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![]() | An algorithm for interpolation using Ronen's pyramid | ![]() |
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In seismology we often deal with instruments spaced more widely than they should be, more widely than they should be for typical data processing such as Fourier transform, more widely than is suitable for data display. Fundamentally something is lost, but that does not detract from our goal of regular spaced data on a dense enough mesh. Given regular data, finding a PEF is a linear problem. Given a PEF, interpolating data is a linear problem (demonstrated by many examples in Claerbout's free on-line book, ``Image Estimation by Example''). In both cases we are minimizing the energy of filtered data. A more generalized approach minimizes energy in the filtered data where some unknowns are in the PEF while others are missing data values among the knowns. This minimization is nonlinear (because the PEF multiplies missing data).
The main difficulty of trying to utilize Ronen's pyramid in practice
is the issue of bringing the -space to the
-space.
On first glance it seems to require interpolation.
On trial, this interpolation seems to need to be done extremely carefully.
An alternate approach, which we take here, is to sample the
-space very densely.
This, of course, introduces many locations not touched directly by the data.
We have traded the interpolation problem for a missing data problem,
nonlinear because we must estimate this new missing data
at the same time we estimate the PEF.
We'd like to come up with a reliable pyramid method of interpolating
aliased data that is devoid of low frequency information.
An attractive feature is that
the pyramid concept does not require the original data on a regular mesh in
-space.
Our method will build a dense regular mesh in model
-space.
The problem is non-linear because of the product of unknowns, the PEF multiplying the missing data. Here we approach the problem by multistage linear least squares which can be iterated to solve the nonlinear problem. In any nonlinear problem the initial guess must be ``near enough''. Hopefully the proposed method will not demand unaliased low frequency information.
Imagine five unevenly spaced traces on the -axis.
The data space is defined as
at five known values of the coordinate
.
Define a model space
that is dense
(many uniformly spaced points) on
-space.
We are interested in fitting
Next let us upgrade
.
At each
from the model space
,
make an operator
for convolution over the
axis.
Simultaneously for all
we find the regression for an upgraded PEF
(which is constant over
).
Although we are planning to iterate, we will never change
.
From solution of the regression above we have
the vector
which we use to make the filter operator
.
Use it in place of
in the regression for
above.
Iterate to get an
that is improved over
. Call it
. Iterate.
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![]() | An algorithm for interpolation using Ronen's pyramid | ![]() |
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