The potential of FFNN and MLP-FFA approaches in prediction of Manning coefficient in ripple and dune bedforms Open Access

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).

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.

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.

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.

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). ɛ_i is the random number vector. This parameter is obtained from a Gaussian distribution (ɛ_i 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 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(s−1)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(s−1)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%.

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(s−1)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).

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(s−1)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.

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.

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.

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(s−1)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(s−1)D₅₀³]^0.5 led to better accuracy. In the ripple state, the model with parameters Vy/(g(s−1)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.