2.6 Wavelet analysis
“Morlet wavelet” by Wang et al. (2014) and Zhang et al. (2015) was
used to conduct periodicity analysis on the annual and seasonal
ETref variations. Continuous wavelet analysis (CWA) has
localized in terms of time and frequency domains compared to moving
filter analysis and Fourier analysis. It can be used to assess locations
on time series dataset. For certain \(\varphi(t)\), the wavelet
transforms, wavelet coefficient and wavelet variance of meteorological
time series dataset \(\ \ \ f(k.t)\) (k =1, 2, ……, N,
which denotes the time series data) were defined as the following Eqs.
(9-11):
\(w_{f}\left(a,b\right)=\ \left|a\right|^{-1/2}t\sum_{k=1}^{n}{f(k.t)\varphi(\frac{k.t-b)}{\alpha}}\)……………………………………..(9)
\(w_{f}\left(a,b\right)=\ \left|a\right|^{-1/2}t\sum_{k=1}^{n}{f\left(k\right).e^{\text{ict}}}.e^{-t2/2}\)……………………………
………(10)
\(Var(a)=\ \int_{\infty}^{+\infty}{\left|w_{f}(a,b)\right|^{2}\text{db}}\)…………………………………………….
………(11)
Where, a=scale factor representing the cycle length (periodicity), b=
time factor representing the time location), Wf (a, b) =
the transformation coefficient, Var(a)= the variance of continuous
wavelet analysis c = the constant value
The transformation coefficient of CWA can be attained by
Wf (a, b). The information about strength and phase of
scale signals at various time period could be represented by the bar.
Negative value (anti-phase) presents higher period whereas positive
value (in-phase) denotes lower period. If Wf (a, b) = 0,
it equals to the breakpoint. The variation rises significantly with the
increasing of Wf (a, b).
After “Morlet wavelet” for CWA, then, we employed wavelet transform
coherence (WTC) to check the co-relationships of meteorological
parameters that influence the variation in annual ETref (Rahman and
Islam 2019; Uddin et al. 2020). WTC quantifies the covariance magnitude
between two time-series which varies from 0 to 1
(\(0\leq R^{2}\leq 1)\). 0 refers to no coherence at all whereas 1
refers to perfect coherence. WTC is employed to analyze the
co-relationship between two parameters. The calculated Eq. (12) for WTC
is outlined as follows:
\(R^{2}\left(m,n\right)=\frac{\left|S(s^{-1}W_{\text{xy}}(s)\right|^{2}}{S(s^{-1}\left|W_{x}\left(s\right)\right|^{2\ }.\ \ S(s^{-1}\left|W_{y}(s)\right|^{2}}\)……………………………….
(12)
where \(x\) and \(y\) are two time series with their respective wavelet
transforms \(W_{x}(s)\) and \(W_{y}(s)\). Monte-Carlo (MC) simulation is
applied for statistical significance in this study.