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Next: Discussion Up: Berryman: MESA Previous: An information theory criterion

Examples

One good way to study various choices of operator length is to use unexpanded checkshot data. Two traces of this type are shown in Figure 2. By choosing a large window, we obtain a good estimate of the spectrum of the pulse. Then, by choosing smaller and smaller windows which pinch down on the pulse, we should expect the positions of the major peaks to remain unchanged while resolution becomes more difficult.

The first column of Figure 3 gives the ordinary power spectrum of Figure 2(a) computed using five different length windows. In seconds, the windows are from top to bottom (0.0,2.0), (0.1,0.5), (0.15,0.45), (0.15,0.25), and (0.175,0.225). No taper is included in the Fourier transform of the trace. We see that the resolution is quite good for the two second window, but the resolution gets progressively worse until it is almost nonexistent for the 50 ms window.

In contrast, the first column of Figure 4 gives the maximum entropy spectrum of Figure 2(a) with the operator length $ M = 2N/\ln 2N$. We see that the two second window MESA spectrum is essentially the same as the ordinary spectrum. However, as the number of data points decreases from 500 (for a two second window) to 13 (for a 50 ms window), we see that MESA is still able to resolve the peaks at 10 Hz and 30 Hz.

The first column of Figures 5-7 give examples of the results obtained from MESA for the trace of Figure 2(a) with other choices of operator length. Choosing $ M = N/2$ in Figure 5 gives acceptable results for all but the smallest window where the 10 Hz peak has moved towards 20 Hz. Also, the computation time was increased for the longest three windows. Choosing $ M = N$ in Figure 6 demonstrates the fact that choosing a longer operator does not lead to improved results. Here the single peak at 10 Hz has been split into two spurious peaks for the shortest two windows. The longest three windows give very spiky spectra and (although they properly indicate where the spectrum lies) they do not give useful power spectra. This Figure shows how the variance in (32) can dominate the bias and produce useless power spectra. Finally, we choose $ M = N/\ln N$ in Figure 7 to demonstrate the effect of choosing a slightly different functional form for $ M$. The spectrum of Figure 4 mimics that of Figure 3 better than Figure 7 in all cases except possibly for the 100 ms window where Figure 7 gives a stronger peak at 30 Hz. For the smallest window, Figure 7 does not have its peak at 10 Hz as it should.

Figure 2(b) is also an unexpanded checkshot trace which is translated in time from that in Figure 2(a). The second columns of Figures 3-7 were computed as before, but the input trace was Figure 2(b).

As a final note, we wish to point out that it has been observed in general that $ M = 2N/\ln 2N$ is in fact an upper bound on the operator lengths one would obtain from either (35) or (36). For example, using the trace of Figure 2, both the relative error coefficient test and the FPE have a series of minima for $ 120 < M < 135$ with an absolute minimum at $ M = 132$ for both criteria. For comparison, we find $ M = 145$ for $ N = 500$ using (49) and $ M = 127$ using (47). It is encouraging that the arguments of the subsection on An Information Theory Criterion give estimates for the operator length so close to those of Equations (35) and (36) without the added complication of monitoring a performance parameter.

FIG2a FIG2b
FIG2a,FIG2b
Figure 2.
Two checkshot traces: (a): The first two seconds of an unexpanded checkshot trace. The windows indicated in the Figure are repectively: $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. (b) Same as previous case for a slightly different trace. [NR]
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FIG3
FIG3
Figure 3.
The ordinary power spectrum of the traces in Figure 2 as a function of frequency (0-60 Hz). Windows from top to bottom are $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. [NR]
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FIG4
FIG4
Figure 4.
The maximum entropy power spectrum of the traces in Figure 2 as a function of frequency (0-60 Hz) with operator length $ M = 2N/\ln(2N)$. Windows same as in Figure 3. Windows from top to bottom are $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. [NR]
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FIG5
FIG5
Figure 5.
The maximum entropy power spectrum of the traces in Figure 2 as a function of frequency (0-60 Hz) with operator length $ M = N/2$. Windows same as in Figure 3. Windows from top to bottom are $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. [NR]
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FIG6
FIG6
Figure 6.
Same as Figure 5 with $ M = N$. Windows from top to bottom are $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. [NR]
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FIG7
FIG7
Figure 7.
Same as Figure 5 with $ M = N/\ln N$. Windows from top to bottom are $ (0.0,2.0)$, $ (0.1,0.5)$, $ (0.15,0.45)$, $ (0.15,0.25)$, and $ (0.175,0.225)$ in seconds. [NR]
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Next: Discussion Up: Berryman: MESA Previous: An information theory criterion

2009-04-13