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RMO function with arbitrary scaling of velocity

The expressions of the derivative of $z_\gamma$ with respect to arbitrary perturbations of individual velocity components (i.e. VV, VH, and VN) are slightly more complex than with respect to because the wavefronts are deformed when the velocity components are unevenly perturbed. These derivatives can be expressed as:
         \begin{eqnarray}
\frac{\partial z_\gamma}{\partial \rho_V_V}
&=&

-\frac{z_\xi}{...
 ...al \rho_V_N} 
\left(1 + \tan \gamma\tan \widetilde{\gamma}\right).\end{eqnarray} (21)
(22)
(23)

The partial derivatives of the RMO function $\Delta z_{\rm RMO}$are directly derived from the partial derivatives of $z_\gamma$, taking into account that for flat reflectors only the vertical velocity component VV influences the image depth of normal incidence. The derivatives of $\Delta z_{\rm RMO}$ can thus be written as follows:
         \begin{eqnarray}
\frac{\partial \Delta z_{\rm RMO}}{\partial \rho_V_V}
&=&

-\fr...
 ...al \rho_V_N} 
\left(1 + \tan \gamma\tan \widetilde{\gamma}\right).\end{eqnarray} (24)
(25)
(26)

The expressions for the derivatives of the slowness function with respect to the perturbation parameters depend on the particular form chosen to approximate the slowness function. Appendix C presents a particular approximation to the VTI group slowness function and derives the corresponding partial derivatives to be substituted in equations 21-23 and in equations 24-26. I used the same approximation to the VTI group slowness for the numerical experiments shown in this paper.


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Next: Conversion of depth errors Up: Anisotropic residual moveout for Previous: RMO function with uniform
Stanford Exploration Project
11/1/2005