Caveats in the computation of velocity spectra

Eventually, we illustrate the caveats in the computation of anisotropic velocity spectra and demonstrate that as few approximations as possible should be made in order to attain accurate anisotropic parameter estimates. As equation 5 shows, the predicted RMO is a function of both phase and group aperture angles. Phase aperture angles are computed by applying post-processing slant-stacks on the prestack migrated image (equation 3). Group aperture angles have to be derived from phase aperture angles. Figure  illustrates the velocity spectra computed when approximating the group aperture angles with the phase aperture angles. It shows the inaccuracy of the estimates of the anisotropic parameters for large velocity perturbations. Figure  illustrates the velocity spectra computed when the first-order derivatives in equation 5 are computed around the migration velocity model and not the velocity model whose semblance we compute. The group aperture angles are computed from phase aperture angles using equation 11. Figure  illustrates the inaccuracy of the anisotropic parameter estimates for large velocity perturbations. It indicates that when computing velocity model semblances, first-order derivatives of the RMO functions have to be estimated independently for each velocity model.

 

Comb-VelSpec-phase_centrd
Figure 5 Velocity spectra obtained when the group aperture angles are approximated as equal to the phase aperture angles. The data have been modeled with a constant anisotropic velocity model (Taylor Sand) and then 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 different semblance panels are computed for different 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 $\rho_V_N$ normalized by the correct perturbation values.

 

Comb-VelSpec-group_approx_centrd
Figure 6 Velocity spectra obtained when the first-order derivatives in equation 5 are computed around the migration velocity model. The group aperture angles are computed from phase aperture angles using equation 11. The data have been modeled with a constant anisotropic velocity model (Taylor Sand) and then 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 different semblance panels are computed for different 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 $\rho_V_N$, normalized by the correct perturbation values.