Modulated Haar Wavelet Analysis of Climatic Background Noise
- 130 Downloads
- 1 Citations
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.
Keywords
Climatic background noise Wavelet power spectrum Wavelet phase Haar waveletReferences
- 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) CrossRefGoogle 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) MATHMathSciNetCrossRefGoogle 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) CrossRefGoogle 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) CrossRefGoogle 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) CrossRefGoogle Scholar
- 8.Plaut, G., Ghil, M., Vautard, R.: Interannual and interdecadal variability in 335 years of central England temperatures. Science 268, 710–713 (1995) CrossRefGoogle Scholar
- 9.Torrence, C., Compo, G.P.: A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61–78 (1998) CrossRefGoogle Scholar