[72] In the present study we used “Singular System Analysis” (also known as “Singular Spectrum Analysis,” SSA [*Broomhead and King*, 1986; *Elsner and Tsonis*, 1996; *Golyandina et al.*, 2001; *Ghil et al.*, 2002]) as basis for the time-frequency model-data comparison. The application of SSA is justified because of some theoretical advantages compared to other techniques (discussed by *Mahecha et al.* [2007]), and favored by its applicability to fragmented time series. Note however that the question of how to separate superimposed subsignals from a single data stream has been a matter of debate for decades, and is still not resolved [*Ghil et al.*, 2002]. Other methods might be equally applicable in the context of model-data comparisons.

[73] SSA aims at extracting subsignals of a given time series *X*(*t*), *t* = 1, …, *N* belonging to characteristic scales of variability. Initially, an embedding dimension has to be defined a priori. This is a window of length *P* which in this study was set equally for all time series to 3 years. Sliding the window along the time series leads to a trajectory matrix consisting of the sequence of *K* = *N* – *P* + 1 time-lagged vectors of the original series. The *P* dimensional vectors of the trajectory matrix **Z** are set up as described in equation (B1) [*Golyandina et al.*, 2001].

Based on the *K* × *P* trajectory matrix **Z** a *P* × *P* covariance matrix **C** = {*c*_{i}_{,j}} is built:

For constructing the covariance matrix, various approaches have been reported in the literature and we refer the reader to the specialized literature for understanding different variants of SSA [*Vautard and Ghil*, 1989; *Golyandina et al.*, 2001; *Ghil et al.*, 2002]. The entries of the resulting *P* × *P* matrix represent the captured lag covariance. This is used to determine the orthonormal basis by solving equation (B3),

where, **E** is a *P* × *P* matrix of the eigenvectors *E*_{i}, also called empirical orthogonal functions (EOFs) of **C**. The matrix **Λ** contains the respective eigenvalues in the diagonal, sorted by convention in descending order *diag*(**Λ**) = (*λ*_{1}, …, *λ*_{P}), where *λ*_{1} ≥ *λ*_{2} ≥ … ≥ *λ*_{P}. It can be shown that due to the properties of the covariance matrix **C**, preserving symmetry and being real valued and positive semidefinite, all eigenvectors and eigenvalues are real valued, where the latter are nonnegative scalars. The eigenvalues are proportional to the fraction of explained variance corresponding to each EOF. In analogy to the well known Principal Component Analysis, the decomposition allows the construction of principal components (PCs) as generated time series representing the extracted orthogonal modes (equation (B4)). This is why SSA is often also called a “PCA in the time domain.”

As it can be seen in equation (B4), the principal components are obtained by simply projecting the time series onto the EOFs. This projection constructs a set of *P* time series of length *K*.

[74] The last step in SSA is the reconstruction of the time series through the principal components *A*^{κ}(*t*), see equation (B5). The original signal can be fully or partially reconstructed. This is a selective step, and the analyst has to decide which *A*^{κ}(*t*) are combined so that one obtains an interpretable combination of principal components. This enables signal-noise separation and the reconstruction of specifically selected frequency components, as illustrated by equation (B5).

In this reconstruction procedure, *κ* is an index set determining the selection of modes used for the reconstruction, *M*_{t} is a normalization factor, and the corresponding extension for the series boundaries are given by *L*_{t} and *U*_{t} (definitions for the boundary terms are given in Table B1; a comprehensive derivation can be found in the work of *Ghil et al.* [2002]).

##### B1. SSA Gap-Filling Procedure

[75] *Kondrashov and Ghil* [2006] introduced an iterative SSA gap-filling strategy. Their method allows a time series reconstruction for fragmented time series, and thus is a tool for gap filling. Here we outline the fundamental steps of the SSA gap-filling algorithm:

[76] 1. Center the time series to zero mean, the latter being estimated from present data only.

[77] 2. Initiate an inner-loop iteration by applying SSA of the zero-padded time series. The leading reconstructed component (identified through its highest eigenvalue) is used to fill the values in the gaps. This leads to a new estimate of the time series mean, which is used for recentering the time series. The initially zero-padded values are set to their reconstructions. This procedure is carried out based on the computed and recomputed reconstructed components until a convergence criterion (in terms of the RMSE) is met.

[78] 3. After the first inner-loop iteration meets the convergence criterion, switch to an outer-loop iteration. This is the natural extension of the described procedure above, achieved by simply adding a second (third, etc.) additionally reconstructed component to inner-loop iteration.

##### B3. Subsignal Confidence Boundaries

[80] It has been reported that despite of orthogonal base functions (the EOFs) the accuracy of subsignals separability is not warranted [*Golyandina et al.*, 2001]. This methodological uncertainty has to be strictly distinguished from the effective model-data disagreement. We quantified the separation inaccuracy by a surrogate technique: For each subsignal in a frequency bin the corresponding residual was retained. For each residual a set of surrogates is generated. Since most of the analysis will focus on the coarse bins, 500 surrogates were created in case of the coarse binning, and 20 in case of the fine binning. A surrogate is a time series which resembles the original counterpart (here the residuals) in two fundamental aspects: the distribution and spectral properties. The latter warrants as identical autocorrelation structure. We followed the technique proposed by *Schreiber and Schmitz* [1996] known as “Iterative Amplitude Adjusted Fourier Transform, IAAFT” and we refer the reader to the original paper for more details.

[81] One problem inherent to IAAFT is that when the difference of start and end points are large, the corresponding spectral power of the “jump” is spread over all frequencies. This affects especially smooth time series with low powers in the high frequencies. The corresponding surrogate time series appear more noisy in all frequency ranges compared to the reference [*Schreiber and Schmitz*, 2000]. Fortunately, in our set up a precise definition of the spectral content is provided by the frequency binning (Table 2). Thus in a final step, surrogates undergo itself SSA, warranting that the surrogates accurately match the frequency structure of the residuals.

[82] After a set of surrogates has been generated and added to the subsignal of interest, the latter are reextracted. Any subsignal is thus replaced by an array of subsignals and their deviations quantify the extraction uncertainty. All analyses in this paper rely on this array instead of a single subsignal and form the basis for confidence envelopes for any estimated metric. Figure S1 conceptually summarizes the procedure in a flow chart.