## 5 Discussion  
The parameter inversion model established in this paper combined the misfit and regularization terms into an objective function to find the optimized bed roughness that minimizes the objective function, which represents a balance between the goodness of fit to the observed data and the required degree of regularization. As a result, the iterative process performed well in our parameter sensitivity section, avoiding the calculation failure that commonly occurs if the regularization term is excluded. From the perspective of optimization theory [24], the roughness distribution between different river channels can be great, as long as the difference between the observed data and the model prediction is minimized. However, this is impractical, as most rivers have smooth bed roughness distribution along them besides during floods or other extreme events [25]. This is illustrated by the relatively smooth variation in the inversed roughness results along Kootenai River, as shown in Figure 11. In this study, a simple rule is proposed to produce a smoothing matrix for river networks or a single-channel river. An example of the rule application is illustrated in Figure 1.  
The addition of the regularization term to the objective function increased the complexity of the parameter inversion model (Equation (1)), which in turn increased the computation time required to solve the model [26], as shown in Table 2. However, the increased computational cost is a trade-off that must be considered when using this approach for accurately estimating bed roughness. For the plain river network in this study, a regularization weight of 100 significantly reduced the roughness inversion errors. The overall computational cost was still minimal; thus, the addition of the regularization term was acceptable. Different to plain rivers, mountainous rivers typically exhibit high gradient roughness distribution upstream, and a low gradient downstream [27]; the regularization weight coefficient could be smaller, and computational cost may be higher. Plain and mountainous rivers can be compared regarding their optimized parameter setting in the future. Figure 13: Forecasting errors between the measured water level and the forecasted values by using the inversed roughness from this paper and from the Federal Emergency Management Agency [23]. Our numerical experiment shows that the new parameter inversion model had good capacity for spatial resolution in large river networks. There were 41 river sections in the simulated river network distributed in 4 corners of the whole river network (Figure 3). Our testing shows only 3 measurement stations with 30 observational data were sufficient to produce a good roughness inversion result (Figure 4). These stations were at Nodes 3, 7, and 26, located in the central, top-left-corner, and bottom-right-corner zones, respectively (Figure 3). This indicates that, although the shortage of observational data would cause the underdetermination problem when the amount of observed data is insufficient to determine the number of model parameters in practice, our inversion model was capable of performing reliable parameter inversion for river networks. In practice, the bed roughness is usually estimated by an engineer who uses all their experience and intuition [19]. High uncertainty thus exists in initial roughness that may impact inversion model efficiency and accuracy, particularly for underdetermination inversion models. However, our model produced reliable roughness inversion results with a low computational cost regarding initial roughness values from 0.02 to 0.03 (Table 3). Although the observational noise inevitably impacted the quality of observational data, its impact on roughness-inversion accuracy was insignificant (Table 3). For the observational noise of less than 0.01 m in the field, the roughness RMSE was less than the small value of 0.0014.