## 3 Materials and Methods  
### Topmodel  
TOPMODEL simulates the runoffs using the water balance due to precipitation, estimated infiltration (regarded as a saturation deficit), and the loss by evapotranspiration. Figure 2 shows the basic concept of the TOPMODEL scheme. \(P\) represents the precipitation, \(a_{i}\) is the slope area per unit contour length, \(q_{i}\) is the runoff per unit width, \(S_{rz}\) is the root-zone deficit, and \(S_{uz}\) is the local water storage in the unsaturated zone. \(q_{v}\) is the flux of water that locally enters the water table (per unit area), and \(Q\) represents the drainage of the watershed unit. The formula that serves as the basis of TOPMODEL is based on the continuity equation and Darcy's Law. The model equation is developed through several important assumptions as follows. First, the flow of water in the watershed is assumed to be at a steady state. Second, the hydraulic gradient of the saturated soil in the watershed can be expressed as tan \(\beta\), which represents the surface gradient. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Watershed** & **Year** & \
\begin{tabular}{c} **Rainfall–Runoff** \\ **Event Time (m.dd.hh)** \\ \end{tabular} &
\begin{tabular}{c} **Durations of** \\ **Events (h)** \\ \end{tabular} & **Name of Events** \\ \hline \multirow{4}{*}{Dongkok} & 1996 & 6. 16. 23–6. 17. 22 & 16 & DK960616 \\  & 8. 26. 22–8. 27. 19 & 17 & DK960826 \\ \cline{2-5}  & 1997 & 7. 15. 18–7. 16. 21 & 17 & DK970715 \\ \cline{2-5}  & 1998 & 8. 15. 14–8. 17. 10 & 16 & DK980815 \\ \hline \multirow{4}{*}{Ieemokjung} & 1991 & 5. 25. 13–5. 26. 17 & 26 & IM910525 \\ \cline{2-5}  & 1992 & 9. 24. 6–9. 24. 24 & 17 & IM920924 \\ \cline{2-5}  & 1994 & 6. 29. 23–7. 1. 23 & 42 & IM940629 \\ \cline{2-5}  & 1994 & 8. 1. 3–8. 2. 4 & 19 & IM940801 \\ \cline{2-5}  & 1996 & 7. 4. 13–7. 5. 5 & 13 & IM960704 \\ \cline{2-5}  & 8. 25. 13–8. 27. 24 & 17 & IM960825 \\ \cline{2-5}  & 1999 & 7. 28. 3–7. 29. 24 & 42 & IM990728 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selected rainfall–runoff events in the TOPMODEL application. Figure 2: Basic concept of the TOPMODEL scheme (modified from [5; 14; 19]). Third, the saturation permeability coefficient (\(T_{0}\), m\({}^{2}\)/h) exponentially decreases from the soil surface to the groundwater surface. These assumptions indicated that the most important factor on the storage and groundwater of the watershed is the topographic index, also known as topographic wetness index, i.e., (\(ln(\frac{a}{tang})\)) [26; 27; 28; 29; 30]. The saturated hydraulic conductivity of soil (\(K_{\mathrm{s}}\), m/h) can be expressed as a negative exponential function with respect to depth, as follows:  
\[K_{\mathrm{s}}=K_{0}\exp{(-fz)} \tag{1}\]  
where \(z\) (m) is the underground water depth, \(K_{0}\) (m/h) is the coefficient of permeability of the soil at the ground surface, and \(f\) (1/m) is the coefficient ratio of the \(K_{\mathrm{s}}\) change depending on underground water depth \(z\). \(f\) was assumed to be constant in the entire watershed. The hydraulic gradient in the saturated layer can be expressed as a topographic slope in the watershed, and the behavior of the saturated layer follows Darcy's Law. Therefore, the flow rate per unit width \(q_{i}\) (m\({}^{2}\)/h) at point \(i\) is expressed as follows:  
\[q_{i}=T_{i}(z_{i})tan\beta_{i} \tag{2}\]  
where \(tan\beta_{i}\) is the slope of the surface at location \(i\), and \(T_{i}(z_{i})\) is the transmissivity coefficient. \(T_{i}(z_{i})\) in Equation (2) can be obtained by integrating Equation (1) as follows:  
\[T_{i}(z_{i})=\int\limits_{z_{i}}^{z(t)}K_{\mathrm{s}}(x)dx=\frac{K_{0}}{f}[ \exp{(-fz_{i})}-\exp{(-fz)}]=\frac{1}{f}[K_{\mathrm{s}}(z_{i})-K_{\mathrm{s}}( z)] \tag{3}\]  
Generally, when depth \(z\) of the groundwater is large, the saturation permeability coefficient \(K_{\mathrm{s}}(z)\) (m/h) is negligible, because it is very small compared with saturation permeability \(K_{\mathrm{s}}(z_{i})\) at depth \(z_{i}\).