Synthetic-data examples of RMO functions in ADCIGs

To verify the accuracy of the RMO functions derived in this section, we perform several numerical tests using synthetic data modeled and migrated using an anisotropic source-receiver modeling program and its adjoint. As in Biondi (2005b), this program performed depth extrapolation by numerically solving the following dispersion relation:  

$$k_z= \frac{\omega}{V_V} \sqrt{\frac {\omega^2 - {V_H}^2k_x^2} {\omega^2 + \left({V_N}^2-{V_H}^2\right)k_x^2} },$$ (37)

where $\omega$ is the temporal frequency, and k_x and k_z are the horizontal and vertical wavenumbers, respectively. This dispersion relation corresponds to the following slowness function Fowler (2003):

$$\begin{eqnarray} 2 S^2_{\rm VTI}\left(\theta\right) &=& S^2_{\rm Ell}\left(\thet... ...a\right) + {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta } ,\end{eqnarray}$$ (38)

where,

$$\begin{eqnarray} S^2_{\rm Ell}\left(\theta\right) = {S_V}^2\cos^2 \theta+ {S_H}^2\sin^2 \theta,\end{eqnarray}$$ (39)

is the elliptical component.

We tested the theory with the set of anisotropic Thomsen parameters of the Taylor Sand ($\epsilon=0.110, \delta=-0.035 \rightarrow \eta=.155$) described in Tsvankin (2001).

Figures 8 to 10 show examples of the application of the RMO function expressed in equation (30) when perturbing the velocity uniformly ($\rho_V=.95$). In the different examples, we didn't approximate the values of $\gamma$ but computed them from the values of $\widehat{\gamma}$.

Figure 8 shows that for flat reflectors, the RMO function we derived accurately tracks the actual RMO function when the perturbations are sufficiently small to be within the range of accuracy of the linearization. This is consistent with the results presented in Biondi (2005b). The relative lack of accuracy at large aperture angle is due to the limited offset range we used for the modeling and the migration.

The top panel of Figure 9 shows that for a $15^{\circ}$dipping reflector the RMO function we derived accurately tracks the actual RMO function (the relative lack of accuracy at large aperture angle is due to the limited offset range we used). However the bottom panel shows that the use of the RMO function given in Biondi (2005b) (i.e. assuming that the reflector is flat) gives comparable results.

The top panel of Figure 9 shows that for a $30^{\circ}$ dipping reflector, the RMO function we derived perfectly matches the actual RMO (we used a larger offset range). The bottom panel shows that the RMO function given in Biondi (2005b) (i.e. assuming that the reflector is flat) gives poor results. It justifies a posteriori the need for generalizing to dipping reflectors the concepts of quantitatively relating perturbations in anisotropic parameters to the corresponding reflector movements in anisotropic ADCIGs.

 

Rho-95-Alpha-0
Figure 8 ADCIGs obtained for a flat reflector when a constant anisotropic velocity was perturbed by $\rho_V=.95$. Superimposed onto the images is the RMO function computed using the RMO function we derive in this paper (equation (30)).

 

Rho-95-Alpha-15
Figure 9 ADCIGs obtained for a $15^{\circ}$ dipping reflector when a constant anisotropic velocity was perturbed by $\rho_V=.95$. Superimposed onto the images are the RMO functions computed: a) using the RMO function we derived in this paper (equation (30)); b) assuming that the reflector is flat, i.e. using the equation derived in Biondi (2005b).

 

Rho-95-Alpha-30
Figure 10 ADCIGs obtained for a $30^{\circ}$ dipping reflector when a constant anisotropic velocity was perturbed by $\rho_V=.95$. Superimposed onto the images are the RMO functions computed: a) using the RMO function we derived in this paper (equation (30)); b) assuming that the reflector is flat, i.e. using the equation derived in Biondi (2005b)