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This section illustrates some limitations of the norm
for the estimation of shaping filters.
In figure , a simple 1-D problem is shown.
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 that minimizes
the objective function
| |
(9) |

where is the matrix representing the convolution with
the time series for the multiple model (Figure
b) and the time
series for the data (Figure a).
Now, if the filter is estimated with enough degrees of freedom
(enough coefficients) to minimize equation (),
we obtain for the estimated primaries, i.e., ,
Figure a, and for the estimated multiples, i.e., , Figure b.
The estimated primaries do not resemble the primary in Figure a.
I show the corresponding shaping filter in Figure .
**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
norm. (b) The noise estimated with the norm.

This filter is not an unit spike at *lag*=0 as expected. The problem stems from
the least-squares criterion which yields an estimated signal that has, by definition,
minimum energy. In this 1-D case, the total energy in the estimated
signal (Figure a) is *e*_{2}=2.4, which is less than the
total energy of the primary alone
(*e*_{2}=4). This is a fundamental problem if we use the norm to
estimate the shaping filter. In the next section, I show that the
norm should be used if amplitude differences exist between primaries
and multiples.

** Next:** Shaping filters and the
** Up:** Principles of norm and
** Previous:** Principles of norm and
Stanford Exploration Project

5/5/2005