2.2 Wavelet Analysis
The concepts of wavelet analysis have been reviewed in previous publications (Daubechies, 1990; Labat, 2005; Torrence & Comp, 1998). Briefly, similar to Fourier analysis, wavelet analysis extracts frequency information (called scales) from time series. Wavelet analysis also reveals the timing of the features. For this work, we adopted the algorithm of wavelet transform from Torrence and Compo (1998). The wavelet power spectrum is then defined as\(\left|W_{n}\left(s\right)\right|^{2}\) and the phase angle is\(atan2\left(\frac{\text{Imag}\left\{W_{n}\left(s\right)\right\}}{\text{Real}\left\{W_{n}\left(s\right)\right\}}\right)\), where atan2 is the four-quadrant inverse tangent function,\(\text{Real}\left\{W_{n}\left(s\right)\right\}\) is the real part of the continuous wavelet transform \(W_{n}\left(s\right)\), and\(\text{Imag}\left\{W_{n}\left(s\right)\right\}\) denotes the imaginary part. The wavelet power spectrum provides insight into the temporal-scale variability of the time series.
The global wavelet spectrum is defined as the time-averaged wavelet spectrum over all the local wavelet spectra:
\({\overset{\overline{}}{\mathbf{W}}}^{\mathbf{2}}\left(\mathbf{s}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{N}}\sum_{\mathbf{n=0}}^{\mathbf{N-1}}\left|\mathbf{W}_{\mathbf{n}}\left(\mathbf{s}\right)\right|^{\mathbf{2}}\). (1)
where N is the number of points in the time series.
Given two time series X and Y , with wavelet transforms\(W_{n}^{X}\left(s\right)\) and \(W_{n}^{Y}\left(s\right)\), the cross-wavelet spectrum is defined as:
\(\mathbf{W}_{\mathbf{n}}^{\mathbf{\text{XY}}}\left(\mathbf{s}\right)\mathbf{=}\mathbf{W}_{\mathbf{n}}^{\mathbf{X}}\left(\mathbf{s}\right)\mathbf{W}_{\mathbf{n}}^{\mathbf{Y}^{\mathbf{*}}}\left(\mathbf{s}\right)\)(2)
where \(W_{n}^{Y^{*}}\left(s\right)\) is the complex conjugate of\(W_{n}^{Y}\left(s\right)\). The cross-wavelet power is\(\left|W_{n}^{\text{XY}}\left(s\right)\right|\), and indicates covariance between the time series at all time scales. The wavelet coherence is defined as the square of the cross-spectrum,\(\left|W_{n}^{\text{XY}}\left(s\right)\right|^{2}\) normalized by the individual power spectra. The wavelet coherence ranges between 0 and 1 and provides a qualitative estimator of the temporal evolution of the degree of linearity of the relationship between two signals on a given temporal scale. The coherence phase is therefore defined as\(\text{atan}\left(\frac{\text{Imag}\left\{W_{n}^{\text{XY}}\left(s\right)\right\}}{\text{Real}\left\{W_{n}^{\text{XY}}\left(s\right)\right\}}\right)\).
To derive the phase, \(\phi\), the difference in timing between two time series at their maximum power spectrum, we multiplied the coherence phase in proportion to 2\(\pi\) by their coherent period at their maximum cross wavelet power spectrum band:
\(\mathbf{\phi=atan}\mathbf{2}\left(\frac{\mathbf{\text{Imag}}\left\{\mathbf{W}_{\mathbf{n}}^{\mathbf{\text{XY}}}\left(\mathbf{s}_{\mathbf{\max}}\right)\right\}}{\mathbf{\text{Real}}\left\{\mathbf{W}_{\mathbf{n}}^{\mathbf{\text{XY}}}\left(\mathbf{s}_{\mathbf{\max}}\right)\right\}}\right)\frac{\mathbf{S}_{\mathbf{\max}}}{\mathbf{2}\mathbf{\pi}}\)(3)
where \(\mathbf{s}_{\mathbf{\max}}\) means the time scale (or period) of the maximum cross wavelet power spectrum band. And the values range of\(atan2\) is [–\(\pi\), \(\pi\)]. Although the phase difference has been defined when the cross-wavelet analysis method was developed, there are few studies on its application and to the best of the authors’ knowledge, there was no previous research used this technique to analyze the relationships among P , ET , and TWSA in Amazon.
We performed wavelet analyses for ET , P , and TWSAand their coherences for four spatial resolutions: (1) averaged across the Amazon basin, (2) for 33 watersheds within the Amazon basin, (3) for individual 1 km scale grid cells, and (4) averaged for three zones in the Amazon basin. The underlying resolutions of P , ET , andTWSA are 0.25-degree, 1 km, and 1-degree respectively as described in Section 2.1, but P and TWSA were interpolated to the same 1 km resolution as ET to calculate the 1 km scale phases. Comparisons among these spatial averaging units allow us to analyze the behavior of the coherences at different spatial scales. Hereafter, we define the phases between ET and P at the different spatial scales as\(\phi_{ET-P}^{\text{Amazon}}\),\(\ \phi_{ET-P}^{\text{subbas}},\)\(\phi_{ET-P}^{1km}\), and \(\phi_{ET-P}^{\text{Zone}}\), and the phases between ET and TWSA as\(\phi_{ET-TWSA}^{\text{Amazon}}\),\(\phi_{ET-TWSA}^{\text{subbas}},\ \phi_{ET-TWSA}^{1km}\), and\(\phi_{ET-TWSA}^{\text{Zone}},\) for the whole Amazon basin, sub-basins, the individual grid cells, and three zones, respectively. A positive \(\phi_{ET-P}^{\text{Amazon}}\) would mean that ETsignal leads that of P .
For simplicity and convenience, the time scales and wavelet power spectrum are presented using the 2-based logarithmic scale in all figures shown in the results sections below.