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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