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The strong primary in Figure can be seen as an outlier
that attracts much of the solver's efforts during the filter estimation.
filterl2
Figure 3 Shaping filter estimated for the
1D problem with the norm. This filter is not a single
spike at lag=0.

 
filterl1
Figure 4 Shaping filter estimated for the
1D problem with the norm. This filter is a single spike at
lag=0.

 
Consequently, some of the signal leaks into the
noise. Because the norm is robust to outliers, I propose
estimating the filter coefficients with it. This insensitivity to large
``noise'' has a statistical interpretation: robust measures are
related to longtailed density functions in the same way that is related to the shorttailed gaussian density function Tarantola (1987).
In this section, I show 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
 
(10) 
The function in equation () is singular where any residual
component vanishes implying that the derivative of is not
continuous everywhere. Unfortunately, most of our optimization
techniques, e.g., conjugategradient or Newton methods,
assume that the first derivative of the objective function is
continuous in order to find its minimum. Therefore, specialized
techniques have been developed to either minimize or approximate the
norm.
For instance, various approaches based on a linear programming viewpoint have been used
with success, e.g., Barrodale and Roberts
1980.
Another idea is to minimize a hybrid norm
with an iteratively reweighted leastsquares (IRLS) method
Bube and Langan (1997); Gersztenkorn et al. (1986); Scales and Gersztenkorn (1988).
Alternatively, the Huber norm is utilized.
This technique yields a good approximation of the norm.
Therefore, the objective function we minimize becomes
 
(11) 
The minimization of the Huber norm is performed with a quasiNewton
method (Appendix ). The only parameter that
needs to be set is in equation (). Consistent with
the strategy of Chapter , I set
 
(12) 
From now on, I call norm the Huber norm. In text section, I
show that the norm leads to the desired result for
the filter estimation problem.
Next: Results on a simple
Up: Principles of norm and
Previous: Shaping filters and the
Stanford Exploration Project
5/5/2005