Insights into the prediction capability of roughness coefficient in current ripple bedforms under varied hydraulic conditions

Accurate prediction of the flow resistance (i.e., roughness coefficient) in open channel hydraulics has a significant effect on flow conditions and can be identified as a crucial part of designing and operating hydraulic structures. It is way more complicated to determine roughness since it is affected by several factors including irregular channel bed properties, bed material, vegetation, cross-sectional, plan form variability, 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 rivers 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 1969). In the case of fine sediment, ripples are formed, while coarser sediments and higher subcritical velocity (Froude number <1) will usually form dunes (Figure 1). Ripples refer to triangular sand waves with small dimensions, typically shorter than about 0.6 meters 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, many analytical and semi-empirical approaches have been presented in order to predict the total roughness coefficient owing to bedform roughness (Meyer-Peter & Müller 1948; Einstein & Barbarossa 1952; Tylor & Brooks 1962; Raudkivi 1967; Richardson & Simons 1967; Smith 1968; Van Rijn 1984; Karim 1995; Yang et al. 2005; Van der Mark et al. 2008). There are also a large number of studies characterizing the effect of the Reynolds and Froude numbers on the roughness coefficient (Rouse et al. 1963; Brownlie 1983; Colosimo et al. 1988). Utilizing the Froude number as an independent parameter, Ugarte & Madrid (1994) prepared an expression for Manning's n. Afzalimehr & Anctil (1998) conducted dimensional analysis on different dimensionless parameters with the Froude number as one of them, which had a significant effect on friction factor. García Díaz (2005) suggested that the Froude number is inversely proportional to Manning's n. Zhang et al. (2010) studied a curve-fitting relationship between Manning's n with the Reynolds and Froude numbers. However, the existing relationships for the 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 the 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 that is capable of predicting roughness coefficient for the channel with different types of bedform is deemed a crucial problem. In the recent decade, AI (artificial intelligence) methods have been introduced as reliable tools in providing persistent success in various fields of hydraulic engineering. In AI models we are looking for a learning machine capable of finding an accurate approximation of a natural phenomenon, as well as expressing it in the form of an interpretable equation. However, this bias towards interpretability creates several new issues. The computer-generated hypotheses should take advantage of the already existing body of knowledge about the domain in question. However, the method by which we express our knowledge and make it available to a learning machine remains rather unclear (Babovic 2009). More recently, AI methods including artificial neural network (ANN), support vector machine (SVM), gene programming (GP), and group method of data handling (GMDH) have been applied for modeling flow resistance and bedform dimensions of alluvial channels. The effectiveness of GP-based approaches was revealed in the experiments of developing a formula for the description of vegetation-induced roughness (Babovic & Keijzer 2000; Keijzer & Babovic 2002; Giustolisi 2004; Baptist et al. 2007). Azamathulla et al. (2013) accurately solved a high nonlinear relationship between Manning's n and input parameters of the Froude and Reynolds numbers, width to depth ratio, bed slope of channel, and relative roughness through the gene expression programming (GEP) approach. Roushangar et al. (2017) offered a useful prediction method based on the least squares support vector machine (LSSVM) coupled with particle swarm optimization (PSO). Compared with semi-empirical equations, their hybrid model enjoyed higher performance when it came to predicting the Manning and Darcy–Weisbach roughness coefficients in open channels with dune bedforms. In another investigation into the application of AI methods on the modeling characteristics of dune bedforms, Roushangar et al. (2018a, 2018b) developed GEP-based equations for the prediction of the Manning's roughness coefficient and relative dune height. Javadi et al. (2015) found that SVM surpasses ANN in terms of predicting dune bedform dimension. Qaderi et al. (2017) used a combination of GMDH with shuffled complex evolution (SCE) and harmony search (HS) in simulating bedform dimensions, and concluded that the developed hybrid models outperform all other empirical approaches for predicting bedform dimensions. Roushangar et al. (2018a, 2018b) applied extreme learning machine (ELM) in order to find the nonlinear interaction among different input variables for the prediction of coefficient of friction of overland flows. More recently, Saghebian et al. (2020) presented the applicability of Gaussian process regression (GPR) for the prediction of total and bedform resistance of dune bed channels.

A great deal of previous research into the application of AI methods for predicting flow resistance of alluvial channels shows that much attention has been paid to dunes' bedform and, to the best of our knowledge, there is a lack of research on the comprehensive study of predicting the roughness coefficient in channels with ripple bedform. Therefore, the present study aims to investigating the generalization capability of SVM and GPR as effective kernel-based techniques for modeling the total Manning's coefficient (which includes both grain friction and form resistance) in channels with ripple bedform. The proposed techniques were developed using experimental datasets. Performance of the different input combinations was evaluated in both training and testing phases and under two scenarios through several statistical measures. Moreover, in order to undertake the comprehensive assessment of the bedform's role in modeling the Manning's coefficient, prediction results of the employed methods for channels with dune bedform were discussed using four experimental datasets (one original experiment was performed by the author and three other experiments by other researchers). In the last stage, the most influential parameters in predicting the total Manning's coefficient were determined using sensitivity analysis.

Due to the fact that employing more datasets from varied hydraulic conditions can challenge the ML methods and enjoy more reliable evaluation, a total of 26 related data sources with the appropriate experimental data were explored and the relevant data were considered for the modeling process. As a result, there are 840 records for open channels with ripple bedforms. The sources of data as well as the ranges of measured and calculated parameters are presented in Table 1. It should be pointed out that the measured values of the Manning's coefficient are calculated from Manning's formula which is simple in form and is well confirmed by much practical experience.

Artificial neural networks are a family of machine learning algorithms originally inspired by biological neural networks that can be employed to approximate any measurable function with an arbitrary number of inputs (Tayfur 2014). The feed-forward neural network (FNN) with back propagation (BP) is a widely known utilized strategy in water resources engineering issues (Karami et al. 2012; Li et al. 2018). The employed ANN is the common FFNN algorithm with three layers of input, hidden, and target (Figure 2). The Levenberg–Marquardt preparing calculation (Hagan & Menhaj 1994) was utilized, and the mean square error (MSE) between the calculated and observed values served as the cost function. Different numbers of parameters were used as input and the optimum number of neurons was obtained through trial and error process. Furthermore, in the proposed work, the tan-sigmoid was used as an activation function in the hidden and output layers.

Considering and ⁠, different kernel functions can be defined as shown in Table 3. In this table, represents the length scale parameter, represents the signal standard deviation, and represents the separate length scale. In the present study, in order to tune the related hyper parameters, a standard gradient descent optimizer was utilized through maximizing the log marginal likelihood.

For the purpose of predicting Manning's coefficient, due to training and testing goals, data were divided into training set (75% of total data) and testing set (remaining 25% of data). As a result, there are 630 measurements for training and 210 measurements for testing.

The performances of the employed ANN, SVM, and GPR methods were compared with each other and the associated results appear in Table 5 representing the evaluation indices of the applied models. In the first step, it was attempted to express the Manning's coefficient modeling process through variables based on hydraulic characteristics. To achieve this goal, the employed ML approaches were fed with double input variables (models I and II). According to the results of Table 5, it can be seen that introducing the Froude number with depth to width ratio yielded better prediction accuracy. Considering the results of the modeling process with hydraulic characteristics, ANN confirms its superiority over kernel-based approaches with respect to statistical indices (R = 0.747, NSE = 0.548, RMSE = 0.073, and e = −0.541) for the testing part.

It is apparent from the results that the model (V) with four input parameters including Reynolds number (Re), the ratio of depth to width of channel (y/B), the ratio of the hydraulic radius to median grain diameter (R/D₅₀) and grain Froude number (⁠⁠) has the best performance for prediction of roughness coefficient in alluvial channels with ripple bedforms. The obtained results indicate that the implementation of model (V) as input combination of the SVM method provided very good outcomes (R = 0.880, NSE = 0.775, RMSE = 0.051, and e = 0.687), superior to the other machine learning methods employed, while ANN generated poor results. According to NSE values, when comparing model (I) and model (II), considering Re (in model II) increases the model accuracy by approximately 28% (ANN), 8% (SVM), and 6.7% (GPR). It can indicate the merits of each input combination and sensitivity of employed kernel-based approaches to input parameters. It can be seen that omitting and introducing relative discharge (⁠⁠) in model (VI) reduces the performance of ANN by approximately 9%. On the other hand, considering the obtained results of GPR and SVM approaches, models (V) and (VI) show similar potential, indicating more flexibility and generalization capability of kernel-based approaches in quantification of Manning's roughness coefficient. Figure 3 shows the variation of statistical parameters of NSE via the numbers of neurons in the hidden layer (fed with model (V)). Taking into account the NSE values, significant variability can be seen throughout the performance of the employed ANN approach. ANN performance ranges between NSE = 0.330 (for 1 neuron) and NSE = 0.719 (for 21 neurons). It is observed that the best network structure is 4-21-1. Furthermore, it can be seen that for solving the objective problem with employed datasets, the ANN model with numbers of neurons in the hidden layer less than 21 leads to over-fitting.

Based on the value of the logarithmic transformation variable, GPR presents relatively better prediction accuracy (e = 286) in comparison to SVM (e = 0.687). With the aid of standard gradient descent optimizer, the best values of related parameters for the different kernels were achieved as the length scale parameter (⁠⁠) ranging from 0.0240 to 0.1320 and the signal standard deviation (⁠⁠) ranging from 0.0664 to 0.1149.

Figure 4 depicts the scatter plots between the observed data and model results. The results of previous works (Roushangar et al. 2017; 2018a; Roushangar and Shahnazi 2020 and Saghebian et al. 2020) show that the best results for dune-bed channel studies for prediction of Manning's roughness coefficient were obtained from GEP (R = 0.866, NSE = 0.742, and RMSE = 0.0035), least squares SVM (R = 0.839, NSE = 0.705, and RMSE = 0.0036), and GPR (R = 0.784 and NSE = 0.715), respectively. Hence, it seems that hydraulic conditions governing ripple bedforms provide better predictive ability for machine learning approaches in comparison to channels with dune bedforms.

In order to assess the prediction capability of the Manning's coefficient under varied hydraulic conditions, different intervals of the Reynolds number were considered based on trial and error. Then, the best input combination (Re, y/b, R/D₅₀, and ⁠) was rerun for selected data categories. Results of the testing parts are plotted in Figure 5, which shows a clear ascending trend in performance of kernel-based approaches. These findings indicate that the performance of SVM and GPR approaches with the selected best input combination tends to be more robust with increasing Reynolds number. Prediction process for Reynolds number greater than 26,000 (which includes 339 data points) gave the most accurate results with SVM (R = 0.941, NSE = 0.884, RMSE = 0.036, and e = 0.326) and GPR (R = 0.945, NSE = 0.890, RMSE = 0.035, and e = 0.928). According to the obtained results, the ANN method showed less stability and presented a poor performance in predicting the Manning's coefficient in different intervals of the Reynolds number. In addition, for a Reynolds number greater than 11,000 (which includes 88% of employed data), SVM demonstrated satisfactory outcomes with R = 0.873, NSE = 0.761, RMSE = 0.048, and e = 0.618.

Since the Froude number is an effective parameter for illustrating the hydraulic properties of rivers, it may be beneficial to check out the effectiveness of the best input combination for the prediction of Manning's coefficient in different intervals of this parameter. In accordance with the obtained results, prediction of Manning's coefficient with lower Froude number (Fr < 0.25) increases the modeling performance of GPR and ANN by 4.2% and 5.5%, respectively. On the contrary, hydraulic conditions governing the flow with the higher Froude number (Fr > 0.55) decreased the modeling accuracy with R = 0.539, NSE = 0.267, and RMSE = 0.033 for SVM and R = 0.532, NSE = 0.279, and RMSE = 0.032 for GPR. Generally, it can be inferred that Manning's coefficient in channels with ripple bedforms has better prediction capability in flows with lower Froude number. Results of predicted Manning's coefficient in different intervals of the Froude number are presented in Figure 6.

The functionality of the employed ML approaches was investigated in different ranges of the R/D₅₀ as the most influential parameter in determination of Manning's coefficient of ripple bedform channels. As shown in Figure 7, variation of the R/D₅₀ from 50 to 400 caused fluctuation in performance of ANN from NSE = 0.444 to NSE = 0.756, SVM from NSE = 0.629 to NSE = 0.806, and GPR from NSE = 0.692 to NSE = 0.807. Results revealed that the GPR model offered a more consistent performance with the variation of R/D₅₀ values. Moreover, as can be seen in Figure 7, when the depth is larger compared with bed material size so that R/D₅₀ > 450, the performance of the employed ML approaches decreases significantly.

In SVM, model behavior is largely dependent on the RBF kernel parameter (γ), which can lead to under-fitting and over-fitting in the prediction process (Roushangar & Shahnazi 2019). Figure 8 illustrates the statistical indices via gamma values of the SVM model (fed with model (V)). From the figure, it can be seen that the best fitting gamma values are obtained when ⁠. When gamma is small (in our example ⁠), the SVM model tends to memorize all the training data but is capable of generalizing unseen data: hence, only trained data points can be predicted. Thus, for solving the objective problem with the employed datasets, SVM model with gamma values less than 50 leads to over-fitting. According to the output, the optimum value of RBF kernel function was assessed as 300 for model (V). It is worth noting that statistical indices can show different behavior with variations of gamma value for different input combinations.

Since a well-advised application of kernel-based modeling methods is to find an appropriate kernel function and tune associated hyper parameters, various kernel functions were used as a core tool of the employed GPR methods. Table 6 lists the results of statistical indices of different kernels for model (III) as the best input combination for prediction of the Manning's roughness coefficient.

Attending to the NSE values, the trivial variability can be seen throughout the considered kernel functions. GPR performance ranges between 0.722 (for rational quadratic kernel) and 0.798 (for ARD Matern 3/2 kernel). In addition to exponential kernel, the constructed GPR structure with squared exponential kernel seems to be robust since their performance in the test set is encouraging. The comparison demonstrated that the Matern 3/2 Matern 5/2 kernel functions performed equally well. Moreover, based on the results of Table 6, utilization of automatic relevance determination (ARD) kernels improved the effect of a good nonlinear interpolation of the employed GPR method and increased the global average accuracy by approximately 3% in terms of NSE values. As mentioned before, the result proved that SVM–RBF had the most success rate values (R = 0.880, NSE = 0.775, and RMSE = 0.051). It was followed by polynomial kernel function (R = 0.513, NSE = 0.108, and RMSE = 0.102) and linear kernel (R = 0.271, NSE = 0.038, and RMSE = 0.106). The results showed that the SVM with sigmoid kernel function achieved the worst performance for both training and testing phases (R = 0.116, NSE = −0.00056, and RMSE = 0.149).

In the last step, a simple sensitivity analysis of the input parameters on the prediction accuracy of employed ML approaches for the Manning's roughness coefficient of channel with ripple bedforms is presented. The selected input combination for the analysis is model (III), as its prediction accuracy was proved to be the best in the previous sections. In addition, the ANN model with Levenberg–Marquardt learning algorithm, SVM with RBF kernel, and GPR with exponential kernel have also been selected for the sensitivity analysis of inputs. The sensitivity analysis was implemented by successively omitting each input from the model (V). Consequently, the statistical behavior of the eliminated input is reduced in terms of employed criteria such as R and NSE, allowing the prediction models to quantify the effect of excluded input on the prediction targets. Figure 9 shows the results of the sensitivity analysis. In this figure ΔNSE stands for the values of percent reduction of NSE pertaining to each excluded parameter.

In view of the analysis results, depicted in Figure 9, it can be deduced that R/D₅₀ is the most sensitive input parameter and plays a significant role in the prediction process. This conclusion is reached using ANN, GPR, and SVM methods with the highest ΔNSE values (15.3%, 21.7%, and 18.1%, respectively). The Reynolds number is the second most effective parameter on the Manning's coefficient of ripple bedforms. With its elimination ΔNSE increased to 12.5%. In the case of SVM, given the ΔNSE measure, the two parameters of y/B and Re had a similar effect on prediction accuracy, while the parameter y/B has the least effect on prediction accuracy of ANN. It can be observed that different ML approaches use different degrees of input parameter features for appropriate modeling of the relationship between input and output parameters. It can be seen that, in the case of the ANN method, the parameter y/b had the least impact on the modeling process, but, on the other hand, this parameter was the second most effective parameter on the modeling accuracy of the employed SVM method.

Although there have been some useful steps towards the application of ML approaches in determining the characteristics of dune bedforms, the effectiveness of the aforementioned approaches for modeling flow resistance in alluvial channels with ripple bedforms remains elusive. In this regard, three ML approaches, namely, GPR, ANN, and SVM, were employed in this study for the Manning's coefficient prediction. An extensive dataset consisting of 840 experimental samples from 24 sources was used in order to interpret their embodied knowledge by employed techniques. Six input combinations were tested, based on flow and sediment characteristics. All the variables were used in non-dimensional form in order to ensure dimensional consistency between inputs and outputs. Regarding the results, the model (V) with parameters Re, y/b, R/D₅₀, and was the most accurate model. SVM yielded the best results in all the employed statistical indices (R = 0.880, NSE = 0.775, RMSE = 0.051, and e = 0.687). Moreover, the proper utilization of ML techniques in different intervals of the Reynolds number and relative roughness dictates more stability and generalization capability of GPR in predicting the Manning's coefficient under varied hydraulic conditions. It was deduced that the prediction of the Manning's coefficient for Reynolds values of greater than 26,000 was more precise than those with lower Reynolds values. In addition, the obtained results demonstrated that complicated hydraulic conditions governing channel systems caused a considerable decrease in the performance of the employed ML approaches. However, it should be noted that the employed GPR, SVM, and ANN are data-driven models and the ML-based models are data sensitive, so further studies using data ranges beyond this study and field data should be conducted in order to prove the merits of the proposed models to estimate roughness coefficient of ripple bedforms in real flow conditions.

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