Specifically, we first carried out an MEM analysis of the original monthly "raw" data (shown as dots in Figure 3), with only a linear trend subtracted. The data set consisted of 1620 points. Spectral estimates were obtained for order M (number of poles; see Press et al. (1989)) equal to 750. The periods of the 24 most prominent spectral peaks found are listed in the first column of Table 1 for frequencies greater than 0.6 cycles per year. Since the use of such a large number of poles invites criticism that we have exceeded the limits of the MEM method, we carried out a second analysis in which we first removed the lowest and highest frequencies from the record by subtracting the very stiff curve "1" shown in Figure 4 from the flexible spline shown in Figure 3. The resulting detrended and slightly smoothed temperature record is shown in Figure 5, panel 1. It can be seen by comparison with Figure 3 that almost all of the oscillatory character in the original record with periods between 1 and 30 years is retained in panel 1 of Figure 5. We then calculated the MEM spectrum on the basis of this slightly

TABLE 1 MEM Spectra of Monthly Global Surface Temperature, 1855-1989

Raw Data Spectrum*

(years)

Smoothed

(years)

Amplitude for 24-Harmonic Fit**

(°C)

Non-parametric Multitaper

(years)

41.3

31.40

0.017

 

22.10

21.69

0.039

20.5

15.28

15.59

0.038

15.5

10.07

12.15

0.017

 

9.18

10.22

0.036

 

7.509

9.29

0.037

9.1

6.643

7.610

0.014

 

6.042

6.670

0.026

 

5.218

6.043

0.037

6.1

4.740

5.227

0.040

5.2

4.382

4.758

0.041

4.8

4.128

4.436

0.023

 

3.756

4.170

0.026

4.0

3.546

3.746

0.022

3.8

3.283

3.548

0.034

3.5

3.122

3.290

0.025

 

2.867

3.127

0.022

 

2.716

2.866

0.027

 

2.318

2.727

0.009

 

2.241

2.320

0.023

 

1.998

2.242

0.017

 

1.885

1.997

0.020

 

1.796

1.884

0.014

 

 

1.797

0.017

 

* From Ghil and Vautard (1991, Figure 2). Their plot also shows an unlabeled, strong peak near 10.2 years.

** From a fit of sine and cosine functions to the smoothed data. The amplitudes are peak to trough.

smoothed record with data points spaced three months apart, beginning with January 1855. The data set consisted of 540 points. Given the degree of smoothing of this record, more closely spaced data would not have contributed any significant additional information, as perhaps they could have done if the raw data had been used. As shown in the second column of Table 1, the power spectrum of the slightly smoothed record, with M set equal to only 250, gives almost the same frequencies for peaks as the spectrum based on the raw data with M set equal to 750.

A check was also made by creating a subset of points, reducing the raw data by a factor of 3 (selecting values for every third month only, beginning with the first). With a linear trend removed as before, we obtained nearly the same spectrum with M set equal to 275 as in the previous two cases.

We then computed the spectrum using the slightly smoothed data spaced three months apart, but with M reduced from 250 to 150. Figure 6 shows this spectrum (indicated by a dashed line) has fewer and less sharp peaks than that computed with M equal to 250 (solid line). In particular, in the decadal-frequency region a single peak with a period of 9.7 years replaces a pair of peaks with periods of 9.3 and 10.2 years. This single peak, having a frequency of approximately the mean of the two peaks in the prior case, shows relative stability in frequency despite the change in M value.

As demonstrated by a synthetic example of a sample of points from the sum of two sinusoids (Press et al., 1989), the MEM method at low pole numbers does not resolve closely spaced oscillations, whereas when MEM is used with high pole numbers, spectral lines may split where there is no firm basis for expecting multiple peaks. These are typically exaggerated in regions of the spectrum having low power spectral density. In our particular example of global temperature, both spectra in Figure 6 indicate high power in the decadal region, whether that power is represented by a single oscillation or a pair. We chose to accept the spectrum showing a pair of peaks, because the spline fit to the temperature data in the decadal band (shown in Figure 4, panel 2) also indicated that the decadal signal vanished in the middle of the record, and this is better represented by a pair of decadal oscillations.

Apart from the changes in frequency caused by the merger of spectral lines as the M value decreased, we did not find noticeable changes in the position of the spectral peaks as we varied our method of analysis. We found, however, that the peaks shifted somewhat in frequency if we used only limited portions of either version of the record—for example, by deleting the data before 1875 that exhibit more high-frequency oscillations than the subsequent record. Nevertheless, for peaks with periods of 6 years or greater, all of the spectra examined were similar to the two versions summarized in Table 1.



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