** Next:** Angle domain results
** Up:** Migration results
** Previous:** Migration results

I first show in Figure a comparison between the
migration result of the Marmousi dataset () and the
remodeled data (). We notice that the migration of the
remodeled data (Figure b) lowers the amplitudes in the
upper part of the model. Therefore, we expect the filters to correct
for this difference. Figure displays few
estimated filters for the Marmousi result. The filters are ten by ten
with 40 patches in depth and 80 along the midpoint axis. I shown only a fifth of these
filters in both axes. It is interesting to notice that these filters
have their highest value at zero-lag, meaning that we have a strong
amplitude correction with few kinematic changes. The zero-lag values
are also larger at the top of the model, as anticipated. Looking more
closely at these filters, we see that the coefficients follow the
structure of the Marmousi model (upper right corner).
Having estimated the filters in equation (12), I
apply them to to obtain an improved image.
To validate this approach I show in Figure a the
result of five conjugate gradient (CG) iterations with the Marmousi
data. This results show higher amplitudes at the top but with
inversion artifacts. This problem should be addressed with a
proper regularization scheme Prucha et al. (1999).
In Figure b, I show the corrected image with
the approximated Hessian . The amplitude behavior is very
similar to Figure a, without the inversion
artifacts. Additionally, the cost is much lower.

I show in Figure the ratio of the envelope of Figure
b and a.
This Figure illustrates that the effects of the non-stationary
filters, i.e, the Hessian, are stronger on the top of the model.

**m1.m2
**

Figure 1 (a) Migration result of
the Marmousi dataset, i.e., in equation (9).
(b) Migration result of the remodeled data, i.e.,
in equation (9).

**filter-minv0
**

Figure 2 Each cell represents
a non-stationary filter with its zero-lag coefficient in the
middle. A fifth of the filters are actually shown in both
directions. Each filter position corresponds roughly to a similar area
in the model space (Figure a). After close
inspection of the filter coefficients, these filters seem to
follow the structure of the Marmousi model. They are also stronger at
the top of the model, as expected.

**inv.ampcorr
**

Figure 3 (a) Model estimated
after five iterations of CG. The model is noisy because no
regularization has been applied. (b) Model estimated after applying
the adaptive filters to . The amplitude behavior is
similar to (a) without the artifacts and with fewer iterations.

**amplitude0
**

Figure 4 Ratio of the envelopes
of Figures a and b. Brighter
colors correspond to higher values. The main effect
of the filters is clearly visible at the top.

** Next:** Angle domain results
** Up:** Migration results
** Previous:** Migration results
Stanford Exploration Project

7/8/2003