## 2 Materials and Methods
### Methods  
#### 2.3.1 Standardized Precipitation Index (SPI)  
Drought indices are critical for drought assessment because they facilitate a clear understanding of the complex interrelationships across multiple study areas. The SPI is a parameter solely based on precipitation data. The SPI is not directly normalized by the arithmetic mean and standard deviation in the calculation process. Instead, it is firstly fitted to the probability distribution suitable for describing the characteristics of precipitation, such as gamma distribution, in order to accurately portray its non-negativity, right skewedness, and zero-value characteristics [44]. Subsequently, it is standardized by transforming the probability values to the quantile of standard normal distribution through the cumulative distribution function [45]. This method fully considers the non-normal characteristics of precipitation data, and its theoretical basis lies in eliminating the influence of the original distribution through the probability distribution transformation. The process of enabling precipitation data at different spatial and temporal scales can be presented in a standardized and comparable form. This ensures the statistical reasonableness and reliability of drought assessment results. The SPI compares the precipitation for a particular period with the total precipitation for the same period over all recorded years, facilitating the temporal analysis of drought phenomena. The SPI is calculated as follows:  
Let the amount of precipitation at a given time (t) be denoted as x\({}_{0}\):  
\[\text{SPI}=\text{S}\Bigg{\{}\text{t}-\frac{(\text{c}_{2}\text{t}-\text{c}_{1 })\text{t}+\text{c}_{0}}{\big{[}\big{(}\text{d}_{3}\text{t}^{2}+\text{ d}_{2}\text{t}+\text{d}_{1}\big{)}\text{t}+1.0\big{]}}\Bigg{\}} \tag{1}\]  
\[(1.0\ \geq\ \text{F}\ >\ 0.5,\ \text{S}=1;\ \text{F}\ \leq\ 0.5,\ \text{S}=-1) \tag{2}\]  
\[\text{t}=\sqrt{\ln\frac{\text{T}}{\text{F}^{2}}} \tag{3}\]  
where F is the probability that the random variable x \(<\) x\({}_{0}\); c\({}_{0}\), c\({}_{1}\), c\({}_{2}\), d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\) are constants with values of c\({}_{0}\) = 2.515517, c\({}_{1}\) = 0.802853, c\({}_{2}\) = 0.010328, d\({}_{1}\) = 1.432788, d\({}_{2}\) = 0.189269, and d\({}_{3}\) = 0.001308, respectively. These values are derived from the classical polynomial approximation method found in mathematical manuals, which converts cumulative probabilities into normalized indices by simplifying the formula. The drought classifications for SPI are shown in Table 1.  
The nature of drought is the cumulative effect of precipitation shortages. Because the SPI calculation is based on precipitation sequences in different cumulative time windows, it can analyze multiple time scales. In drought studies, selecting the multi-time-scale SPI to calculate drought characteristics can provide a more comprehensive assessment of the drought situation in the study area. The shorter-time-scale SPI can quickly respond to precipitation changes and capture short-term drought events. But it only reflects changes over the last 30 days. In contrast, the longer-time-scale SPI can reflect the cumulative impacts of drought on water resources and assess long-term drought risks [46; 47]. Additionally, the multi-time-scale SPI can smooth short-term fluctuations, reduce misjudgments, integrate multiple factors, enhance the spatial and temporal comparability of drought characteristics, and facilitate the comparison and reflection of the spatial and temporal evolution of drought patterns across different regions. Using only a single time scale would fail to capture the immediate, persistent, and cumulative characteristics of meteorological droughts simultaneously, leading to a one-sided understanding of the drought formation mechanism and evolutionary process. Therefore, this study calculated SPIs for 30-, 60-, 90-, 180-, 270-, and 360-day time scales.