omega_0$ function of $\omega$ and determine the integration boundaries for $\omega_0$.We start with the expression for $\omega$:

$$\omega^2 = \omega_0^2 + {{\omega_0^2 v_h^2} \over {\omega_0^2-v_y^2}}$$ (17)

and after reducing

$$\omega^2 \omega_0^2 - \omega^2 v_y^2 = \omega_0^4 - \omega_0^2 v_y^2 + \omega_0^2 v_h^2$$

and grouping

$$\omega_0^4 - \omega_0^2 (\omega^2 + v_y^2 - v_h^2) +\omega^2 v_y^2 = 0$$

we have

$$\omega_0^2 = {1 \over 2} (\omega^2+v_y^2-v_h^2 \pm {\sqrt {(\omega^2+v_y^2-v_h^2)^2-4 \omega^2 v_y^2}}).$$ (18)

The discriminant $\Delta$ is

$$\Delta = (\omega-v_y-v_h)(\omega-v_y+v_h) (\omega+v_y-v_h)(\omega+v_y+v_h).$$

From the conditions on k_z, $\Delta$ is always positive and therefore $\omega_0^2$ is always real within the $\omega$ limits.

The integration limits for $\omega_0$ are found by starting with the limits for $\omega$ :

$$\omega = \mid v_h \mid + \mid v_y \mid$$

and after we square both sides

$$\omega^2 = v_h^2+v_y^2+2\mid v_h v_h \mid$$

and replacing $\omega^2$ in the equation for $\omega_0^2$ we have

$$\begin{array} {lcl} \omega_0^2 & = & \displaystyle{ {1 \over... ...\ \\ & = & \displaystyle{ v_y^2+\mid v_h v_y \mid}.\end{array}$$ (19)

The integration in $\omega_0$ is done from $(-\infty,-\sqrt{v_y^2+\mid v_h v_y \mid})$ and from $(\sqrt{v_y^2+\mid v_h v_y \mid},\infty)$. After changing the order of integration from $\omega_0$ to k_h, the integration boundaries for k_h become

$$k_h \in (-{2 \over v}{{\omega_0^2-v_y^2} \over {\mid v_y \mid}}, {2 \over v}{{\omega_0^2-v_y^2} \over {\mid v_y \mid}}).$$ (20)