## 2 Materials and Methods
### TOPMODEL Model  
#### 2.3.1 Topmodel Model Principle  
TOPMODEL model is based on the principle of variable runoff area. The model determines the size and location of the runoff source area through soil water content. The whole hydrological process is mainly described by water balance and Darcy law, and the model is simplified by using three critical assumptions:  
(1) Assumption 1: The watershed has a saturated layer area with a stable water supply.  
(2) Assumption 2: The relationship between soil hydraulic conductivity and water scarcity is exponentially decreasing, so [60]  
\[T_{i}=T_{0}\times e^{-\frac{z_{i}}{z_{min}}} \tag{17}\]  
where \(T_{0}\) is the saturated water conductivity of the basin, and it is considered that the saturated water conductivity is evenly distributed in the whole catchment. \(S_{zm}\) is the maximum water storage depth in the unsaturated zone. \(T_{i}\) is the water conductivity at point \(i\) of the watershed.  
(3) Assumption 3: The hydraulic gradient on the saturated area of the basin is similar to the slope of the surface topography. This assumption is more suitable for the actual situation of saturated groundwater. On the basis of Darcy law, any underground runoff can be expressed as [60]  
\[q_{i}=T_{i}\tan(\beta_{i})=Ra_{i} \tag{18}\]where \(q_{i}\) is the soil discharge per unit of water passing width at point \(i\) of the basin, \(\tan(\beta)\) is the local slope of the watershed surface, \(R\) is the discharge per unit area of the watershed, and \(a_{i}\) is the catchment area per unit width at point \(i\) of the watershed.  
By combining Equations (1) and (2), the following equation can be obtained [60]:  
\[z_{i}=-S_{zm}\ln\frac{Ra_{i}}{T_{0}\tan\beta_{i}} \tag{19}\]  
Therefore, the average water surface depth of the whole basin is [60]  
\[\overline{z}=\frac{1}{A}\int_{A}z_{i}\mathrm{d}A=\frac{S_{zm}}{A}\int_{A} \left(-\ln\frac{Ra_{i}}{T_{0}\tan\beta_{i}}\right)\mathrm{d}A \tag{20}\]  
Since the saturated water conductivity \(T_{0}\) is considered to be uniformly distributed in the basin in the TOPMODEL model, it can be obtained from Equations (3) and (4) [60]:  
\[z_{i}=\overline{z}-S_{zm}(\ln\frac{a_{i}}{\tan\beta_{i}}-\lambda^{*}) \tag{21}\]  
where \(A\) is the area of the whole watershed, \(\overline{z}\) is the average surface depth of the watershed, \(\ln\frac{a_{i}}{\tan\beta_{i}}\) is the topographic index value at point \(i\) of the watershed, and \(\lambda^{*}\) is the mean value of the basin topographic index.  
On the basis of the above three assumptions, the soil in the watershed was divided into three water-bearing zones: the vegetation root zone, unsaturated soil zone, and saturated underground zone. The hydrological process in the basin is as follows: precipitation first penetrates the root zone of vegetation, and part of the water is stored in this zone and involved in evaporation. In contrast, the rest directly penetrates the unsaturated zone of the soil. Then, the amount of water in the unsaturated soil zone penetrates the groundwater zone vertically at the rate of \(q_{v}\). As a result, regression flow is generated if the groundwater surface rises and emerges at a low-lying confluence. Finally, in the groundwater area, the groundwater flows in saturated soil through lateral movement. Therefore, in the TOPMODEL model, saturated soil flow and saturated overland flow constitute the total runoff of the watershed.