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A matrix operation like
arises whenever we travel from one space to another and back again.
The inverse of this matrix arises when we ask to return from the other
space with no approximations.
In general, can be complicated beyond comprehension,
but we have seen some recurring features.
In some cases this matrix turned out to be a diagonal matrix
which is a scaling function in the physical domain.
With banded matrices, the
matrix is also a banded matrix,
being tridiagonal for operators of both
(22)
and (21).
The banded matrix for the derivative operator (22)
can be thought of as the frequency domain weighting factor .We did not examine for the filter operator,
but if you do,
you will see that the rows (and the columns) of
are the
**autocorrelation**
of the filter.
A filter in the time domain is simply a weighting function
in the frequency domain.
The tridiagonal banded matrix for linearly-interpolated NMO
is somewhat more complicated to understand,
but it somehow represents
the smoothing inherent to the composite process of NMO
followed by adjoint NMO,
so although we may not fully understand it,
we can think of it as some multiplication in the spectral domain
as well as some rescaling in the physical domain.
Since
clusters on the main diagonal,
it never has a ``time-shift'' behavior.

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Stanford Exploration Project

10/21/1998