The residual moveout is defined as the difference between the reflector movement at finite aperture angle and the reflector movement at normal incidence. From equation 38 the partial derivative of with respect to is equal to the following expression:
Figure shows ADCIGs when an anisotropic velocity was perturbed by .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 from by applying equation 1, whereas the dashed line was computed by approximating as equal to .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 (). 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. ). 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 (solid lines), and by using in place of (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.