Appendix C - Derivatives of VTI slowness function with respect to the perturbation parameters

In this Appendix I present the analytical expressions for the derivatives of the VTI group slowness function expressed in equation 6 in the main text. These derivatives are necessary for the numerical computation of the RMO functions.

The VTI slowness function can be approximated as Fowler (2003):

$$\begin{eqnarray} S^2_{\rm VTI}\left(\theta\right) &=& \frac { {S_V}^2\cos^2 \the... ...ht) + {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta } } {2},\end{eqnarray}$$ (57)

where

$$S^2_{\rm Ell}\left(\theta\right) = {S_V}^2\cos^2 \theta+ {S_H}^2\sin^2 \theta$$ (58)

is the elliptical component.

The derivatives are then written as:

$$\begin{eqnarray} \left. \frac{\partial S_{\rm VTI}\left(\theta\right)}{\partial ... ...\right) + {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta } },\end{eqnarray}$$ (59) (60) (61)

where the derivatives of the elliptical component with respect to $\rho_V_V$ and $\rho_V_H$ are:

		(62)
	(63)