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The overall results may seem disappointing: the simple stacking of the input data give
similar results to the stacking of multiple-free gathers using *l*^{2}, *l*^{1} and *l*^{1}
with *l*^{1} regularization. Nonetheless, data with more complex multiples would
certainly lead to different conclusions.
In addition, these data demonstrate that when the gathers are not particularly noisy,
which is the case here, the *l*^{1} norm and the *l*^{2} norm behave similarly.
The *l*^{1} norm with *l*^{1} regularization produces expected sparse velocity panels
with very bad convergence properties, meaning that we need to think about new strategies
to improve it. Claerbout (2000, Personal communication) recently suggested that I minimize
where **R** is the Prediction Error Filter (PEF) of the residual. The PEF
would help to obtain Independent Identically Distributed (IID) variables in
the residual. Another idea is to minimize
where *F*.*T*_{2-D} is the 2-D Fourier Transform. The idea behind this last equation is that
the far offset data, which the *l*^{1} norm with *l*^{1} regularization does not fit very well, creates
(almost) mono-frequency patterns in the residual that would map in a very localized
area of the Fourier space with high amplitudes (two points for a perfectly
mono-frequency event with one slope). The Huber norm with an appropriate threshold
would treat these focused energies as outliers and get rid of them.
**stack_IN
**

Figure 9 Stacked section of the input data with multiples.

**stack_L2
**

Figure 10 Stacked section after multiple suppression
using the *l*^{2} norm.

**comp_stack_L2
**

Figure 11 Difference between the stacked section
with multiples and the stacked section without multiples using the *l*^{2} norm.

**stack_L1
**

Figure 12 Stacked section after multiple suppression
using the *l*^{1} norm.

**comp_stack_L1
**

Figure 13 Difference between the stacked section
with multiples and the stacked section without multiples using the *l*^{1} norm.

**stack_L1L1
**

Figure 14 Stacked section after multiple suppression
using the *l*^{1} norm with *l*^{1} regularization.

**comp_stack_L1L1
**

Figure 15 Difference between the stacked section
with multiples and the stacked section without multiples using the *l*^{1} norm with *l*^{1}
regularization.

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** Previous:** NMO-Stacking process
Stanford Exploration Project

4/27/2000