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A systematic wavelet-based exploratory analysis of climatic variables

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Nonstationarity and time scale dependence are essential features of the climate system that may be dealt with simultaneously using wavelet analysis. In this study, we present a systematic analysis of a set of climate system variables, which include both natural and anthropogenic contributions, using wavelet-based exploratory methods. The tools of the continuous wavelet transform, the wavelet spectrum, coherence, and phase offer a comprehensive assessment of the characteristic modes of variability of climate system forcings and of the scale-based relationships of anthropogenic and natural climate variables with global surface temperature. Shorter-term variations in global surface temperature are associated with internally generated natural climate variability and external climate forcings, while longer-term variations are strongly related to human-induced changes only. In this respect, a long-term component of the net radiative forcing of human activities longer than 30 years displays a statistically significant relationship with global warming and cooling periods identified in the climate change literature.

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  1. In contrast, solar irradiance and variables representing internal fluctuations in the climate system, like SOI and PDO, are characterized by, respectively, quasi-periodic (Bouwer 1992) and irregular cyclical patterns.

  2. The aim of extracting from a time series unobserved components within specified frequency bands can be also pursued by using frequency domain methods. Although in principle, spectral analysis is appropriate because of its ability to simultaneously estimate the contribution of different cyclical components, in practice its application is greatly limited by the requirement that the series needs to be stationary. Kuo et al. (1990) and Thomson (1997) are examples of applications of frequency domain analysis to the relation between temperature and forcing variables.

  3. This mismatch between the characteristic or intrinsic time scale and the scales of data analysis may be only partially addressed by applying cointegration and error correction methods, as they may alleviate issues associated with nonstationarity such as stochastic trends and spurious regressions, but may just separate out two timescales in time series data, the short- and the long-run (Engle and Granger 1987, Johansen 1988).

  4. A brief technical introduction to wavelet analysis is provided in the Appendix.

  5. Specifically, using the wavelet power spectrum, we can detect the dominant modes of variability in the data, and using the wavelet coherence and phase, we can uncover the relation of external climate forcings and natural internal climate variability with global surface temperature on a scale-by-scale basis.

  6. Our dataset includes anthropogenic and natural forcings, such as CO2 emissions and aerosol-induced cooling (Hansen et al. 2010), solar irradiance changes (Mann et al. 1998), volcanic activity (Fyfe et al. 2013; Santer et al. 2014), and internal modes of variability of the climate system, like NAO (Hurrel 1995), SOI (Power and Kociuba 2011), and IPO (Meehl et al. 2010; Dai et al. 2015).

  7. The North Atlantic Oscillation (NAO) index is based on the surface sea-level pressure difference between the Subtropical (Azores) High and Subpolar Low. The Southern Oscillation index (SOI) measures the difference in air pressure between Tahiti and Darwin, Australia, and is an indicator of the starting time and strength of an El Nino, which occurs irregularly every 2 to 7 years. The Interdecadal Pacific Oscillation index is a variability pattern of SST fluctuations, and sea level pressure changes in the entire Pacific basin.

  8. The analysis has been performed using the MatLab package developed by Grinsted et al. (1994). MatLab programs for performing the bias-rectified wavelet power spectrum (Liu et al. 2007) and the cumulative areawise significance test (Schulte 2015) are available at and, respectively.

  9. Since annual observations cover a period slightly more than 150 years, the extended sample allows to satisfactorily investigate interannual (2–8 years), quasi-decadal (8–16), interdecadal (16–30 years), and multidecadal (32–64 years) timescales.

  10. As to the choice of the null model, the partial autocorrelation plot of the global surface temperature presented below shows clear statistical significance for lags 1, with lags 3 and 4 being at the borderline of statistical significance. Thus, an AR(1) process can be considered an acceptable noise background model for the global surface temperature, as it is for numerous climatic time series data given the persistence (memory) of climatic processes.

  11. The statistical significance of the results obtained through wavelet power analysis was first assessed by Torrence and Compo (1998) using pointwise significance test. However, since pointwise significance testing results are just artifacts of multiple testing (Maraun and Kurths 2004), an areawise significance test has been developed by Maraun et al. (2007).

  12. Temperature and radiative forcing series can be characterized as stationary processes around nonlinear deterministic trends with time-ordered breaks (Estrada et al. 2013) or unit root (stochastic trending) processes (Kaufmann et al. 2013).

  13. The potency of detection of volcanic eruptions measured in Teragrams was 21.864 for Krakatoa in 1883 and 30.094 for Mount Pinatubo in 1991.

  14. However, being the product of two non-normalized wavelet spectra, the cross-wavelet can identify the significant cross-wavelet spectrum between two time series, although there is no significant correlation between them.

  15. See Figure 4 in the Appendix for the interpretation of phase differences.

  16. The contribution from anthropogenic sulphate aerosols is less clear and subject to a high range of uncertainty.

  17. Nonstationarity of the statistical properties of historical climatic variables invalidates the use of standard inference procedures and calls for methodologies that account for data nonstationarity both individually and jointly.

  18. The DWT is only one of the various filtering/smoothing techniques that may be used to capture patterns of observed data without imposing any rigid functional form to the relationship and yielding a less variable signal where noise is reduced. With respect to various techniques of smoothing like loess, kernel, or splines are, at varying degree, sensitive to irregular data, wavelets have the ability to handle complex irregular signals. Indeed, signals with substantial rapid oscillations cannot be handled effectively by smoothing procedures, and in these circumstances, the wavelet approach is a logical choice.

  19. The decomposition process is iterative. The analysis begins with a decomposition of the signal into an approximation and a detail, but in each successive step, only the approximations are decomposed. At the end of such a procedure, the signal is broken down into many higher- and one lower-resolution components.

  20. Wavelet multi-resolution analysis attains a nonparametric approximate decomposition of a signal into the sum of a structural and a noise component that can provide an effective solution to the signal-to-noise problem (Ramsey et al. 2010).

  21. Santer et al. (2011) show that the separation of the distributions of unforced and forced trends is virtually complete for 30-year trends

  22. A 4-level decomposition produces four wavelet details vectors D1, D2, D3, D4, each associated with oscillatory variations on various timescales, 2–4, 4–8, 8–16, and 16–32 years, respectively, and one wavelet smooth vector, S4 capturing fluctuations longer than 32 years

  23. This transform is a compromise between the CWT, with continuous variations in scale, and DWT where the power of the transform is highly localized. The MODWT is highly redundant, so that the transformations at each scale are not orthogonal, but the offsetting gain is that applying the transform leaves the phase invariant, a very useful property in analyzing transformations, and the transform is not restricted to limitations imposed by the dyadic expansion used by the DWT.

  24. A negative contemporaneous correlation coefficient equal to − 0.049 emerges for the 1850–1910 period. Spearman’s rank correlation coefficients and Kendall’s tau provide results qualitatively similar to Pearson’s correlation coefficients for the whole sample and each separate sub-sample.

  25. In a strict sense, only the mid-century period can be considered a “cooling” period.

  26. According to Kaufmann et al. (2006, p.271): “The radiative forcing of anthropogenic sulfur emissions increases at about the same rate as greenhouse gases between 1944 and 1976. As a result, there is relatively little net increase/decrease in total anthropogenic radiative forcing and therefore, global surface temperature.”


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Gallegati, M. A systematic wavelet-based exploratory analysis of climatic variables. Climatic Change 148, 325–338 (2018).

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