Estimation of the horizontal and NMO velocities

Since the vertical velocities cannot be estimated from RMO transformations in anisotropic ADCIGs, we computed the semblance values for $\rho_V_H$ and $\rho_V_N$only. Figure  illustrates the velocity spectra computed at the image depth for the various ADCIGs illustrated in Figure . The various semblance panels are computed for various aperture angle ranges ($30^{\circ}$, $45^{\circ}$ and $60^{\circ}$). When computing the semblance of a velocity model, the first-order derivatives in equation 5 are computed around that velocity model, not around the one that was used for the migration. The group aperture angles are computed from phase aperture angles using equation 11. The parameterization of the estimated anisotropic parameters is done with the perturbations in the horizontal ($\rho_V_H$) and NMO velocities ($\rho_V_N$). For visualization purposes, Figure  illustrates the same semblance panels, but this time, the axes $\rho_V_H$ and $\rho_V_N$have been normalized by the correct perturbations in the anisotropic migration velocities. As a consequence, in Figure , the true velocities lie in the center of the semblance panels. Several conclusions can be drawn from Figure .

• Effect of the aperture angle: As the range of the aperture angle increases, so does the residual moveout and the information about the true velocity model. As a consequence, the estimates of the anisotropic parameters become more accurate. Furthermore, as the range of the aperture angle increases, the relative constraints on the horizontal and NMO velocities become more even.
• Accuracy of the estimation method: For aperture angles around $30^{\circ}$, Figure  illustrates that the velocity perturbations are unresolved (especially $\rho_V_H$). For aperture angles larger than 45$^{\circ}$, the largest semblance values are found close to the center of the semblance panels, demonstrating the accuracy of the estimation method. However, the largest semblance values are not exactly centered, but slightly shifted toward the upper left, indicating that the anisotropic migration velocities tend to be underestimated. This is consistent with Figure  that illustrates that the predicted RMO functions underestimate the absolute value of the true RMOs.
• Convergence of the estimation method: Table 1 displays the values of $\rho_V_H-1$ and $\rho_V_N-1$ for the maximum semblance velocity models, as a function of the aperture angle range. These quantities are equivalent to the percentage error in V_H and V_N estimates. Table 1 indicates that starting from an isotropic model or a uniform perturbation of $\rho_V=0.9$, the estimation method improves the estimates of V_H and V_N by reducing the percentage errors to less than 2%. However, it also shows that for a velocity model uniformly perturbed by $\rho_V=0.99$, the estimation method may neither converge nor improve the accuracy of the migration velocity model since for a $60^{\circ}$ aperture angle range, the estimation error is still of the order of 1% (i.e. the starting velocity model error).
• Trade-off between the anisotropic migration velocities: Figure  shows a negative trade-off between $\rho_V_H$ and $\rho_V_N$, because a perturbation in V_H can be balanced by a perturbation of the opposite sign in V_N, and result in a very small net change in the predicted RMO function. This negative trade-off has an important practical consequence: it justifies the joint-estimation of $\rho_V_H$ and $\rho_V_N$. The successive estimation of $\rho_V_H$ and $\rho_V_N$ may lead to a much poorer convergence rate.

 

Comb-VelSpec-group
Figure 3 Velocity spectra obtained at the image depth when data modeled with a constant anisotropic velocity model (Taylor Sand) have been migrated using: a) a velocity uniformly perturbed by $\rho_V=0.99$, b) a velocity uniformly perturbed by $\rho_V=0.9$, and c) an isotropic velocity with the correct vertical velocity. The various semblance panels are computed for various aperture angle ranges ($30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ from left to right). The parameterization of the estimated anisotropic velocity model is done with the perturbations in the horizontal ($\rho_V_H$) and in the NMO velocity ($\rho_V_N$). The correct perturbation values are: a) $\rho_V_H=0.99$ and $\rho_V_N=0.99$, b) $\rho_V_H=0.9$ and $\rho_V_N=0.9$, and c) $\rho_V_H=0.905$ and $\rho_V_N=1.037$.

 

Comb-VelSpec-group_centrd
Figure 4 Velocity spectra obtained at the image depth when data modeled with a constant anisotropic velocity model (Taylor Sand) have been migrated using: a) a velocity uniformly perturbed by $\rho_V=0.99$, b) a velocity uniformly perturbed by $\rho_V=0.9$, and c) an isotropic velocity with the correct vertical velocity. The various semblance panels are computed for various ranges of aperture angles ($30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ from left to right). The parameterization of the estimated anisotropic velocity model is done with $\rho_V_H$ and $\rho_V_N$ by the correct perturbation values.

 

Velocity model perturbation Initial perturbation 3c|Range of aperture angles

[0,30] [0,45] [0,60]

$$\% \left(-1.0\%,-1.0\% \right) \left(-3.5\%,+5.0\% \right) \left(-2.5\%,+2.0\% \right) \left(-1.0\%,+0.5\% \right)$$

$$\% \left(-10.0\%,-10.0\% \right) \left(-4.5\%,+7.0\% \right) \left(-3.0\%,+0.0\% \right) \left(-1.5\%,-1.5\% \right)$$

$$\left(-9.5\%,+3.7\% \right) \left(+1.5\%,-3.5\% \right) \left(-2.0\%,+1.5\% \right) \left(+0.5\%,+0.0\% \right)$$ Isotropic model with right V_V