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.