Bulletin of the American Meteorological Society: Vol. 79, No. 1, pp. 61–78.

A Practical Guide to Wavelet Analysis

Christopher Torrence and Gilbert P. Compo
Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado
Keywords: Wavelet analysis, significance and confidence testing, red noise power spectrum, El Niño-Southern Oscillation.

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 formulae 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 time scales. 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.

  1. Introduction
  2. Data
  3. Wavelet Analysis
    1. Windowed Fourier Transform
    2. Wavelet Transform
    3. Normalization
    4. Wavelet Power Spectrum
    5. Wavelet Functions
    6. Choice of Scales
    7. Cone of Influence (COI)
    8. Wavelet Scale and Fourier Frequency
    9. Reconstruction
  4. Theoretical Spectrum and Significance Levels
    1. Fourier Red Noise Spectrum
    2. Wavelet Red Noise Spectrum
    3. Significance Levels
    4. Confidence Interval
    5. Stationarity
  5. Smoothing in Time and Scale
    1. Averaging in Time (Global Wavelet Spectrum)
    2. Averaging in Scale
  6. Extensions to Wavelet Analysis
    1. Filtering
    2. Power Hovmöller
    3. Cross-Wavelet Spectrum
    4. Wavelet Coherence and Phase
  7. Summary
  8. Acknowledgments
  9. References