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Modulated Haar Wavelet Analysis of Climatic Background Noise

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Abstract

We model background noise in time series of climate data as AR(1) red noise. To extract significant features of these time series, we use comparisons of wavelet power spectrum time series with that of AR(1) red noise. We improve on earlier work which has so far only relied on empirical formulas for the distribution of AR(1) red noise in Morlet wavelet power spectrum; see Torrence and Compo (Bull. Am. Meteorol. Soc. 79:61–78, 1998). Although wavelet phase distributions do play an important role in analysis of causal relations in climate time series, up to now, the wavelet phase distribution of AR(1) red noise has not been rigorously established for wavelet transforms. In this paper, we derive the formulas for both the wavelet power spectrum distribution, and the wavelet phase distribution of modulated Haar wavelet background noise. We give rigorous proofs.

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References

  1. IPCC: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  2. Chui, C.K.: An Introduction to Wavelet. Academic Press, San Diego (1992)

    Google Scholar 

  3. Mann, M.E., Lees, J.M.: Robust estimation of background noise and signal detection in climatic time series. Clim. Change 33, 409–445 (1996)

    Article  Google Scholar 

  4. Narasimha, R., Bhattacharyya, S.: A wavelet cross-spectral analysis of solar-ENSO-rainfall connections in the Indian monsoons. Appl. Comput. Harmon. Anal. 28(3), 285–295 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kutzbach, J.E., Bryson, R.A.: Variance spectrum of Holocene climatic fluctuations in the North Atlantic Sector. J. Atmos. Sci. 31, 1958–1963 (1974)

    Article  Google Scholar 

  6. Allen, M.R., Smith, L.A.: Investigating the origins and significance of low-frequency modes of climate variability. Geophys. Res. Lett. 21, 883–886 (1994)

    Article  Google Scholar 

  7. Gilman, D.L., Fuglister, F.J., Mitchell, J.M.: On the power spectrum of red noise. J. Atmos. Sci. 20, 182–184 (1963)

    Article  Google Scholar 

  8. Plaut, G., Ghil, M., Vautard, R.: Interannual and interdecadal variability in 335 years of central England temperatures. Science 268, 710–713 (1995)

    Article  Google Scholar 

  9. Torrence, C., Compo, G.P.: A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61–78 (1998)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Global Change and Earth System Science, Beijing Normal University, Beijing, 100875, China

    Zhihua Zhang

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Palle E. T. Jorgensen

Authors
  1. Zhihua Zhang
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  2. Palle E. T. Jorgensen
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Corresponding author

Correspondence to Zhihua Zhang.

Additional information

This research is partially supported by National Key Science Program for Global Change Research No. 2013CB956604 and No. 2015CB953600; Fundamental Research Funds for the Central Universities (Key Program) No. 105565GK; Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry; and Polar Climate and Environment Key Laboratory.

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Zhang, Z., Jorgensen, P.E.T. Modulated Haar Wavelet Analysis of Climatic Background Noise. Acta Appl Math 140, 71–93 (2015). https://doi.org/10.1007/s10440-014-9979-8

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  • DOI: https://doi.org/10.1007/s10440-014-9979-8

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