Wavelet Coherence & Cross-Wavelet Transform

Squared Coherency

 = 0.1, 0.05, 0.01

Wavelet Coherence R2 looks like definition of correlation coefficient (localized correlation coefficient in frq-time space) 𝑹 𝒏 𝟐 𝐬 = 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔 𝟐 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿 𝒔 𝟐 ∗𝑺 𝒔 −𝟏 𝑾 𝒏 𝒀 𝒔 𝟐 Calculate Coherence from Wavelets of each time series S is smoothing operator, in time and frequency W X , W Y are the wavelets for each time series W XY is the cross wavelet Similar to a correlation coefficient varying in time/frequency Get 95% confidence level using chi-squared s-1 is used to convert to energy density (s = scale, n = time index)

Example

Example U, m/s

Wavelet Coherence Period, days

Phase of Coherence Phase arrows pointing: right: in-phase left: anti-phase down: X leading Y by 90° up: Y leading X by 90° 𝝓 𝒏 𝐬 = 𝐭𝐚𝐧 −𝟏 𝒊𝒎𝒂𝒈 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔 𝒓𝒆𝒂𝒍 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔

Coherence and Phase Days in 2010 right: in-phase; left: anti-phase; down: X leading Y by 90°; up: Y leading X by 90°

Cross-Wavelet Transform The Cross-Wavelet Transform of two time series xn and yn indicates common power and relative phase in time-frequency space The Cross-Wavelet Transform is given by: W XY = W X W Y*; * is complex conjugate |W XY | = cross-wavelet power arg (W XY ) = relative phase between xn and yn

Wavelet Coherence & Cross-Wavelet Tips Normally distributed histograms Consider transforming the time series, if the probability density functions of the time series are far from Gaussian. Know what to expect from the results, could find statistically significant links by chance Caution with results within the COI If significant coherence, phase lag should vary slowly and be consistent The significance level of the WTC has to be determined using Monte Carlo methods

References Grinsted, A., Moore, J.C., Jevrejeva, S. (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, doi:10.5194/npg-11-561-2004 [pdf] Jevrejeva, S., Moore, J.C.,  Grinsted, A. (2003) Influence of the Arctic Oscillation and El Niño-Southern Oscillation (ENSO) on ice conditions in the Baltic Sea: The wavelet approach, J. Geophys. Res., 108(D21), 4677, doi:10.1029/2003JD003417 [pdf] Torrence, C., Compo, G.P. (1998) A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79, 61–78 [pdf] Torrence, C., Webster, P. (1999)  Interdecadal changes in the ESNO-Monsoon System, J.Clim., 12, 2679–2690