We explore the capability of singular spectrum analysis (SSA) to extract time-variable seasonal from continual GPS observations and demonstrate the statistical assessment on the colored noise (in particular the first-order autoregressive AR(1) noise) using Monte Carlo SSA (MCSSA) methodology. We provide example applications to ~ 15-year vertical coordinate time series for 36 globally distributed International GNSS Service (IGS) sites. We find the SSA-filtered seasonal signals can easily pass the confidence interval and hypothesis tests of MCSSA. However, maximum likelihood estimate (MLE) results show that 72% of sites have their flicker noise amplitudes reduced after removing SSA-filtered annual signal, implying that the SSA-filtered seasonal signals may contain an artificial signal driven by colored noise. Therefore, the AR(1) null hypothesis noise model may be misleading in surrogate data tests for GPS seasonal signals. Moreover, comparison between SSA-filtered GPS annual signals and joint geophysical model predictions (non-tidal atmospheric loading + non-tidal ocean loading + hydrological loading) confirms that seasonal signals are resulting from a combination of mass loading and systematic error.

Singular systems (or singular spectrum) analysis (SSA) was originally proposed for noise reduction in the analysis of experimental data and is now becoming widely used to identify intermittent or modulated oscillations in geophysical and climatic time series. Progress has been hindered by a lack of effective statistical tests to discriminate between potential oscillations and anything but the simplest form of noise, that is, "white" (independent, identically distributed) noise, in which power is independent of frequency. The authors show how the basic formalism of SSA provides a natural test for modulated oscillations against an arbitrary "colored noise" null hypothesis. This test, Monte Carlo SSA, is illustrated using synthetic data in three situations: (i) where there is prior knowledge of the power-spectral characteristics of the noise, a situation expected in some laboratory and engineering applications, or when the "noise" against which the data is being tested consists of the output of an independently specified model, such as a climate model; (ii) where a simple hypothetical noise model is tested, namely, that the data consists only of white or colored noise; and (iii) where a composite hypothetical noise model is tested, assuming some deterministic components have already been found in the data, such as a trend or annual cycle, and it needs to be established whether the remainder may be attributed to noise. The authors examine two historical temperature records and show that the strength of the evidence provided by SSA for interannual and interdecadal climate oscillations in such data has been considerably overestimated. In contrast, multiple inter- and subannual oscillatory components are identified in an extended Southern Oscillation index at a high significance level. The authors explore a number of variations on the Monte Carlo SSA algorithm and note that it is readily applicable to multivariate series, covering standard empirical orthogonal functions and multichannel SSA.

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

The availability of time series of the evolution of the properties of physical systems is increasing, stimulating the development of many novel methods for the extraction of information about their behaviour over time, including whether or not they arise from deterministic or stochastic dynamical systems. Surrogate data testing is an essential part of many of these methods, as it enables robust statistical evaluations to ensure that the results observed are not obtained by chance, but are a true characteristic of the underlying system. The surrogate data technique is based on the comparison of a particular property of the data (a discriminating statistic) with the distribution of the same property calculated in a set of constructed signals (surrogates) which match the original data set but do not possess the property that is being tested. Fourier transform based surrogates remain the most popular, yet many more options have since been developed to test increasingly varied null hypotheses while characterizing the dynamics of complex systems, including uncorrelated and correlated noise, coupling between systems, and synchronization. Here, we provide a detailed overview of a wide range of surrogate types, discuss their practical applications and demonstrate their use in both numerically simulated and real experimental systems. We also compare the performance of various surrogate types for the detection of nonlinearity, synchronization and coherence, coupling strength between systems, and the nature of coupling. A MatLab toolbox for many of the surrogate methods is provided.

Oscillatory modes with the period of approximately 7–8 yr were detected in monthly time series of sunspot numbers, geomagnetic activity aa index, NAO (North Atlantic Oscillation) index and near-surface air temperature from several mid-latitude European locations. Instantaneous phases of the modes underwent synchronization analysis and their statistically significant phase coherence, beginning from 1950s, has been observed. Thus the statistical evidence for a coupling between solar/geomagnetic activity and climate variability has been obtained from continuous monthly data, independent of the season, however, confined to the temporal scale related to oscillatory periods about 7–8 yr.

Characterizing ecosystem-atmosphere interactions in terms of carbon and water exchange on different time scales is considered a major challenge in terrestrial biogeochemical cycle research. The respective time series currently comprise an observation period of up to one decade. In this study, we explored whether the observation period is already sufficient to detect cross-relationships between the variables beyond the annual cycle, as they are expected from comparable studies in climatology. We investigated the potential of Singular System Analysis (SSA) to extract arbitrary kinds of oscillatory patterns. The method is completely data adaptive and performs an effective signal to noise separation. We found that most observations (Net Ecosystem Exchange, NEE, Gross Primary Productivity, GPP, Ecosystem Respiration, R-eco, Vapor Pressure Deficit, VPD, Latent Heat, LE, Sensible Heat, H, Wind Speed, u, and Precipitation, P) were influenced significantly by low-frequency components (interannual variability). Furthermore, we extracted a set of nontrivial relationships and found clear seasonal hysteresis effects except for the interrelation of NEE with Global Radiation (R-g). SSA provides a new tool for the investigation of these phenomena explicitly on different time scales. Furthermore, we showed that SSA has great potential for eddy covariance data processing, since it can be applied as a novel gap filling approach relying on the temporal correlation structure of the time series structure only.

Detection and extraction of quasi-oscillatory dynamical modes from instrumental records of geophysical data became a useful tool in analyzing variability of observed phenomena reflected in complex, multivariate geophysical signals. Using the extension of the Monte Carlo singular system analysis (MC SSA), based on evaluating and testing regularity of dynamics of the SSA modes against the colored noise null hypothesis, we demonstrate detection of oscillatory modes with period of about 96 months in the long-term records of aa index as well as in the records of surface air temperature from several mid-latitude European locations and in the North Atlantic Oscillation index.