Poststack land data multiple removal example

In this section we attenuate in the poststack domain surface-related multiples with shaping filters that we estimate with the $\ell^2$- and $\ell^1$-norm. These filters are non-stationary. Figure 10a shows the multiple-infested data. Figure 10b displays the multiple model computed with the Delft modeling approach Kelamis et al. (1999). Note that for this gather, the amplitude differences between the primaries and the multiples are not very strong. Our 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. We specifically focus on the event at 1.6s in Figure 10. This event is a primary that we want to preserve during the subtraction.

Figures 12 displays the estimated signal when the non-stationary shaping filters are computed with the $\ell^2$ and $\ell^1$-norm. The amplitude of the primary at 1.6s is well preserved with the $\ell^1$-norm in Figure 11a. However, the amplitude of this primary is attenuated with the $\ell^2$-norm as displayed in Figure 11b. Figure 10 shows a comparison between the subtracted noise with the $\ell^1$ (Figure 10a) and the $\ell^2$-norm (Figure 10b). We conclude that the $\ell^2$-norm tends to subtract too much energy.

This last example proves that the estimation of shaping filters can always be done with the $\ell^1$-norm. The good thing about our inversion scheme and the objective function in equation (3) is that only one parameter ($\epsilon$) controls the $\ell^1-\ell^2$ behavior. Thus we can decide to switch from one norm to another very easily. In Figure 13, I show a histogram of the input data and of the estimated noise with the $\ell^1$ and $\ell^2$-norms. The theory predicts that the distribution of the $\ell^2$ result should be gaussian and that distribution of the $\ell^1$ result should be exponential. Figure 13 corroborates this.

 

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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.

 

hist7636 Figure 13 Histograms of the input data and of the estimated noise with the $\ell^1$- and $\ell^2$-norms. As predicted by the theory, the density function with the $\ell^1$-norm is much narrower than with the $\ell^2$-norm.