(A) Extraction of annual seasonality anomaly (*Φ*(*t*,n)) from long term Lake Baikal surface temperature time series. (B) The time series is mean-subtracted and then a sliding window of varying length, in this case 60 months (*n* = 60), is extracted sequentially as the window is passed down the length of the series. (C) At each point in time (*t*), the small window is Fourier transformed and the magnitude and phase spectra of the harmonic components are estimated. The annual seasonality is the peak at 0.083^{ ^} cycles month^{−1}, and *Φ*(1971.83^{ ^},60) is, in this example, the phase of the annual harmonic for a window 60 months long, centered at October 1971. (D and E) Time series of lake temperature (gray) compared to a single annual harmonic with a single phase estimated for the entire 58 year time series (black) to indicate the local phase anomaly, with examples of “early” and “late” seasons indicated by phase. The 60 month period 1955–1960 (D) has negative *Φ* values, indicating that lake temperature variations are advanced relative to the long term seasonality. This is observable as particularly cold Fall seasons (water temperature falling). The 60 month period 1996–2001 (E) has positive *Φ* values, indicating that lake temperature variations are delayed relative to the long term seasonality. This is observable as particularly cold Spring seasons (water temperature rising). (F) Time series of *Φ*(*t*,60) assembled by repeatedly estimating the phase of the annual harmonic as the 60 month window is passed down the data set. This time series would then be pre-whitened (noise addition) to reduce auto-correlation for cross-correlation analysis.

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