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.