# Water Level Simulation in River Network by Data Assimilation Using Ensemble Kalman Filter
## 6 Experimental Evaluation
### Results and Discussion
#### 6.4.2 Parameter Sensitivity  
To verify the effectiveness of EnKF with different key parameter values, Figure 9 shows the RMSE values of EnKF compared to EKF at different observation noise levels. The EnKF algorithm exhibited much lower simulation errors at observation noise level values of 0.0005 and 0.005, regardless of the observation stations. The EKF algorithm had a slightly lower simulation error at Hongqiao station and Liushi station, which may be due to the impractically high value of observation noise in practice. Figure 8: Water level simulation errors among different data treatment schemes at (**a**) Hongqiao station, (**b**) Lecheng station, and (**c**) Liushi station. The solid black line represents the data that were not treated via data assimilation; the green dashed line represents the data that were treated via EnKF; the red dashed line represents the data that were treated via EKF. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Treatment Scheme**} & **Calculation** & \multicolumn{3}{c}{**RMSE (m)**} & \multicolumn{3}{c}{**Filtering Time (h)**} \\ \cline{2-7}  & **Cost (s)** & **Hongqiao** & **Lecheng** & **Liushi** & **Hongqiao** & **Lecheng** & **Liushi** \\ \hline Without data assimilation & 204.3 & 0.1735 & 0.0655 & 0.1246 & / & / & / \\ EKF & 9335.4 & 0.0113 & 0.0187 & 0.0090 & 20 & 14 & 20 \\ EnKF & 4243.6 & 0.0026 & 0.0016 & 0.0044 & 5 & 6 & 8 \\ Ratio of EnKF to EKF & 45\% & 23\% & 9\% & 49\% & 25\% & 43\% & 40\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of evaluation metric results for EKF and EnKF in Case 2. Similarly, EnKF's calculation costs and the time required for EnKF to reach stability after the beginning of filtering at different observation noise levels were also consistently much lower than those of EKF, particularly at practical observation noise levels (0.0005 and 0.005), as shown in Table 9. The time for the EnKF to reach stability after the beginning of filtering was 27% to 81% of that for EKF, and the calculation cost for EnKF was 27% to 81% to that for EKF. The above testing comparison between EnKF and EKF showed the great advantages of the EnKF-based data assimilation model when it was applied to the complex river network. Indeed, both EnKF and EKF have their advantages in predicting water levels. However, EnKF was more effective, especially in terms of computational effort and the time to reach stable prediction. The computational characteristics of the Kalman filter (e.g., a short online estimation time, small storage size, and convenient error analysis) are assessed to determine its suitability for real-time correction processing of dynamic systems. For a nonlinear dynamic system of a complex river network, the EKF algorithm is often used, but there are still some problems with this approach, mainly: (1) it is difficult to calculate the prediction error variance matrix, and its accuracy cannot be guaranteed, as verified by our experiments in which data assimilation based on EKF produced much higher simulation errors than EnKF-based data assimilation. (2) It is necessary to calculate both the tangent state operator and the covariance of dynamic noise [12], which makes the algorithm structure complex and causes a large amount of calculations. The calculation cost of EKF is almost twice as high as that of EnKF. (3) The hydraulic calculation of river networks is a strongly nonlinear problem. The use of the EKF algorithm requires one to omit the terms above the first-order derivative; however, the omitted terms could precisely reflect the nonlinear effect of the strong nonlinear problem. (4) The EKF algorithm contains an assumption in the application, that is, the error obeys a normal distribution; therefore, it is not suitable to be applied to a system with a non-normal distribution of errors. (5) EKF is not capable of addressing horizontal error correlations in the model or measurement errors in large systems due to computational limitations [24].