Abstract.Many scientists have made use of the wavelet method in analyzing time series, often using popular free software. However, at present there are no similar easy to use wavelet packages for analyzing two time series together. We discuss the cross wavelet transform and wavelet coherence for examining relationships in time frequency space between two time series. We demonstrate how phase angle statistics can be used to gain confidence in causal relationships and test mechanistic models of physical relationships between the time series. As an example of typical data where such analyses have proven useful, we apply the methods to the Arctic Oscillation index and the Baltic maximum sea ice extent record. Monte Carlo methods are used to assess the statistical significance against red noise backgrounds. A software package has been developed that allows users to perform the cross wavelet transform and wavelet coherence

[1] We analyze the Permanent Service for Mean Sea Level (PSMSL) database of sea level time series using a method based on Monte Carlo Singular Spectrum Analysis (MC-SSA). We remove 2-30 year quasi-periodic oscillations and determine the nonlinear long-term trends for 12 large ocean regions. Our global sea level trend estimate of 2.4 ± 1.0 mm/yr for the period from 1993 to 2000 is comparable with the 2.6 ± 0.7 mm/yr sea level rise calculated from TOPEX/Poseidon altimeter measurements. However, we show that over the last 100 years the rate of 2.5 ± 1.0 mm/yr occurred between 1920 and 1945, is likely to be as large as the 1990s, and resulted in a mean sea level rise of 48 mm. We evaluate errors in sea level using two independent approaches, the robust bi-weight mean and variance, and a novel ''virtual station'' approach that utilizes geographic locations of stations. Results suggest that a region cannot be adequately represented by a simple mean curve with standard error, assuming all stations are independent, as multiyear cycles within regions are very significant. Additionally, much of the between-region mismatch errors are due to multiyear cycles in the global sea level that limit the ability of simple means to capture sea level accurately. We demonstrate that variability in sea level records over periods 2-30 years has increased during the past 50 years in most ocean basins.

[1] Variability in time series of ice conditions in the Baltic Sea is examined within the context of atmospheric circulation represented by the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO) winter indices using the wavelet approach. We develop methods of assessing statistical significance and confidence intervals of cross-wavelet phase and wavelet coherence. Cross-wavelet power for the time series indicates that the times of largest variance in ice conditions are in excellent agreement with significant power in the AO at 2.2-3.5, 5.7-7.8, and 12-20 year periods, similar patterns are also seen with the Southern Oscillation Index (SOI) and Niño3 sea surface temperature (Niño3) series. Wavelet coherence shows in-phase linkages between the 2.2-7.8 and 12-20 year period signals in both tropical and Arctic atmospheric circulation and also with ice conditions in the Baltic Sea. These results are consistent with GCM simulations showing dynamical connections between high-latitude surface conditions, tropical sea surface temperatures mediated by tropical wave propagation, the wintertime polar vortex, and the AO and with models of sea ice and oceanic feedbacks at decadal periods.

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