Estimation of the vertical velocity

First, we show that for flat reflectors and VTI media, vertical velocities cannot be estimated from reflection seismic data only. These results are consistent with Tsvankin (2001), who shows that P-wave signatures depend only on two combinations of V_V, $\delta$ and $\epsilon$, and that vertical velocities cannot be estimated from reflection seismic data only.

For elliptical media, we demonstrate that the first-order derivative of the image-point depth with respect to vertical velocity perturbations is independent of the aperture angle so that the residual moveout is independent of the vertical velocity perturbations:

$$\begin{eqnarray} \frac{\partial z_{\widetilde{\gamma}}}{\partial \rho_V_V} = -z_\xi,\end{eqnarray}$$ (8)

$$\begin{eqnarray} \frac{\partial \Delta z_{\rm RMO}}{\partial \rho_V_V} = \left. ... ...amma}}}{\partial \rho_V_V} \right\vert _{\widetilde{\gamma}=0} =0.\end{eqnarray}$$ (9)

As a consequence, vertical velocity perturbations cannot be estimated from RMO transformation.

For anelliptical media, we show numerically that the first-order derivative of the RMO with respect to vertical velocity perturbations is not significantly different from zero and does not allow the estimation of vertical velocity perturbations.