# Water Level Simulation in River Network by Data Assimilation Using Ensemble Kalman Filter
## 5 Parameter Analysis
### Observation Noise Level and Standard Deviation of the Initial Ensemble  
Table 1 shows the test scheme. The results calculated based on the true value of the roughness were regarded as the precise water level values, and the observation noise (i.e., error) was added to the precise water level values to produce the observation values. The observation noise levels \(\delta\) (0.0, 0.0001, 0.0005, 0.001, 0.005, 0.01, and 0.05 m) were analyzed. Figure 2: Illustration of the simulated water river network. The standard deviation of the initial ensemble was 0.0, 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, and 0.1 m. The average water depth value was 2.5 m; correspondingly, the relative observation noise level values were 0%, 0.004%, 0.02%, 0.04%, 0.2%,0.4%, and 2.0%. The relative standard deviation of the initial ensemble was 0%, 0.004%, 0.02%, 0.04%, 0.2%,0.4%, 2.0%, and 4.0%. A constant ensemble size of 100 was tested. The performance of the EnKF algorithm with different parameters was assessed on the basis of the root-mean-square-error (RMSE) of the water level, which was calculated based on the simulated and precise values. When the observation noise level \(\delta\) equaled 0.0, except for standard deviation = 0.0 (i.e., equivalent to no data assimilation), the data assimilation failed. This is because the gain matrix \(K\) decreased continuously during the data assimilation process, implying that the weight of using new information to correct the predicted values decreased, i.e., the tracking performance of the system decreased. It was thus necessary to use an observation noise level of more than zero to ensure a certain disturbance level and prevent data assimilation failure. Figure 3 shows that the RMSE values changed greatly with different settings for the parameters of standard deviation, \(\sigma\), and observation noise level, \(\delta\), at Stations 1 and 2. When \(\sigma\) = 0.0, the sample data in the initial ensemble were constant numbers, which were the initial conditions of the water level at each node. In this case, the simulation results were not affected by the observation noise or the ensemble size, and the results were equivalent to the results without data assimilation. The RMSE values decreased significantly with increasing standard deviation, particularly at the low level of observation noise. This indicates that a relatively high standard deviation and low observation noise level could lead to a good data assimilation result. For example, at Station 1, a negligible RMSE value of 0.0011 m was produced when \(\sigma\) = 0.1, \(\delta\) = 0.0001, whereas a high RMSE value of 0.1563 m was obtained when \(\sigma\) = 0.0001, \(\delta\) = 0.05. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Observation Noise Levels/m**} & \multicolumn{8}{c}{**Standard Deviation \(\sigma\)/m**} \\ \cline{2-10}  & **0.0** & **0.0001** & **0.0005** & **0.001** & **0.005** & **0.01** & **0.05** & **0.1** \\ \hline
0 & \(\surd\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\
0.0001 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
0.0005 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
0.001 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
0.005 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
0.01 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\
0.05 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ \hline \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 1: Combination of standard deviation and observation noise levels for parameter analysis.