Shaping filters and the -norm

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Shaping filters and the $\ell^2$ -norm

h This section illustrates some limitations of the $\ell^2$ -norm for the estimation of shaping filters. In figure 1, we display a very simple 1D problem. On the top we have four events corresponding to one primary (on the left) and three multiples (on the right). Note that the primary has higher amplitude than the multiples. On the bottom we show a multiple model that exactly corresponds to the real multiples. Our goal is to estimate one shaping filter ${\bf f}$ that minimizes the objective function

$\begin{displaymath} e({\bf f})=\Vert{\bf d}-{\bf Mf}\Vert^2_2 ,\end{displaymath}$

(1)

where ${\bf M}$ is the matrix representing the convolution with the time series for the multiple model (Figure 1b) and ${\bf d}$ the time series for the data (Figure 1a).

Now, if we estimate the filter ${\bf f}$ with enough degrees of freedom (enough coefficients) to minimize equation (1), we obtain for the signal Figure 2a, and for the noise Figure 2b. The estimated signal does not resemble the primary in Figure 1a. We show the corresponding shaping filter in Figure 3. This filter is not the single spike at lag=0 that we desire. The problem stems from the least-squares criterion which yields an estimated signal that has, by definition, minimum energy. In this 1D case, the total energy in the estimated signal (Figure 2a) is e=2.4, which is less than the total energy of the primary alone (e=4). This is the fundamental problem if we use the $\ell^2$ -norm to estimate the shaping filter. In the next section, we show that the $\ell^1$ norm should be used if amplitude differences exist between primaries and multiples.

datmul
Figure 1 (a) The data with one primary on the left, and three multiples on the right. (b) The multiple model that we want to adaptively subtract from (a).

1Dl2
Figure 2 (a) The signal estimated with the $\ell^2$ -norm. (b) The noise estimated with the $\ell^2$ -norm.

filterl2
Figure 3 Shaping filter estimated for the 1D problem with the $\ell^2$ -norm. This filter is not a single spike at lag=0.

Next: Shaping filters and the Up: A simple 1D problem Previous: A simple 1D problem

Stanford Exploration Project
6/7/2002