Uniform scaling of velocity

The derivative with respect to the perturbation component has the following particularly simple form:  

$$\frac{\partial z_\gamma}{\partial \rho_V} = z_\xi \left( 1 + \tan \gamma\tan \widetilde{\gamma} \right),$$ (37)

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

$$\frac{\partial S\left(\gamma\right)}{\partial \rho_V} = -S\left(\gamma\right),$$ (38)

that leads to the derivative ${\partial L}/{\partial \rho_V}$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 38 the partial derivative of $\Delta z_{\rm RMO}$ with respect to is equal to the following expression:  

$$\frac{\partial \Delta z_{\rm RMO}}{\partial \rho_V} = z_\xi \tan \gamma\tan \widetilde{\gamma}.$$ (39)

When the medium is isotropic, and the phase angles are equal to the group angles, the RMO expression in equation 40 becomes the RMO expression introduced by Biondi and Symes (2003). The dependency of equation 40 from the group angles makes its use in RMO analysis somewhat less convenient, because it requires the transformation of phase angles (measured directly from the image) into group angles by applying equation 1. The computational cost of evaluating equation 1 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 40.

Figure  shows ADCIGs when an anisotropic velocity was perturbed by $\rho_V=.99$.The four panels correspond to four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale. Superimposed onto the images are the RMO functions computed using equation 40. The solid line was computed by computing $\tan \gamma$ from $\tan \widetilde{\gamma}$by applying equation 1, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.The RMO curves computed using the correct group angle perfectly match the residual moveout of the images. On the contrary, when the phase angles are used instead of the group angles, significant errors are introduced even for such a small perturbation in the parameters ($\rho_V=.99$). It is interesting to notice that the errors are larger for the rock types exhibiting strong unelliptical anisotropy (Taylors Sand and GreenLight River Shale) than for the strongly anisotropic but quasi-elliptical rock (Mesa Clay Shale).

The expression for the RMO function derived in equation 40 is based on a linearization, and thus when the the perturbations in velocity parameters are large it is not as accurate as it is when the perturbations are small (e.g. $\rho_V=.99$). Figure  illustrates this fact by showing a similar experiment as the one shown in Figure , but with a perturbation 10 times larger; that is, with .As in Figure , the four panels correspond to four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale, and the lines superimposed onto the images are the RMO functions computed by using the correct values for $\tan \gamma$ (solid lines), and by using $\tan \widetilde{\gamma}$ in place of $\tan \gamma$ (dashed lines). With large perturbations, the predicted RMO functions differ from the actual RMO functions at wide aperture angles even when the correct values of the group angles are used in equation 40. However, even with such large perturbations the predicted RMO functions are still useful approximations of the actual RMO functions. In particular, it can be observed that the predicted RMO function correctly approximates the differences in shape of the actual RMO function among the rock types. These shape variations are related to the variations in shape of the wavefronts, which are reflected in the predicted RMO function through the variations in the mapping from phase angles to group angles.

 

Quad_Aniso-rho.99_overn
Figure 7 ADCIGs obtained when a constant anisotropic velocity was perturbed by $\rho_V=.99$for four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale. Superimposed onto the images are the RMO functions computed using equation 40. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 1, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.

 

Quad_Aniso-rho.9_overn
Figure 8 ADCIGs obtained when a constant anisotropic velocity was perturbed by for four rock types: a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale, and d) GreenLight River Shale. Superimposed onto the images are the RMO functions computed using equation 40. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 1, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.