Shaping filters and the $\ell^1$-norm

We prove that the $\ell^1$-norm solves the problem highlighted in the preceding section. Now our goal is to estimate one shaping filter ${\bf f}$ that minimizes the objective function  

$$e({\bf f})=\vert{\bf d}-{\bf Mf}\vert _1 .$$ (2)

To achieve this, the shaping filter is estimated iteratively using a nonlinear conjugate gradient solver (NLCG) as described in Claerbout and Fomel (2001). The objective function we actually minimize is  

$$e({\bf f})=\Vert{\bf W}({\bf d}-{\bf Mf})\Vert^2_2,$$ (3)

with

$${\bf W} = {\bf diag} \left( \frac{1}{(1+r_i^2/\epsilon^2)^{1/4}} \right),$$ (4)

where r_i is the residual for one component of the data space, and $\epsilon$a constant we choose a priori. Equation (3) is minimized with the standard iteratively re-weighted least-squares approach (Nichols (1994); Bube and Langan (1997); Guitton (2000)) The objective function in equation (3) amounts to the $\ell^1$ measure when $r_i/\epsilon$ is large and amounts to the $\ell^2$ measure when $r_i/\epsilon << 1$ with a smooth transition between the two.

In Figure 4, we display the result of the adaptive subtraction when the $\ell^1$-norm is utilized to estimate the shaping filter [equation (3) with a small $\epsilon$]. The estimated signal in Figure 4a is perfect, and so is the estimated noise. It is easy to check that the energy in Figure 4a (e=2) is less than the energy in Figure 2a (e=3.2) if we use the $\ell^1$ norm. Figure 5 shows the shaping filter associated with the $\ell^1$-norm. This filter is a spike at lag=0. This simple 1D example demonstrates that the $\ell^1$ should be utilized each time significant amplitude differences exist between multiples and primaries. In the next section, we show another synthetic example where internal multiples are attenuated.

 

1Dl1
Figure 4 (a) The signal estimated with the $\ell^1$-norm. (b) The noise estimated with the $\ell^1$-norm.

 

filterl1 Figure 5 Shaping filter estimated for the 1D problem with the $\ell^1$-norm. This filter is a single spike at lag=0.