Application of the cross wavelet transform and wavelet coherence to geophysical time series

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 coher- ence for examining relationships in time frequency space be- tween two time series. We demonstrate how phase angle statistics can be used to gain confidence in causal relation- ships 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 (http://www.pol.ac.uk/home/research/waveletcoherence/). As we are interested in extracting low s/n ratio signals in time series we discuss only CWT in this paper. While CWT is a common tool for analyzing localized intermittent os- cillations in a time series, it is very often desirable to ex- amine two time series together that may be expected to be linked in some way. In particular, to examine whether re- gions in time frequency space with large common power have a consistent phase relationship and therefore are sug- gestive of causality between the time series. Many geophys- ical time series are not Normally distributed and we suggest methods of applying the CWT to such time series. From two CWTs we construct the Cross Wavelet Transform (XWT) which will expose their common power and relative phase in time-frequency space. We will further define a measure of Wavelet Coherence (WTC) between two CWT, which can find significant coherence even though the common power is low, and show how confidence levels against red noise back- grounds are calculated. We will present the basic CWT theory before we move on to XWT and WTC. New developments such as quanti- fying the phase relationship and calculating the WTC sig- nificance level will be treated more fully. When using the methods on time series it is important to have solid mecha- nistic foundations on which to base any relationships found, and we caution against using the methods in a "scatter-gun" approach (particularly if the time series probability density functions are modified). To illustrate how the various meth- ods are used we apply them to two data sets from meteo- rology and glaciology. Finally, we will provide links to a MatLab software package.