** Next:** Meet the Toeplitz matrix
** Up:** WEIGHTED ERROR FILTERS
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It is a common but questionable practice to apply **AGC** to echo soundings
before filter analysis.
A better practice is first to analyze according to the laws of physics
and only at the last stage to apply gain functions
for purposes of statistical estimation and final display.
Here we will examine correct and approximate ways
of setting up deconvolution problems with gain functions.
Then we will use CG to solve the proper formulation.
Solving problems in the time domain offers
an advantage over the frequency domain
because in the time domain it is easy to control
the interval where the solution should exist.
Another advantage of the time domain arises when
weighting functions are appropriate.
I have noticed that people sometimes use Fourier solutions
inappropriately, forcing themselves to use uniform weighting
when another weighting would work better.
Since we look at echos, it is unavoidable that we apply gain functions.
Weighting is always justified on the process *outputs*,
but it is an approximation of unknown validity on
the data that is *input* to those processes.
I will clarify this approximation by an equation
with two filter points and an output of four time points.
In real-life applications, the output is typically 1000-2000 points
and the filter 5-50 points.
The valid formulation of a filtering problem is

| |
(18) |

The weights *w*_{t} are any positive numbers we choose.
Typically the *w*_{t} are chosen so that the residual components
are about equal in magnitude.
If, instead, the weighting function is applied to the *inputs*,
we have an approximation that is somewhat different:

| |
(19) |

Comparing the weighted output-residual equation (18) to
the weighted input-data equation (19), we note that
their right-hand columns do not match.
The right-hand column
in (18)
is (0, *w*_{2} *x*_{1}, *w*_{3} *x*_{2}, *w*_{4} *x*_{3})'
but
in (19)
is (0, *w*_{1} *x*_{1}, *w*_{2} *x*_{2}, *w*_{3} *x*_{3})'.
The matrix
in (19)
is a simple convolution,
so some fast solution methods are applicable.

** Next:** Meet the Toeplitz matrix
** Up:** WEIGHTED ERROR FILTERS
** Previous:** Automatic gain control
Stanford Exploration Project

10/21/1998