An accurate prediction of roughness coefficient is of substantial importance for river management. The current study applies two artificial intelligence methods namely; Feed-Forward Neural Network (FFNN) and Multilayer Perceptron Firefly Algorithm (MLP-FFA) to predict the Manning roughness coefficient in channels with dune and ripple bedforms. In this regard, based on the flow and sediment particles properties various models were developed and tested using some available experimental data sets. The obtained results showed that the applied methods had high efficiency in the Manning coefficient modeling. It was found that both flow and sediment properties were effective in modeling process. Sensitivity analysis proved that the Reynolds number plays a key role in the modeling of channel resistance with dune bedform and Froude number and the ratio of the hydraulic radius to the median grain diameter play key roles in the modeling of channel resistance with ripple bedform. Furthermore, for assessing the best-applied model dependability, uncertainty analysis was performed and obtained results showed an allowable degree of uncertainty for the MLP-FFA model in roughness coefficient modeling.

  • FFNN and MLP-FFA methods were selected to identify influential parameters for prediction of roughness coefficient in alluvial channel.

  • Experimental datasets were used to feed the utilized models.

  • Uncertainty analysis was performed to evaluate the best-applied model dependability.

  • Results showed desirable performance of the applied models in roughness coefficient modeling.

Accurate prediction of the open channels flow resistance has a significant effect on flow conditions and can be identified as a crucial part of designing and operating hydraulic structures. Determination of the friction coefficient in open channel hydraulics is a substantial and intransitive issue in the design and operation of hydraulic structures, the calculation of water depth, flow velocity and an accurate characterization of energy losses. Assessments of hydraulic resistance, or flow resistance, are not a trivial matter, due to the multitude of factors influencing roughness (e.g., bed material, bedforms, cross-sectional and plan form variability, vegetation etc.). The problem of predicting flow resistance and roughness coefficient depends to a large extent on the bedform. Assessments of the bedform such as dunes and ripples in river and marine environments require information on the instability mechanism, as the development of the bedform is inversely proportional to the lag between bed shear stress, sediment transport and bed elevation. When the tractive force is sufficient to begin the sediment transport, an initially flat bed will be unstable and deformed into irregular features (Kennedy 1963). In the case of fine sediment, ripples are formed, while coarser sediments and higher subcritical velocity will usually form dunes (Figure 1). Ripples refer to triangular sand waves with small dimension, typically shorter than about 0.6 meter and higher than about 60 mm, whereas dunes are associated with larger dimensions formed in natural streams (Engelund & Fredsoe 1982).

Figure 1

Different types of bedforms in alluvial open channels.

Figure 1

Different types of bedforms in alluvial open channels.

Close modal

In the past half century, several analytical and semi empirical approaches have been presented in order to predict the total roughness coefficient owing to bedform roughness (Meyer-Peter & Müller 1948; Raudkivi 1967; Richardson & Simons 1967; Smith 1968; Van Rijn 1984; Yang et al. 2005; Van der Mark et al. 2008). However, the existing relationships for prediction of roughness coefficient related to bedforms differ from each other and no universal equation for roughness coefficient was established. This can be due to the complicated process of interaction between a large number of variables, 3D nature of bedform development and also the lag in adjustment of bedform in reaction to changing flow conditions (Karim 1999). Concerning the point emphasized in the foregoing discussion, development of a flexible and robust methodology is deemed a crucial problem for predicting roughness coefficient in the channel with different types of bedform.

In the recent decades several artificial intelligence (AI) methods [e.g., Artificial Neural Networks (ANNs), Neuro-Fuzzy models (NF), Genetic Programming (GP), Gene Expression Programming (GEP), and Support Vector Machine (SVM), Feed-Forward Neural Network (FFNN), and Kernel Extreme Learning Machine (KELM)] have been used for investigating the complex hydraulic and hydrologic phenomena. Real-time hydrologic forecasting (Yu et al. 2004), prediction of groundwater quality (Yang et al. 2017), hydraulic jump investigation (Roushangar et al. 2018a), longitudinal dispersion coefficients computing in natural streams (Azamathulla & Wu 2011), rainfall–runoff process modeling (Komasi & Sharghi 2016), monthly streamflow modeling (Zhu et al. 2018), predicting energy dissipation in culverts (Roushangar et al. 2019), atmospheric temperature estimating (Azamathulla et al. 2018), investigation of discharge coefficient of trapezoidal labyrinth weirs (Norouzi et al. 2019), modeling the energy depreciation due to the use of vertical screen in the downstream of inclined drops (Norouzi Sarkarabad et al. 2019), predicting vertical drop hydraulic parameters in the presence of dual horizontal screens (Daneshfaraz et al. 2021), and investigating effects of hydraulic characteristics, sedimentary parameters, and mining of bed material on scour depth of bridge pier groups (Majedi-Asl et al. 2021) are some of the examples.

Machine learning, a branch of artificial intelligence, deals with the representation and generalization using data learning techniques. Representation of data instances and functions evaluated on these instances are part of all machine learning systems. Generalization is the property that the system will perform well on unseen data instances; the conditions under which this can be guaranteed are a key object of study in the subfield of computational learning theory. There is a wide variety of machine learning tasks and successful applications. In general, the task of a ML algorithm can be described as follows: given a set of input variables and the associated output variable(s), the objective is learning a functional relationship for the input-output variables set.

Computation of the river stage and flow velocity relies on the determination of bedform roughness; therefore, knowledge of the bedform is very important. Predicting the roughness coefficient is a complex phenomenon due to the nonlinearity and uncertainties of the process and the existing regression models do not show desired accuracy and output of them are often associated with large errors. In a nonlinear method the resistance factor is not separated into grain roughness and bedform roughness, as in the linear superposition approaches. Instead, it is kept as a single factor. In linear modeling of Manning roughness coefficient in movable beds, it is necessary that initially, the plan bed resistance be calculated and then, the form resistance investigated. This is a complex and time-consuming process and sometimes for this purpose there are not enough data in published literature. The aim of this study was to predict the total friction factor based on hydraulic and sediment parameters using nonlinear approaches. Therefore, two AI models (i.e. FFNN, MLP-FFA) were applied to predict the roughness coefficient in channels with ripple and dune bedforms and also investigate the best input models and effective parameters for each state. Different inputs combinations were developed and their performance was evaluated considering two scenarios. In scenario 1, only hydraulic characteristics were used for Manning roughness coefficient modeling and in scenario 2, both hydraulic and sediment characteristics were applied. In addition, uncertainty analysis (UA) was performed to investigate the applied models dependability.

Experimental data used in the study

Due to the fact that employing more data sets from varied hydraulic conditions can challenge the AI methods and enjoy more reliable evaluation, several related experimental data sets were considered for modeling process. In this paper, the following data sets corresponding to the friction coefficient in open channels with ripple and dune bedforms selected from published literature were used:

  • (I)

    Williams (1970) organized several experiments that were made in channels with different widths and water depths in laboratories in Washington, DC. Sediment transport rates, grain size, water depth, and channel width were measured. Furthermore, water discharge, mean velocity, slope (energy gradient), and bedform characteristics were considered as the dependent variables.

  • (II)

    As a part of the research program of the Water Resources Division of the U.S. Geological Survey, a project was organized at Colorado State University between 1956 and 1961 to determine the effects of the size of bed material, temperature of flow, and the fine sediment in the flow of the hydraulic and transport variables. The investigations for each set cover flow phenomena ranging from a plane bed with no sediment movement to violent anti-dunes (Guy et al. 1966).

  • (III)

    The U.S. Army Corps of Engineers Waterways Experiment Station in 1935 organized several experiments to research about sediment transport.

  • (IV)

    Saghebian (2018) studied friction factor in alluvial channels. Several dune bedform experiments were performed in a 10 m long, 0.5 and 1 m wide, and 0.8 m height rectangular flume at the hydraulic laboratory of Tabriz University. The flume had glass walls and metal floor. Sediment particles used in the experiments were sand with specific gravity of 2.65 and uniform average diameters of 0.15 and 0.27 mm. Water flow was supplied by a pump, re-circulating between the upstream and downstream. The original flume had ratchet screw jacks for adjusting the slope of the flume. To measure the water depth, a point gauge was used with accuracy of 0.1 mm. The point gauge was able to move along the length and width of the channel and measure the bedform height and water depth in the entire of the channel. By changing the flow depth and discharge, the average velocities, Froude numbers, dune height, and other parameters were calculated.

  • (V)

    Singh (1960) investigated bed-load transport in channels with bedforms. He organized 100 experiments that were made in channels with different widths and water depths at Imperial College London. Sediments with grain size of 0.62 mm were used during the studies.

The ranges of measured and calculated parameters are presented in Table 1. It should be pointed out that Manning's formula was used for Manning's coefficient (n) values calculating. The used variables in this table are: channel width (b), mean grain diameter (D50), Flow depth (y), Froude number (Fr = V/[g × y]1/2) in which V is flow velocity and g is gravitational acceleration, and Reynolds number (Re = VR/ν) in which R is hydraulic radius and ν is kinematic viscosity.

Table 1

Details of the used data

ResearcherParameters
No. of data
b (mm)D50 (mm)FrRey (mm)n
Dune bedform 
Williams (1970)  76.2–1,118 1.35 0.34–0.84 11,932–101,920 87.1–222 0.0091–0.0201 89 
Guy et al. (1966)  609, 2,438 0.19–0.93 0.25–0.65 46,800–255,500 91.4–405 0.015–0.038 114 
Saghebian (2018)  500, 1,000 0.15, 0.27 0.19–0.49 23,561–47,238 190–370 0.015–0.031 65 
WSA (1935)  705,736 0.18–0.47 0.3–0.72 19,061–66,432 65.5–208 0.0127–0.0249 61 
Ripple bedform 
Guy et al. (1966)  609–2,438 0.18–0.5 0.14–0.36 14,505–47,429 80–310 0.007–0.019 44 
WSA (1935)  730 0.18–0.47 0.11–0.73 5,218–61,013 10–260 0.003–0.051 215 
Singh (1960)  250–750 0.62 0.27–0.85 3,100–36,129 10–200 0.008–0.024 100 
ResearcherParameters
No. of data
b (mm)D50 (mm)FrRey (mm)n
Dune bedform 
Williams (1970)  76.2–1,118 1.35 0.34–0.84 11,932–101,920 87.1–222 0.0091–0.0201 89 
Guy et al. (1966)  609, 2,438 0.19–0.93 0.25–0.65 46,800–255,500 91.4–405 0.015–0.038 114 
Saghebian (2018)  500, 1,000 0.15, 0.27 0.19–0.49 23,561–47,238 190–370 0.015–0.031 65 
WSA (1935)  705,736 0.18–0.47 0.3–0.72 19,061–66,432 65.5–208 0.0127–0.0249 61 
Ripple bedform 
Guy et al. (1966)  609–2,438 0.18–0.5 0.14–0.36 14,505–47,429 80–310 0.007–0.019 44 
WSA (1935)  730 0.18–0.47 0.11–0.73 5,218–61,013 10–260 0.003–0.051 215 
Singh (1960)  250–750 0.62 0.27–0.85 3,100–36,129 10–200 0.008–0.024 100 

Feed-Forward Neural Network (FFNN)

Artificial Neural Networks (ANNs) are a family of machine learning algorithms originally inspired by biological neural networks that can be employed to approximate any measurable function with arbitrary number of inputs (Tayfur 2012). The FFNN with back propagation (BP) is widely known utilized strategy in water resources engineering issues. The employed ANN is the common FFNN algorithm with three layers of input, hidden and target. The Levenberg–Marquardt preparing calculation (Tayfur 2012; Najafi et al. 2018) was utilized, and the Mean Square Error (MSE) between the calculated and observed values served as the cost function.

Hybrid Multilayer Perceptron Firefly Algorithm (MLP-FFA)

MLP-FFA method is the integration of Multilayer Perceptron (MLP) and Firefly Algorithm (FFA) models. The MLP is a class of feed-forward artificial neural network. The FFA as a swarm intelligence optimization method is conceptually based on the fireflies motion. This method is applied in strategic searching for optimal parameters of the MLP model. The MLP-FFA modeling flowchart is shown in Figure 2. Formulating the objective function and modified the light intensity is the main aim of MLP-FFA model designing. Equations (1)–(3) can be applied for calculating the light intensity I(r), attractiveness η, and the Cartesian distance between any two i and j fireflies:
(1)
(2)
(3)
where xi,k,, d, I(r), Io, respectively are the kth component of the spatial coordinate xi of the ith firefly, the coefficient of light absorption, the given problem dimensionality; the light intensity at distance r, and initial light intensity from a firefly. Also, η(r) and ηO are the attractiveness η at a distance r and r = 0, respectively. The next firefly i coordinate is expressed as below:
(4)
(5)
Figure 2

A schematic view of the MLP-FFA method.

Figure 2

A schematic view of the MLP-FFA method.

Close modal

In Equation (5), the attraction effect is shown using the first term, and randomization is shown using the second term. The χ is randomization coefficient. The χ amounts varies from 0 to 1 (for the current study χ = 0.5). ɛi is the random number vector. This parameter is obtained from a Gaussian distribution (ɛi is 0.96 in this study).

In general, artificial intelligence methods such as FFNN and MLP-FFA are relatively new important methods and are capable of adapting themselves to predict any variable of interest via sufficient inputs. The training of these methods is fast and has high accuracy. The FFNN can be applied to many problems, as long as there is some data. It can be applied to problems, for which analytical methods do not yet exist. If there is a pattern, then neural networks should quickly work it out, even if the data are noisy. Also, the MLP model is a distinct item of ANN that contains a parallel information processing system with neurons that are organized in the input, hidden and output layers. In its elementary form, where an optimizer is not used with the MLP model, it contains input, hidden and output layers. The objective of the back-propagation algorithm is to reduce the global error between predictor and target variables in training stage. Overall, the FFA is an optimization algorithm that is known to yield better performance that is attributed to identifying the global minimum within the feature data sets. In this paper, the optimal values for the weights determined by the MLP model were computed by the FFA algorithm where the final model aimed to optimize the magnitude of the weights dependent on the features that were present in the training data set.

In order to assess the applied models capability, the Determination Coefficient (DC), Correlation Coefficient (R), and Root Mean Square Error (RMSE) statistical criteria were used. The DC measures the fraction of the variance in the data explained by the model. The R provides information for the linear dependence between measured and predicted values, and the RMSE describes the average difference between predicted and measured values corresponding to the predicted values. These statistical criteria are formulated as:
(6)
(7)
(8)
where no, np, , and show the observed, predicted, mean observed, and mean predicted values, respectively and ND is data number. Also, all utilized variables were scaled between 0 and 1 in order to eliminate the input and output variables dimensions.

Simulation and models development

In order to implement employed FFNN and MLP-FFA for prediction of the Manning roughness coefficient, the most significant step is to find the best architecture of the model, which can properly simulate the relationship between input and output variables. It is assumed that the roughness coefficient (n) as a dependent variable can be described as below (Roushangar et al. 2018b):
(9)
where s is the specific gravity. In order to determine the output variables (friction coefficient) under two different scenarios, a set of dimensionless variables was developed as input combinations and listed in Table 2. For predicting the Manning's coefficient, due to training and testing goals, data were divided into training set (80% of total data) and testing set (remaining 20% of data). In addition, the applied models were run via coding in MATLA software. Figure 3 shows applied scenarios considered in this study.
Table 2

Considered models for the Manning's n prediction

Investigation of roughness coefficient
Flow properties
Flow and sediment properties (HS)
ModelInputsModelInputs
F1 Re FS1 R/D FS6 Vy/[g(s 1)D503]0.5, R/D50 
F2 Fr FS2 Re, R/D50 FS7 Vy/[g(s 1)D503]0.5, Re 
F3 y/b FS3 Fr, R/D50 FS8 Re, Vy/[g(s − 1)D503]0.5, R/D50 
F4 Re, y/b FS4 V/[g(s1)D50] 0.5   
F5 Fr, y/b FS5 Vy/[g(s1)D503] 0.5   
Investigation of roughness coefficient
Flow properties
Flow and sediment properties (HS)
ModelInputsModelInputs
F1 Re FS1 R/D FS6 Vy/[g(s 1)D503]0.5, R/D50 
F2 Fr FS2 Re, R/D50 FS7 Vy/[g(s 1)D503]0.5, Re 
F3 y/b FS3 Fr, R/D50 FS8 Re, Vy/[g(s − 1)D503]0.5, R/D50 
F4 Re, y/b FS4 V/[g(s1)D50] 0.5   
F5 Fr, y/b FS5 Vy/[g(s1)D503] 0.5   
Figure 3

Schematic view of the considered scenarios.

Figure 3

Schematic view of the considered scenarios.

Close modal

In this section of the paper, the capability of FFNN and MMLP-FFA methods were used in forecasting the Manning's n. It should be noted that each artificial intelligence method has its own settings and parameters. The optimal values of these parameters are needed for achieving the desired results. In FFNN and MLP-FFA modeling the network topology has direct effects on its computational complexity and generalization capability. Therefore, the appropriate structure of these methods should be selected. In this study, various networks were tried for determining the hidden layer node numbers. For this aim, different number of neurons (i.e. 2, 3, 5, 7, and 9) in hidden layer was tested. Also, it should be noted that large numbers of classical models have been developed based on experimental data sets, which describe the complex phenomenon of the flow resistance process in channels with bedforms. However, these existing equations rely on a limited database, untested model assumptions, and a general lack of field data, and do not show the same results under variable flow conditions. These issues cause uncertainty in the prediction of roughness coefficient; therefore, in this study, several data sets were mixed to model the roughness coefficient in channels with ripple and dune bedforms. The use of a wide range of data can lead to more reliable results. The performances of employed AI methods were compared with each other and the obtained results were listed in Tables 3 and 4. In the first scenario, expression of the Manning's coefficient modeling process was attempted through variables based on flow characteristics. According to the results obtained for dune bedform (Table 3), it could be seen that the combination of the Reynolds number with depth to width ratio yielded better prediction accuracy. In the second scenario, the flow and sediment properties were considered as inputs in n modeling. According to the results, the model with variables Re, R/D50, and Vy/[g(s1)D503]0.5 yielded to best consequences for the Manning roughness coefficient modeling. Also, the Re and R/D50 combination showed desirable accuracy in the Manning coefficient predicting. It could be seen that adding Vy/[g(s1)D503]0.5 and R/D50 to input combinations improved the modeling accuracy. Considering the results of modeling process under two scenarios, MLP-FFA confirmed its superiority over FFAA approach with respect to statistical indices (having higher R and DC and lower RMSE) for the testing set. Figure 4 shows the scatter plots of predicted Manning roughness coefficient vs. observed values for test series of superior models. According to this figure, the range of error distribution for observed and predicted values was approximately between 10 and 14%.

Table 3

The results obtained for the Manning's n modeling in dune bedform channel (test series)

ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.70 0.50 0.0058 FS1 FFNN 0.72 0.69 0.0062 
 MLP-FFA 0.70 0.54 0.0055  MLP-FFA 0.72 0.70 0.0061 
F2 FFNN 0.56 0.45 0.0069 FS2 FFNN 0.85 0.75 0.0049 
 MLP-FFA 0.58 0.55 0.0058  MLP-FFA 0.89 0.79 0.0046 
F3 FFNN 0.51 0.43 0.0062 FS3 FFNN 0.69 0.65 0.0067 
 MLP-FFA 0.65 0.51 0.0059  MLP-FFA 0.70 0.66 0.0066 
F4 FFNN 0.88 0.73 0.0039 FS4 FFNN 0.88 0.70 0.0061 
 MLP-FFA 0.89 0.78 0.0033  MLP-FFA 0.89 0.72 0.0057 
F5 FFNN 0.60 0.49 0.0066 FS5 FFNN 0.67 0.53 0.0068 
 MLP-FFA 0.63 0.52 0.0064  MLP-FFA 0.71 0.59 0.0067 
     FS6 FFNN 0.68 0.55 0.0073 
      MLP-FFA 0.68 0.60 0.0067 
     FS7 FFNN 0.82 0.71 0.0059 
      MLP-FFA 0.84 0.73 0.0055 
     FS8 FFNN 0.90 0.87 0.0032 
      MLP-FFA 0.92 0.89 0.0031 
ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.70 0.50 0.0058 FS1 FFNN 0.72 0.69 0.0062 
 MLP-FFA 0.70 0.54 0.0055  MLP-FFA 0.72 0.70 0.0061 
F2 FFNN 0.56 0.45 0.0069 FS2 FFNN 0.85 0.75 0.0049 
 MLP-FFA 0.58 0.55 0.0058  MLP-FFA 0.89 0.79 0.0046 
F3 FFNN 0.51 0.43 0.0062 FS3 FFNN 0.69 0.65 0.0067 
 MLP-FFA 0.65 0.51 0.0059  MLP-FFA 0.70 0.66 0.0066 
F4 FFNN 0.88 0.73 0.0039 FS4 FFNN 0.88 0.70 0.0061 
 MLP-FFA 0.89 0.78 0.0033  MLP-FFA 0.89 0.72 0.0057 
F5 FFNN 0.60 0.49 0.0066 FS5 FFNN 0.67 0.53 0.0068 
 MLP-FFA 0.63 0.52 0.0064  MLP-FFA 0.71 0.59 0.0067 
     FS6 FFNN 0.68 0.55 0.0073 
      MLP-FFA 0.68 0.60 0.0067 
     FS7 FFNN 0.82 0.71 0.0059 
      MLP-FFA 0.84 0.73 0.0055 
     FS8 FFNN 0.90 0.87 0.0032 
      MLP-FFA 0.92 0.89 0.0031 
Table 4

The results obtained for Manning's n modeling in ripple bedform channel (test series)

ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.67 0.50 0.0088 FS1 FFNN 0.85 0.78 0.0050 
 MLP-FFA 0.70 0.54 0.0079  MLP-FFA 0.87 0.79 0.0048 
F2 FFNN 0.72 0.69 0.0071 FS2 FFNN 0.73 0.60 0.0080 
 MLP-FFA 0.74 0.72 0.0067  MLP-FFA 0.76 0.64 0.0073 
F3 FFNN 0.70 0.67 0.0073 FS3 FFNN 0.78 0.65 0.0072 
 MLP-FFA 0.71 0.69 0.0069  MLP-FFA 0.79 0.69 0.0068 
F4 FFNN 0.68 0.65 0.0076 FS4 FFNN 0.82 0.70 0.0059 
 MLP-FFA 0.69 0.68 0.0072  MLP-FFA 0.83 0.74 0.0055 
F5 FFNN 0.87 0.76 0.0049 FS5 FFNN 0.83 0.72 0.0056 
 MLP-FFA 0.88 0.79 0.0045  MLP-FFA 0.84 0.75 0.0053 
     FS6 FFNN 0.88 0.80 0.0044 
      MLP-FFA 0.88 0.82 0.0043 
     FS7 FFNN 0.68 0.59 0.0084 
      MLP-FFA 0.73 0.61 0.0079 
     FS8 FFNN 0.81 0.79 0.0047 
      MLP-FFA 0.80 0.81 0.0045 
ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.67 0.50 0.0088 FS1 FFNN 0.85 0.78 0.0050 
 MLP-FFA 0.70 0.54 0.0079  MLP-FFA 0.87 0.79 0.0048 
F2 FFNN 0.72 0.69 0.0071 FS2 FFNN 0.73 0.60 0.0080 
 MLP-FFA 0.74 0.72 0.0067  MLP-FFA 0.76 0.64 0.0073 
F3 FFNN 0.70 0.67 0.0073 FS3 FFNN 0.78 0.65 0.0072 
 MLP-FFA 0.71 0.69 0.0069  MLP-FFA 0.79 0.69 0.0068 
F4 FFNN 0.68 0.65 0.0076 FS4 FFNN 0.82 0.70 0.0059 
 MLP-FFA 0.69 0.68 0.0072  MLP-FFA 0.83 0.74 0.0055 
F5 FFNN 0.87 0.76 0.0049 FS5 FFNN 0.83 0.72 0.0056 
 MLP-FFA 0.88 0.79 0.0045  MLP-FFA 0.84 0.75 0.0053 
     FS6 FFNN 0.88 0.80 0.0044 
      MLP-FFA 0.88 0.82 0.0043 
     FS7 FFNN 0.68 0.59 0.0084 
      MLP-FFA 0.73 0.61 0.0079 
     FS8 FFNN 0.81 0.79 0.0047 
      MLP-FFA 0.80 0.81 0.0045 
Figure 4

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of dune bedform channel.

Figure 4

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of dune bedform channel.

Close modal

In the next part of the study, the developed models were tested for channels with ripple bedforms. Results are listed in Table 4 and shown in Figure 5. According to the results, it could be seen that in the first scenario, using the Froude number as the only input parameter yielded to better outcomes compared to the Reynolds number. In the state of modeling based on flow properties, the model F5 with input parameters of Fr, y/b was superior and the use of depth to width ratio increased the correlation between the observed and predicted values and caused a relative improvement in the accuracy of the models. In the second scenario, the input combination of Vy/(g(s1)D503)0.5, R/D50 variables had the best performance for the roughness Manning coefficient modeling in channels with ripple bedforms. Obtained results indicated that the implementation of the model FS6 as input combination of MLP-FFA method provided very good outcomes (R = 0.88, DC = 0.82, and RMSE = 0.0043).

Figure 5

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of ripple bedform channel.

Figure 5

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of ripple bedform channel.

Close modal

Combined data

For estimating the Manning roughness coefficient in channels with different bedforms, and in order to assess the applied methods capability considering wide range of data; the used data series were mixed. Then, developed models were analyzed for the new data series. The results are shown in Table 5 and Figure 6. Based on the results obtained for the mixed data, the model FS8 with parameters Re, R/D50, and Vy/[g(s1)D503]0.5 was superior. It was found that the desirable accuracy was not achieved for the mixed data series via the used models, especially for higher values of n. However, it should be taken in account that the mixed data results are capable of covering a wider range of data and in this case without regarding the bedform types, roughness coefficient can be studied.

Table 5

The results obtained for Manning's n modeling using mixed data (test series)

ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.57 0.43 0.0107 FS1 FFNN 0.72 0.66 0.0061 
 MLP-FFA 0.60 0.46 0.0096  MLP-FFA 0.74 0.67 0.0059 
F2 FFNN 0.61 0.59 0.0087 FS2 FFNN 0.62 0.51 0.0098 
 MLP-FFA 0.63 0.61 0.0082  MLP-FFA 0.65 0.54 0.0089 
F3 FFNN 0.60 0.57 0.0089 FS3 FFNN 0.66 0.55 0.0088 
 MLP-FFA 0.60 0.59 0.0084  MLP-FFA 0.67 0.59 0.0083 
F4 FFNN 0.74 0.65 0.0060 FS4 FFNN 0.70 0.60 0.0072 
 MLP-FFA 0.75 0.67 0.0055  MLP-FFA 0.71 0.63 0.0067 
F5 FFNN 0.58 0.55 0.0093 FS5 FFNN 0.71 0.61 0.0068 
 MLP-FFA 0.59 0.58 0.0088  MLP-FFA 0.71 0.64 0.0065 
     FS6 FFNN 0.69 0.67 0.0057 
      MLP-FFA 0.68 0.69 0.0055 
     FS7 FFNN 0.58 0.50 0.0102 
      MLP-FFA 0.62 0.52 0.0096 
     FS8 FFNN 0.75 0.68 0.0054 
      MLP-FFA 0.75 0.70 0.0052 
ModelMethodScenario 1
ModelMethodScenario 2
RDCRMSERDCRMSE
F1 FFNN 0.57 0.43 0.0107 FS1 FFNN 0.72 0.66 0.0061 
 MLP-FFA 0.60 0.46 0.0096  MLP-FFA 0.74 0.67 0.0059 
F2 FFNN 0.61 0.59 0.0087 FS2 FFNN 0.62 0.51 0.0098 
 MLP-FFA 0.63 0.61 0.0082  MLP-FFA 0.65 0.54 0.0089 
F3 FFNN 0.60 0.57 0.0089 FS3 FFNN 0.66 0.55 0.0088 
 MLP-FFA 0.60 0.59 0.0084  MLP-FFA 0.67 0.59 0.0083 
F4 FFNN 0.74 0.65 0.0060 FS4 FFNN 0.70 0.60 0.0072 
 MLP-FFA 0.75 0.67 0.0055  MLP-FFA 0.71 0.63 0.0067 
F5 FFNN 0.58 0.55 0.0093 FS5 FFNN 0.71 0.61 0.0068 
 MLP-FFA 0.59 0.58 0.0088  MLP-FFA 0.71 0.64 0.0065 
     FS6 FFNN 0.69 0.67 0.0057 
      MLP-FFA 0.68 0.69 0.0055 
     FS7 FFNN 0.58 0.50 0.0102 
      MLP-FFA 0.62 0.52 0.0096 
     FS8 FFNN 0.75 0.68 0.0054 
      MLP-FFA 0.75 0.70 0.0052 
Figure 6

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of mixed data.

Figure 6

Scatter plots of predicted Manning roughness coefficient vs. observed values for test series of mixed data.

Close modal

Sensitivity analyzing

To investigate the impacts of different parameters of the MLP-FFA-best models on the Manning roughness coefficient, sensitivity analysis was performed. The significance of each variable was assessed by omitting the variable. Figure 7 revealed that in prediction of the roughness Manning coefficient in the dune bedform and mixed data states, Re was the most efficient parameter and in the ripple bedform state, Fr (in scenario 1) and R/D50 (in scenario 2) were the dominant parameters.

Figure 7

Relative significance of each input parameter of the MLP-FFA best models.

Figure 7

Relative significance of each input parameter of the MLP-FFA best models.

Close modal

Validation of proposed FFNN and MLP-FFA models using separately data sets

In this part, each experimental data set was separately used to evaluate the performance of applied methods in order to compare the differences between experimental setups and their impacts on the obtained results. The models FS8 (in dune bedform) and FS6 (in ripple bedform) from scenario 2 were selected for this aim. Williams (1970), Guy et al. (1966), Saghebian (2018), WSA (1935), and Singh (1960) studied flow resistance considering different conditions. The ranges of their experiments were listed in Table 1. According to the evaluation criteria (R, DC, and RMSE) listed in Table 6, it could be seen that the efficiency of the applied methods for each data set was different. In the case of dune bedform, the data set from Guy et al. (1966) and in the case of ripple bedform, the data set from WSA (1935) led to better predictions. This issue confirmed that the artificial intelligence methods were data sensitive and data with different ranges could lead to different accuracy. Although, the used methods approximately showed the same results and they were successful in modeling process using different range of data sets, however, mixing of several datasets and applying a wide range of data could lead to more reliable results.

Table 6

Validation of proposed models for the selected data sets (test series)

Data setMethodScenario 2
RDCRMSE
Dune bedform 
Williams (1970)  FFNN 0.87 0.84 0.0036 
MLP-FFA 0.88 0.86 0.0034 
Guy et al. (1966)  FFNN 0.95 0.92 0.0027 
MLP-FFA 0.98 0.94 0.0024 
Saghebian (2018)  FFNN 0.91 0.89 0.0030 
MLP-FFA 0.92 0.91 0.0028 
WSA (1935)  FFNN 0.89 0.87 0.0032 
MLP-FFA 0.92 0.90 0.0029 
Ripple bedform 
Guy et al. (1966)  FFNN 0.85 0.79 0.047 
MLP-FFA 0.85 0.80 0.045 
WSA (1935)  FFNN 0.87 0.82 0.042 
MLP-FFA 0.92 0.85 0.040 
Singh (1960)  FFNN 0.83 0.77 0.051 
MLP-FFA 0.86 0.79 0.048 
Data setMethodScenario 2
RDCRMSE
Dune bedform 
Williams (1970)  FFNN 0.87 0.84 0.0036 
MLP-FFA 0.88 0.86 0.0034 
Guy et al. (1966)  FFNN 0.95 0.92 0.0027 
MLP-FFA 0.98 0.94 0.0024 
Saghebian (2018)  FFNN 0.91 0.89 0.0030 
MLP-FFA 0.92 0.91 0.0028 
WSA (1935)  FFNN 0.89 0.87 0.0032 
MLP-FFA 0.92 0.90 0.0029 
Ripple bedform 
Guy et al. (1966)  FFNN 0.85 0.79 0.047 
MLP-FFA 0.85 0.80 0.045 
WSA (1935)  FFNN 0.87 0.82 0.042 
MLP-FFA 0.92 0.85 0.040 
Singh (1960)  FFNN 0.83 0.77 0.051 
MLP-FFA 0.86 0.79 0.048 

The results of uncertainty analysis

The UA was applied to determine the uncertainty of the best MLP-FFA model. In this study, the Monte Carlo UA method was used. In the UA method, two elements are used to test the robustness and to analyze the models uncertainty (Noori et al. 2015). The first one is the percentage of the studied outputs which are in the range of 95PPU and the next one is the average distance between the lower and upper uncertainty bands. The upper and lower bands are considered 2.5% and 97.5% probabilities of the cumulative distribution, respectively. Two important indices should be used in the proper confidence level. Firstly, the 95PPU band brackets most of the observations. Secondly, the average distance between the upper and lower parts of the 95PPU (d-Factor) be smaller than the observed data standard deviation (Abbaspour et al. 2007). In order to account the input uncertainties, the two mentioned indices were used. According to Abbaspour et al. (2007), the average width of the confidence interval band can be calculated as follows:
(10)
where σx and are the observed data standard deviation and the average width of confidence band, respectively. The 95PPU can be calculated as:
(11)
where, 95PPU shows 95% predicted uncertainty; k is the observed data number, XU and XL are the upper and lower uncertainty bands, and Xreg is the current registered data. The obtained results for the UA are shown in Figure 8. Based on the d-Factor and 95PPU indices amounts, it was observed that for both bedform channels both observed and predicted amounts were within the 95PPU band in most of the cases. Also, it was found that the amount of d-Factors for training and test data sets were less than the observed data standard deviation. Therefore, based on the results, the roughness coefficient modeling via MLP-FFA model showed a desirable degree of uncertainty.
Figure 8

Uncertainty analysis for the best model of the MLP-FFA method.

Figure 8

Uncertainty analysis for the best model of the MLP-FFA method.

Close modal

Roughness coefficient needs to be adequately quantified owing to its significant influence on the performance of hydraulic structures and river management. In the present study, the performance of two artificial intelligence methods was evaluated in order to predict the Manning friction coefficient in open channels with ripple and dune bedforms. In order to develop FFNN and MLP-FFA models, two scenarios were considered and the impact of different modeling for the estimation of the friction coefficient was evaluated. The obtained results revealed that scenario 2 which used both flow and sediment parameters as inputs in modeling of the Manning coefficient was more accurate than scenario 1 which applied only flow parameters as inputs. In both bedform types, it was observed that adding Vy/[g(s1)D503]0.5 and R/D50 to input combinations enhanced the models efficiency. For both dune and mixed data sets, the model with parameters Re, R/D50, and Vy/[g(s1)D503]0.5 led to better accuracy. In the ripple state, the model with parameters Vy/(g(s1)D503)0.5, R/D50 was superior. Based on the obtained results from sensitivity analysis, it was observed that in the dune bedform state, Re was the most efficient parameter and in the ripple bedform state, Fr (in the scenario 1) and R/D50 (in the scenario 2) had a more dominant role in modeling process. Also, the dependability of the best applied models was assessed via UA. Based on the results, an allowable degree of uncertainty was found for the MLP-FFA model in the Manning roughness coefficient modeling in channels with different bedforms. It should, however, be noted that FFNN and MLP-FFA are data-driven models and the FFNN- and MLP-FFA-based models are data sensitive, so further studies using data ranges from this study and field data should be carried out to find the merits of the models to estimate roughness coefficient in the real condition of flow.

All relevant data are included in the paper or its Supplementary Information.

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