Synthetic-data examples of RMO function in ADCIGs

To verify the accuracy of the RMO functions derived in this section I performed several numerical tests using synthetic data modeled and migrated using an anisotropic source-receiver migration and modeling program. This program performed depth extrapolation by numerically solving the following dispersion relation:  

$$k_z= \frac{\omega}{V_V} \sqrt{\frac {\omega^2 - {V_H}^2k_x^2} {\omega^2 + \left({V_N}^2-{V_H}^2\right)k_x^2} },$$ (29)

where $\omega$ is the temporal frequency, and k_x and k_z are respectively the horizontal and vertical wavenumbers. This dispersion relation corresponds to the slowness functions in equation 36 Fowler (2003), which was used to compute the RMO functions according to the theory developed above.

To test the theory under realistic and diverse anisotropic conditions, in the numerical examples I used three sets of anisotropic Thomsen parameters representing three different rocks described by Tsvankin (2001):  

 $$\begin{array} {l} \bullet\;{\rm Taylor\;Sand:}\;\;\epsilon=0... ...n=0.0975,\;\;\delta=-0.11,\;\;\rightarrow \eta=.266.\end{array}$$

The GreenLight River Shale is derived from the Green River Shale described by Tsvankin (2001) by halving the anisotropic parameters ($\epsilon$ and $\delta$), because the strong anelliptic nature of the original one ($\eta=.74$) causes the group-slowness approximation in equation 36 to break down. Consequently, the kinematic computations based on ray tracing, and thus on group velocity and angles, become inconsistent with wavefield migration based on the dispersion relation in equation 29. Notice that the GreenLight River Shale is still the most anelliptic among the set of rocks I am using.

The first set of numerical experiments tests the RMO equation with uniform scaling of velocity expressed in equation 20. In addition to the three anisotropic cases described above, this RMO function is tested also for the special case of isotropic velocity. The second set tests the generalized RMO functions expressed in equations 24-26. Only the three anisotropic cases are tested because there is no meaningful isotropic case to test the generalized RMO function. In all the synthetic-data examples I plot the correct RMO curve computed by applying either equation 18 or equations 24-26, and the approximate RMO curve computed using an ``isotropic'' approximation and ignoring the distinction between the group aperture angle $\gamma$and the phase aperture angle $\widetilde{\gamma}$.

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 20. The solid line was computed by computing $\tan \gamma$ from $\tan \widetilde{\gamma}$by applying equation 31, 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 anelliptic 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 20 is based on a linearization, and thus when 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 $\rho_V=.9$.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 20. 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 5 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 20. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 31, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.

 

Quad_Aniso-rho.9_overn
Figure 6 ADCIGs obtained when a constant anisotropic velocity was perturbed by $\rho_V=.9$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 20. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 31, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.

Figures  and  show examples of the application of the generalized RMO functions expressed in equations 24-26. As in Figures -, I show the ADCIGs for three different anisotropic rock types, but, differently from the previous figures, not for the isotropic case. The order of the rock types is the same as in Figures -; that is: panels a) correspond to Taylor Sand, panels b) to Mesa Clay Shale, and panels c) to GreenLight River Shale. Furthermore, as in Figures -, one figure (Figure ) shows the ADCIG obtained with a smaller perturbation than the ADCIGs shown in the other figure (Figure ). The ADCIGs shown in Figure  were obtained by performing isotropic migration on the synthetic data modeled assuming anisotropic velocity. The ADCIGs shown in Figure  were computed by scaling by .25 the parameter perturbations used to compute Figure . 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).

The predicted RMO functions accurately track the actual RMO functions when the parameter perturbations are sufficiently small to be within the range of accuracy of the linearization at the basis of the derivation of equation 20 (Figure ). But even when the perturbations are large (Figure ) and cause a substantial RMO (up to 30% of the reflector depth) the predicted RMO functions are excellent approximations of the actual RMO functions.

The RMO functions associated with the two strongly anelliptic rocks (Taylor Sand and GreenLight River Shale) exhibit a characteristic oscillatory behavior; the events at narrow-aperture angles are imaged deeper than the normal incidence event, whereas the events at wide-aperture angles are imaged shallower. This oscillatory behavior is well predicted by the analytical RMO function introduced in equations 24-26.

In contrast, the approximation of the group angles with the phase angles (dashed lines in the figures) seriously deteriorates the accuracy of the predicted RMO functions. Notice that, in contrast with the uniform-perturbation case illustrated in Figures - , the dashed lines are different among the panels, because the derivatives of the slowness function with respect to the perturbation parameters depend on the anisotropic parameters of the background medium.

 

Trio_Aniso-iso_overn
Figure 7 ADCIGs obtained when data modeled with an anisotropic velocity have been migrated using an isotropic velocity. The anisotropic data were modeled assuming three rock types: a) Taylor Sand, b) Mesa Clay Shale, and c) GreenLight River Shale. Superimposed onto the images are the RMO functions computed using equation 20. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 31, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.

 

Trio_Aniso-scaled_overn
Figure 8 ADCIGs obtained when data modeled with an anisotropic velocity have been migrated using a less anisotropic velocity; that is, with anisotropic parameters obtained by scaling by .25 the parameter perturbations used to compute Figure . The anisotropic data were modeled assuming three rock types: a) Taylor Sand, b) Mesa Clay Shale, and c) GreenLight River Shale. Superimposed onto the images are the RMO functions computed using equation 20. The solid line was computed when $\tan \gamma$ was derived from $\tan \widetilde{\gamma}$by applying equation 31, whereas the dashed line was computed by approximating $\tan \gamma$ as equal to $\tan \widetilde{\gamma}$.