Roughness Inversion of Water Transfer Channels from a Data-Driven Perspective

Abstract

Manning’s roughness coefficient ($n_c$) is an important parameter characterizing the flow capacity of water transfer channels, and it is also an important and sensitive parameter in one-dimensional (1D) flow simulation. This study focused on the roughness inversion for datasets with different sequence lengths, time steps and anomalous data points. A case study was performed with the datasets of the Shandong Jiaodong Water Transfer Project under steady-state conditions. For sequence lengths, the datasets of 6, 12, 24, 40, 88, and 142 h were selected, all with a time step of 1 min. Subsequently, the time step was changed to 5, 10, 15, 30, 60, and 120 min for the 40 h dataset mentioned above. Finally, the flow data point under a certain moment was selected and changed by 10%, 20%, 30%, and 40% respectively. The results show that there is a quadratic relationship between the $n_c$ value and the objective function value and the optimal $n_c$ value is $n_c=-b/2a$. It is recommended that the $n_c$ value retains four decimal places and is inverted using high-frequency and cleaned datasets.

1. Introduction

Interbasin water transfer projects are designed to deliver water to one or more cities for domestic and industrial use through open channels. Manning’s roughness coefficient ($n_c$) is an important parameter describing the flow capacity of open channels, and it is likely to deviate from the design value under the influence of various factors such as construction quality, operation time and environmental changes. For example, the roughness of open channels will be increased to some extent in cases of water weed growth, dead material gathering, substrate siltation and algae growth, which will reduce the water transfer capacity of these channels. Even a slight change in $n_c$ will be reflected in water level, flow, and other conditions, which will eventually affect the formulation of scheduling plan. In addition, $n_c$ is an important and sensitive hydraulic parameter in the numerical simulation of open channel hydrodynamics [1]. Therefore, there is a need for accurate estimation of $n_c$ for water supply channels.

Empirical formulas and parametric inversion methods are the most widely used methods for $n_c$ estimation. There are numerous factors affecting $n_c$ [2,3], including surface roughness, channel irregularity and alignment, flow unsteadiness, and vegetation around the section [4,5,6,7]. As it is impossible to measure $n_c$ accurately in practice, many empirical formulas have been proposed [8,9,10,11], but most of them are site-dependent with poor transferability [1]. Parameter inversion methods are capable of determining parameters based on control equations and measured data, and they can be classified into two types depending on whether a hydrodynamic model is utilized. For parameter inversion without using a hydrodynamic model, a large amount of water monitoring data is directly back-calculated or fitted to obtain $n_c$ values. However, this method is no longer applicable when the data are limited in sample size or representativeness due to short monitoring time or small monitoring location. In this case, a hydrodynamic model is introduced to minimize the deviation between simulated and measured values, and $n_c$ values are determined by manual trial and error or intelligent algorithm. The manual trial-and-error method is a common method for $n_c$ inversion [12], the objective of which is to adjust the $n_c$ values to render the calculated results closer to the observed results based on experience. The accuracy of this method depends largely on manual experience, which makes it less suitable for complex river networks [13]. Many attempts have been made to overcome these problems. In the 1970s, the $n_c$ values of rivers and channels were determined by the influence coefficient method, extreme value method [14], and Newton–Rephson method [15]. In the 1990s, some direct solution methods were proposed. Lal. [16] obtained a better inversion effect by using the singular matrix decomposition method. Later, modern optimization algorithms were introduced to improve the speed and accuracy of the solution. Many gradient optimization methods, such as Lagrangian multipliers [17], limited-memory quasi-Newton [18,19], and sequential quadratic programming algorithms [20], and common intelligent optimization methods, such as particle swarm optimization (PSO) [12] and genetic algorithms (GAs) [21], have been widely used in the inversion of stratification. In more recent years, data assimilation methods have emerged for roughness inversion [22,23].

It should be noted that the influence of the water level and flow dataset itself on the inversion of roughness is not considered in previous methods. This may be more problematic for water transfer channels, because compared with natural rivers, the regulation of control buildings, such as gates and pumps, in water transfer channels can cause dramatic changes in water level or flow over a short period of time. For instance, the flow rate is doubled at the moment of opening the pumping units. The regulation process can be recorded in real time by the monitoring system at the control section, which means that the dataset with a shorter time step is more capable of describing the change process of water level and flow. However, most available datasets are of low frequency with a long time step (e.g., 120 min) for privacy and security reasons [24,25,26]. Compared with minute-level datasets, low-frequency datasets are less accurate to describe water level and flow changes and may even lead to large cumulative deviations. Take the 120 min time step dataset as an example. In one circumstance, when the flow is 15 m³/s at 0:00 and 2:00 on the same day, is there any change in flow between 0:00 and 2:00? If so, the magnitude and duration of the change may also vary substantially over time. In another circumstance, when the flow is 15 m³/s at 0:00 and 4:00 on the same day and that at 2:00 is 20 m³/s, then the data at 2:00 may be abnormal or normal due to gate or pump regulation. If it is a normal value, the regulation process may also vary substantially, and if it is an abnormal data or caused by a short period of gate or pump regulation, it has a greater impact on roughness inversion compared to the first circumstance. This is related to the encryption treatment by the model. As the model computation time step is usually shorter than the boundary time step, the dataset needs to be encrypted, which may cause deviations between low-frequency and minute-level datasets and consequently the roughness inversion. Thus, it is necessary to investigate how this deviation affects the roughness inversion.

The aim of this paper is to study roughness inversion for datasets with different sequence lengths, time steps and anomalous data points, which serve to allow us to dig deeper into the characteristics and patterns of roughness inversion from a data-driven perspective. In order to minimize the influence of uncertainty factors, only the measured datasets of a single channel with no control structures during the non-regulation period are selected, and the minute-level datasets are considered as real-time and used as reference. A one-dimensional hydrodynamic model is used. The remainder of this paper is organized as follows. Section 2 describes the inversion model and encryption processing, Section 3 presents the study area and data-driven approach, Section 4 describes and analyzes the results, and Section 5 summarizes the main conclusions of this study.

2. Inversion Model and Encryption Processing

The problem of roughness estimation can be solved by establishing an inversion model. The objective of this model is to determine the $n_c$ value by minimizing the differences between calculated and observed water levels at each boundary time step downstream of the channel. This section will introduce the inversion model and encryption processing.

2.1. One-Dimensional Hydrodynamic Model

The calculated water levels are determined from the solution of the 1D hydrodynamic model. This model is based on the Saint-Venant equations, including continuity and momentum equations [24,27]:

$$B\frac{\partial Z}{\partial t}+\frac{\partial Q}{\partial x}=q,$$

(1)

$$\frac{\partial Q}{\partial t}+\frac{\partial }{\partial x}\left(\frac{\alpha Q^2}{A}\right)+gA\frac{\partial Z}{\partial x}+gA{S}_f=0,$$

(2)

where $B$ is the channel cross-sectional width, m; $Z$ is the water level, m; $t$ is the time, s; $Q$ is the discharge, m³/s; $x$ is the distance along the channel, m; $q$ is the lateral inflow per unit length of channel, m²/s; $\alpha$ is the momentum correction coefficient; $A$ is the wetted cross-sectional area, m²; $g$ is the gravitational acceleration, m/s²; and $S_f$ is the friction slope, which can be calculated from the following equation:

$$S_f=\frac{n_c^{2}Q\left|Q\right|}{A^2{R_c}^{4/3}},$$

(3)

where $n_c$ is the roughness coefficient, s/m^1/3; $R_c$ is the hydraulic radius of the channel section, m, defined by $R_c=A/P$, where $P$ is the wetted perimeter, m; and other symbols are the same as before.

In this study, Equations (1) and (2) are discretized by the Preissmann four-point linearized implicit difference with weights, and the discrete grid form is shown in Figure 1.

In this format, the continuous function $f(x,t)$ and its time and space derivations are taken as follows [28]:

$$f\left(x,t\right)\left|M=\frac{\theta }{2}\right.\left(f_{i+1}^{n+1}+f_i^{n+1}\right)+\frac{1-\theta }{2}\left(f_{i+1}^n+f_i^n\right)$$

(4)

$$\frac{\partial f}{\partial x}\left|M=\theta \frac{f_{i+1}^{n+1}-f_i^{n+1}}{\Delta x}\right.+\left(1-\theta \right)\frac{f_{i+1}^n-f_i^n}{\Delta x}$$

(5)

$$\frac{\partial f}{\partial t}\left|M=\frac{f_{i+1}^{n+1}+f_i^{n+1}-f_{i+1}^n-f_i^n}{2\Delta t}\right.$$

(6)

where $f$ denotes the function value; subscripts $i$ and $i+1$ denote the $i^{th}$ and $({i+1)}^{th}$ section, respectively; superscripts $n$ and $n+1$ denote the $n^{th}$ and ${(n+1)}^{th}$ moment, respectively; $∆t$ is the time step, s; $∆x$ is the spatial step, m; $\theta$ is the weighting factor, $0\le \theta \le 1.0$. The scheme is unconditionally stable for $0.5\le \theta \le 1.0$. In this study, the value of $\theta$ is set to 0.75. $∆x$ is automatically divided according to a set of $∆t$ ($∆x=∆t\sqrt{g{h}^0}$, where $h^0$ is the initial water depth of the control section downstream of the calculated section) to further improve the stability of the format by controlling the phase error generated by the numerical dispersion.

Substituting the above approximations into Equations (1) and (2) yields the following linear discretized equations [28]:

$$-Q_i^{n+1}+C_i{Z}_i^{n+1}+Q_{i+1}^{n+1}+C_i{Z}_{i+1}^{n+1}=D_i$$

(7)

$$E_i{Q}_i^{n+1}-F_i{Z}_i^{n+1}+G_i{Q}_{i+1}^{n+1}+F_i{Z}_{i+1}^{n+1}={\Phi }_i$$

(8)

where $Q_i^{n+1}$ and $Z_i^{n+1}$ are the flow and water level of the $i^{th}$ section at the ${(n+1)}^{th}$ moment, respectively; and $C_i$, $D_i$, $E_i$, $G_i$, $F_i$, and ${\Phi }_i$ can be calculated from the hydraulic parameters and the $n^{th}$ moment of the hydraulic elements.

For a water transfer project with $N_s$ cross sections, the system contains a total of 2$N_s$ unknown quantities ($N_s$ water level and $N_s$ flow), and therefore 2$N_s$ mutually independent equations are required to obtain the solutions. $N_s$ cross sections produce a total of $N_s$ − 1 buildings. There are 2$N_s$ − 2 control equations, and two additional upstream and downstream external boundary conditions are required to form a closed set of equations. For the simulation of historical processes, the water level and flow at the boundary are obtained by the monitoring system, and the boundary combinations of the model can be: (1) upstream flow and downstream water level; (2) upstream water level and downstream flow; (3) upstream water level and downstream water level. In this paper, the second combination is chosen, and a catch-up method is used to solve equations.

In this study, the objective function of the roughness inversion model is considered as the sum of the squares of the deviations between calculated and observed water levels [1]:

$$F\left(n_c\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left({\sum }_{i=0}^{i=N_b}{(Z_i^c-Z_i^o)}^2\right)$$

(9)

where $N_b$ is the number of boundary sequences, $N_b=\frac{Et-St}{Bs}$; $Et$ and $St$ are the end time and start time of the boundary sequences, respectively, minute; $Bs$ is the time step of the boundary sequence, minute; $Z_i^c$ and $Z_i^o$ are the $i^{th}$ calculated and observed water levels, respectively, m.

2.2. Encryption Processing

The boundary dataset needs to be encrypted when the model calculation time step does not coincide with the boundary time step. The former (e.g., 1 min and 5 min) is usually shorter than the latter (e.g., 60 min and 120 min). The simplest encryption method is linear processing. Assuming that the calculation time step is 1 min and the boundary time step is 5 min, Figure 2 shows the linear encryption processing of the flow.

Figure 2 shows the cumulative deviation in the water volume before and after encryption. The water volume during the 0–5 min time step is the sum of the areas of all rectangles and right-angled trapezoids for the minute-level dataset (original dataset in Figure 2) and the area of the red right-angled trapezoid for the 5 min boundary dataset, respectively, and the difference between them is the cumulative deviation in water volume (the green shaded area in Figure 2).

The inversion results of $n_c$ values can be affected by time step, anomalous data points, and sequence length of the boundary dataset. As can be seen in Figure 2, the data point at each moment and the number of graphs within the time step of the boundary may affect the size of the cumulative deviation in water volume for the corresponding time period. The outliers directly affect the data values at each moment, and the boundary time step determines the number of graphs within each time step. Thus, the time step and anomalous values of the boundary dataset affect the cumulative deviation in water volume within a single time step. The cumulative deviations of the water volume at each time step can cause deviations in the water level at that time step, while multiple consecutive water level deviations can cause changes in the objective function values and thus the inversion results of $n_c$ values.

3. Study Area and Methods

3.1. Study Area

The Shandong Jiaodong Water Transfer Project is an important part of China’s South–North Water Transfer Project, which is a long-distance, interbasin, cross-region water transfer project aiming to alleviate water shortage and achieve optimal allocation of water resources in the Jiaodong area of Shandong Province, China. The project has 39 water diversion gates and covers the cities of Qingdao, Dongying, Yantai, Weifang and Weihai. In the past 33 years of operation, the project has diverted a total of 11.674 billion cubic meters of water and thus plays a critical role in the social and economic development of the Jiaodong area.

As seen in Figure 3, the Jiaodong Water Transfer Project consists of the Yellow River Water Diversion Project and the Jiaodong Area Yellow River Water Diversion Project. The project is intended to divert water from the Yellow River to the Qing, which begins from the inlet gate of Dayuzhang and passes through Wangdao pumping station, Songzhuang pumping station, Wangnou pumping station, Songzhuang gate, Tingkou pumping station, and Jihongtan pumping station to Jihongtan reservoir along the route. The total length of the project is about 265 km. Both Wangdao Pumping Station and Yubeihe Inverted Siphon have minute-level data, and thus the 5.233 km channel is used as the study area, where the bottom width, side slope coefficient, upstream and downstream bottom elevation are 11.2 m, 2.5, 1.56 m and 0.53 m, respectively.

A float-type water level meter and Doppler ultrasonic flow meter were installed behind the Wangdao pumping station and in front of the Yubeihe inverted siphon for real-time (minute-level) monitoring of water level and flow, respectively. Notably, the float-type water level meter was installed in the water level well in order to reduce the influence of waves and thus to measure water levels more accurately. All monitoring data can be displayed at different time intervals (i.e., minute level, 30 min level, 60 min level, and 120 min level).

3.2. Methods

The datasets with different sequence lengths and time steps under steady-state conditions (pumping stations or gates are not regulated and the water level and flow of the channel remain relatively stable) were collected from 18:00 of 10 November 2022. To validate the effect of sequence length, the datasets with a time step of 1 min and time lengths of 6, 12, 24, 40, 88, and 142 h were selected for roughness inversion, which were denoted as S6-1, S12-1, S24-1, S40-1, S88-1, and S142-1, respectively. Subsequently, the S40-1 datasets with time steps of 5 min (S40-5), 10 min (S40-10), 15 min (S40-15), 30 min (S40-30), 60 min (S40-60), and 120 min (S40-120) were used to validate the effect of time step.

As the flow is likely to change more significantly over a short period of time than the water level, this study only explores the effect of flow data with different percentages of anomalous values on roughness inversion. S40-1, S40-30, S40-60 and S40-120 datasets were selected for the study. To ensure that each dataset had the same flow data point, the flow value (17.5 m³/s) at 22:00 of November 10 was selected for the study. Taking S40-1 as an example, the best $n_c$ values were found after increasing 10%, 20%, 30% and 40% deviation from 17.5, respectively, which were denoted as S40-1-0.1, S40-1-0.2, S40-1-0.3 and S40-1-0.4, respectively. The same operation was performed for other datasets and named according to the same rule.

For a single channel, the upstream boundary of the hydrodynamic model is usually the water level and the downstream boundary is the flow. In this study, manual trial and error is used to find the optimal $n_c$ value to better explore the roughness inversion characteristics.

4. Results and Discussion

Figure 4 and Figure 5 show the S40-1 dataset of water level and flow upstream and downstream of the channel, respectively.

It is found that the upstream water level fluctuates more drastically than the downstream water level and the upstream flow is more stable than the downstream flow, which may be related to the installation location and calibration of the monitoring system. The upstream water level meter is installed in the pool behind the pumping station. Since the pool is very small, the minute-level water level fluctuates greatly, while the downstream water level meter is installed about 100 m before the gate and the water level fluctuates little under steady-state conditions. The upstream flow meter is installed about 150 m downstream of the pool, and the flow is stable and measured accurately under steady-state conditions, while the downstream flow meter installed about 150 m upstream of the gate, which is newly installed and still in the calibration period, and thus the measured flow data fluctuate more significantly.

4.1. Sequence Length

The S40-1 dataset is discussed here as an example to save space. The $n_c$ value is increased from 0.012 to 0.0132 at a step of 0.0002 based on experience, and the simulated and measured water levels before the downstream gate for each $n_c$ value are shown in Figure 6.

It is seen in Figure 6 that the simulated water levels exhibit approximately the same variation pattern at different $n_c$ values except for the first 60 data points, and it can be assumed that the $n_c$ value can only change the magnitude of simulated water levels without affecting the overall variation pattern. The average difference in simulated water levels between two adjacent $n_c$ values is 0.01 m. As far as the first 60 data points are concerned, the water level curve at each $n_c$ value is axially symmetric about the water level curves at one $n_c$ value. This means that there must be an optimal $n_c$ value that can minimize the objective function when the measured value is known. Therefore, there may be a quadratic relationship between the $n_c$ value and the objective function. The function expression can be quickly fitted by Excel, and the fitting results for S6-1, S12-1, S24-1, S40-1, S88-1, and S142-1 datasets are shown in Figure 7.

Figure 7 reveals that: (1) the $R^2$ of each fitted curve is over 0.99, indicating perfect fit; (2) given the same $n_c$ value, the value of the objective function increases with the increase in sequence length; and (3) each curve is essentially quadratic, with the opening facing upward. Therefore, it is concluded that near the best $n_c$ value, there is a quadratic relationship between the $n_c$ value and the objective function, which is independent of the length of the dataset. For the quadratic function, the $y$ value is the maximum (minimum) value when $x=\frac{-b}{2a}$. Therefore, the optimal $n_c$ value is equal to $\frac{-b}{2a}$. Taking S6-1 as an example, the optimal $n_c$ value is $n_c=\frac{-b}{2a}=0.012683124$. When the $n_c$ values are retained to three, four and five decimal places, the optimal $n_c$ values and the corresponding objective function values for each dataset are shown in

Table 1. The optimal $n_c$ values and objective function values for each dataset with different decimal places.

Scheme ID	Three Decimal Places (Function Values)	Four Decimal Places (Function Values)	Five Decimal Places (Function Values)
S6-1	0.013 (0.1121)	0.0127 (0.0040)	0.01268 (0.0038)
S12-1	0.013 (0.1914)	0.0127 (0.0165)	0.01272 (0.0145)
S24-1	0.013 (0.3158)	0.0129 (0.2538)	0.01288 (0.2521)
S40-1	0.013 (0.6180)	0.0128 (0.4567)	0.01284 (0.4446)
S88-1	0.013 (1.5459)	0.0128 (1.1387)	0.01283 (1.1216)
S142-1	0.013 (6.6240)	0.0126 (2.8486)	0.01264 (2.7848)

.

Table 1 shows that the variation in the optimal $n_c$ value with the length of the dataset depends on the number of decimal places of the $n_c$ value. When three decimal places are retained, the optimal $n_c$ value is 0.013 for each dataset, which is independent of the length of the dataset; when four decimal places are retained, the optimal $n_c$ value varies with the length of the dataset for some datasets; when five decimal places are retained, the optimal $n_c$ value varies with the length of the dataset, but there is no correlation between them. Because of the quadratic relationship between the $n_c$ value and the objective function, two different optimal $n_c$ values would be obtained if two different lengths of datasets were fitted with different formulas, and then the optimal $n_c$ value for one dataset may not be the optimal $n_c$ value for the other dataset.

The objective function values in Table 1 and the quadratic function curves indicate that the $n_c$ values should be kept to four decimal places. For a given quadratic function, reducing the number of decimal places of the $n_c$ value moves the $n_c$ value away from $n_c=\frac{-b}{2a}$. If the number of decimal places of the $n_c$ value is too small, it may deviate too much from the optimal $n_c$ value, resulting in too large an objective function value and consequently substantial deviation between simulated and observed water levels, while if the number of decimal places of the $n_c$ value is too large, the objective function value will not change much. In Table 1, the deviation in the objective function value in the last two columns is smaller than that in the first two columns, and the values in the last two columns are closer to the minimum objective function value, which means that the objective function value will not change much when four or five decimal places are retained, but changes more dramatically when three or four decimal places are retained. Therefore, the $n_c$ value should be retained to four decimal places.

The first 85 data points in Figure 6 are shown in Figure 8.

As can be seen in Figure 8, the simulated water levels at different $n_c$ values are approximately the same for the first 18 min and then the difference becomes more pronounced. However, the variation trend is basically the same from the 25th minute. It is possible that the first 18 min is the hydraulic response time of the channel and the 19–25 min step is the warmup period of the simulation. The results will be further verified in Section 4.2.

4.2. Time Step

For all datasets (S40-5, S40-10, S40-15, S40-30, S40-60, and S40-120), the $n_c$ values are increased at a step of 0.0002 from 0.0124 to 0.0132, and the results are curve-fitted (dotted line) for each dataset using Excel. The fitted results are shown in Figure 9, and the fitted equations, R², and optimal $n_c$ values for each curve are shown in

Table 2. The fitted curve formulas and optimal $n_c$ values for different datasets.

Scheme ID	Fitted Curve Formulas	R2	$\mathbf{Optimal} {\mathit{n}}_{\mathit{c}}$
S40-5	y = 1,500,371.11x2 − 38,500.94x + 247.09	1	0.0128
S40-10	y = 746,138x2 − 19,170x + 123.18	0.9986	0.0128
S40-15	y = 492,742x2 − 12,674x + 81.534	0.9987	0.0129
S40-30	y = 244,725x2 − 6306x + 40.642	0.9989	0.0129
S40-60	y = 122,774x2 − 3164.1x + 20.397	0.9989	0.0129
S40-120	y = 59,201x2 − 1535.8x + 9.9656	0.9995	0.0130

.

Figure 9 shows that the objective function value decreases as the size of the dataset decreases due to the increase in time step. Table 2 shows that the $R^2$ value is over 0.99 for each dataset, indicating perfect fit, and the optimal $n_c$ value gradually increases from 0.0128 at the minute level to 0.013 at the 120 min level, indicating that the time step has an influence on the inversion of the $n_c$ value when the length of the dataset sequence is given, and the longer the time step is, the higher the optimal $n_c$ value will be.

The first few simulated data points for different $n_c$ values for each dataset in Table 2 are shown in Figure 9 and Figure 10.

Figure 10 shows that the number of data points with the same simulated water level ($k$) for each $n_c$ value in different datasets is 4, 2, 2, 1, 1 and 1, respectively, and its relationship with the hydraulic response time ($∆t$) and the time step of the boundary ($Bs$) is described as $k=∆t/Bs+1$. The corresponding hydraulic response time ranges in each subplot in Figure 10 are 15–20 min, 10–20 min, 15–30 min, within 30 min, within 60 min, and within 120 min, respectively. As the time step increases, the range of estimated hydraulic response time becomes larger and the estimation accuracy decreases, and there is only one data point with the same simulated water level when the time step exceeds the hydraulic response time corresponding to the minute level. It is also found that the warmup period is very short and has a similar pattern.

4.3. Abnormal Data Points

The flow value at 22:00 of 10 November is 17.5, 10%, 20%, 30%, and 40% of which are 1.75, 3.5, 5.25 and 7, respectively. The optimal $n_c$ values for datasets with different abnormal flow data points are shown in

Table 3. The optimal $n_c$ values for datasets with different abnormal flow data points.

Deviation Percentage	Deviation (m3/s)	Datasets and Roughness Values
		S40-1	S40-30	S40-60	S40-120
0%	0	0.0128	0.0129	0.0129	0.0130
10%	1.75	0.0128	0.0129	0.0129	0.0129
20%	3.5	0.0128	0.0129	0.0128	0.0128
30%	5.25	0.0128	0.0128	0.0127	0.0126
40%	7	0.0128	0.0128	0.0126	0.0123

, and the simulated water levels at the optimal $n_c$ value for each dataset are shown in Figure 11. The deviations in flow and simulated water level for different datasets are shown in

Table 4. The deviations in storage volume and simulated water level for different datasets.

Deviation Percentage	Deviation (m3/s)	S40-1		S40-30		S40-60		S40-120
		SV(m3)	WL(m)	SV(m3)	WL(m)	SV(m3)	WL(m)	SV(m3)	WL(m)
0%	0	0	0	0	0	0	0	0	0
10%	1.75	0.875	0.02	26.25	0.03	52.5	0.04	105	0.06
20%	3.5	1.75	0.04	52.5	0.07	105	0.09	210	0.12
30%	5.25	2.625	0.06	78.75	0.1	157.5	0.14	315	0.18
40%	7	3.5	0.09	105	0.14	210	0.19	420	0.27

Note: SV is the storage volume deviation and WL is the water level deviation.

, and the fitted curves are shown in Figure 12.

Table 3 shows that the inversion results are less affected by flow anomalies at a shorter time step of the dataset. The optimal $n_c$ values of dataset S40-1 are the same (0.0128) irrespective of the deviation percentage, indicating that the deviation in flow values has little effect on the roughness inversion of the minute-level data. The optimal $n_c$ values of dataset S40-30 are 0.0129 at deviation percentages less than 30% and decreased slightly to 0.0128 at deviation percentages of 30% and 40%, indicating that a deviation of 30% or more in flow value has a small effect on the roughness inversion of 30 min data. The optimal $n_c$ value of dataset S40-60 is 0.0129 at a deviation percentage of 10% and decreases to 0.0126 with increasing deviation percentage, indicating that a deviation of over 10% in flow value has a great impact on the roughness inversion of 60 min data. The optimal $n_c$ values of dataset S40-120 vary from 0.013 without flow deviation to 0.0123 at 40% deviation percentage, with a large difference of 0.007, and the difference in adjacent optimal $n_c$ values increases with increasing deviation percentage with a variation of 0.001, 0.001, 0.002, 0.003, respectively, indicating that the deviation in individual flow values has a significant impact on the roughness inversion of 120 min data. At a deviation percentage of 40%, the optimal $n_c$ value is 0.0128 for the minute-level dataset and 0.0123 for the 120 min level dataset. Figure 5 shows that the water level deviation for the two $n_c$ values is mostly in the range of 0.03–0.035 m, indicating that the simulated water level at the optimal $n_c$ value of the 120 min level dataset is deviated by 3 cm on average from the measured water level, and this deviation becomes more pronounced with the increase in flow deviation percentage. Hence, the minute-level data should be used in the roughness inversion to minimize the influence of abnormal flow data points.

Figure 11 shows that for any dataset, the deviation in simulated water level at 22:00 increases with the increase in flow deviation percentage, and at the same deviation percentage, the longer the time step is, the larger the deviation will be. This is further verified by the values in the WL column of Table 4. Combined with Table 4, the larger the deviation in the simulated water level at 22:00 is, the greater the impact on the roughness inversion will be. This is because subsequent simulated water levels need to deviate from the observed ones in the opposite direction to minimize the objective function. As the simulated water level at 22:00 is decreased, the $n_c$ value needs to be reduced to appropriately increase the simulated water level. It is also found that the single-point flow deviation can cause water level fluctuation in subsequent moments, and the extent of the impact varies for different datasets, which needs to be further verified.

The SV and WL column data for each dataset are shown in Table 4. Both increase linearly with increasing deviation percentage and time step. For the S40-1 dataset, the deviation in storage volume in one minute is 0.875 m³ at a flow deviation of 1.75, that is, 0.875 m³ or more water flows out per minute. Accordingly, the water level before the gate is decreased by 0.02 m. The water levels of S40-30, S40-60 and S40-120 datasets are decreased by 0.03 m, 0.04 m, and 0.06 m, respectively, indicating that the longer the time step is, the larger the WL value will be. As the deviation percentage increases, the maximum WL value of each dataset is 0.03 m, 0.04 m, 0.06 m, and 0.09 m, respectively, which indicates that the longer the time step is, the more sensitive the WL value is to flow deviation.

Figure 12 shows that the flow deviation is linearly related to the water level deviation. The R² values of the fitted curves for each dataset are 0.9918, 0.9976, 0.9983 and 0.9918, respectively. Flow correction can be reversed based on the known curve equation. When there are no abnormal measured water levels and no manual regulation for a period of time, and when the simulated water level at a given moment deviates from the measured one by 15%, the flow data can be corrected by 15%. If artificial regulation is performed but the specific regulation process is unknown, then the degree of water deviation and the actual situation of the project can be inferred from the change magnitude and time of the flow.

5. Conclusions

This paper explores the roughness inversion of water transfer channels from the data-driven perspective, and the objective function is to minimize the deviation between simulated and measured water levels. The effects of sequence length, time step and percentage of anomalous flow data of the dataset on roughness inversion are investigated. A case study is performed with the datasets of a single channel of the Jiaodong Water Transfer Project. The main conclusions are summarized as follows. (1) There is a quadratic relationship between the $n_c$ value and the objective function value, and the optimal $n_c$ value can be quickly obtained by the curve-fitting formula. (2) At a fixed boundary time step, the sequence length has an effect on the $n_c$ value when the effective decimal place of the $n_c$ value is not considered. This study suggests that the $n_c$ value should be retained to four decimal places. (3) At a fixed sequence length, the boundary time step has little effect on the $n_c$ value in the absence of anomalous flow data, but has a significant effect in the presence of anomalous flow data. The greater the deviation in abnormal data or the time step, the greater the impact on the $n_c$ value. Thus, an accurate and effective method is needed to clean the water level and flow data by removing abnormal values and filling in missing values.

This paper has some limitations that should be noted: (1) Section 2.2 analyzes how the sequence length, time step, and anomalous flow data affect roughness inversion without a rigorous theoretical basis; and (2) the water level and flow data under steady-state conditions are selected in this study, where there are no abnormal values as no artificial regulation is performed. Single-point flow abnormalities are analyzed, and it is necessary to investigate the effect of multipoint abnormal flow or water-level data on roughness inversion. In addition to roughness inversion under artificial regulation, the inversion of the regulation process should also be considered in future research.