Enhancing the Accuracy of Water-Level Forecasting with a New Parameter-Inversion Model for Estimating Bed Roughness in Hydrodynamic Models

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The new parameter-inversion model for estimating bed roughness in hydrodynamic Models is potentially usefully for the accurate flood water forecasting in reality.

Abstract

The accurate and efficient estimation of bed roughness using limited historical observational data is well-established. This paper presents a new parameter-inversion model for estimating bed roughness in hydrodynamic models that constrains the roughness distribution between river sections. The impact of various factors on the accuracy of inversed roughness was analyzed through a numerical experiment with the number of measurement stations, observed data amount, initial bed roughness, observational noise, and the weight of the regularization term. The results indicate that increasing the number of measurement stations and the amount of observed data significantly improves the robustness of the model, with an optimal parameter setting of 3 stations and 30 observed data. The initial roughness had little impact on the model, and the model showed good noise resistance capacity, with the error significantly reduced by controlling the smoothness level of inversed roughness using a small weight of the regularization term (i.e., 100). An experiment conducted on a real river using the calibrated model parameters shows a forecasted water level RMSE of 0.041 m, 31% less than that from the Federal Emergency Management Agency. The proposed model provides a new approach to estimating bed roughness parameters in hydrodynamic models and can help in improving the accuracy of water-level forecasting.

1. Introduction

The numerical simulation of river hydraulics plays a crucial role in various applications, such as engineering design, urban flood control, and water-environment and -resource planning. In this context, the roughness parameter in hydraulic modeling is particularly important, as its reasonableness significantly impacts the frictional resistance that water experiences as it flows through the channel, and thus influences the model’s output results, e.g., flow discharge and water level [1]. Hence, determining roughness is a prerequisite for the hydraulic, water-quality, and scour analyses of rivers.

Researchers developed real-time hydrodynamic-correction models of river systems with the simultaneous real-time correction of channel-bed roughness and hydraulic-state variables such as the water level by using the Kalman filtering technique and its variants [2,3]. These techniques are used to improve the accuracy of river flow forecasts by accounting for the discrepancies between observed and simulated flow conditions by integrating current observations into the model in real time to adjust and improve the predictions [4]. Real-time correction models are especially beneficial for special hydrodynamic modeling where model predictions can be affected by rapidly changing environmental conditions such as tides, waves, and storms [5,6]. However, it is particularly challenging for real-time models that require a high temporal or spatial resolution in large river networks, as the data that need to be processed and assimilated can be substantial. Real-time correction models may not always be feasible or practical in some applications due to limitations in data availability, accessibility, or timeliness.

The direct measurement of roughness coefficients requires knowledge of the physical properties of the overland environment, and can be rather challenging to handle spatially varying coefficients, rendering practical measurements unfeasible [7]. Given this limitation, and considering the importance of roughness calibration in engineering applications, theoretical research on roughness identification and optimization based on historical observational water-flow data has attracted the attention of many researchers [8,9]. The calibrated roughness is then used as input into hydrodynamic models. Existing methods mainly focus on optimization algorithms to minimize the difference between hydraulic simulation and observed values. A common method that calibrates roughness is trial-and-error due to scarce hydraulic information in practice. For example, Boulomytis et al. [10] used field observations to estimate Manning’s roughness values, and applied the Hydrologic Engineering Center’s River Analysis System (HEC-RAS) model for calibration. The calibration process involved a trial-and-error approach, and the findings showed that the estimation and calibration methods were efficient. However, empirical or trial-and-error methods used to calibrate roughness suffer from several drawbacks. The selection of parameters is time-consuming [11] and depends on researchers’ knowledge or experience, which lacks clarity [12]. Further, the workload associated with determining parameters for a large number of river sections is high, requiring highly skilled technical personnel. Additionally, the lack of systematic theoretical guidance undermines the accuracy of the results and the reliability of rate determination, hindering assessing their accuracy. Consequently, further research is required to address this significant issue that still poses a challenge for engineering designers.

To overcome the shortcomings of trial-and-error methods, many researchers developed intelligent optimization methods to calibrate the roughness [13,14]. For example, Drisya et al. [15] developed an automated calibration framework for estimating Manning’s roughness coefficient using a genetic algorithm that was formulated as an optimization problem with a heuristic search procedure. The study suggested that multiobjective GA using pooled and balanced aggregated function statistics can be used to optimize multiple objectives. Additionally, a coupled optimization–simulation model that integrated the particle swarm optimization (PSO) algorithm and InfoWorks Integrated Catchment Modelling (ICM) software was established to solve the objective function and hydraulic process. The proposed method was applied to estimating the roughness values and showed better performance in model calibration [16]. The nature of these intelligent models makes them require a large amount of observational data, which may pose challenges in practical river measurement systems.

Parameter-inversion models, on the other hand, can be less computationally intensive, as they use model calibration techniques to adjust the model parameters to better fit historical data, but require high-quality historical data for model calibration. The challenging task of identifying parameters in certain instances can be attributed to the ill-posed nature of the inverse problem [17]. This is because roughness inversion results could theoretically be influenced by initial conditions such as the starting values of the model parameters [18], particularly for underdetermined inversion models, which refers to a common situation in which the amount of the observed data is insufficient to determine the number of the model parameters in practice. For complex river networks, insufficient hydrological observations and underdetermined inversion models often lead to model calculation failure, requiring manual intervention to ensure reasonable results. It is, thus, important to develop parameter-inversion models with high robustness to historical data in terms of quality and quantity.

To provide a more accurate and reliable method for calibrating riverbed roughness, which is a critical parameter for hydraulic simulations, this study proposes a new inversion model of bed roughness based on historical data considering the regularization term for controlling the continuous variation in river roughness along the flow direction. The model is solved through an iterative process. A simulated river network with four levels of river channels is used for optimizing the model parameters through analyzing the impact of the number of measurement stations, the amount of the observed data, initial bed roughness, observational noise, and the weight of the regularization term on the accuracy of the inversed roughness. An experiment was carried out on a real river in North America using the model after parameter calibration, and its results were compared with those from the literature. The potential application of the model for hydraulic simulations is discussed. The outline of this study is as follows: In Section 2, we present the procedure for establishing the inversion model of bed roughness. In Section 3, we outline how we applied the inversion model in a simulated river network, and analyze the impact of various parameters on the accuracy of inversed roughness, including the number of measurement stations, the amount of the observed data, the initial bed roughness, observational noise, and the weight of the regularization term. Section 4 focuses on the calibration process of the inversion model established in this study, and compares the results with those from other researchers. Lastly, in Section 5, we provide a conclusion on the basis of our findings.

2. Model Development

2.1. Inversion Model of Bed Roughness

In this paper, an inversion model for estimating riverbed roughness is established that could take into account the observational error and spatial distribution of the roughness in the whole river network. The measurement error of the observational data set d_obs was assumed to have normal distribution with variance in C_d. The inversion model seeks to minimize an objective function that measures the difference between the observed data and the model simulations. The following objective function, consisting of two terms (a misfit term and a regularization term), was developed:

$$\begin{array}{ll}F& =\min .\frac{1}{2}||\mathit{g}(\mathit{n})-{\mathit{d}}_{obs}|{|}_{Cd}^2+\frac{1}{2}||\mathit{L}\mathit{n}|{|}_{Cn}^2\\ & =\min .\frac{1}{2}{[\mathit{g}(\mathit{n})-{\mathit{d}}_{obs}]}^T{C}_d^{-1}[\mathit{g}(\mathit{n})-{\mathit{d}}_{obs}]+\\ & \hspace{1em} \frac{1}{2}{\mathit{n}}^T{\mathit{L}}^T{\mathit{C}}_n^{-1}\mathit{L}\mathit{n}\end{array}$$

(1)

where $\frac{1}{2}{‖\mathit{g}\left(\mathit{n}\right)-{\mathit{d}}_{obs}‖}_{C_d}^2$ is the misfit term that measures the difference between the observed data and the model simulations, and regularization term $\frac{1}{2}{‖\mathit{L}\mathit{n}‖}_{C_n}^2$ (also named the smoothing term) constrains the solution to be smooth in order to avoid overfitting or underfitting. It is introduced to control the trade-off between fitting the data well and achieving smooth roughness distribution. ${\mathit{C}}_n^{-1}$ is the weight of the smoothing term. The smaller ${\mathit{C}}_n$ is, the greater the weight of the smoothing term, i.e., forcing the spatial distribution of the roughness to be as smooth as possible. L is the spatial variance rate of roughness, n is the river roughness, and $\mathit{g}\left(\mathit{n}\right)$ are hydraulic states such as water level that can be obtained by using Saint Venant equations that are often used to describe one-dimensional flow [19]. The continuous equation is written as follows:

$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q_L$$

(2)

The momentum equation is expressed as follows:

$$\frac{\partial Q}{\partial t}+\frac{\partial }{\partial x}(\frac{\alpha Q^2}{A})+gA\frac{\partial Z}{\partial x}+gA\frac{Q\left|Q\right|}{K^2}=q_L{v}_x$$

(3)

where A is the cross-sectional area; t is time; Q is the average flow of the cross section; x is the displacement along the water flow; q_L is the lateral streamflow per unit length; a is the momentum correction coefficient; g is the acceleration of gravity; Z is the water level of a section; K is the flow modulus, K = AR^2/3/n, where R is the hydraulic radius; v_x is the velocity component of the lateral stream flow along the water flow, usually taken as v_x = 0. A space-time discretization procedure is performed for the water flow in a river system because the Saint Venant equations are nonlinear hyperbolic partial differential and cannot be solved analytically bu using the initial conditions [19]. The details of the development of the discrete equations by using the finite difference method can be found in [20].

The spatial variance rate of roughness L is used to measure the variation degree of roughness $\mathit{n}={[n_1,n_2,n_3,\cdots n_m]}^T$ over the length of a river channel, where m is the river section number, and each element of the vector corresponds to the local roughness of the corresponding river segment. In this paper, the spatial variance rate of roughness is represented by smoothing matrix ${\mathit{L}}_{m\times m}$, which is typically constructed on the basis of the assumption that the bed roughness varies smoothly in space and that nearby locations are more likely to have similar roughness values than distant locations are. The detailed approach to developing the smoothing matrix is as follows.

If there is a river segment with number i and its adjacent segments $j_1,j_2,j_3\cdots j_N$ where $N$ is the number of segments adjacent to river segment $i$, then the elements of smoothness matrix $l_{ii}$ are taken as $N$, and the elements of $l_{i{j}_1},l_{i{j}_2},l_{i{j}_3}\cdots l_{i{j}_N}$ are taken as −1, and the remaining elements are taken as 0. After performing the above operation for all the river segments in turn, corresponding smoothness matrix ${\mathit{L}}_{m\times m}$ can be obtained. Then, the cumulative value of the roughness difference between the adjacent river segments can be expressed with ${\mathit{L}}_{m\times m}\mathit{n}$, which quantifies the total change in roughness from one river segment to the next, and is often used to characterize the spatial variability of the roughness in a river reach. As an example, Figure 1 illustrates specific smoothing matrix ${\mathit{L}}_{4\times 4}$ for a simulated river network.

2.2. Method for Roughness Solution

The optimization model for roughness inversion was solved through an iterative process. The whole solution process is shown in Figure 2:

(1)

Initial roughness n⁰ was set according to the river section characteristics.

(2)

A 1D hydrodynamic river model was utilized to calculate the hydrodynamic simulation values g(n^k−1) for n^k−1 conditions. The current iteration step is denoted by k, starting from 1.

(3)

The gradient of the objective function (Equation (1)) can be obtained with the following equation:

$$\nabla F({\mathit{n}}^k)=\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}({\mathit{C}}_d^{-1}[\mathit{g}({\mathit{n}}^k)-{\mathit{d}}_{obs}])+{\mathit{L}}^T{\mathit{C}}_n^{-1}\mathit{L}{\mathit{n}}^k$$

(4)

Combining $\mathit{g}({\mathit{n}}^k)\approx \mathit{g}({\mathit{n}}^{k-1})+\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}({\mathit{n}}^k-{\mathit{n}}^{k-1})$ with Equation (4), we can obtain the following:

$$\begin{array}{ll}\nabla F({\mathit{n}}^k)& =\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}({\mathit{C}}_d^{-1}[\mathit{g}({\mathit{n}}^k)-{\mathit{d}}_{obs}])+{\mathit{L}}^T{\mathit{C}}_n^{-1}\mathit{L}{\mathit{n}}^k\\ & =\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}({\mathit{C}}_d^{-1}[\mathit{g}({\mathit{n}}^{k-1})+\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}({\mathit{n}}^k-{\mathit{n}}^{k-1})-{\mathit{d}}_{obs}])+{\mathit{L}}^T{\mathit{C}}_n^{-1}\mathit{L}\mathit{n}\end{array}$$

(5)

where $\frac{\partial \mathit{g}({\mathit{n}}^k)}{\partial {\mathit{n}}^k}$ is the gradient of g (n) with respect to n^k at the k-th iteration, which can be replaced with sensitivity matrix G^k. Therefore, Equation (5) can be simplified as follows:

$$\nabla F({\mathit{n}}^k)={\mathit{G}}^k({\mathit{C}}_d^{-1}[\mathit{g}({\mathit{n}}^{k-1})+{\mathit{G}}^k({\mathit{n}}^k-{\mathit{n}}^{k-1})-{\mathit{d}}_{obs}])+{\mathit{L}}^T{\mathit{C}}_n^{-1}\mathit{L}\mathit{n}$$

(6)

where sensitivity matrix G^k is solved using a single-factor perturbation method.

(4)

For the convex programming problem without boundary constraints, in this paper, the gradient at the extremum point was 0 when the roughness was equal to the optimization value, i.e., $\nabla F({\mathit{n}}^k)=0$. The optimal solution of roughness at the $k$ iteration is obtained as follows:

$$\begin{array}{ll}{\mathit{n}}^k=& {({({\mathit{G}}^k)}^T{C}_d^{-1}{\mathit{G}}^k+{\mathit{L}}^T{C}_n^{-1}\mathit{L})}^{-1}\{{({\mathit{G}}^k)}^T{C}_d^{-1}\\ & [{\mathit{d}}_{obs}-g({\mathit{n}}^{k-1})]+{({\mathit{G}}^k)}^T{C}_d^{-1}{\mathit{G}}^k{\mathit{n}}^{k-1}\}\end{array}$$

(7)

(5)

The roughness correction at the k-th iteration can also be obtained as follows:

$$\begin{array}{ll}\Delta {\mathit{n}}^k=& \{{[{({\mathit{G}}^k)}^T{C}_d^{-1}{\mathit{G}}^k+{\mathit{L}}^T{C}_n^{-1}\mathit{L}]}^{-1}{({\mathit{G}}^k)}^T{C}_d^{-1}{\mathit{G}}^k-\\ & \mathit{I}\}{\mathit{n}}^{k-1}+{[{({\mathit{G}}^k)}^T{C}_d^{-1}{\mathit{G}}^k+{\mathit{L}}^T{C}_n^{-1}\mathit{L}]}^{-1}{({\mathit{G}}^k)}^T{C}_d^{-1}\\ & [{\mathit{d}}_{obs}-\mathit{g}({\mathit{n}}^{k-1})]\end{array}$$

(8)

(6)

To enhance the stability of the iterative calculation, the roughness of the k-th iteration is denoted by ${\mathit{n}}^k={\mathit{n}}^{k-1}+\rho \Delta {\mathit{n}}^k$, where $\rho \in (0,1)$. The iteration process was repeated from Steps (2) to (6) until the convergence criterion had been met, which is defined as Max (Δn^k) < 0.0001, meaning that the iteration terminates when the maximal correction value of the roughness is below 0.0001. The solution flowchart is depicted in Figure 2.

In addition, the objective equation shows that, if the variance level is $C_n>>C_d$, misfit term $\frac{1}{2}{‖\mathit{g}\left(\mathit{n}\right)-{\mathit{d}}_{obs}‖}_{C_d}^2$ is much greater than regularization term $\frac{1}{2}{‖\mathit{L}\mathit{n}‖}_{C_n}^2$, and the model optimization follows the direction of minimizing the squared error; if the variance level is $C_n<<C_d$, the misfit term is much smaller than the regularization term, and the model optimization follows the direction of the smoothest roughness.

In the above inversion model of bed roughness and solution approach, the key model parameters include observational noise parameter C_d, the weight of the smoothing term (${\mathit{C}}_n^{-1}$), the inputs of initial roughness, the amount of the observed data, and the number of the measurement stations. These parameters are examined through the following parameter analysis to optimize the parameter setting of the roughness inversion model.

3. Model Application in a Simulated River Network

3.1. River Network Description

A complex unsteady-flow river network with four levels of river channels from the literature [21] was used for case analysis as shown in Figure 3. In order to test the spatial detection ability of the developed model for sudden changes in roughness, various roughness values were set in different river channels. The roughness of River Channel 8 was set to 0.026, the roughness of River Channel 24 was 0.02, and the roughness values of the rest were 0.023. These values are regarded as the true values of roughness. River Channel 24 is far away from River Channel 8 (Figure 3), and there were a total of 41 river sections for roughness inversion.

Sensitivity analyses were then conducted to test the robustness of the model, and identify the most important factors affecting the results and their optimized data range. By performing this, the uncertainty and potential errors in the roughness inversion results can be minimized, and the accuracy and reliability of the model can be improved. Since the amount of the observed data in practice is normally insufficient when so many bed roughness parameters are required at a number of river sections (i.e., 41 in this simulated river network) for inversion, the model simulation is an underdetermined problem for solving parameters. Regulation techniques that constrain the model parameters (i.e., the regularization term in Equation (1)) are used to render roughness estimation more reasonable and stable.

3.2. 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 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. RMSE values of roughness and water level at the measurement nodes.

Measurement Station Number	Roughness RMSE	Water-Level RMSE of Node 3	Water-Level RMSE of Node 7	Water-Level RMSE of Node 26	Water-Level RMSE of Node 14	Water-Level RMSE of Node 24
1	0.0018	1.358 × 105	/	/	/	/
2	0.0012	8.182 × 106	9.292 × 106	/	/	/
3	0.0011	7.461 × 106	3.038 × 106	1.083 × 105	/	/
4	0.0010	5.466 × 106	3.104 × 106	1.300 × 105	3.550 × 106	/
5	0.0008	3.448 × 106	3.178 × 106	4.967 × 106	1.404 × 106	2.952 × 106

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:

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{({\mathit{S}}_i-{\mathit{M}}_i)}^2}{N}}$$

(9)

where ${\mathit{M}}_i$ is the measurements, ${\mathit{S}}_i$ is the predicted values, and $\mathit{N}$ 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.

Table 2. Roughness RMSE and water-level RMSE at Nodes 3, 7 and 26 with different observational data.

Observational Data	Calculation Cost (s)	Roughness RMSE	Water-Level RMSE
			Node 3 (m)	Node 7 (m)	Node 26 (m)
10	26.9	0.00104467	7.46 × 106	3.04 × 106	1.08 × 105
20	34.9	0.00103155 (↓ 1.256%)	5.28 × 106 (↓ 29.2%)	3.23 × 106 (↓ −6.3%)	1.04 × 105 (↓ 4.4%)
30	40.2	0.00103153 (↓ 0.002%)	3.37 × 106 (↓ 36.2%)	2.27 × 106 (↓ 29.6%)	3.82 × 106 (↓ 63.1%)

Note: the ↓ means the decreasing percentage.

shows that the roughness RMSE and water-level RMSE of the observational points generally decreased as the amount of observational data increased. The roughness RMSE decreased slightly, only 1.256%, when the observational data points increased from 10 to 20. A further increase in observational data caused a negligible decrease in roughness RMSE. This is also reflected by the spatial distribution of the inversed roughness at all 41 nodes (Figure 7). The radar map of 20 observational data overlapped with that of 30 observational data. However, the water-level RMSE significantly decreased when the observational data points increased from 20 to 30. For example, there was a 36.2% and 63.1% decrease in water-level RMSE at Nodes 3 and 26, respectively. This observation indicates that water-level forecasting is rather sensitive to variations in bed roughness. Care should be taken to invert the bed roughness at each node. Although the model calculation cost increased by 49% with observational data from 10 to 30, this calculation cost (less than 1 min) was still low given that a complex river network was used in this study.

The personal computer used for analysis was a Windows 10 system equipped with a 3.2 GHz Intel Core i7-8700K processor and 16.0 GB DDR4 RAM. We obtained the optimized model with 3 measurement stations and 30 observational data points.

3.3. Impact of Initial Bed Roughness as Input into the Developed Model

A stable mathematical or numerical-solution model should be able to produce similar or even the same results under different initial conditions with disturbances. The solution is unique rather than invalid or scattered due to the selection of initial values with disturbance. To stabilize the developed model in this paper, the model was comprehensively tested with the inputs of different initial conditions, i.e., the initial roughness in this paper.

The initial roughness of each river section was the same and set at 0.02, 0.025, and 0.03 for the comparative analysis of the roughness inversion results. We used 3 stations at Nodes 3, 7, and 26 with 30 observational data points each. We then set $C_d{}^{1/2}$ = 0.0001 m and $C_n{}^{1/2}$ = 0.01. Results in

Table 3. RMSE of roughness and water level at Nodes 3, 7, and 26 with different initial roughness values.

Initial Roughness	Calculation Cost (s)	Roughness RMSE	Water-Level RMSE at Node 3 (m)	Water-Level RMSE at Node 7 (m)	Water-Level RMSE at Node 26 (m)
0.02	40.2	0.00104467	7.46 × 106	3.04 × 106	1.08 × 105
0.025	39.2	0.00104360	4.90 × 106	4.40 × 106	5.68 × 106
0.03	44.6	0.00104886	5.92 × 106	5.40 × 106	6.36 × 106

show that the model calculation cost and roughness RMSE were both insensitive to the variation in initial roughness. The radar map (Figure 8) also shows that the inversed individual section roughness generally overlapped in circumstances with different initial roughness values. Due to the sensitivity to the water level, the tiny variations in roughness caused gentle fluctuations in the water-level RMSE values at different nodes. Thus, sensitivity analysis indicates that our model had good stability capacity when 3 measurement stations and 30 observational data for each were used.

3.4. Impact of Observational Noise and the Weight of the Regularization Term

In practice, the observational errors of hydraulic state variables are inevitable. A robust inversion model should be able to produce inversion results that are closer to the real solution within a certain observational error range without large errors due to small observational perturbations [17]. In addition, the above parameter optimization shows that the regularization term is critical to controlling the smoothness of the inversed roughness distribution and thus maintain stable model calculation. In this section, the impact of the observational noise and the weight of the regularization term on the model inversion performance is explored.

To analyze the impact of observational noise, observational-noise levels ${\mathit{C}}_d^{1/2}$ = 0.001, 0.005, 0.01, and 0.05 m were analyzed given the weight of regularization term ${\mathit{C}}_n^{-1/2}$ = 100 (i.e., ${\mathit{C}}_n^{1/2}=0.01$). To analyze the impact of the weight of the regularization term, regularization term weights ${\mathit{C}}_n^{-1/2}$ = 20, 100, 200, and 1000 m⁻¹ were analyzed given observational-noise level ${\mathit{C}}_d^{1/2}$ = 0.005. Three measurement stations at Nodes 3, 7, and 26 were used, and each had 30 consecutive observations. For each parameter combination scheme, a total of 100 random inversion tests were performed. The average roughness RMSE values were calculated and are shown in

Table 4. Average roughness RMSE with different observational-noise levels and regularization weights.

Impact of Observational Noise			Impact of Regularization Term Weight
${\mathit{C}}_{\mathit{n}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$ (m−1)	${\mathit{C}}_{\mathit{d}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$ (m)	Average Roughness RMSE	${\mathit{C}}_{\mathit{n}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$ (m−1)	${\mathit{C}}_{\mathit{d}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$ (m)	Average Roughness RMSE
100	0.001	0.001209	20	0.005	0.002820
	0.005	0.001347	100		0.001347
	0.01	0.001402	200		0.001128
	0.05	0.001586	1000		0.001250

.

As expected, the roughness RMSE gently increased with increasing observational noise, indicating the model had good noise resistance capacity (Figure 9a). For example, with a 10-fold increase in observational-noise level from 0.001 to 0.01 m, the average roughness RMSE only increased by 16%. The weight of the regularization term was very effective in reducing roughness RMSE by 52% when its value increased from 20 to 100, but the roughness RMSE stabilized with a further increase in regularization weight (Figure 9b). Therefore, the weight of the regularization term was set to 100 (i.e., ${\mathit{C}}_n^{1/2}=0.001$).

Our parameter optimization study shows that the measured data with a certain amount of noise could inevitably increase the roughness inversion errors, while the error could be significantly reduced by controlling the smoothness level of inversed roughness by using a small weight of the regularization term, e.g., 100 in the simulated case of this study.

4. Experimental Evaluations

In this section, we outline how the developed inversion model was used on a real river to test its capacity for bed roughness estimation and water-level forecasting. The bed roughness was first inversed and then compared with other researchers’ estimation results. The estimated roughness was then input into the model water-level forecasting.

4.1. Data Description

This study focused on a 24 km stretch of the transboundary Kootenai River that flows through Montana in the United States, as shown in Figure 10. One of three hydrological stations was located upstream, midstream, or downstream. The measured water volume rate at the Tribal Hatchery station was used as the boundary condition for the upstream entrance, and the measured water level at the Klockmann Ranch station was used for the downstream exit. Water-level measurements from the Bonners Ferry and Tribal Hatchery stations were used to ascertain the bed roughness. The data used in the study were obtained from the official website of the United States Geological Survey.

Table 5. Basic information of Kootenai River.

Section Number	River Length/m	River Width/m	Average Slope
1	1595	265–432	0.0004
2	4634	173–319
3	10,327	188–343
4	1567	151–289
5	1543	106–197
6	4627	155–299

presents the basic information about the selected section of the river.

The river bank roughness coefficient was 0.06 during the parameter inversion [22]. The main channel roughness coefficient was unknown and was estimated by carrying out parameter inversion using the developed model. The parameter calibration results in Section 2 show that the initial roughness values of the main channel for all sections were 0.03 with a practical observational noise value of 0.005 m (i.e., $C_d{}^{1/2}$ = 0.005 m) and a small regularization term weight of 100 (i.e., ${\mathit{C}}_n^{-1/2}$ = 100).

Table 6. Observations of volume flow in Kootenai River.

Date	18 August 2003	19 August 2003	20 August 2003	21 August 2003	22 August 2003	23 August 2003	24 August 2003	25 August 2003	26 August 2003
Daily mean discharge (m3/s)	532.4	535.2	529.6	532.4	470.1	410.6	410.6	407.8	410.6

shows the observations of the mean daily discharge from 18 to 26 August 2003 that were used as the input data for parameter estimation.

4.2. Roughness Inversion and Water-Level Forecasting

As there were no measured bed-roughness values in practice, we compared our inversed roughness values established from the river and floodplain model by the Federal Emergency Management Agency [23]. Figure 11 shows that most river sections had bed roughness close to 0.035, and Section 2 had a much lower value of around 0.03. Compared to the estimation results from the Federal Emergency Management Agency [23], our inversed roughness values were slightly lower, particularly at Section 3 and Section 6.

In order to verify the inversion accuracy, measured water-level data from 25 May to 17 June 2003 at the Bonners Ferry station were used to compare the model forecasting results of our model. Generally, the forecasting errors produced by this work approached zero with the elapsed time, while those produced by the Federal Emergency Management Agency [23] deviated from zero with the elapsed time, as illustrated in Figure 12. The forecasting water-level RMSE in this work was 0.041 m, which is 31% less than that from Federal Emergency Management Agency (i.e., RMSE = 0.060 m) as shown in Figure 13.

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.

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. The experimental evaluation also shows that, at the observational-noise level of 0.005 m, the water-level forecasting error approached zero and outperformed the work from the Federal Emergency Management Agency.

Bed roughness could be influenced by various factors, including the size and shape of bed sediments [28], the degree of sorting and the roundness of bed sediments, the flow rate [29] and depth [30] of the river, and the presence of obstructions in the river channel [27]. For example, Berenbrock and Bennett showed that Manning’s roughness decreased with a higher flow rate [22]. Future work can be carried out to compare the developed model’s roughness inversion capacity and water-level forecasting performance at different conditions that may influence bed roughness.

6. Conclusions

Due to the importance of the bed-roughness parameter in the hydrodynamic model and its challenging estimation, we developed a new parameter inversion model for estimating bed roughness by constraining the roughness distribution between river sections. The model was solved using an iterative process, and a simulated river network with four levels of river channels was used for optimizing the model parameters, followed by an experimental evaluation.

The impact of various factors on the accuracy of inversed roughness was analyzed, including the number of measurement stations, the amount of the observed data, initial bed roughness, observational noise, and the weight of the regularization term. The numerical experimental results demonstrate that increasing the number of measurement stations and the amount of the observed data significantly improved the robustness of the model, with an optimal parameter setting of 3 stations and 30 observed data at each station. In contrast, the initial roughness had little impact on the model. The RMSE of the initial roughness slightly decreased by 1.256% with the increase in data from 10 to 20, and the results were similar to those of the increase in data from 20 to 30. The RMSE of the water level significantly decreased with the increase in data from 20 to 30, indicating that water-level forecasting is rather sensitive to variations in bed roughness. The parameter analysis also shows that the model had good noise resistance capacity, and the error could be significantly reduced by controlling the smoothness level of the inversed roughness by using a small weight of the regularization term, e.g., 100.

An experiment was conducted on a real river, Kootenai, using the calibrated model parameters. The results were compared with those from the literature. The forecasted water-level RMSE in this study was 0.041 m, which was 31% less than that from the Federal Emergency Management Agency (i.e., RMSE = 0.060 m). Overall, the proposed model provides a new approach for estimating bed roughness parameters in hydrodynamic models and improves the accuracy of water-level forecasting. Future work could compare the model’s roughness inversion capacity and water-level forecasting performance under varying bed roughness conditions.