Figure 3

The Blackman-Tukey spectrum of the entire temperature reconstruction shown in Figure 1. A first-order Markov null spectrum was used for constructing the a priori 95 percent confidence level (dashed line). The four spectral periods of interest here are all statistically significant on the basis of these confidence levels, with the 31-year term being the weakest. These results are consistent with the split spectra shown in Figure 2.

of the PCs using high-resolution, maximum-entropy spectral analysis (Marple, 1987).

As in classical spectral analysis based on the Blackman-Tukey approach (Jenkins and Watts, 1968), the number of lags used in computing the ACF for SSA is somewhat arbitrary, being a tradeoff between resolution and stability. Initial experiments indicated that about 300 lags (13.1 percent of the series length) was the minimum needed to simultaneously resolve the 31-, 56-, 80-, and 200-year oscillations. In fact, for lag windows over the range 300 to 700, these four oscillations were always found in the four leading EOF pairs after suitable low-pass prefiltering to remove unwanted high-frequency variance. However, spectral analyses of their PCs indicated that about 400 lags ( 17.5 percent) were needed to cleanly separate the closely spaced 56- and 80-year oscillations. When only 300 lags were used, the spectra of these oscillations were somewhat contaminated by each other's power. In contrast, the 31-year term alone could be reasonably well resolved using only 100 lags (4.3 percent). For this reason, 100 lags were chosen to isolate the 31-year term, while 400 lags were used to isolate the longer oscillations.

Figure 4 shows the four even EOFs estimated as described above. The average period of each EOF is very similar to that obtained from the spectral analysis of its respective waveform. The EOFs are remarkably regular, given that they are based solely on the data and are not constrained to be sinusoidal. Figure 5 shows the normalized power spectra estimated by the maximum entropy method, which confirms the strongly periodic nature of these oscillations. These spectra are purely descriptive and cannot be

Figure 4

The four even empirical orthogonal functions (EOFs) estimated by singular spectrum analysis representing the oscillatory modes in the temperature reconstruction.

readily tested for statistical significance The identified periods are very close to those found earlier (compare Figures 2 and 3 and Cook et al., 1992). Each spectrum has some degree of side-lobe power in the form of small secondary peaks, which may reflect a degree of amplitude and phase modulation in the waveform. However, they are small enough to be relegated to noise at this stage of analysis. The 31-year oscillation has the most complicated spectrum, with distinct secondary peaks of 29 and 35 years. This may reflect some drift or instability in the mean period over time.

The waveforms upon which the maximum entropy spectra are based are shown in Figure 6. The 204-year waveform



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