## Abstract

Aiming to describe spatio-temporal climate variability on decadal-to-centennial time scales and longer, we analyzed a data set of 26 proxy records extending back 1,000–5,000 years; all records chosen were calibrated to yield temperatures. The seven irregularly sampled series in the data set were interpolated to a regular grid by optimized methods and then two advanced spectral methods—namely singular-spectrum analysis (SSA) and the continuous wavelet transform—were applied to individual series to separate significant oscillations from the high noise background. This univariate analysis identified several common periods across many of the 26 proxy records: a millennial trend, as well as oscillations of about 100 and 200 years, and a broad peak in the 40–70-year band. To study common NH oscillations, we then applied Multichannel SSA. Temperature variations on time scales longer than 600 years appear in our analysis as a dominant trend component, which shows climate features consistent with the Medieval Warm Period and the Little Ice Age. Statistically significant NH-wide peaks appear at 330, 250 and 110 years, as well as in a broad 50–80-year band. Strong variability centers in several bands are located around the North Atlantic basin and are in phase opposition between Greenland and Western Europe.

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## Acknowledgments

It is a pleasure to thank two anonymous reviewers for detailed and constructive comments. MG acknowledges support from U.S. National Science Foundation grants DMS-1049253 and OCE-1243175, and U.S. Department of Energy grant DE-SC0006694.

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## Appendices

### Appendix 1: Singular Spectrum Analysis (SSA)

The SSA methodology involves three basic steps: (1) embedding a time series of length \(N\) in a vector space of dimension \(M\)—for the choice of \(M\), see Vautard et al. (1992) and Ghil et al. (2002; 2) computing the \(M\times M\) lag-covariance matrix \(C_{\mathrm{D}}\) of the data—see the two different approaches of Broomhead and King (1986) and Vautard and Ghil (1989); and (3) diagonalizing \(C_{\mathrm{D}}\):

where \(\Lambda _{\mathrm{D}} = {\mathrm{{diag}}}(\lambda _1, \lambda _2, \ldots \lambda _M)\), with \(\lambda _1, \lambda _2, \ldots \lambda _M>0\) the real, positive eigenvalues of the symmetric matrix \(C_{\mathrm{D}}\), and \(E_{\mathrm{D}}\) is the \(M\times M\) matrix having the corresponding eigenvectors \(\mathbf {E_k}\), \(k=1, \ldots M,\) as its columns.

For each \(\mathbf {E_k}\) we construct the time series of length \(N-M+1\), called the \(k\)-th principal component (PC); this PC represents the projection of the original time series on the eigenvector \(\mathbf {E_k}\), also called empirical orthogonal function (EOF). Each eigenvalue \(\lambda _k\) gives the variance of the corresponding PC; its square root is called a singular value.

Given a subset \(\mathcal {K}=(k^*_1, k^*_2, \ldots k^*_K)\) of eigenvalues, it is possible to extract time series of length \(N\), by combining the corresponding PCs. These time series are called reconstructed components (RCs) and capture the variability associated with the \(K\) eigenvalues of interest. A nonlinear, data-adaptive trend is typically identified by the largest eigenvalue and the associated EOF that has no zero. An oscillatory mode is associated with a pair of nearly equal eigenvalues, while the corresponding EOFs are in quadrature, like a pair of anharmonic, data-adaptive sine and cosine.

In order to reliably identify the trend and oscillations in a series, the Monte Carlo method (MCSSA) is used (Allen and Smith 1996). In this approach, one assumes a model for the analyzed time series and one determines this model’s parameters using a maximum-likelihood criterion; one refers to this model commonly as the null hypothesis.

Then a Monte Carlo ensemble of surrogate time series is generated from the null hypothesis, and SSA is applied to the data as well as the surrogates, in order to test whether it is possible to distinguish the original time series from the ensemble of surrogates. Since a large class of geophysical processes generate series with larger power at lower frequencies (Ghil et al. 2002), we assume AR(1) noise in evaluating evidence for trend and oscillations. This is done to avoid overestimating the significance of the low-frequency peaks in the spectrum, by underestimating the amplitude of the stochastic component of the time series in the lower part of the frequency range (Allen and Smith 1996; Ghil et al. 2002).

### Appendix 2: Multichannel SSA (MSSA)

In order to reconcile the different information items contained in each of the data sets under study, it is useful to apply the SSA analysis to a combination of all the available measurements, and not just to each one separately. Doing so may allow us to extract a subset of oscillations that are common to all the time series, thus establishing which periodicities are the most dominant for the whole data set.

Multichannel Singular Spectrum Analysis [MSSA; Keppenne and Ghil (1993); Plaut and Vautard (1994)] is a multivariate extension of the SSA method, with each *channel* corresponding to one of the time series of interest. As in the univariate case, this data-adaptive filtering technique can identify nearly periodical oscillating modes within a specified spectral window. This method is equivalent, in principle, to extended empirical orthogonal function analysis [EEOF: Weare and Nasstrom (1982)], but in MSSA the focus is on the temporal structure of the variability, whereas in EEOF it is the spatial variability that is emphasized.

Basically, MSSA decomposes a multichannel time series **X**
\(_{l,i}\), with \(i=1, \ldots , N\) representing time and \(l=1, \ldots , L\) the channel number, into an orthonormal, data-adaptive space-time structure whose elements represent eigenvectors grand cross-covariance matrix of size \(LM \times LM\), where \(M\) is the window width. The computation thus results in a set of eigenvectors \(\mathbf{E}^k\), called space-time EOFs (ST-EOFs), and their associated space-time PCs (ST-PCs) \(a^k\), computed by projecting **X** onto \(\mathbf{E}^k\). The \(L \times M\) real eigenvalues \(\lambda _k\), each associated with the \(k\)-th eigenvector \(\mathbf{E}^k\), equal the variance in the \(a^k\) direction.

The MSSA expansion of the original data series is thus given by

Each reconstructed component (RC) allows one to reconstruct the dynamical behavior in **X** that belongs to \(\mathbf{E}^k\) (Plaut and Vautard 1994; Ghil et al. 2002).

MSSA shares with single-channel SSA the ability to identify robust modulated oscillations in the data with period smaller than \(M\) by means of pairs of subsequent eigenvectors that are in phase quadrature, that is, such that the cross-correlation between these pairs is maximum at a lag of roughly one quarter the associated period. In order to select an oscillatory mode, it is required that the corresponding eigenvalues are nearly equal and that the two ST-PCs share aprroximately the same frequency (Vautard et al. 1992; Plaut and Vautard 1994).

As in the case of single-channel SSA, the presence of an eigenvalue pair is not sufficient to conclude that they represent an oscillation in the data. The significance of such tentatively identified oscillations has to be additionally checked, because also pure white- or red-noise processes are able to randomly generate pairs of eigenvectors satisfying the above-mentioned criteria (Vautard and Ghil 1989; Vautard et al. 1992; Allen and Smith 1996).

Allen and Robertson (1996) suggested two different significance tests for the presence of oscillations at low signal-to-noise ratios in multivariate data, based on the same null hypothesis. In the first test, the lag-covariance matrix is computed from the original, observed data, whereas in the second, the lag-covariance matrix is computed from the surrogate data generated by Monte Carlo methods. The subsequent procedure of projecting the data and the noise surrogates onto the ST-EOF basis is similar in both tests. The second test is more robust, since the former implicitly assumes the existence of a signal before any signal has been detected (Allen and Smith 1996).

We have applied both of these tests to determine whether the modes associated with pairs of eigenvalues correspond to actual oscillatory modes. The following step, given the detection of frequencies with significantly more power than expected from red noise, is to reconstruct the multivariate structure associated with each of them. Variability on time scales longer than \(M\) is described by the nonlinear, data-adaptive trend component, which however may also represent an oscillation with a time scale larger than \(M\) (Ghil and Vautard 1991; Ghil et al. 2002).

### Appendix 3: Continuous Wavelet Transform (CWT)

The Wavelet Transform (WT) allows an evolutionary spectral analysis of a series in the time-scale plan (Foufoula-Georgiou and Kumar 1994; Percival and Walden 2000). The concept of scale is typical of this method: the scale is a time duration that can be properly translated into a Fourier period, and hence a frequency. The Continuous Wavelet Transform (CWT) in spectral applications is discretized by computing it at all available time steps and on a dense set of scales (Torrence and Compo 1998).

The square modulus of the transform expresses spectral density as a function of time and frequency. A filtered version of the signal can then be reconstructed selecting only the contributions from a given set of periods. By time averaging the CWT at each value of the period (scale), the Global Wavelet Spectrum (gws)—and the corresponding significance levels, using a background spectrum of red noise—can be computed, thus obtaining a time-averaged spectral estimate comparable with those obtained by classical methods. However, CWT is a multiresolution analysis: frequency resolution is high at low frequencies and poor at high frequencies (Torrence and Compo 1998). It is, therefore, particularly suited for determining the frequency of the oscillations and to reconstruct them accurately in the low-frequency range of the spectrum (Tables 5, 6).

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Taricco, C., Mancuso, S., Ljungqvist, F.C. *et al.* Multispectral analysis of Northern Hemisphere temperature records over the last five millennia.
*Clim Dyn* **45, **83–104 (2015). https://doi.org/10.1007/s00382-014-2331-1

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### Keywords

- Space-and-time domain analysis
- Multi-scale analysis of time series
- Climatic oscillations
- Recent paleoclimate
- Temperature proxy records
- Past two millennia
- Solar forcing