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Figure 3 shows the result of the inversion for one CMP. Since
the Mobil AVO dataset does not include very complex structures with strong
velocity contrasts, this panel illustrates what happens for all the gathers.
The left panel shows the input data. The other panels display the reconstructed
data using the different schemes. Note that the *l*^{2} and *l*^{1} inversions give
similar results and that the *l*^{1} regularization doesn't converge as well.
Figure 4 highlights this difference
between the different problems. The best convergence is achieved using least-squares and
the worst is achieved with the *l*^{1} regularization. In my implementation, however, the
*l*^{1} problem with or without regularization was solved using twice as many iterations as
with *l*^{2}. Figure 5 shows the differences between the input data and
the remodeled data. It appears that the *l*^{1} norm with *l*^{1} regularization
encounters some difficulties in fitting the far offset data. Note
that the *l*^{1} norm and the *l*^{2} norm are both comparable. This is expected since
the data are not strongly noisy.
Differences arise in favor of the *l*^{1} regularization when we look at the model space
(Figure 6), however. The *l*^{1} and *l*^{2} results are again very similar and the
*l*^{1} norm with *l*^{1} regularization appears spikier. This result is consistent with the theory
(see Theory section). The spiky result is then used to define the limit between
primaries and multiples (black line in the right panel of Figure 6).
A mask is defined accordingly and the primaries are muted out in the model space.
The next step consists of remodeling the multiples back
in the data space, applying the hyperbola superposition principle (operator **H**).
Figure 7 shows the predicted multiples. Note that for the
three inversion schemes, some primaries remain. This is particularly annoying to us
in our attempt to produce true amplitude multiple-free gathers.
Figure 8 displays the result of the multiple attenuation process.
The three methods display similar results. Nonetheless, at far offset, the
*l*^{1} regularization shows more energetic events. This is consistent with Figure
5 where we showed that the *l*^{1} regularization was unable
to fit this part of the data.

**comp_dat
**

Figure 3 Left: input data. Middle-left: *l*^{2} reconstructed data.
Middle-right: *l*^{1} reconstructed data. Right: *l*^{1} with *l*^{1} regularization reconstructed data.

**residual
**

Figure 4 Comparison of the convergence for different inversion
schemes for one CMP gather.

**diff
**

Figure 5 Left: input data. Middle-left: *l*^{2} residual. Middle-right: *l*^{1} residual. Right: *l*^{1} with *l*^{1} regularization residual.

**comp_scan
**

Figure 6 Left:*l*^{2} model. Middle: *l*^{1} model. Right: *l*^{1} with *l*^{1} regularization. The line shows the limit of the muting process that separates ``guessed''
multiples on the left from ``guessed'' primaries on the right for the spiky model.

**comp_mult
**

Figure 7 Predicted multiples. Left: *l*^{2} multiples. Middle: *l*^{1}
multiples. Right: *l*^{1} with *l*^{1} regularization multiples.

**comp
**

Figure 8 Gathers after multiple attenuation. Left: input data with multiples. Middle-left: *l*^{2} multiple attenuation. Middle-right: *l*^{1} multiple attenuation. Right: *l*^{1} with *l*^{1} regularization multiple attenuation.

** Next:** NMO-Stacking process
** Up:** Marine Data Results
** Previous:** Computing aspects
Stanford Exploration Project

4/27/2000