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deconvolution
For solving the unknowninput problem,
we put the known filter f_{t} in a matrix of downshifted columns .Our statement of wishes is now to find x_{t} so that
.We can expect to have trouble finding unknown inputs x_{t}
when we are dealing with certain kinds of filters,
such as bandpass filters.
If the output is zero in a frequency band,
we will never be able to find the input in that band
and will need to prevent x_{t} from diverging there.
We do this by the statement that we wish
,where is a parameter that is small
and whose exact size will be chosen by experimentation.
Putting both wishes into a single, partitioned matrix equation gives
 
(37) 
To minimize the residuals and ,we can minimize the scalar
.This is
 

 (38) 
We solved this minimization
in the frequency domain
(beginning from equation (4)).
Formally the solution is found just as with equation (35),
but this solution looks unappealing in practice
because there are so many unknowns and because
the problem can be solved much more quickly
in the Fourier domain.
To motivate ourselves to solve this problem in the time domain,
we need either to find an approximate solution method that is
much faster, or to discover that
constraints or timevariable weighting functions
are required in some applications.
This is an issue we must be continuously alert to,
whether the cost of a method is justified by its need.
Next: KRYLOV SUBSPACE ITERATIVE METHODS
Up: From the frequency domain
Previous: Unknown filter
Stanford Exploration Project
4/27/2004