Bivariate Wavelet Analysis for Environmental Time Series
Introduction
Environmental time series often contain variability across multiple temporal scales. Examples include ocean temperature, atmospheric CO₂, plankton abundance and aerosol concentrations.
Fourier analysis decomposes a signal into frequencies but does not indicate when those frequencies occur. Wavelet analysis introduces time localization and therefore allows transient oscillations and scale dependent relationships to be analysed.
Fourier transform
Let (x(t)) be a signal.
The Fourier transform is
\[ \hat{x}(f)= \int_{-\infty}^{\infty} x(t)e^{-2\pi i f t}\,dt \]
The power spectrum is
\[ E(f)=|\hat{x}(f)|^2 \]
The Fourier kernel
\[ e^{-2\pi i f t} \]
oscillates indefinitely in time. Fourier analysis therefore identifies frequencies but does not indicate when those frequencies occur.
Time localization
To analyse oscillations that evolve in time we introduce a localized oscillatory function called a wavelet.
A wavelet can be translated in time and dilated in scale.
Continuous wavelet transform
Let ((t)) be a wavelet function.
The continuous wavelet transform of (x(t)) is
\[ W_x(a,b)= \frac{1}{\sqrt{a}} \int_{-\infty}^{\infty} x(t) \psi^*\!\left(\frac{t-b}{a}\right) dt \]
where
(a) : scale parameter
(b) : time translation
(^*) : complex conjugate.
Large scale corresponds to long period oscillations.
Small scale corresponds to short period oscillations.
Morlet wavelet
The Morlet wavelet is defined as
\[ \psi(t)= \pi^{-1/4} e^{-i\omega_0 t} e^{-t^2/2} \]
The function contains
a complex oscillation
a Gaussian envelope.
The Gaussian envelope ensures temporal localization because
\[ e^{-t^2/2} \rightarrow 0 \quad \text{as} \quad |t|\rightarrow \infty \]
Wavelet power spectrum
The local variance of the signal at time (b) and scale (a) is
\[ P_x(a,b)=|W_x(a,b)|^2 \]
This quantity is called the wavelet power spectrum.
Cross wavelet transform
For two signals (x(t)) and (y(t))
\[ W_{xy}(a,b)= W_x(a,b) W_y^*(a,b) \]
This quantity measures joint variability across time and scale.
Wavelet coherence
Wavelet coherence is a normalized measure of correlation in time scale space
\[ R^2(a,b)= \frac{|S(W_{xy}(a,b))|^2} {S(|W_x(a,b)|^2)\,S(|W_y(a,b)|^2)} \]
where (S()) denotes smoothing in time and scale.
The coherence coefficient satisfies
\[ 0 \le R^2 \le 1 \]
Phase relationships
Because the Morlet wavelet is complex it provides phase information.
The phase difference between two signals is
\[ \phi(a,b)= \tan^{-1} \left( \frac{\operatorname{Im}(W_{xy}(a,b))} {\operatorname{Re}(W_{xy}(a,b))} \right) \]
Interpretation
right direction indicates signals vary together
left direction indicates opposite behaviour
upward direction indicates the first signal leads
downward direction indicates the second signal leads.
References
Torrence C., Compo G. 1998
A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society.
Grinsted A., Moore J. C., Jevrejeva S. 2004
Application of cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics.