## 2 Materials and Methods
### Methods
#### 2.3.2 Standardized Precipitation Evapotranspiration Index (SPEI)  
The Standardized Precipitation Evapotranspirationigration Index (SPEI) developed in [21] is a meteorological drought index that combines precipitation and potential evapotranspiration values over multiple timescales (1, 3, 6, and 12 months). Based on the climatic water balance, the SPEI compares monthly precipitation (P) and potential evapotranspiration (PET), and then aggregates these values into standard deviations from the mean [53]. Compared to the SPI, the SPEI is more effective for assessing drought severity because it accounts for the impact of extreme temperatures and variability in the context of global warming [21; 54]. Using the SPEI at different timescales captures various aspects of drought. SPEI-1 and SPEI-3 assess short-term humidity conditions and estimate seasonal precipitation as indicators of meteorological drought [55]. SPEI-6 characterizes medium-term agricultural droughs and evaluates river and reservoir levels [55], while SPEI-12 reflects hydrological drought by considering the effects of prolonged water shortage in rivers, lakes, and reservoirs [55]. We calculated the SPEI based on the difference between precipitation and evapotranspiration, with PET estimated using [PERSON]'s method [56], which relies on monthly and annual temperature data [57]. This method requires fewer meteorological inputs than alternatives like Hargreaves, which also involve minimum and maximum temperatures and solar radiation [58]. [PERSON]'s method, primarily based on temperature, is widely used in regions with limited meteorological data, such as our study area. The difference (\(D_{i}\)) between precipitation (\(P_{i}\)) and potential evapotranspiration (\(PET_{i}\)) for each month \(i\) was calculated (Equation (1)), and then aggregated at different timescales \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistic Metrics** & **Formula** & \
\begin{tabular}{c} **Range** \\ **Value** \\ \end{tabular} &
\begin{tabular}{c} **Optimal** \\ **Value** \\ \end{tabular} & **Unit** \\ \hline BIAS & \(BIAS\)\(=\)\(\frac{1}{N}\sum_{i=1}^{N}(C_{i}-O_{i})\) & \(-\infty\) to \(\infty\) & 0 & - \\ \(r\) & \(r\)\(=\)\(\frac{\sum_{i=1}^{N}(C_{i}-\overline{C})^{2}(O_{i}-\overline{O})}{\sqrt{\sum_{i=1} ^{N}(C_{i}-\overline{C})^{2}}}\) & \(-1\) to \(1\) & \(-1\) or \(1\) & - \\ RMSE & \(RMSE\)\(=\)\(\sqrt{\frac{1}{N}\sum_{i=1}^{N}(C_{i}-O_{i})}^{2}\) & \(0\) to \(\infty\) & 0 & mm \\ MAE & \(MAE\)\(=\)\(\frac{1}{N}\sum_{i=1}^{N}|C_{i}-O_{i}|\) & \(0\) to \(\infty\) & 0 & mm \\ \(R^{2}\) & \(R^{2}\)\(=\)\(1-\frac{\sum_{i=1}^{N}(O_{i}-\overline{C})^{2}}{\sum_{i=1}^{N}(O_{i}- \overline{O})^{2}}\) & \(0\) to \(1\) & 1 & - \\ \hline \hline \end{tabular} where \(C_{i}\) represents the rainfall estimates from CHIRPS, \(O_{i}\) is the observed rainfall from rain gauges, \(\overline{C}\) is the mean of the estimated rainfall, \(\overline{O}\) is the mean of the observed rainfall, and \(N\) is the number of samples.  
\end{table} 
Table 2: Statistical metrics used for evaluation. (\(D_{k}\)), where \(k\) is the number of months or the timescale of interest, and \(n\) is the month of calculation (Equation (2)) [21]. \[D_{i}=P_{i}-PET_{i} \tag{1}\]  
\[D^{k}=\sum_{i=0}^{k-1}P_{n-i}-PET_{n-i},n\geq k \tag{2}\]  
Aggregating the monthly differences between precipitation (\(P\)) and potential evapotranspiration (\(PET\)) can lead to significant seasonal variations in the SPEI series. To reduce these seasonal effects, we applied the log-logistic probability distribution function, as represented by Equation (3), to transform the \(D\) series into a standard normal distribution with a mean of 0 and a standard deviation of 1. The parameters \(\alpha\), \(\beta\), and \(\gamma\) representing the distribution's scale, shape, and origin, respectively, were used in this transformation [21, 59]. \[FX=\left[1+\big{(}\frac{\alpha}{X-\gamma}\big{)}^{\beta}\right]^{-1} \tag{3}\]  
Table 3 presents the categories for the SPEI classes.