*2.3. Methods*

2.3.1. Calculation of the Standard Precipitation Evapotranspiration Index (SPEI)

The difference between precipitation and potential evapotranspiration (*PET*), *PD*, is a key parameter for SPEI calculation. *PET* was calculated using the Thornthwaite method because fewer meteorological elements are required [9], as follows:

$$PD\_{\bar{i}} = P\_{\bar{i}} - PET\_{\bar{i}} \tag{1}$$

$$PET = 16.0 \times \left(\frac{10Ti}{H}\right)^A \tag{2}$$

$$A = 6.75 \times 10^{-7} H^3 - 7.71 \times 10^{-5} H^2 + 1.79 \times 10^{-2} H + 0.492 \tag{3}$$

where *Pi* is the monthly precipitation of the *i*-th month, *PETi* the monthly evapotranspiration, *A* a constant, and *H* the annual heat index. The log-logistic function based on three parameters (*α*, *β*, *γ*) was used to perform the normal fitting to the time series of *PDi* and compute the probability distribution function *F*(*x*). The log-logistic probability distribution function is given as below:

$$F(\mathbf{x}) = \left[1 + \left(\frac{\alpha}{\mathbf{x} - \gamma}\right)^{\beta}\right]^{-1} \tag{4}$$

where α is the scale parameter, *β* the shape parameter, and *γ* the position parameter; all are obtained by the linear-moment method.

The probability distribution function was standardized to obtain the cumulative probability *Q* (Equation (5)):

$$Q = 1 - F(\mathbf{x})\tag{5}$$

and the SPEI value was then calculated as:

$$SPEI = \begin{cases} \left. w - \frac{a\_0 + a\_1 w + a\_2 w^2}{1 + d\_1 w + d\_2 w^2 + d\_3 w^3} \right\}, w = \sqrt{-2\ln(Q)} (Q \le 0.5) \\ - (w - \frac{a\_0 + a\_1 w + a\_2 w^2}{1 + d\_1 w + d\_2 w^2 + d\_3 w^3}), w = \sqrt{-2\ln(1 - Q)} (Q \ge 0.5) \end{cases} \tag{6}$$

where the constants *a*<sup>0</sup> = 2.515517, *a*<sup>1</sup> = 0.802853, *a*<sup>2</sup> = 0.010328, *d*<sup>1</sup> = 1.432788, *d*<sup>2</sup> = 0.189269, and *d*<sup>3</sup> = 0.001308. The degree of drought (Table 1) was classified according to the local climate conditions [21,34].

**Table 1.** Drought classification based on SPEI.


#### 2.3.2. Trend Analysis

The univariate linear regression equation (Equation (7)) was used for trend analysis to calculate the variation trend of SPEI during the growing seasons from 2000 to 2018:

$$\theta\_{slops} = \frac{n \times \sum\_{i=1}^{n} \left( i \times SPEI\_i \right) - \sum\_{i=1}^{n} i \sum\_{i=1}^{n} SPEI\_i}{n \times \sum\_{i=1}^{n} i^2 - \left( \sum\_{i=2}^{n} i \right)^2} \tag{7}$$

where *n* (*n* = 19) is the length of time series and *θ*slope is the slope in the linear regression equation. *θ*slope > 0 indicates that the drought trend is reduced; otherwise, the drought is aggravated. The variation trend of SPEI was divided into five levels based on the standard deviation (STD), i.e., significant degradation (*θ*slope < −STD), slight degradation (−STD < *θ*slope < −0.5STD), substantially unchanged (−0.5STD < *θ*slope < 0.5STD), slight improvement (0.5STD < *θ*slope < STD), and significant improvement (*θ*slope > STD).

#### 2.3.3. Geodetector

Geodetector is a spatial statistical model based on spatial autocorrelation theory to reveal the spatial differentiation of geographic elements and their driving factors [30]. We mainly used the factor detector, ecological detector, and interactive detector within the model. The factor detector quantifies the contribution of influencing factors to dependent variables, and it is calculated as follows:

$$q = 1 - \frac{SSW}{SST} \tag{8}$$

$$SSW = \sum\_{h=1}^{l} N\_h \sigma\_h^2 \, \text{SST} = N\sigma^2 \tag{9}$$

where *SSW* is the sum of factor variances over all layers and *SST* is the total sum of variance, where *h* = 1, ... *l* is the number of strata of the dependent variable or independent variable; *Nh* and *N* are the number of units in class *h* and the whole region, respectively; and *σ<sup>h</sup>* 2 and *σ*<sup>2</sup> are the variances of the dependent variable for the units in class h and the whole region, respectively. The larger the *q*-value is, the stronger the explanatory power of the factor to the drought phenomenon. The effective range of *q* is [0, 1].

The ecological detector uses an *F* test to measure the significant difference of the impact of different influencing factors on the spatial distribution of drought. The *F* value is determined as follows:

$$F = \frac{N\_{n=1}(N\_{n=2} - 1)\sigma\_{n-1}^2}{N\_{n=2}(N\_{n=1} - 1)\sigma\_{n-2}^2} \tag{10}$$

where *Nn*=1 and *Nn=*<sup>2</sup> refer to the sample size of two random factors, and *F* reflects the significance level.

The interaction detector was used to identify whether two driving factors, *x*<sup>1</sup> and *x*2, increase or decrease the explanatory power of the drought index SPEI when they work together (Table 2).



In addition to the influences of meteorological variables, droughts are also affected by other factors including geographic location, topography, soil, land cover type, human activities, etc. Land cover type affects runoff, infiltration, and evapotranspiration of surface water through water absorption [35]. We selected 12 potential drought driving factors as follows: Mean Air Temperature (MAT), Mean Precipitation (MP), Mean Wind Speed (MWS), and Mean Sunshine Duration (MSD) during the growing season, representing the meteorological conditions; Percent of Sand (POS) in soil, representing soil texture; elevation, slope, and slope aspect, representing topographic conditions; Distance to rivers (DTR), representing potential water availability; Distance to Prefecture Cities (DTC), Land-Use and Land-Cover Change (LUCC), and Average of population Density (AOPD), representing human factors that can transform and regulate the local environment [36]. These factors are easy to be quantified [25,37,38]. Since Geodetector can only handle discrete variables, the 12 variables need to be discretized individually. The LUCC factors were divided into 36 grades according to the land use type conversion maps from 2000 to 2018, the slope and aspect were divided into 9 grades, and each of the other 10 factors was divided into 6 grades by the Jenks Natural Breaks Classification Method (Figure 2).
