# Water Level Simulation in River Network by Data Assimilation Using Ensemble Kalman Filter
## 5 Parameter Analysis
### Ensemble Size  
As the computational resources are limited for realistic and large-scale hydraulic applications, the ensemble size should be small. We thus tested four ensemble sizes, n = 50, 100, 150, and 200. The observation noise level \(\delta\) was taken to be 0.0001 m and 0.05 m, and the standard deviation \(\sigma\) was taken to be 0.0001 m and 0.1 m. Table 2 presents the parameters and their values.  
Tables 3 and 4 show the RMSE values with different ensemble sizes at certain observation noise levels and standard deviation values. Generally, the RMSE values decreased with increasing ensemble size for the low observation noise level (i.e., \(\delta\) = 0.001, \(\sigma\) = 0.0001 or 0.1), whereas for large observation noise level of 0.05, either the RMSE values were constant or the calculation failed, indicating that the data assimilation process was not improved by increasing the ensemble size. As shown in Table 5, computational cost increased significantly with ensemble size, and a large ensemble size could result in an extensive cost in terms of computational resources. Thus, we determined that the ensemble sample size could be set at a medium level (i.e., 100 or 150). Based on the sensitivity analysis of the observation noise level and the standard deviation of the initial ensemble, we suggest that a relatively low observation noise level (e.g., from 0.0001 to 0.01 m) and a relatively high standard deviation (e.g., from 0.01 to 0.1 m), along with a medium ensemble sample size (e.g., from 100 to 150), should be used to produce a good data assimilation with low RMSE values.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Ensemble Size m**} & \multicolumn{4}{c}{\(\delta\)**/m and \(\sigma\)/m**} \\ \cline{2-5}  & \(\delta\) = 0.0001, & \(\delta\) = 0.0001, & \(\delta\) = 0.05, & \(\delta\) = 0.05, \\  & \(\sigma\) = 0.0001 & \(\sigma\) = 0.1 & \(\sigma\) = 0.0001 & \(\sigma\) = 0.1 \\ \hline
50 & 0.01015 & 0.00038 & 0.05223 & 0.01627 \\
100 & 0.00643 & 0.00051 & 0.05223 & 0.00807 \\
150 & 0.00305 & 0.00023 & 0.05223 & / \\
200 & 0.00233 & 0.00029 & 0.05223 & / \\ \hline \end{tabular} Note: \(\surd\) means the calculation was successful; \(\times\) means the calculation was unsuccessful due to the failure of convergence and data assimilation.  
\end{table}
Table 2: Combination of ensemble size, standard deviation, and observation noise level.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Ensemble Size m**} & \multicolumn{4}{c}{\(\delta\)**/m and \(\sigma\)/m**} \\ \cline{2-5}  & \(\delta\) = 0.0001, & \(\delta\) = 0.0001, & \(\delta\) = 0.05, & \(\delta\) = 0.05, \\  & \(\sigma\) = 0.0001 & \(\sigma\) = 0.1 & \(\sigma\) = 0.0001 & \(\sigma\) = 0.1 \\ \hline
50 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
100 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
150 & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) \\
200 & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) \\ \hline \end{tabular} Note: \(\surd\) means the calculation was successful; \(\times\) means the calculation was unsuccessful due to the failure of convergence and data assimilation.  
\end{table}
Table 3: RMSE values with different ensemble sizes at fixed \(\delta\) and \(\sigma\) values for Station 1.