Modeling total resistance and form resistance of movable bed channels via experimental data and a kernel-based approach

Assessment of flow resistance is 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.). Flow resistance in alluvial channels can be due to two roughnesses: (1) the grain (or friction) roughness, which in turn depends on size of the bed grain, and (2) the shape roughness, which depends on the shape and dimensions of the bedform as well as the depth of flow (Rouse 1965; Morvan et al. 2008). According to Kazemipour & Apelt (1983) and Talebbeydokhti et al. (2006), almost 90% of the total base flow resistance may be caused by form resistance; therefore, the form roughness should not be overlooked.

Total roughness coefficient can be developed in the form of a linear separation concept. The linear separation of the Manning roughness coefficient is expressed in two parts: (i) grain resistance (skin roughness), (ii) form resistance (shape roughness). Thus, the total bed roughness coefficient can be expressed as: n = f (n′, n″), where n′ is the skin resistance and n″ is the form resistance that is due to bedform drag (form drag) or roughness bedform.

In recent years artificial intelligence approaches (e.g. Artificial Neural Networks (ANNs), Neuro-Fuzzy models (NF), Genetic Programming (GP), Gene Expression Programming (GEP), and Support Vector Machine (SVM), Gaussian Process Regression (GPR)) have been used for the assessment of the accuracy of complex hydraulic and hydrologic phenomena, such as prediction of groundwater levels (Amaranto et al. 2018), estimation of hydraulic jump energy dissipation in channels with rough elements (Roushangar & Ghasempour 2018), prediction of flow resistance in alluvial channels (Roushangar et al. 2018), prediction of pile group scour in waves (Ghazanfari-Hashemi et al. 2011), computing longitudinal dispersion coefficients in natural streams (Azamathulla & Wu 2011), real-time hydrologic forecasting (Yu et al. 2004), side weir discharge coefficient (Azamathulla et al. 2017), and prediction of non-cohesive sediment transport in circular channels (Roushangar & Ghasempour 2017). Machine learning, a branch of artificial intelligence, deals with the representation and generalization of physical phenomena using a data learning technique. 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 an 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.

In this study, first, the impact of hydraulic and sediment parameters on total and form roughness coefficient was assessed by using the experimental data carried out at the Hydraulic Laboratory of University of Tabriz. Then, these data were combined with several available data sets in the literature, and the capability of the GPR as a kernel-based approach was investigated for modeling roughness coefficient in channels with dune bedforms. The models were defined considering various input combinations alternatives, specifically based on hydraulic characteristics and sediment properties, in order to evaluate the most appropriate input combination for roughness coefficients modeling. Finally, a sensitivity analysis was performed to find the most significant parameters in the modeling processes.

Kernel-based approaches, such as GPR, are a relatively new and important method based on the different kernel types, which are based on statistical learning theory. Such models 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. GPRs can model non-linear decision boundaries, and there are many kernels to choose from. They are also fairly robust against overfitting, especially in high-dimensional space. However, the appropriate selection of kernel type is the most important step in the GPR due to its direct impact on the training and classification precision. In fact, these methods are memory intensive, trickier to tune due to the importance of picking the right kernel, and do not scale well to larger datasets. In these models the proper behavior of the system can be predicted, although its intrinsic structure and behavior cannot be characterized.

GPR models are based on the assumption that adjacent observations should convey information about each other. Gaussian processes are a way of specifying a priori directly over function space. This is a natural generalization of the Gaussian distribution, whose mean and covariance are a vector and matrix, respectively. The Gaussian distribution is over vectors, whereas the Gaussian process is over functions. Thus, due to prior knowledge about the data and functional dependencies, no validation process is required for generalization, and GP regression models are able to understand the predictive distribution corresponding to the test input (Rasmussen & William 2006). A GP is defined as a collection of random variables, any finite number of which has a joint multivariate Gaussian distribution. Considered input space of n-dimensional vectors to an output space of real-valued targets, in which n pairs (x_i, y_i) are drawn independently and identically distributed. For regression, assume that ⁠; then, a GP on is defined by a mean function and a covariance function ⁠.

To find the hyperparameters, the partial derivative of Equation (8) can be obtained with respect to and k, and minimization can be achieved by gradient descent. For more details about GP regression and different covariance functions, readers are referred to Kuss (2006). The optimal value of capacity constant (C), the size of error- intensive zone (ɛ), and kernel parameter (γ) in SVM and Gaussian noise in GPR are required due to their high impact on the accuracy of mentioned regression approaches. In this study, optimization of these parameters has been performed by a systematic grid search of the parameters using cross-validation on the training set. In this grid search a normal range of parameters settings are investigated. First, optimized values of C and ɛ for a specified γ were obtained and then γ was changed. Statistical parameters (R, DC, and MAPE) were used to find optimums. The values of C, ɛ and kernel parameter (γ) which lead to the highest R and DC and lowest MAPE, were selected as optimum amounts.

In order to study the variation of roughness coefficients in open channels with dune bedform, several dune bedform experiments were performed in a 10 m long, 0.5 and 1 m wide, and 0.8 m high rectangular flume at the hydraulic laboratory of Tabriz University (Saghebian 2018). The flume had glass walls and a 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. In these experiments, discharge rate was controlled by a valve in the discharge pipe of the pump, and sediment was re-circulated together with water. The original flume had ratchet screw jacks for adjusting the slope of the flume. In this research, the flume slope was variable from 0 to 0.5%. 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 channel. By changing the flow depth and discharge, the average velocities, Froude numbers, dune height, and other parameters were calculated. In this study, at first several experiments were performed in the state of channel without bedform. Then, the friction coefficients of the channel's walls and bed were determined. To determine the form resistance, the effective roughness coefficient was extracted using the composite channel roughness equations. Finally, the walls roughness coefficients were subtracted from effective roughness coefficient and form resistance was obtained. Figure 1 shows a view from the channel, and pebbles used in downstream of the channel for reducing the turbulence of the flow.

Together with these experiments, the data sets of laboratory experiments of roughness coefficient in open channels with dune bedforms carried out by Guy et al. (1966), Williams (1970) and Roushangar (2010) were used. The ranges of various parameters used in the experiments are listed in Table 1. 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. 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. Guy et al. (1966) studied the effects of the bed material size, flow temperature, and the fine sediment within the flow on the hydraulic and transport variables at Colorado State University. The investigations for each set covered flow phenomena ranging from a plane bed with no sediment movement to violent anti-dunes. Roushangar (2010) organized several dune bedforms experiments that were made in a 5 m long, 0.5 m wide, and 0.25 m high rectangular flume in the hydraulic laboratory of Caen University. Natural quartz sand was used as sediment particles in the experiments.

It should be noted that at first D’ in Equation (3) was solved by trial and error, followed by the determination of given in this equation. Then could be found from Equation (1), since it is straightforward to find ⁠. Finally, Equation (11) is used to solve the n″ value.

According to Figure 2, it can be seen that the correlation of the n with the Reynolds number is slightly better than the Froude number. For n″, there is no desired correlation between this parameter and Re and Fr and, with increasing the Froude number which leads to dune bedform variation, the form friction coefficient decreased. It seems that the total and bedform roughness coefficients could not only depend on hydraulic parameters. According to Figure 2, the Shields parameter (θ) shows the best correlation with total and form resistance among the studied parameters, and with increasing the Shields parameter values, the values of n and n″ increase (due to the direct relationship between the energy line slope and the flow resistance coefficient). Also, it could be seen that n and n″ did not show a significant correlation with Vy/[g(s–1) D₅₀³]^0.5. This issue indicates that there is an uncertainty in using the hydraulic and sediment parameters as the only input variables in the roughness coefficient estimating process.

To investigate the impact of the form resistance on the total roughness coefficient, the variations of the skin resistance, form resistance and total resistance versus hydraulic and sediment parameters are presented in Figure 3. According to Figure 3(a), it can be seen that the roughness variations due to the form resistance are almost in agreement with the variations of the total Manning roughness coefficient, and it can be concluded that the performance of the n″ is similar to n. From the results, it was found that the skin resistance variations are in a small range, meaning that bedform resistance dominates the flow. This issue also could be deduced from Figure 3(b). According to this figure, the roughness variations due to the bedform are almost between 40–80% of the total roughness coefficient, while the n′ variation percentage is between 20 and 60%.

In the design of the GPR model, it is necessary to select the appropriate type of kernel function. A number of kernels are discussed in the literature. In this study, for determining the best performance of GPR and selecting the best kernel function, the model SH10 was predicted using various kernels. Figure 4 indicates the results of statistical parameters of different kernels for this model. According to the results, using the kernel function of Pearson led to better prediction accuracy. Pearson kernels were used as a core tool of GPR which was applied for the rest of the models.

For evaluating the impact of flow features on total and bedform roughness coefficient, several models using only hydraulic characteristics as input variables were developed. Then, the developed models were trained and tested using the GPR. The results of the developed models are listed in Table 3 and shown in Figure 5. According to the obtained results, in both n and n′ modeling, the model H4 with input parameters of Re and y/b shows better performance than the others from the RMSE, CC, and DC viewpoints (i.e. highest CC and DC and lowest RMSE). It seems that using parameter y/b as the input parameter caused an increase in model efficiency. Also, it could be seen that using Reynolds number as the only input parameter yielded to the desired accuracy compared to the case where Froude number was used as the only input. Figure 5 represents the scatter plot of the observed and predicted total and bedform roughness coefficients for the best input combination. Additionally, based on trial and error, data sets were divided into two groups in terms of the Reynolds number (Re < 80,000 and Re > 80,000). Then, the best input combination (Re, y/b) was rerun for both data categories. The results are represented in Table 3 (last two rows). These findings demonstrate that when the Reynolds numbers is less than 80,000, the model tends to be more accurate in predicting the Manning's coefficient. In higher ranges of the Reynolds number (Re > 80,000) the model accuracy decreased.

Flow and sediment characteristics were employed in the establishment of input combination to evaluate the significance of the bedform and sediment features for the modeling of the total and bedform roughness coefficients. The results in Table 4 reveal that the model HS10 including Re, R/D₅₀ and Vy/[g × (s–1)D₅₀³]^0.5 as inputs parameters yielded better predictions. According to the results, it could be inferred that the ratio of hydraulic radius to the sediment diameter and Reynolds number led to the desired results in predicting n and n″ roughness coefficients. Therefore, both hydraulic and sediment characteristics have a significant effect on the prediction process. It could be seen that adding parameters Vy/[g × (s–1)D₅₀³]^0.5 and R/D₅₀ to Re caused an increase in models efficiency. Based on the results, it was found that the developed models with flow and bedform characteristics performed relatively weakly in comparison with developed models with only flow characteristics. Comparison between Tables 3 and 4 indicates that the applied methods for developed models based on flow and sediment properties yielded better predictions compared to using only flow parameters. Figure 6 presents the scatter plot of the observed and n and n″ roughness coefficients for the best input combination.

To investigate the impacts of different parameters of the GPR-best models on total and bedform roughness coefficients, a sensitivity analysis was performed. In order to evaluate the effect of each independent parameter, the model was run with all input parameters and then, one of the input parameters was eliminated and the GPR model was re-run. The MAPE performance criteria was used as indication of the significance of each parameter. The obtained results are shown in Figure 7. Based on obtained results and according to Roushangar et al. (2018), it can be seen that Re and Vy/[g × (s–1)D₅₀³]^0.5 had the key role in the modeling process.

In the present study, the capability of the GPR model as a kernel-based approach was verified for predicting the total and bedform roughness coefficients in alluvial channels. In this regard, an experimental study was conducted at the Hydraulic Laboratory of University of Tabriz to investigate the impact of hydraulic and sediment parameters on total and bedform roughness coefficients. Then, these data were combined with other laboratory data sets and the capability of the GPR was assessed in modeling the roughness coefficients. Different input combinations based on flow and sediment characteristics were considered in order to develop the GPR models. The experimental study showed that the Reynolds number has a better correlation with n roughness coefficient in comparison with Froude number for the tested conditions in the experiments. It was found that the variation of Manning roughness coefficient due to the bedform were almost in agreement with the variations of the total Manning roughness coefficient, and the variations of skin resistance were in a small range. The roughness variations due to the bedform were almost between 40–80% of the total roughness coefficient. It was shown that the total and bedform roughness coefficients cannot be represented only by means of hydraulic parameters, without including the sediment characteristics. Also, the obtained results from the developed models proved the desired capability of the GPR method in the modeling process. The obtained results revealed that for predicting the roughness coefficient, the model which took the advantages of flow and sediment characteristics performed more successfully than the others. Regarding the total and bedform roughness coefficients with sediment feature characteristics, the model named HS10 with parameters Re, R/D₅₀, and Vy/[g× (s–1)D₅₀³]^0.5 was the most accurate model. These results showed that for predicting the total and bedform roughness coefficients under only hydraulic characteristics, including (y/b) as an input parameter significantly improved the efficiency of the models. According to the conducted sensitivity analysis, it was found that the Re and Vy/[g× (s–1)D₅₀³]^0.5 played the most important roles in predicting the total and bedform roughness coefficients in alluvial channels. However, it should be noted that the used method is a data-driven model and the GPR-based model is data sensitive, so further studies using data ranges out of this study and field data should be carried out to determine the merits of the model to estimate roughness coefficient in the real conditions of flow.