*2.3. Methods*

The SPEI is a drought index based on the probability model, which was constructed by combining the potential evapotranspiration (PET) with the SPI [58]. Given the input and output of water resources, the calculation results of the SPEI mainly depicted the excess or deficit of water in an ecosystem within a certain period. The computational procedure of the SPEI can be divided into the following—calculation of the potential evapotranspiration (PET) based on the Thornthwaite method; computation of the difference value *D* between the precipitation and evapotranspiration, and finally, normalization of the value *D*. The specific calculation methods were as follows:

Firstly, the potential evapotranspiration was calculated and the difference between the potential evapotranspiration and precipitation was defined as:

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

where *Di* is the difference between monthly precipitation *Pi* and potential evapotranspiration *PETi*.

Secondly, the *Di* data series was normalized—the SPEI, which is similar to SPI, adopts the log-logistic of three parameters to normalize the cumulative values of the sequence of *Di* data. The calculation formula was:

$$F(\mathbf{x}) = \left[ 1 + \left( \frac{\alpha}{\mathbf{x} - \mathbf{y}} \right)^{\beta} \right]^{-1} \tag{2}$$

where *F(x)* is the probability distribution function, α, β, and γ represent the respective ratio, shape and source parameter, which could all be estimated by the linear distance method.

Finally, the cumulative probability *P* for a given time scale was derived, and then the normalized value of SPEI was calculated. The equations were as follows:

$$P = 1 - F(\mathbf{x})\tag{3}$$

When *P*<0.5

$$\mathcal{W} = \sqrt{-2\ln(p)}\tag{4}$$

$$SPEI = \mathcal{W} - \frac{\mathbb{C}\_0 + \mathbb{C}\_1 \mathcal{W} + \mathbb{C}\_2 \mathcal{W}^2}{1 + d\_1 \mathcal{W} + d\_2 w^2 + d\_3 w^3} \tag{5}$$

When *P*>0.5

$$P = 1 - P\tag{6}$$

$$SPEI = -\left(\mathcal{W} - \frac{\mathbb{C}\_0 + \mathbb{C}\_1 \mathcal{W} + \mathbb{C}\_2 \mathcal{W}^2}{1 + d\_1 \mathcal{W} + d\_2 \mathcal{W}^2 + d\_3 \mathcal{W}^3}\right) \tag{7}$$

The constants included *C0* = 2.515517, *C1* = 0.802853, *C2* = 0.010328, *d1* = 1.432788, *d2* = 0.189269, and *d3* = 0.001308. A negative SPEI value indicates dryness whereas a positive value represents wetness. The Table 1 lists the SPEI-based drought index classification criteria [27].


**Table 1.** SPEI meteorological drought index classification [27].

One-dimensional linear regression was employed to analyze SPEI and NDVI in the study area to describe the spatio-temporal trends of SPEI and NDVI between 1982 and 2015 [59]. The calculation formula was:

$$slope = \frac{n\sum\_{i=1}^{n} \left(i \times \mathbb{C}\_{i}\right) - \sum\_{i=1}^{n} i \times \sum\_{i=1}^{n} \mathbb{C}\_{i}}{n \times \sum\_{i=1}^{n} i^{2} - \left(\sum\_{i=1}^{n} i\right)^{2}}\tag{8}$$

where *Slope* represents the changing trends of NDVI and SPEI, *n* is the study temporal interval, *n* = 34, and *Ci* represents SPEI or NDVI for the year *i*. A significance test was performed on the changing trends of NDVI and SPEI (*P* < 0.01 indicates an extremely significant change, *P* < 0.05 indicates a significant change, and *P* > 0. 05 indicated the change is not significant).

The correlation coefficient (*R*) was used to investigate the linear relationship between NDVI and SPEI at the pixel scale in this study, which was defined as:

$$R = \frac{\sum\_{i=1}^{n} \left[ (\mathbf{x} - \overline{\mathbf{x}})(\mathbf{y} - \overline{\mathbf{y}}) \right]}{\sqrt{\sum\_{i=1}^{n} (\mathbf{x}\_i - \overline{\mathbf{x}})^2 \sum\_{i=1}^{n} (y\_i - \overline{y})^2}} \tag{9}$$

where *xi* and *yi* represent the respective annual SPEI and NDVI values for the year *i*, *x* represents the mean annual SPEI values, and *y* represents the mean annual NDVI values. The significance test was used to illustrate the correlation between SPEI and NDVI (*P* < 0.01 indicates an extremely significant correlation, *P* < 0.05 indicates a significant correlation, and *P* > 0.05 indicates the correlation is not statistically significant).

Four statistical indicators were used in this study to evaluate the performance of the GLDAS-NOAH outputs in the YZR basin, which were the Pearson correlation coefficient (*R*), mean bias (*MB*), root-mean-square error (*RMSE*), and Nash-Sutcliffe efficiency coefficient (*NSE*). The Pearson correlation analysis was used to reflect the strength of the linear relationship between the compared datasets. The *MB* and *RMSE* revealed the degree of deviation of the paired data. The *MB* provided information on the absolute overestimation or underestimation of the two paired datasets, whereas the *RMSE* was a good reflection of the procedural precision. The *NSE* ranged from [−∞, 1], and the credibility of the simulation was much higher when it was approaching 1. These statistical indicators were defined as follows:

$$MB = \frac{1}{n} \sum\_{i=1}^{n} (x\_i - y\_i) \tag{10}$$

$$RMSE = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left(\mathbf{x}\_i - y\_i\right)^2} \tag{11}$$

$$NSE = 1 - \frac{\sum\_{i=1}^{n} (x\_o^i - x\_m^i)^2}{\sum\_{i=1}^{n} (x\_o^i - \overline{x\_o})^2} \tag{12}$$

where *n* is the number of the data, and *xi* nd *yi* are the observed data and GLDAS-NOAH data, respectively. In Equation (12), *xio* and *xim* separately represent the observed value and model simulated value of the variable; *xo* is the average value of the observed data. Generally, if the monthly *NSE* > 0.5 and monthly *R* > 0.77 (corresponding to the determination coefficient *R*<sup>2</sup> > 0.6), the model performance was considered to be acceptable [60].

As an effective and practical statistical method recommended by the World Meteorological Organization, the Mann-Kendall nonparametric test was applied to detect the significance of the trend. The detailed information can be obtained in [61–63]. In this study, the 0.1 significance level was used.
