Analytical evaluation of the tangent plane to the impulse response

The expression for the generalized impulse response of prestack anisotropic migration leads to the analytical evaluation of the offset dip and midpoint dip along the planes tangent to the impulse response, as a function of the group angles and velocity. In this section I demonstrate that in the simple case of flat reflectors this analysis leads to exactly the same results as the phase-space analysis presented in the previous section. The derivation of the general relationships expressed in equations 13 and 12, which are valid for an arbitrary reflector's dip, is left to the reader.

By applying elementary analytical geometry, I demonstrate in Appendix A that the derivative of the depth with respect to the subsurface offset, at constant midpoint, is given by:  

$$\left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{m... ...{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha_x} },$$ (25)

and the derivative of the depth with respect to the midpoint, at constant subsurface offset, is given by:  

$$\left. \frac{\partial z_\xi}{\partial m_\xi} \right\vert _{h... ...{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha_x} }.$$ (26)

In the special case of flat reflectors the ${\partial z_\xi}/{\partial \alpha_x}$ and ${\partial h_\xi}/{\partial \gamma}$ vanish, and thus equation 25 simplifies into the following expression:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{\lef... ...\widebar L} - \frac{\partial L}{\partial \gamma}\sin \gamma }. \\ \end{eqnarray}$$ (27)

By substituting into equation 27 the appropriate derivative of the image coordinates and of the half path-length with respect to the angles, all provided in Appendix A, I further simplify the expression into the following:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{\lef... ...{ 1 - \frac{1}{S}\frac{\partial S}{\partial \gamma} \tan \gamma }.\end{eqnarray}$$ (28)

Finally, by applying the transformation from group angles into phase angles expressed in equation 4, I obtain the final result that for flat reflectors the subsurface-offset dip is exactly equal to the tangent of the phase aperture angle $\widetilde{\gamma}$;that is:  

$$\left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{\... ...widebar m_\xi, \;\alpha_x=0}\right)} = \tan \widetilde{\gamma}.$$ (29)