Since the vertical velocities cannot be estimated from RMO
transformations in anisotropic ADCIGs, we computed the semblance
values for and 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 (,
and ). 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 () and NMO velocities (). For visualization purposes,
Figure illustrates the
same semblance panels, but this time, the axes and 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 , Figure illustrates that the
velocity perturbations are unresolved (especially ).
For aperture angles larger than 45, 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 and
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
, 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 , the estimation
method may neither converge nor improve the accuracy of the
migration velocity model since for a 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 and , 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 and . The successive
estimation of and 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 , b) a velocity
uniformly perturbed by
, and c) an isotropic velocity with the correct
vertical velocity. The various semblance panels are computed for
various aperture angle ranges (, and from left to right).
The parameterization of the estimated anisotropic
velocity model is done with the perturbations in the horizontal
() and in the NMO velocity (). The correct perturbation values are: a)
and , b) and
, and c) and .
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 , b) a velocity uniformly perturbed by
, and c) an isotropic velocity with the correct
vertical velocity. The various semblance panels are computed for
various ranges of aperture angles (, and
from left to right). The parameterization of the
estimated anisotropic velocity model is done with and
by the correct perturbation values.
Table 1:
Values of for the maximum semblance velocity
model, as a function of the aperture angle range.