Derivatives of VTI slowness function with respect to the perturbation parameters

In this Appendix I present the analytical expressions for the derivatives of the group slowness function with respect to the velocity-perturbation parameters $\left(\rho_V_V,\rho_V_H,\rho_V_N\right)$.These derivatives depend on the particular form chosen to approximate the slowness function. In this paper I use following approximation of the VTI slowness function Fowler (2003):

$$\begin{eqnarray} S^2_{\rm VTI}\left(\theta\right) &=& \frac { {S_V}^2\cos^2 \the... ...ht) + {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta } } {2},\end{eqnarray}$$ (36)

where

$$S^2_{\rm Ell}\left(\theta\right) = {S_V}^2\cos^2 \theta+ {S_H}^2\sin^2 \theta$$ (37)

is the elliptical component.

The derivatives are then written as:

$$\begin{eqnarray} \left. \frac{\partial S_{\rm VTI}\left(\theta\right)}{\partial ... ...\right) + {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta } },\end{eqnarray}$$ (38) (39) (40)

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

		(41)
	(42)