A Practical Guide to Wavelet Analysis

TLDR

The paper provides a practical, step‑by‑step guide to wavelet analysis using ENSO time‑series examples. The guide details methodological aspects such as comparing wavelet analysis to the windowed Fourier transform, selecting appropriate wavelet bases, addressing edge effects, linking scale to frequency, developing statistical significance tests for white and red noise, and extending the technique to filtering, power Hovmöller, cross‑wavelet spectra, and coherence. The authors demonstrate that smoothing improves confidence in wavelet spectra, provide empirical formulas for its effect, and show that ENSO variance has significantly higher power during 1880–1920 and 1960–90, lower during 1920–60, with a possible 15‑year modulation, while sea‑level pressure exhibits significant 2–8‑year wavelet power variations across longitude and time.

Abstract

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.