Rectified Wavelet Power Spectrum

Why & how to rectify the bias in the wavelet power spectrum?

For a time series comprised of sine waves with the same amplitude but different frequencies the widely adopted wavelet method [e.g., Torrence and Compo, 1998] does not produce a spectrum with identical peaks (see the middle panels of the figure to the right and the FAQs of the wavelet toolbox), in contrast to a Fourier analysis.  This wavelet bias problem is addressed in Liu et al. [2007].  It is demonstrated that a physically consistent definition of energy for the wavelet power spectrum should be the transform coefficient squared divided by the scale it associates. Thus, a bias rectification is proposed, i.e., the wavelet power spectrum should be divided by its scales.

Such adjusted wavelet power spectrum results in a substantial improvement in the spectral estimate (see the bottom panels of the figure to the left), allowing for a comparison of the spectral peaks across the scales/frequencies/periods. This rectification is validated with an artificial time series and a real coastal sea level record at St. Petersburg, Florida. 

Also re-examined is the example of the wavelet analysis of the Niño-3 SST data.  Compared to the original wavelet spectrum, there is a slight adjustment of the relative magnitude of the wavelet spectrum across the frequency domain in the rectified spectrum. The adjusted wavelet spectrum overlays better with the 95% significance contours. The spectral line of the time averaged spectrum also becomes smoother. However, the major peak is still located between the 2- and 8-yr periods. Thus the main conclusion on the Niño-3 SST wavelet analysis in Torrence and Compo [1998] is still correct. That may be the main reason why the bias problem had been overlooked for decades. For details, see Liu et al. [2007].


  • Liu, Y., X.S. Liang, and R.H. Weisberg, 2007: Rectification of the bias in the wavelet power spectrum. Journal of Atmospheric and Oceanic Technology, 24(12), 2093-2102, doi:10.1175/2007JTECHO511.1.
  • Torrence, C., and G.P. Compo, 1998: A practical guide to wavelet analysis. Bulletin of American Meteorological Society, 79, 61-78. (The wavelet toolbox).

Artificial time series (top) comprising of five sine waves, with a unit amplitude and five different periods (1, 8, 32, 128, and 365 days). The original (middle) and rectified (bottom) wavelet power spectra (left column) and time-averaged wavelet power spectra (right column) of the artificial time series. Red and blue indicate high and low wavelet power spectra, respectively. (Addapted from Figure 2 of Liu et al. [2007])

How to implement the bias rectification in wavelet analysis software?

power_no_bias = wave@power/conform(power,wave@scale,0)

gws_no_bias = wave@gws/wave@scale

A frequent mistake is found in referencing the paper as "Liu, Y., X. San Liang, and R.H. Weisberg, 2017" in some literature. Actually, the second author's lastname is "Liang", not "San Liang". The correct citation is "Liu, Y., X.S. Liang, and R.H. Weisberg, 2017". View citations of this work on Google Scholar.

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(Last updated on 7/21/2017)

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