## 2 Materials and Methods
### Definition of Compound Droughs  
The VPD is defined by the difference between saturation (\(e_{s}\)) and actual (\(e_{s}\)) vapor pressure and is an effective indicator of atmospheric drought [6; 8; 9]. The relationship between saturation vapor pressure and air temperature is expressed as follows:  
\[e^{0}(T)=0.6108\cdot\exp\left(\frac{17.27\cdot T}{T+237.3}\right) \tag{1}\]  
where \(e^{0}(T)\) is the saturation vapor pressure (kPa) at an air temperature of T (\({}^{\circ}\)C). The Food and Agriculture Organization recommends the following formula [35] to calculate the saturation vapor pressure (\(e_{s}\)):  
\[e_{s}=\frac{e^{0}(T_{max})+e^{0}(T_{min})}{2} \tag{2}\]  
where \(e_{s}\) is the average saturation vapor pressure at the daily \(T_{max}\) and \(T_{min}\). The actual vapor pressure is calculated as:  
\[e_{s}=e_{s}.\frac{RH}{100} \tag{3}\]  
where \(RH\) is the relative humidity (0-100).  
The standardized precipitation index (SPI), based on a two-parameter gamma distribution, is widely used to identify meteorological drought conditions over a range of various timescales [27; 36; 37; 38]. The SPI at short timescales can reflect SM conditions, and the SPI at long timescales can describe runoff, groundwater, and reservoir conditions. Here, we converted the monthly VPD and root-zone SM into the standardized vapor pressure deficit index (SVPDI) and standardized soil moisture index (SSMI) via a similar SPI calculation method (i.e., the two-parameter gamma distribution). The probability density distribution \(f(x)\) and cumulative distribution \(F(x)\) functions of the gamma distribution can be calculated as follows [37; 39]:  
\[f(x)=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta} \tag{4}\]  
\[F(x)=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_{0}^{\infty}x^{\alpha-1}e^{-x/ \beta}dx \tag{5}\]  
where \(\beta\) is a scale parameter, \(\alpha\) is a shape parameter, \(x\) denotes the monthly data, and \(\Gamma(\alpha)\) is the gamma function. In this study, the SVPDI and SSMI were used to describe the dry and wet conditions of monthly-scale atmospheric and soil conditions in the TRHR. The wet and dry classification criteria for the SVPDI and SSMI are similar to those of the SPI [37], as shown in Table 1.  
Compound atmospheric and soil droughs, referred to here as simply \"compound droughs\", were defined in this study as simultaneously meeting the definition of atmospheric drought and soil drought for the same month at a given grid. Generally, atmospheric droughs can occur when the VPD is high. We adopted 1-month SVPDI \(\geq\) 0.5 as the threshold for an atmospheric drought. Similarly, soil droughs can occur when SM is low. We used  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**SVPDI Value** & **SSMI Value** & **Category** \\ \hline \((-\infty,-1.5]\) & \([1.5,+\infty)\) & Severely wet \\ \((-1.5,-1.0]\) & \([1.0,1.5)\) & Moderately wet \\ \((-1.0,-0.5]\) & \([0.5,1.0)\) & Slightly wet \\ \((-0.5,0.5)\) & \((-0.5,0.5)\) & Normal \\ \([0.5,1.0)\) & \((-1.0,-0.5]\) & Mild drought \\ \([1.0,1.5)\) & \((-1.5,-1.0]\) & Moderate drought \\ \([1.5,+\infty)\) & \((-\infty,-1.5]\) & Severe drought \\ \hline \hline \end{tabular}
\end{table}
Table 1: Wet and dry (drought) categories according to SVADI/SSMI values.  
1-month SSMI \(\leq-0.5\) as the threshold for a soil drought. We used 12-month SVPDI/SSMI values in December of each year and 5-month SVPDI/SSMI values in September of each year to identify atmospheric/soil droughts throughout each whole year and their GS.