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Anisotropic residual moveout for flat reflectors

The kinematic formulation of the generalized impulse response presented in the previous section enables a simple analysis of the residual moveout (RMO) in ADCIGs caused by errors in anisotropic velocity parameters. For the sake of simplicity, at the present, I limit my analysis to reflections from flat interfaces. However, a generalization of the flat-events analysis to dipping events should be conceptually straightforward, though not necessarily simple from the analytical point of view.

A VTI velocity function, either group or phase, is described by the following vector of three velocities ${\bf V}=(V_V,V_H,V_N)$,as for example used in equations 5, or by the corresponding vector of three slownesses ${\bf S}=(S_V,S_H,S_N)$used in equation 6. I define the perturbations as one multiplicative factors for each of the velocities and one multiplicative factor for all velocities; that is, the perturbed velocity $_\rho{\bf V}$ is defined as:
_\rho{\bf V}=
\rho_V\left(\rho_V_VV_V,\rho_V_HV_H,\rho_V_NV_N\right).\end{displaymath} (30)
The velocity-parameter perturbations is thus defined by the following four-components vector $\hbox{{<tex2html_image_mark\gt ... = $\left(\rho_V,\rho_V_V,\rho_V_H,\rho_V_N\right)$.

For flat reflectors, the transformation to angle domain maps an image point at coordinates $(z_\xi,h_\xi)$ into an image point with coordinates $(z_\gamma,\widetilde{\gamma})$according to the following mapping:
\frac{\partial z_\xi}{\pa...
 ...\vert _{m_\xi=\widebar m_\xi}=
z_\xi-h_\xi\tan \widetilde{\gamma}.\end{eqnarray} (31)
The partial derivative of the angle-domain depth $z_\gamma$with respect to the i-th component in the perturbation vector can be expressed as follows:
\frac{\partial z_\gamma}{\partial \rho_{i}}
\frac{\partial ...{\gamma}}
\frac{\partial \widetilde{\gamma}}{\partial \rho_{i}}.\end{eqnarray}
In Appendix B I demonstrate that the terms multiplying the partial derivatives with respect to the angles are zero, and equation 34 simplifies into:  
\frac{\partial z_\gamma}{\partial \rho_{i}}
 ...{\partial L}{\partial S} 
\frac{\partial S}{\partial \rho_{i}},\end{displaymath} (34)
\frac{\partial z_\gamma}{\partial L}=
\frac{\partial z_\xi}{...
\cos \gamma+ \sin \gamma\tan \widetilde{\gamma},\end{displaymath} (35)
\frac{\partial L}{\partial S\left(\gamma\right)}=

-\frac{z_\xi}{S\left(\gamma\right)\cos \gamma},\end{displaymath} (36)