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Finally, let us look at the helix from the view of matrices
and numerical analysis.
This is not easy because the matrices are so large.
Discretize the (x,y)-plane to an array
and pack the array into a vector of components.
Likewise pack the Laplacian operator into a matrix.
For a plane, that matrix is shown in equation ().
The two-dimensional matrix of coefficients for the Laplacian operator
is shown in (),
where,
on a Cartesian space, h=0,
and in the helix geometry, h=-1.
(A similar partitioned matrix arises from packing
a cylindrical surface into a array.)
Notice that the partitioning becomes transparent for the helix, h=-1.
With the partitioning thus invisible, the matrix
simply represents one-dimensional convolution
and we have an alternative analytical approach,
one-dimensional Fourier Transform.
We often need to solve sets of simultaneous equations
with a matrix similar to ().
The method we use is triangular factorization.
Although the autocorrelation has mostly zero values,
the factored autocorrelation from ()
has a great number of nonzero terms,
but fortunately they seem to be converging rapidly (in the middle)
so truncation (of the middle coefficients) seems reasonable.
I wish I could show you a larger matrix, but all I can do is to pack
the signal into shifted columns of
a lower triangular matrix like this:
If you will allow me some truncation approximations,
I now claim that the laplacian represented by the
matrix in equation ()
is factored into two parts
which are upper and lower triangular matrices
whose product forms the autocorrelation seen in ().
Recall that triangular matrices
allow quick solutions of simultaneous equations by backsubstitution.
That is what we do with our
deconvolution program.
Next: GEOESTIMATION: EMPTY BIN EXAMPLE
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Stanford Exploration Project
7/5/1998