## 3 Model Application in a Simulated River Network
### Impact of the Number of Measurement Stations and the Amount of Observed Data  
In reality, there are relatively few available observational hydrologic-state data in a river system; thus, a robust parameter inversion model should be able to produce accurate results even when the amount of information is small. In addition, inversion results could approach the true values as measurement information increases. Observational data depend on the number of measured stations in a river system and the amount of observational data at each station. In this section, the influence of the number of measurement stations on the inversed roughness and water-level prediction performance is tested, followed by the influence of the amount of observed data. Five schemes with increasing measurement stations were tested: (1) one station at Node 3 (located in the first-class river); (2) two stations at Nodes 3 and 7 (located in the secondary river channel); (3) three stations at Nodes 3, 7, and 26 (located in the tertiary  
Figure 3: Schematic diagram of the river network structure. river channel); (4) four stations at Nodes 3, 7, 26, and 14 (located in the secondary river channel); (5) five stations at Nodes 3, 7, 26, 14, and 24 (located in the tertiary river channel). Each observational point was monitored at an interval of 5 min, and 10 consecutive observations were performed at each station. The initial roughness of each river was 0.02, \({C_{d}}^{1/2}\) = 0.0001 m (basically ignoring the observational error), \({C_{n}}^{1/2}\) = 0.01. Table 1 shows the variations in the root mean square error (RMSE) values of roughness and water level at the measurement nodes with the increasing number of measurement stations. As an evaluation metric for estimating forecasting accuracy, RMSE can be calculated as follows:  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}\left(\textbf{S}_{i}-\textbf{M}_{i}\right)^{2}} {N}} \tag{9}\]  
where \(\textbf{M}_{i}\) is the measurements, \(\textbf{S}_{i}\) is the predicted values, and \(\textbf{N}\) is the number of available measurements for analysis. Generally, the water-level RMSE values of all nodes (i.e., 3, 7, 26, 14, and 24) decreased with the increasing number of measurement stations. This was probably due to the contribution of the inversed roughness value that was close to the true value. Indeed, the roughness RMSE decreased significantly with the increasing number of measurement stations, as illustrated in Figure 4. The roughness RMSE decreased hugely when the stations increased from 1 to 2, and gently decreased with a further increase in stations from 2 to 5. To further elucidate the influence of the number of measurement stations on inversed roughness, the spatial distribution of inversed roughness at all river segments is plotted in Figure 5. For the scenario with only one measurement station at Node 3, the inversed roughness at several river segments deviated from the true value (i.e., segments 18, 19, 14, 34, and 37). With the increasing number of measurement stations, the inversed roughness gradually approached the true value at different segments, particularly when there were more than 3 stations. We present the distribution map of the inversed values of roughness when the observational stations were 3, as shown in Figure 6. Interestingly, with 2-5 measurement stations, the model calculation failed when the regularization term in Equation (1) was excluded from the objective function, indicating the importance of the smoothing matrix for optimizing the roughness. Next, the influence of the amount of observed data on the inversed roughness and water-level prediction performance was tested when three measurement stations at Nodes 3, 7, and 26 were used. The initial roughness of each river was 0.02, \({C_{d}}^{1/2}\) = 0.0001 m, \({C_{n}}^{1/2}\) = 0.01. Water-level observational points were monitored at an interval of 5 min, and the inversion tests were conducted for 10, 20, and 30 consecutive observations at each measurement station.