## 2 Materials and Methods
### Method  
#### 2.3.1 SPEI  
SPEI is a powerful and versatile tool for monitoring drought [33; 34]. Unlike traditional indices, such as SPI, which rely solely on precipitation data, the SPEI incorporates PET [37]. This allows it to capture the combined effects of precipitation deficits and temperature-induced water loss [35]. In comparison to scPDSI, the SPEI offers the advantage of being multi-scalar, which is essential for effective drought analysis and monitoring [33; 37]. In the context of global climate change and rising temperatures, the SPEI surpasses traditional indices by integrating both precipitation and temperature-driven evapotranspiration [55]. The SPEI can be applied across various temporal and spatial scales, making it suitable for drought monitoring in diverse climate zones and agricultural regions [3; 7; 56]. Its flexibility positions it as a crucial tool in a range of drought assessment and early warning systems [37; 55].  
Generally, higher SPEI values signify wetter conditions, whereas lower SPEI values denote drier conditions [57]. Based on the drought classification standard of the SPEI index, SPEI values are divided into five categories [58] (Table 1). When the SPEI value is less than or equal to \(-0.5\), it indicates the occurrence of drought. On the contrary, there is no drought.  
The SPEI index is calculated as follows [33]:  
Calculate the difference between precipitation and potential evapotranspiration:  
\[D_{i,j}=P_{i,j}-PET_{i,j} \tag{1}\]  
where \(i\) is the year, \(j\) is the month, \(P\) is precipitation, and \(PET\) is potential evapotranspiration.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**SPEI** & **SPEI \(\leq\)\(-\)2.0** & \(\begin{array}{c}-\)**2.0 \(<\) SPEI** \\ \(\leq\)\(-\)1.5** \\ \end{array} & \(\begin{array}{c}-\)**1.5 \(<\) SPEI** \\ \(\leq\)\(-\)1.0** \\ \end{array} & \(\begin{array}{c}-\)**1.0 \(<\) SPEI** \\ \(\leq\)\(-\)0.5** \\ \end{array} & \(\begin{array}{c}-\)**1.0 \(<\) SPEI** \\ \(-\)0.5 \(<\) SPEI** \\ \end{array} \\ \hline Drought & Extreme & Severe & Moderate & Mild & No Drought \\ Severity & Drought & Drought & Drought & Drought & No Drought \\ \hline \hline \end{tabular}
\end{table}
Table 1: SPEI drought classification.  
Construct cumulative water surplus/deficit series for different time scales of the study data:  
\[D_{i,j}^{k}=\sum_{i=0}^{k-1}\bigl{(}P_{i,j-1}-PET_{i,j-1}\bigr{)} \tag{2}\]  
where \(D_{i,j}^{k}\) is the cumulative value of the difference between precipitation and potential evapotranspiration at time scale \(k\).  
The probability density function of a three-parameter log-logistic distributed variable is expressed as  
\[f(x)=\frac{\beta}{\alpha}(\frac{x-\gamma}{\alpha})^{\beta-1}[1+\left(\frac{x- \gamma}{\alpha}\right)^{\beta}]^{-2} \tag{3}\]  
where \(\alpha\) is the scale parameter, \(\beta\) is the shape parameter, and \(\gamma\) is the position parameter.  
The probability distribution function of the \(D\) series, according to the log-logistic distribution, is given by  
\[F(x)=\int_{0}^{x}f(t)dt=[1+\left(\frac{\alpha}{x-\gamma}\right)^{\beta}]^{-1} \tag{4}\]  
SPEI is calculated as follows [33]:  
\[\begin{cases} SPEI=\omega-\frac{C_{0}+C_{1}\omega+C_{1}\omega^{2}}{1+\omega d_{1 }+d_{2}\omega^{2}+d_{2}\omega^{3}},\ \omega=\sqrt{-2\ln p}\ p\leq 0.5\\ SPEI=\frac{C_{0}+C_{1}\omega+C_{1}\omega^{2}}{1+\omega d_{1}+d_{2}\omega^{2}+d _{2}\omega^{3}}-\omega,\omega=-\sqrt{-2\ln(1-p)}\ p\leq 0.5\end{cases} \tag{5}\]  
where \(\mathrm{P}=1-\mathrm{F}(x)\), \(c_{0}=2.515517\), \(c_{1}=0.802853\), \(c_{2}=0.010328\), \(d_{1}=1.432788\), \(d_{2}=0.189269\), \(d_{3}=0.001308\).