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Now for the paradox.
Imagine in Figure 8.
Will we see the same limit as results from the ensemble average?
Here are two contradictory points of view:

For increasing n,
the fluctuations on the nonzero autocorrelation lags get smaller,
so the autocorrelation should tend to an impulse function.
Its Fourier transform, the spectrum, should tend to a constant.

On the other hand, for increasing n,
as in Figure 3,
the spectrum does not get any smoother,
because the FTs should still look like random noise.
We will see that the first idea contains a false assumption.
The autocorrelation does tend to an impulse,
but the fuzz around the sides cannot be ignoredalthough
the fuzz tends to zero amplitude,
it also tends to infinite extent,
and the product of zero with infinity here
tends to have the same energy as the central impulse.
To examine this issue further,
let us discover how these autocorrelations decrease
to zero with n (the number of samples).
Figure 9 shows the autocorrelation samples as a function of n
in steps of n increasing by factors of four.
Thus increases by factors of two.
fluct
Figure 9
Autocorrelation as a function of number of data points.
The randomnoiseseries (even) lengths are
60, 240, 960.
Each autocorrelation in the figure was normalized at zero lag.
We see the sample variance for nonzero lags of the autocorrelation
dropping off as .We also observe that the ratios between the values for the first nonzero
lags and the value at lag zero roughly fit .Notice also that the fluctuations drop off with lag.
The dropoff goes to zero at a lag equal to the sample length,
because the number of terms in the autocorrelation
diminishes to zero at that lag.
A first impression is that the autocorrelation fits a triangular envelope.
More careful inspection, however, shows
that the triangle bulges upward at wide offsets,
or large values of k
(this is slightly clearer in Figure 8).
Let us explain all these observations.
Each lag of the autocorrelation is defined as
 
(36) 
where (x_{t}) is a sequence of zeromean
independent random variables.
Thus, the expectations of the autocorrelations can be easily computed:
 
(37) 
 (38) 
In Figure 9,
the value at lag zero is more or less (before normalization),
the deviation being more or less the standard deviation
(square root of the variance) of s_{0}.
On the other hand,
for ,as , the value of the autocorrelation is directly
the deviation of s_{k},
i.e., something close to its standard deviation.
We now have to compute the variances of the s_{k}.
Let us write
 
(39) 
So: , where is the sample mean
of y_{k} with nk terms. If , , and we apply
(33) to :
 
(40) 
The computation of is straightforward:
 
(41) 
Thus, for the autocorrelation s_{k}:
 
(42) 
Finally, as , we get
 
(43) 
This result explains the properties observed in Figure 9.
As ,all the nonzero lags tend to zero compared to the zero lag,
since tends to zero.
Then, the first lags
(k<<n) yield the ratio
between the autocorrelations and the
value at lag zero.
Finally, the autocorrelations do not decrease linearly with k,
because of .
We can now explain the paradox.
The energy of the nonzero lags will be
 
(44) 
Hence there is a conflict between the decrease to
zero of the autocorrelations
and the increasing number of nonzero lags,
which themselves prevent
the energy from decreasing to zero.
The autocorrelation does not globally tend to an impulse function.
In the frequency domain, the spectrum is now
 
(45) 
So ,and the average spectrum is a constant,
independent of the frequency.
However, as the s_{k} fluctuate
more or less like ,and as their count in
is increasing with n,
we will observe that will also fluctuate,
and indeed,
 
(46) 
This explains why the spectrum remains fuzzy:
the fluctuation is independent of the number of samples,
whereas the autocorrelation seems to tend to an impulse.
In conclusion,
the expectation (ensemble average) of the spectrum
is not properly estimated by letting in a sample.
Next: An example of the
Up: SPECTRAL FLUCTUATIONS
Previous: SPECTRAL FLUCTUATIONS
Stanford Exploration Project
10/21/1998