## 3 Validation of the Model
### Analysis of Model Results  
Using the parameters above to simulate the temperature of the water in Xulong, the results are shown in Figures 4 and 5:  
Figure 4 shows the temperature duration curves of several typical water depths. It can be seen from the figure that the model fits well for the water temperature of different depths. It can also be seen that the water temperature at the surface is relatively high, and the peak temperature appears at a time similar to that of the river water temperature. With the increase in depth, the variation of water temperature decreases with a time lag phenomenon, and the lag phenomenon becomes more obvious as the depth increases. This is because the heat transfer and diffusion along the depth direction, gradually become weaker. At the bottom of the reservoir, the temperature varies slightly, and remains around 6 \({}^{\circ}\)C all year round.  
Figure 5a clearly demonstrates the changes in the temperature of the water due the variation in depth based on our model. Due to the limited space, Figure 5b only lists the comparison between the simulated and measured data in the month of September. Figure 5c shows the results of three different methods for water temperature. It can be seen from the figure that the method we proposed fits the water temperature better. The two empirical methods are close to an exponential function and cannot accurately describe the variation of the water temperature in the reservoir with depth. After data analysis, the maximum absolute error occurs at 75 m underwater, which is 2.9 \({}^{\circ}\)C. The mean error, mean square error, and root mean square error are 0.8 \({}^{\circ}\)C, 0.9 \({}^{\circ}\)C, and 1.0 \({}^{\circ}\)C, respectively. The NSE of I1 DM is 0.92, very close to 1. NSE is a parameter that characterizes the quality of the model, and NSE ranges from negative infinity to 1. The closer NSE is to 1, the better the quality of the model and the higher the reliability of the model. When NSE is close to 0, it indicates that the results from the model are close to the average level of the observed values. If NSE is much less than 0, the model is not credible. In conclusion, combined with the error data in Figure 5 and Table 3, I1 DM has a better simulation effect on the water temperature in the reservoirs, and can reveal the distribution characteristics and variation trends of the water temperature.  
Figure 4: Temperature at different depths of Xulong. (**a**) Water depth 20 m. (**b**) Water depth 40 m. (**c**) Water depth 100 m. (**d**) Bottom of reservoir.  
\[\text{NSE}=1-\frac{\sum(T_{measured}-T_{simulated})^{2}}{\sum\left(T_{ measured}-\overline{T_{simulated}}\right)^{2}}\]  
Figure 5: Comparison between simulation data and measured data. (**a**) Comparison between simulation data and measured data. (**b**) September. (**c**) Average temperature.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Parameter** & \multicolumn{3}{c}{**Mean Difference**} & \multicolumn{3}{c}{**Root Mean Square**} & \multicolumn{3}{c}{**NSE**} & \multicolumn{3}{c}{**Time**} \\
**Method** & **Xulong** & **Xiluodu** & **Jiexu** & **Sanhekou** & **Xulong** & **Xiluodu** & **Jiexu** & **Sanhekou** & **Xulong** & **Xiluodu** & **Jiexu** & **Sanhekou** & **-** \\ \hline DKYM & 1.7 & 1.1 & 2.0 & 3.0 & 2.5 & 1.7 & 2.6 & 3.6 & 0.89 & 0.81 & 0.64 & 0.20 & \textless{}1 min \\ ZBFM & 2.5 & 1.5 & 1.1 & 2.3 & 3.4 & 2.3 & 1.3 & 2.9 & 0.78 & 0.61 & 0.84 & 0.50 & \textless{}1 min \\ IIDM & 0.8 & 0.8 & 1.0 & 1.0 & 1.0 & 1.4 & 1.0 & 1.5 & 0.92 & 0.86 & 0.91 & 0.88 & \textless{}1 min \\ MIKE3 & 0.7 & 0.8 & 0.9 & 0.9 & 0.9 & 1.1 & 0.9 & 1.3 & 0.93 & 0.93 & 0.94 & 0.90 & \textgreater{}3 h \\ \hline \hline \end{tabular}
\end{table}
Table 3: Analysis of error for the whole year (\({}^{\circ}\)C).