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A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy. Specifying the filter is choosing the interpolation philosophy. Generally the filter is a roughening filter. When a roughening filter goes off the end of smooth data, it typically produces a large transient at the end. Minimizing energy implies a choice for unknown data values at the end to minimize the transient. We examine five cases and then make some generalizations.
A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy. |
Let denote an unknown (missing) value.
The dataset on which the examples are based is
.
Theoretically,
we could adjust the missing
values (each different)
to minimize the energy in the unfiltered data.
Those adjusted values would obviously turn out to be all zeros.
The unfiltered data is data that has been filtered by
an impulse function.
To find the missing values
that minimize energy out of other filters,
we can use subroutine mis1().
Figure 1
shows interpolation of the dataset with
as a roughening filter.
The interpolated data matches the given data where they overlap.
mlines
Figure 1. Top is given data. Middle is given data with interpolated values. Missing values seem to be interpolated by straight lines. Bottom shows the filter ![]() |
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mparab
Figure 2. Top is the same input data as in Figure 1. Middle is interpolated. Bottom shows the filter ![]() |
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mseis
Figure 3. Top is the same input. Middle is interpolated. Bottom shows the filter ![]() |
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moscil
Figure 4. Bottom shows the filter ![]() |
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Figures 1-4
illustrate the rougher the filter,
the smoother the interpolated data,
and vice versa.
Switch attention from the residual spectrum
to the residual.
The residual for Figure 1
is the slope of the signal
(because the filter is a first derivative),
and the slope is constant (uniformly distributed) along the straight lines
where the least-squares procedure is choosing signal values.
So, these examples confirm the idea
that the least-squares method abhors large values
(because they are squared).
Thus, least squares tends to distribute residuals uniformly
in both time and frequency to the extent
allowed by the constraints.
This idea helps us answer the question,
what is the best filter to use?
It suggests choosing
the filter to have an amplitude spectrum
that is inverse to the spectrum we want for the interpolated data.
A systematic approach is given in Chapter ,
but I offer a simple subjective analysis here:
Looking at the data, we see that all points are positive.
It seems, therefore, that
the data is rich in low frequencies;
thus, the filter should contain something like
,
which vanishes at zero frequency.
Likewise, the data seems to contain Nyquist frequency;
so, the filter should contain
.
The result of using the filter
is shown in Figure 5.
Foregoing is my best subjective interpolation
based on the idea that the missing data should look like the given data.
The resulting interpolation and extrapolations are so good that
you can hardly guess which data values are given
and which are interpolated.
We care about this because the goal in geophysical image making is to create an image
that hides locations of our measurements (and missing measurements!).
mbest
Figure 5. Top is the same as in Figures 1 to 4. Middle is interpolated. Bottom shows the filter ![]() ![]() |
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