Poststack land data multiple removal example

In this section I attenuate in the poststack domain surface-related multiples with shaping filters that I estimate with the $\ell^2$ and $\ell^1$ norm. These filters are non-stationary. Figure a shows the multiple-infested data. Figure b displays the multiple model computed with a data-driven modeling approach Kelamis and Verschuur (2000). Note that for this gather, the amplitude differences between the primaries and the multiples are not very strong. My goal is to illustrate the use of the $\ell^1$ norm in a more general case when surface-related multiples are present in the data. I specifically focus on the event at 1.6s in Figure a. This event is a primary that needs to be preserved by the subtraction procedure.

The amplitude of the primary at 1.6s is well preserved with the $\ell^1$ norm in Figure a. Again, $\alpha=max\vert{\bf d}\vert/1000$ gave a satisfying result. However, the amplitude of this primary is attenuated with the $\ell^2$ norm as displayed in Figure b. Figure shows a comparison between the subtracted multiples with the $\ell^1$ (Figure a) and the $\ell^2$ norm (Figure b). I conclude that the $\ell^2$ norm tends to subtract too much energy.

 

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Figure 10 (a) Stack infested with multiples. (b) The multiple model computed with the Delft modeling approach. The subtraction is done poststack.

 

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Figure 11 (a) The estimated primaries with $\ell^1$ norm adaptive subtraction. (b) The estimated primaries with $\ell^2$ norm subtraction. The primary at 1.6s is very attenuated with the $\ell^2$ norm. The $\ell^1$ technique preserves its amplitude better.

 

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Figure 12 (a) The estimated multiples with the $\ell^1$ norm subtraction. (b) The estimated multiples with the $\ell^2$ norm subtraction. The $\ell^2$ norm tends to over-fit some multiples that creates some leaking of primaries in the estimated noise.

This last example proves that the estimation of shaping filters can always be done with the $\ell^1$ norm. An advantage of the inversion scheme with the Huber norm is that only one parameter (i.e., $\alpha$) controls the $\ell^1-\ell^2$ behavior. Thus it is easy to switch from one norm to another. Of course, it remains difficult to assert if the subtracted multiples with the $\ell^1$ norm are more similar to the actual multiples than the subtracted multiples with the $\ell^2$ norm. This judgment is only based on qualitative considerations for few known primary reflectors that need to be preserved.