Discussion

The overall results may seem disappointing: the simple stacking of the input data give similar results to the stacking of multiple-free gathers using l², l¹ and l¹ with l¹ 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 l¹ norm and the l² norm behave similarly. The l¹ norm with l¹ 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

$$f(\bf{m})=\Vert\bf{R}(\bf{Hm}-\bf{d})\Vert$$

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

$$f(\bf{m})=\vert F.T_{2-D}(\bf{Hm}-\bf{d})\vert _{Huber}$$

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 l¹ norm with l¹ 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 l² norm.

 

comp_stack_L2
Figure 11 Difference between the stacked section with multiples and the stacked section without multiples using the l² norm.

 

stack_L1
Figure 12 Stacked section after multiple suppression using the l¹ norm.

 

comp_stack_L1
Figure 13 Difference between the stacked section with multiples and the stacked section without multiples using the l¹ norm.

 

stack_L1L1
Figure 14 Stacked section after multiple suppression using the l¹ norm with l¹ regularization.

 

comp_stack_L1L1
Figure 15 Difference between the stacked section with multiples and the stacked section without multiples using the l¹ norm with l¹ regularization.