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Christopher Torrence and Gilbert P. Compo

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Niño–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence are described.

The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal timescales. Using new datasets that extend back to 1871, the Niño3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880–1920 and 1960–90, and lower power during 1920–60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2–8-yr wavelet power in both longitude and time.

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Christopher Torrence and Peter J. Webster


The El Niño–Southern Oscillation (ENSO) and Indian monsoon are shown to have undergone significant interdecadal changes in variance and coherency over the last 125 years. Wavelet analysis is applied to indexes of equatorial Pacific sea surface temperature (Niño3 SST), the Southern Oscillation index, and all-India rainfall. Time series of 2–7-yr variance indicate intervals of high ENSO–monsoon variance (1875–1920 and 1960–90) and an interval of low variance (1920–60). The ENSO–monsoon variance also contains a modulation of ENSO–monsoon amplitudes on a 12–20-yr timescale.

The annual-cycle (1 yr) variance time series of Niño3 SST and Indian rainfall is negatively correlated with the interannual ENSO signal. The 1-yr variance is larger during 1935–60, suggesting a negative correlation between annual-cycle variance and ENSO variance on interdecadal timescales.

The method of wavelet coherency is applied to the ENSO and monsoon indexes. The Niño3 SST and Indian rainfall are found to be highly coherent, especially during intervals of high variance. The Niño3 SST and Indian rainfall are approximately 180° out of phase and show a gradual increase in phase difference versus Fourier period. All of the results are shown to be robust with respect to different datasets and analysis methods.

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