Application to the isotropic case

The application to the isotropic case is simpler than the anisotropic case because the derivative of the path length is zero, but it is instructive since it verifies known results through a completely different derivation. Substituting equations 51 into equation 49, I obtain:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{m_\x... ...n ^2\alpha_x\tan ^2 \gamma \right] } \nonumber \\ &=& \tan \gamma,\end{eqnarray}$$ (54)

which shows that ${\partial z_\xi}/{\partial h_\xi}$is independent from the dip angle $\alpha_x$.This expression is consistent with the 2-D analysis by Sava and Fomel (2003) and the 3-D analysis by by Biondi and Tisserant (2004).