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Time-series analysis is rich with concepts that
the helix now allows us to apply to many dimensions.
First is the notion of an impulse function.
Observe that an impulse function on the 2-D surface
of the helical cylinder maps to an impulse function
on the 1-D line of the unwound coil.
An autocorrelation function that is an impulse
corresponds both to a white (constant) spectrum in 1-D and
to a white (constant) spectrum in 2-D.
A causal filter in one dimension has
a curious shape on the two-dimensional helix.
In one dimension,
the causal filter has zeros before the ``1.0'' and various values after it.
Say the nonzero filter coefficients on the cylinder
lie within a short distance (two lags) from the ``1.0''.
Extract the little 2-D patch (which is the end of the 1-D filter).
I display it reversed on both axes so the reader can envision it
as crosscorrelation,
first moving down the first seismogram and then down the next.
| |
(205) |
where a,b,c,...,u are adjustable coefficients.
Thus we conclude that the 2-D analog of a 1-D causal filter
has its abrupt beginning along the side of the 2-D filter.
A special causal filter that unites many well established concepts
in time-series analysis is the prediction-error-filter (PEF).
A 2-D PEF, like a 1-D PEF, is a causal filter with adjustable coefficients
as in the array (),
that are adjusted to minimize the filter's output energy
(for a particular input signal).
That the 2-D PEF should have its beginning along a side
(instead of at a corner)
is an abstract idea that I have always found difficult to teach clearly,
until I fell upon the helix explanation.
Here is a brief summary of important ideas in time-series analysis
that the helix makes applicable in higher dimensions:
-
The filter (a, b, c, ..., u) is the negative of the prediction filter.
The filter (1, a, b, c, ..., u) is the prediction-error filter.
-
The method of least-squares is used to find the prediction filter.
This is also called ``autoregression''.
-
Textbooks ()
show that the spectrum of the output of
the PEF tends towards whiteness as the filter length increases.
Thus the spectrum of the PEF itself tends to the inverse of that of its input.
(Noncausal filters do not have white outputs
and cannot be used recursively.)
-
A time series can be decomposed into random impulses (white spectrum)
convolved with a natural wavelet that is the inverse of the PEF.
-
For any power spectrum,
there is a causal wavelet (with that spectrum)
that can be found by ``spectral factorization''.
In the frequency domain this is known as the Kolmogoroff
method.
-
The PEF has the property of ``minimum phase'' which means that
both it and its convolutional inverse are causal.
Thus, we can design stable multidimensional recursive filters
as we do in one dimension.
-
Stable filters can be modeled as layered media where waves
resonate among reflection coefficients bounded in absolute value by unity.
Such models help in PEF estimation by the
Burg method.
For many years it has been true that
our most powerful signal-analysis techniques
are in one-dimensional space,
while our most important applications are in multi-dimensional space.
The helical coordinate system makes a giant step
towards overcoming this difficulty.
The many features of 1-D theory outlined above
are now awaiting multidimensional application.
Next: FINITE DIFFERENCES ON A
Up: Rickett, et al.: STANFORD
Previous: PROGRAM FOR MULTIDIMENSIONAL CONVOLUTION
Stanford Exploration Project
7/5/1998