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## Shaping filters and the -norm

We prove that the -norm solves the problem highlighted in the preceding section. Now our goal is to estimate one shaping filter that minimizes the objective function
 (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
 (3)
with
 (4)
where ri is the residual for one component of the data space, and 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 measure when is large and amounts to the measure when with a smooth transition between the two.

In Figure 4, we display the result of the adaptive subtraction when the -norm is utilized to estimate the shaping filter [equation (3) with a small ]. 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 norm. Figure 5 shows the shaping filter associated with the -norm. This filter is a spike at lag=0. This simple 1D example demonstrates that the 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 -norm. (b) The noise estimated with the -norm.

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

Next: Attenuation of internal multiples Up: A simple 1D problem Previous: Shaping filters and the
Stanford Exploration Project
6/7/2002