RMO function with uniform scaling of velocity

In case of uniform scaling of velocity, the derivative of the imaging depth $z_\gamma$with respect to the perturbation component has the following simple form:  

$$\frac{\partial z_\gamma}{\partial \rho_V} = z_\xi \left( 1 + \tan \gamma\tan \widetilde{\gamma} \right),$$ (18)

because the derivative of the slowness with respect to a uniform scaling of the velocity has the following simple form:

$$\frac{\partial S\left(\gamma\right)}{\partial \rho_V} = -S\left(\gamma\right),$$ (19)

that causes the derivative ${\partial L}/{\partial \rho_V}=-z_\xi/\cos \gamma$to be independent from the ``local'' shape of the anisotropic slowness function. Intuitively, this simplification is related to the fact that the ``shape'' of the wavefronts is not affected by a uniform scaling of the velocity.

The residual moveout $\Delta z_{\rm RMO}$ is defined as the difference between the reflector movement at finite aperture angle $\widetilde{\gamma}$and the reflector movement at normal incidence. From equation 18 the partial derivative of $\Delta z_{\rm RMO}$ with respect to is equal to the following expression:  

$$\frac{\partial \Delta z_{\rm RMO}}{\partial \rho_V} = z_\xi \tan \gamma\tan \widetilde{\gamma}.$$ (20)

When the medium is isotropic, and the phase angles are equal to the group angles, the RMO expression in equation 20 becomes the RMO expression introduced by Biondi and Symes (2003).

The dependency of equation 20 from the group angles increases the complexity of its use because it requires the transformation of phase angles (measured directly from the ADCIGs) into group angles by applying equation 31. The computational cost of evaluating equation 31 is negligible, but its use makes the computations dependent on the local values of the background anisotropic velocity function. On the other hand, the following numerical examples show that substantial errors are introduced when the distinction between the group and phase angles is neglected, and the phase angle is used instead of the group angle in equation 20.