REFERENCES

• Biondi, B., and Ronen, J., 1987, Dip moveout in shot profiles: Geophysics, 52, 1473-1482.

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APPENDIX A

The purpose of this appendix is to express $\omega_0=\omega_0(\omega,k_y,k_h)$ and to find the integration limits for $\omega_0$ given the integration limits for $\omega$. Start with the expression for $\omega$:

$$\omega^2 = \omega_0^2 + {{\omega_0^2 v_h^2} \over {\omega_0^2-v_y^2}}$$ (20)

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

$$\omega_0^2 = {1 \over 2} (\omega^2+v_y^2-v_h^2 \pm {\sqrt {(\omega^2+v_y^2-v_h^2)^2-4 \omega^2 v_y^2}})$$ (21)

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 in equation (12), $\Delta$ is always positive and therefore $\omega_0^2$ is real.

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

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

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

$$\begin{array} {lcl} \omega_0^2 & = & \displaystyle{ {1 \over... ... \\ & = & \displaystyle{ v_y^2+ \mid v_h v_y \mid}.\end{array}$$ (22)