International Journal of Sediment Research 32 (2017) 515–526
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International Journal of Sediment Research journal homepage: www.elsevier.com/locate/ijsrc
Original Research
Predicting characteristics of dune bedforms using PSO-LSSVM Kiyoumars Roushangar a,n, Seyed Mahdi Saghebian a, Dominique Mouaze b a b
Division of Hydraulics, Department of Civil Engineering, University of Tabriz, Tabriz, Iran Morphodynamic Laboratory, University of Caen, France
art ic l e i nf o
a b s t r a c t
Article history: Received 12 September 2016 Received in revised form 14 June 2017 Accepted 12 September 2017 Available online 18 September 2017
Dunes have a large influence on hydraulic roughness, and, thereby, on water levels which could affect the navigability of rivers and performance of hydraulic structures. The present study investigated the variation of geometric and topographic characteristics of dune bedforms and flow features as measured in laboratory studies (data sets from laboratory experiments) to estimate the roughness coefficient and characteristics of dune height. The Least Squares Support Vector Machine (LSSVM), which was optimized using Particle Swarm Optimization (PSO), was used as the Meta model approach to predict the values of interest. Developed models were separated into three categories: modeling using flow characteristics, modeling of flow and bedform characteristics, and modeling by using flow and sediment characteristics. It was found that for estimation of the roughness coefficient in open channels with dune bedforms, models developed based on flow and sediment characteristics performed more successfully. The model with input parameters of flow and grain Reynolds numbers (Re and Rb, respectively) and the ratio of the hydraulic radius (R) to the median grain diameter (D50) yields a squared correlation coefficient (R2) of 0.8609, a coefficient of determination (DC) of 0.7361, and a root mean square error (RMSE) of 0.0034 for a test series of Manning's roughness coefficient which was the most accurate model. Results proved the key role of flow Reynolds number (Re) values as an input feature for all models predicting the roughness coefficient. Accordingly, classic approaches led to poor results in comparison. On the other hand, results obtained for estimated values of relative dune height led to moderate prediction quality, which albeit, outperformed classic approaches. & 2017 International Research and Training Centre on Erosion and Sedimentation/the World Association for Sedimentation and Erosion Research. Published by Elsevier B.V. All rights reserved.
Keywords: Alluvial channel Hydraulic characteristics Dune Roughness coefficient Least Squares Support Vector machine Particle Swarm Optimization
1. Introduction 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 and flow velocity, and the accurate characterization of energy losses. Also, bedforms in a river are created due to flow movement and have a significant effect on bed roughness and resistance to flow. Estimation of flow resistance in open channels with dune bedforms is a complex phenomenon due to the multitude of factors influencing roughness such as bed material, bedforms, plan form variability, and vegetation. The accurate prediction of the geometric characteristics of bedforms is an essential component for estimating the flow resistance and the consequent flow conditions. The complexities and uncertainties of bedform configurations in alluvial channels continue to be a challenge for engineers and those stem n Correspondence to: Department of Hydraulic Engineering, Faculty of Civil Engineering, University of Tabriz, 29 Bahman Ave., Tabriz, Iran. E-mail address:
[email protected] (K. Roushangar).
from the variety of bedform shapes that arise under different flow conditions. Starting from a plane bed without sediment transport, ripples, dunes, and washed-out dunes develop in large experimental flumes as the flow intensity increases in magnitude over a bed of loose sand particles (Fig. 1). A large number of classic friction factor models have been developed, which describe the complex phenomenon of the flow resistance. Azareh et al. (2014), Einstein and Barbarossa (1952), Karim (1995), Meyer Peter and Mueller (1948), Raudkivi (1967), Richardson and Simons (1967), Smith (1968), Taylor and Brooks (1962), van der Mark et al. (2008), van Rijn (1984), and Yang et al. (2005) have all presented expressions for the total friction factor resulting from bedform roughness. Engel and Lau (1980) found the dune length to depth ratio has a significant effect on the friction factor when the dunes are steep. Karim (1999) proposed a new method for predicting relative dune height in a sand-bed stream based on the concept of relating energy loss resulting from form drag to the head loss across a sudden expansion in open channel flows. Tuijnder and Ribberink (2008) concluded that bedforms would be limited when the amount of sediment movement is less than the required volume
http://dx.doi.org/10.1016/j.ijsrc.2017.09.005 1001-6279/& 2017 International Research and Training Centre on Erosion and Sedimentation/the World Association for Sedimentation and Erosion Research. Published by Elsevier B.V. All rights reserved.
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Fig. 1. Types of bedforms in alluvial channels (lower regime), (a). plane bed, (b). ripples, (c). dunes, and (d). washed-out dunes. (after Julien, 2010).
for formation of bedforms. The results of Singh et al. (2011) showed that the flow ratio and components of bed sediment had a significant effect on the multi-scale dynamic, nonlinear degree and the complexity of bedform evolution. Ghoshal and Pal (2014) studied the grain-size distributions of suspended load over a sandgravel bed in a laboratory flume and showed that with an increase of flow velocity, the grain-size distribution curves of suspended load changed the bedform geometry. Heydari et al. (2014) proved that by increasing the Shields number, the ratio of Manning's roughness coefficient related to dune bedforms and the total Manning's roughness coefficient increased with a logarithmic trend. Hanmaiahgari et al. (2017) specified the effect of mobile bedforms on velocities, turbulence intensities, and turbulent stresses in an experimental study. Existing equations for predicting the roughness coefficient rely on limited databases and untested model assumptions, and do not show the same results under variable flow conditions. These issues cause uncertainty in the estimation of the roughness coefficient. Therefore, it is extremely critical to utilize methods which are capable of predicting the roughness coefficient within channels with bedforms under varied hydraulic conditions. Artificial Intelligence (AI) tools especially machine learning approaches, are remarkable forecasting tools which in the recent decade have been implemented in various fields of civil engineering including hydraulic and hydrologic studies. For instance, prediction of total bed load (Chang et al., 2012), computing longitudinal dispersion coefficients in natural streams (Azamathulla & Wu, 2011), prediction of suspended sediment concentration (Kisi et al., 2012), modeling sediment transport (Bhattacharya et al., 2007), and prediction of scour depth downstream of sills (Azamathulla, 2012) are some of such studies. The Support Vector Machine (SVM) approach has been applied in modeling various components of water resources systems. Khan and Coulibaly (2006) applied SVM for long-term prediction of lake water levels. Yang et al. (2009) applied Artificial Neural Networks (ANN) for evaluation of total load sediment transport formulas. Yunkai et al. (2010) analyzed the characteristics of soil erosion in small water basins using SVM and ANN methods. Sivapragasam and Muttil (2005) suggested the use of SVM method in the extrapolation of rating curves. Haghiabi et al. (2016) used ANN and SVM methods for prediction of head loss on a cascade weir. Azamathulla et al. (2016) recently used the SVM method for predicting a side weir discharge coefficient. Santos et al. (2010) optimized channel and plane parameters in an erosion model, using the Particle Swarm Optimization (PSO) approach. Zanganeh et al. (2011) used a PSO based Fuzzy Inference System (ANFIS) model's inherent shortcomings to optimize nondimensional variables to estimate scour depth.
Due to the uncertainties in the roughness coefficient and considering the reviewed literature, there is a lack of comprehensive study on the prediction of the friction coefficient in alluvial channels with dune bedforms using artificial intelligence. Accordingly, the present research is an attempt to extend the previous studies by showing the capability of PSO-LSSVM for the prediction of the friction coefficient and evaluating the performance of the Least Squares Support Vector Machine (LSSVM) approach. The present research also investigates the best input models and determines the parameters most affecting the roughness coefficient in channels with dune bedforms. The outcome of the LSSVM approach was compared with the results obtained from semi-empirical equations. The simulations were done for four different data series obtained from experimental studies in different laboratories, one original experiment done by the first author and data from three other experiments were obtained from the literature. The modeling set up comprises of the Manning and Darcy-Weisbach roughness coefficients and relative dune height (H/h) as objective functions with various inputs in order to find the most appropriate combination of influential hydraulic parameters.
2. Materials and methods 2.1. Dune bedforms From extensive laboratory experiments at Colorado State University by Simons and Richardson (1963, 1966), several types of bedforms have been identified. Flat bed, or plane bed, refers to a bed surface without bedforms. Ripples are small bedforms with wave heights less than a few cm that are only seen in the case of hydraulically smooth bed conditions. Ripple shapes vary from nearly triangular to almost sinusoidal. Dunes are much larger than ripples and are out of phase with the water surface waves. Dunes are often triangular with fairly gentle upstream slopes and downstream slopes approaching the angle of repose of the bed material (Julien, 2010). 2.2. Dune geometry The geometry of dune bedforms is a concern in engineering projects dealing with navigation, flood control, river restoration, hydraulic structures, and resistance to flow. The geometry of dune bedforms refers to the representative dune height (h) and length of dunes (L) as a function of the average flow depth (H), median bed particle diameter (D50), and other flow parameters such as flow mean velocity (v), shear velocity (u ), and the grain Reynolds
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Fig. 2. Characteristics of dune bedforms.
number (Rb). The foreslope and backslope of the dunes can also be counted as a part of the dune geometry (Fig. 2). 2.3. Flow resistance The laws of conservation of energy and momentum must account for hydraulic resistive forces in the calculation of open channel hydraulics. Steady and uniform flow conditions require driving and resisting forces to be balanced; in which, flow is not accelerating or decelerating, so the average channel cross section, slope, and velocity are assumed to be constant under constant discharge conditions. In natural streams velocity or discharge often must be estimated or calculated using other flow parameters, most commonly hydraulic radius, energy slope (or some approximation), and some estimate of channel roughness. Even though natural streams do not strictly comply with uniform flow assumptions, uniform flow conditions often are assumed to simplify velocity and discharge computations. The friction coefficient in river engineering applications is commonly calculated using one of three equations: Manning, Chezy, or Darcy-Weisbach (Yen, 2002). 2.4. Least Squares Support Vector Machine (LSSVM) The LSSVM (Kumar & Kar, 2009) approach is a supervised learning model with associated learning algorithms that analyze data and recognize patterns, and is used for classification and regression analysis by a non-linear function. A LSSVM model represents the samples as points in space, mapped so that the samples of the separate categories are divided by a clear gap that is as wide as possible. As an effective classification method, LSSVM has been proposed on the basis of Statistical Learning Theory (SLT) by Vapnik and Cortes (1995). LSSVM is a non-linear machine working in the high dimensional feature space formed by the non-linear mapping of the n-dimensional input vector X into a K-dimensional feature space (K 4 n) using a function. The SVM method is a relatively new, important method based on the extension of the idea of identifying a line (or a plane or some surface) that separates two classes in a classification. It is based on SLT. This classification method also has been extended to solve prediction problems. It should be noted that data-driven models typically do not really represent the physics of a modelled process; they are just devices used to capture relations between the relevant input and output variables. Therefore, the present study applied the SVM method to predict the variable of interest. Such a model is capable of adapting itself to predict any variable of interest using sufficient inputs. Also, the most important principle of the SVM method is the application of minimizing an upper bound to the generalization error instead of minimizing the training error. Based on this, the SVM method can achieve an optimum network structure (Wang et al., 2013). The SVM neural network applies one hidden layer of non-linear neurons, one-output linear neuron, and a specialized learning procedure leading to the global minimum of the error function and excellent generalization ability of the trained network. Since the Radial Basis Function (RBF) kernel is commonly used in regression problems
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because of its capability, so the RBF kernel function is used in the present study as per Suykens and Vandewalle (1999). The LSSVM model using the RBF kernel is very desirable to apply for prediction of hydrological features since: (I) unlike the linear kernel, the RBF kernel can handle the case when the relation between class labels and attributes is non-linear; (II) tends to give better performance under general smoothness assumptions; and (III) it has fewer tuning parameters than the polynomial and the sigmoid kernels (Noori et al., 2011). In this study the LSSVM approaches have been implemented using MATLAB software. 2.5. Particle Swarm Optimization The Particle Swarm Optimization (PSO) algorithm is an evolutionary optimization algorithm (Kennedy & Eberhart, 1995), where a population of particles or proposed solutions evolves in each iteration, moving towards the optimal solution of the problem. A new population is obtained shifting the positions of the previous population for each iteration. In its movement, each individual is influenced by its neighbor's and its own trajectory. The parameters, or possible set of solutions, are contained in a vector Xi, which is called a “particle” of the swarm and represents its position in the search space of possible solutions. The particle dimension is the number of parameters. The particle position X 0i and its velocity V 0i are randomly obtained. The value of the fitness function is then calculated for each particle and the velocities and positions are updated taking into account these values. The algorithm updates the positions and the velocities of the particles following the equations: ð1Þ V ki þ 1 ¼ ωV ki þ φ1 Gk þ X ki þ φ2 ðI ki þ X ki Þ The velocity of each particle, i, at iteration k, depends on three components: • The previous step velocity term, V ki affected by the constant inertia weight, ω. • The cognitive learning term, which is the difference between the particle's best position so far found (called I ki , local best) and the particle current position X ki . • The social learning term, which is the difference between the global best position found thus far in the entire swarm (called Gk , global best) and the particle's current position X ki . These two last components are affected by φ1 ¼c1 r1 and φ2 ¼ c2 r2 where r1 and r2 are random numbers distributed uniformly in the interval [0,1] and c1 and c2 are constants. The particles of the swarm make up a cloud that covers the whole search space in the initial iteration and the swarm gradually contracts in size as iterations advance, performing the exploration. So in the initial stages the algorithm does an exploration searching for plausible zones and in the final iterations the best solution is improved. The PSO implementation of the algorithm has been refined over the years and many variants have been created. In this paper, the Standard 2011 PSO has been used. It contemplates some improvements in the implementation and the PSO parameters are set to the values:
ω¼
1 and c1 ¼ c2 ¼ 0:5 þ ln 2 2ln2
ð2Þ
The swarm topology defines how particles are connected between them to interchange information with the global best. In the actual Standard PSO each particle informs only K particles, usually three, randomly chosen. Code for the PSO algorithm provided in the MATLAB software has been used in this research.
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Table 1 Details of the utilized experimental data. Researcher
H (mm)
b (mm)
D50 (mm)
Fr
Re
n
f
Data number
Williams (1970) Guy et al. (1966) U. S. Corps of Engineers (1935) Roushangar (2008)
87.1–222 91.4–405 65.5–208 71–145
76.2–1118 609 , 2438 705,736 150
1.35 0.19–0.93 0.18–0.47 0.15, 0.4
0.34–0.84 0.25–0.65 0.3–0.72 0.21–0.40
11,932–101920 46,800–255500 19,061–66432 24,192–45,869
0.0091–0.0201 0.015–0.038 0.0127–0.0249 0.0203–0.0252
0.019–0.059 0.031–0.163 0.032–0.108 0.039–0.063
89 114 61 54
Note: Fr ¼ Froude number ¼ pVffiffiffiffiffi , Re ¼ Reynolds number ¼ ρVR μ , D50:median grain diameter, n:Manning’s roughness coefficient, f:Darcy-Weisbach roughness coefficient, b: gH
channel width, H:flow depth
2.6. Experimental data used in the study In this paper, four kinds of data corresponding to the friction coefficient taken from published literature have been used. Details of the experimental data sets are listed in Table 1.
dimensions and to ensure that all variables receive equal attention during calibration of the model. Therefore RMSE values are to be transformed between 0 and 1 (Nourani et al., 2016).
3. Results and discussion I) Williams (1970) organized several experiments which were made in channels with different widths and water depths in laboratories of Washington, D.C. Sediment transport rates, grain size, water depth, and channel width were measured; water discharge, mean velocity, slope (energy gradient), and bedform characteristics were 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 size of bed material, temperature of flow, and fine sediment in the flow on the hydraulic and transport variables. The investigations of each set covered flow phenomena ranging from a plane bed and 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 (U. S. Army Corps of Engineers, 1935). IV) Sediment and flow variables comprising flow depth and velocity; water surface gradient; and sediment diameter, distribution, and type are the main parameters influencing the bedforms. A rectangular plexiglas flume with 5 m length, 0.15 m width, and 0.25 m height, located in the hydraulic laboratory in Caen University was utilized to do the experiments (Roushangar, 2008).
Selection of various parameters as input combinations can affect the accuracy of the results throughout the modeling process. So selecting appropriate parameters and input combinations has a key role during modeling. For determining the output variables (i.e. friction coefficient), it was attempted to develop the process through a set of dimensionless variables as input combinations. Fig. 3 shows the model combinations considered in this study. The quality of the data driven model does not depend only on the selected input features, it also could be related to the model (LSSVM) parameters. In this study, an optimized LSSVM using the PSO approach was examined to find the best possible estimation for friction factor data of several experiments. An authentic application of this tool is to select an appropriate kernel function and tune related hyper parameters. A number of kernels have been discussed by researchers, but studies indicated the radial basis kernel function in the case of the SVM approach is most effective in the majority of civil engineering applications (Gill et al., 2006; Goel & Pal, 2009; Nourani et al., 2016). For a fair comparison of results, a Radial Basis Function (RBF) kernel, where σ is a kernel specific, and γ is the margin parameter was used with the LSSVM method in this study. The tuned parameter (F) was optimized using PSO. The flowchart of the LSSVM model optimization is shown in Fig. 3. 3.1. Optimization of kernel parameters using PSO
2.7. Performance criteria The statistical criteria, which were used to evaluate the performance of the different proposed models, namely the Root Mean Square Error (RMSE), Square of the Correlation Coefficient (R2), and Nash–Sutcliffe (1970) coefficient of determination (DC) are listed in Table 2. The RMSE is used to quantify modeling accuracy, which generates a positive value by squaring the errors. The RMSE grows from zero for perfect predictions through large positive values as the differences between estimates and observations become increasingly large. These measures are not oversensitive to extreme values (outliers) and are sensitive to additive and proportional differences between model predictions and observations. Therefore, correlation-based measures (e.g., the DC and R2 statistics) can indicate that a model is a good predictor (Legates & McCabe, 1999). R2 measures the accuracy of the overall fit of the predicted values to the measured values. The optimal value of R2 is equal to 1. Obviously, a high value for DC and R2 (up to one) and small value for RMSE indicate high predictive ability of the model. Legates and McCabe (1999) indicated that a hydrological model can be sufficiently evaluated using these statistics. Input and output variables were normalized by scaling between 0 and 1 to eliminate their
In order to improve the LSSVM algorithm's performance, it is important to define the kernel parameters. For this reason, PSO was incorporated in the LSSVM algorithm for friction coefficient estimation. The PSO algorithm could achieve the optimal parameters of the LSSVM algorithm that could lead to the best prediction accuracy. The PSO algorithm herein uses the fittest particles to contribute to the next generation of candidate particles. The procedure can be summarized as follows: Step 1: Form an input matrix based on the selected scenario. Step 2: Set up the LSSVM algorithm and initialize the kernel parameter. Step 3: Initialize parameters m,ω, c1, c2, and θ, in which m is the number of the population, ω is the inertia weight; c1,2 are the weight factors; and θ is the parameter of identification (coefficient of the non-linear rectification equation). The velocity and position of each particle are initialized randomly. The swarm size was set as 20 particles. Step 4: Update each particle's velocity and position. Step 5: Evaluate each particle's fitness. The Mean Square Error (MSE) was used as the fitness function to guide the particle population in searching for the optimum solution. All of the
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Table 2 Statistical evaluation used in this study. Statistical parameter
Expression
Root Mean Square Error (RMSE) 2
Square-Correlation Coefficient (R )
Nash–Sutcliffe (DC)
Eq. number
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ðym −yp Þ RMSE ¼ ∑N N i¼1 0 12 N ∑ y −y ð m m Þñðyp −yp Þ i ¼ 1 A R2 ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ∑N y −ym Þ ñðyp −yp Þ i ¼ 1ð m ∑N
m
(7)
(8)
ðy −y Þ2
DC ¼ 1−∑Ni ¼ 1ðy m−y p Þ2 i ¼ 1
(6)
m
Note: ym and yp denote the measured and predicted values of friction coefficients. y m and y p stand for the mean values of the measured and estimated friction coefficients. N represents the total number of data samples.
features were used in the training process to calculate the error for each particle, so as to generate the training error of particles with training samples. Step 6: Update the personal best position, pbest, and the global best position, gbest. Step 7: If the maximum number of iterations is not achieved or the optimum solution is not acquired, then return to Step 4. In order to avoid overtraining, the validation accuracy curve was observed and the training was stopped when the iteration had the best validation accuracy during the training process.
Step 8: Establish the best LSSVM forecasting model using the optimum solution, and complete the estimation. The ranges of selected values for the optimization process are listed in Table 3. 3.2. Developed models based on flow characteristics In this study, the performance of the PSO-LSSVM approach was investigated for estimating the friction coefficient (Manning and Darcy-Weisbach roughness coefficients). Table 4 lists the performance evaluation indices of the models based on flow characteristics.
Fig. 3. Flowchart of friction coefficient optimization in this study: A. modeling set up, B. LSSVM optimization using PSO.
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A sensitivity analysis was done by evaluating the effect of the input variables based on the results. Selection of various parameter combinations as inputs can affect the accuracy of the results throughout the modeling process. Hence, selection of appropriate variables is a crucial stage in the modeling process. In order to determine the input and output variables, it was attempted to express the process through a minimum set of different variables. Firstly, the models were defined according to flow characteristics. The roughness coefficient (RC) based on flow characteristics could be described as a function of dimensionless variables as follows: RC ¼ f Re; Fr; H=b ð3Þ In which Re, Fr, H, and b stand for the flow Reynolds number, Froude number, flow depth, and channel width, respectively. According to Table 4, the first three combinations were included with only one variable. Results revealed that Re led to the best results among these three combinations for both the Manning and Table 3 Evolution of the number of particles and samples with iteration in this adapted PSO and initial ranges of the hyper parameters of the RBF–SVM-based model. Iteration
Number of particles
Fraction of samples
1-9 10-19 20–29 30–39 40–44 45–50 LSSVM hyper parameters σ γ
20 15 10 10 10 5 Lower limit 0.01 0.01
1/20 1/15 1/10 1/3 1/3 1 Upper limit 100 100
Darcy-Weisbach roughness coefficients. The next 3 combinations were defined with two variables. The combination including Re and H/b yields the best double input results for both outcome variables. Accordingly, in the case of roughness coefficient prediction, the PSO-LSSVM model comprising Re and H/b as the input variables led to the most accurate results for the models with flow characteristics. It was observed that the model for Manning's roughness coefficient (R2 ¼0.8109, DC¼0.6569, and RMSE¼0.0039), demonstrated a better performance than for the Darcy-Weisbach roughness coefficient. According to the results in Table 4, it can be seen that using Reynolds number as the only input parameter yielded the desired prediction accuracy while using only Froude number as input did not yield the desired prediction accuracy. Adding the parameter H/b to the input parameters caused an improvement in model accuracy. It could be deduced that Re and H/b were the most important parameters in modeling the Manning and Darcy-Weisbach roughness coefficients based on flow characteristics. Fig. 4 shows the outcome of modeling using flow characteristics in the verification step. According to the Fig. 4(a), lower values were predicted more accurately in comparison to higher values of Manning's roughness coefficient. Also, the data sets were divided into two groups in terms of the Reynolds number (Reo80,000 and Re480,000). Then, the best input combination (i.e. model with parameters Re and H/b) was considered for both data categories. The results are listed in Table 4 and shown in Fig. 4(b). For predictions of Manning's roughness coefficient, Fig. 4(b) shows the better result for Reynolds number values less than 80,000 in comparison to higher values (Re480,000). It can be deduced that the Reynolds numbers less Table 5 Evaluation of predicted roughness coefficient with bedform characteristics in the verification step.
Table 4 Performance evaluation of models based on flow characteristics.
n n
f
f
Model
R2
DC
RMSE
R2
DC
RMSE
Re Fr H/b Re,H/b Re,Fr Fr,H/b Re4 80,000 Reo 80,000
0.7571 0.3948 0.5959 0.8109 0.7819 0.6648 0.3178 0.8038
0.5403 0.1583 0.3415 0.6569 0.6235 0.4267 0.1183 0.6328
0.0045 0.0062 0.0054 0.0039 0.0041 0.0051 0.0056 0.0027
0.5959 0.3583 0.4195 0.6923 0.5361 0.4637 – –
0.5104 0.2812 0.3695 0.6951 0.4781 0.4122 – –
0.0071 0.0093 0.0078 0.0058 0.0075 0.0069 – –
Re: Reynolds number,Fr: Froude number, H:flow depth, b:channel width,n: Manning’s roughness coefficient, f: Darcy-Weisbach roughness coefficient
Model
R2
DC
RMSE
R2
DC
RMSE
Re,h/L Re,h/H Re,L/H Re,h/H,L/H Re,h/L,L/H h/L,L/H Re,Fr,h/L Fr,h/H,L/H,h/L Re,Fr,h/H Re,h/H,L/H,h/L
0.8179 0.8025 0.8248 0.8077 0.8268 0.5428 0.8168 0.4738 0.8018 0.8228
0.6541 0.6282 0.6708 0.6373 0.6709 0.2845 0.6621 0.2539 0.6269 0.6458
0.0045 0.0047 0.0044 0.0046 0.0044 0.0058 0.0045 0.0067 0.0047 0.6742
0.6741 0.6768 0.6796 0.6721 0.6841 0.3854 0.6722 0.3751 0.6506 0.5969
0.5923 0.5928 0.5992 0.5878 0.6077 0.2304 0.5869 0.2289 0.5122 0.5969
0.0069 0.0069 0.0068 0.0071 0.0068 0.0102 0.0070 0.0108 0.0072 0.0068
h:dune height, H:flow depth, L:dune length
Fig. 4. Scatter plot of (a) predicted vs. observed Manning's roughness coefficient values for the verification data set, (b) predicted and observed Manning's roughness coefficient values vs. Reynolds number values.
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than 80,000 yielded more accurate predictions of Manning's roughness coefficients. 3.3. Developed models based on flow and bedform characteristics Developed models based on flow characteristics concluded with Re and H/b as the input combination which led to the best prediction. In order to consider both flow and bedform features in the modeling process, the models were defined according to flow and bedform characteristics. The RC based on these characteristics could
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be described as a function of dimensionless variables as follows: RC ¼ f Re; Fr; h=L; L=H; h=H ð4Þ where Re and Fr are flow characteristics and h/L, L/H, and h/H are bedform characteristics. In Eq. (4), h, H, and L are dune height, flow depth, and dune length, respectively. The results of the models in the verification step for the friction coefficient with flow and bedform characteristics datasets are listed in Table 5. Among all developed models, the model with parameters Re, h/L, and L/H yielded higher accuracy (R2 ¼ 0.8268, DC¼0.6709, and RMSE¼0.0044 for n). According to the obtained results, it can be inferred that using parameters h/L and L/H as input parameters improved the accuracy of the models. This result indicates that the characteristics of bedforms had an important effect on prediction of the roughness coefficient. According to the statistical criteria in Table 5, application of Manning's roughness coefficient as the objective function for predicted flow resistance led to better results in comparison to the Darcy-Weisbach roughness coefficient as the objective function. Fig. 5 shows the scatterplot obtained from the developed models using flow and bedform characteristics. 3.4. Developed models with flow and sediment characteristics
Fig. 5. Scatter plot of predicted vs. observed Manning's roughness coefficient values for the verification data set including flow and bedform characteristics.
Developed models based on flow characteristics along with modeling using flow and bedform characteristics demonstrated that flow characteristics were capable of yielding accurate predictions. Moreover, to consider the flow and sediment features in roughness coefficient estimation, the flow and sediment
Table 6 Verification statistics of roughness coefficient prediction using flow and sediment characteristics. n
f
Model
R2
DC
RMSE
R2
DC
RMSE
R/D50 Rb V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
0.7389 0.4668 0.5728
0.5456 0.2076 0.3176
0.0045 0.0060 0.0055
0.394 0.3341 0.3811
0.4089 0.3834 0.3931
0.0217 0.0220 0.0219
VH pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3
0.6528
0.4105
0.00518
0.4108
0.3087
0.0216
Re , Fr Re , R/D50 Fr , R/D50 Rb , Re Rb , Fr R/D50 , Rb V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , R/D50
0.7817 0.8398 0.7318 0.8239 0.5118 0.7796 0.7275
0.6235 0.7050 0.5339 0.6780 0.2580 0.5987 0.5269
0.00414 0.0036 0.0046 0.0038 0.0058 0.0042 0.0046
0.4098 0.4933 0.3873 0.4912 0.3523 0.4067 0.3962
0.4032 0.3964 0.4017 0.4121 0.3879 0.3965 0.3710
0.216 0.0202 0.0218 0.0203 0.0219 0.0215 0.0218
V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , Rb
0.6059
0.3657
0.0053
0.3831
0.3971
0.0218
V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , Re
0.8119
0.6781
0.0038
0.4872
0.4051
0.0209
VH pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , Re 3
0.7947
0.6708
0.0038
0.4794
0.4002
0.0209
VH pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , R/D50 3
0.7359
0.5405
0.0045
0.4545
0.3988
0.0212
VH pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , Rb 3
0.6167
0.3764
0.0053
0.3886
0.3643
0.0216
V ffi Re , R/D50 ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.8457
0.7130
0.0036
0.4891
0.3986
0.0208
Re , R/D50 ,Rb VH ffi Re , R/D50 ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
0.8609 0.8478
0.7361 0.7185
0.0034 0.0035
0.4794 0.4762
0.3939 0.3611
0.0203 0.0209
V ffi Rb , R/D50 ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.7549
0.5678
0.0044
0.4642
0.4003
0.0211
VH ffi Rb , R/D50 ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
0.7549
0.5638
0.0044
0.4637
0.3991
0.0211
V VH ffi ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R/D50 ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
0.7308
0.5331
0.0046
0.4635
0.3984
0.0212
V VH ffi,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Rb ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50 gðs−1ÞD350
0.6026
0.3425
0.0054
0.3785
0.3611
0.0217
gðs−1ÞD50 gðs−1ÞD50
gðs−1ÞD50 gðs−1ÞD50 gðs−1ÞD50 gðs−1ÞD50 gðs−1ÞD50 gðs−1ÞD50
gðs−1ÞD50
gðs−1ÞD50
gðs−1ÞD50 gðs−1ÞD50
gðs−1ÞD50
V ffi Rb ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
gðs−1ÞD50
VH ffi ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD350
0.7468
0.5572
0.0044
0.4607
0.3905
0.0212
V VH ffi ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , R/D50 Re,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
0.8457
0.7132
0.0036
0.4792
0.3699
0.0209
V VH ffi ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , Re , R/D50 Rb ,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
0.8438
0.7118
0.0036
0.4748
0.3609
0.0211
gðs−1ÞD50
gðs−1ÞD50
, R/D50
gðs−1ÞD50
gðs−1ÞD50
Rb ¼ grain Reynolds number ¼ U νD50 , U ¼ shear velocity¼
pffiffiffiffiffiffiffiffiffiffi gRSf ,R: hydraulic radius, Sf: energy gradient, ν:kinematic viscosity, D50: median grain diameter, s: Specific gravity.
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K. Roushangar et al. / International Journal of Sediment Research 32 (2017) 515–526
characteristics were used in input combinations. The roughness coefficient (RC) based on the flow and sediment characteristics could be described as a function of dimensionless variables as follows: V VH RC ¼ f ðRe; Fr; R=D50; Rb; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ gðs−1ÞD50 gðs−1ÞD350
ð5Þ
where Rb and s stand for grain Reynolds number and specific gravity. In order to determine the importance of sediment features in modeling of RC, a sensitivity analysis was done for the predictor variables based on the performance of the models. Results listed in Table 6 show that the input combination including Re, R/D50, and Rb led to best prediction of Manning's roughness coefficient. In other words, it can be inferred that the ratio of the hydraulic radius to median diameter and flow and grain Reynolds numbers compose the most effective input combination leading to the best results for predicting Manning's roughness coefficient. On the other hand, the results of modeling for the Darcy-Wiesbach friction coefficient indicated that a combination of grain Reynolds number (Rb) and flow Reynolds number (Re) led to the best results. As can be seen from Table 6, for the case of prediction of the coefficient with sediment features, a comparison between prediction of the Manning and Darcy-Weisbach roughness coefficients found better performance for Manning's roughness coefficient using PSO-LSSVM models. Finally, it is clear that adding Rb to Re and R/D50 caused an improvement in the results for Manning's roughness coefficient. The scatter plot of observed and predicted
Manning's roughness coefficient for the best input combination with flow and sediment characteristics is shown in Fig. 6. 3.5. Discussion about prediction of relative dune height (h/H and h/L) Different studies have been implemented in the field of roughness coefficient prediction. In many practical applications, it was mentioned that it is desirable and sometimes necessary to predict the ratio of dune height to flow depth. This ratio is an important factor in modeling since it can provide useful information for hydraulic engineering research and river management. For this purpose, different parameters were used as input combinations to investigate their roll in modeling relative dune height. Results of models are listed in Table 7. According to the listed performances in Table 7, it is observed that the combination of Re,
Fig. 7. Scatter plot of predicted vs. observed relative dune height values for the verification dataset.
Table 8 Relative significance of each of input parameters of the best PSO-LSSVM models. Best PSO-LSSVM model
Eliminated variable
Flow characteristics Re, H/b
Best PSO-LSSVM model Fig. 6. Scatter plot of predicted vs. observed Manning's roughness coefficient values for verification data set including flow and sediment characteristics.
Table 7 Verification statistics of relative dune height prediction. h/H
Best PSO-LSSVM model h/L
Model
R2
Re Re ,
0.5448 0.2735 0.1403 0.3674 0.2143 0.5259 0.2697 0.1406 0.2428 0.2120
0.0243 0.0246
Re,
0.5929 0.3304 0.1346 0.3858 0.3140
0.0236
Re,
R D50 H b H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ffi , b gðs−1ÞD50
Re,
V , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H R , b D50
V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
0.5619
0.3135
R2
0.6018
DC
,
0.6648 0.4111
RMSE
0.3258 0.0209
Sediment feature characteristics Re, R/D50, Rb
Best PSO-LSSVM model Relative dune height h/L v ffi, Rb,R/D50 Re, H/b, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
0.1363 0.2509 0.0208 0.0245
0.6638 0.4028 0.1272 0.5939 0.3198
gðs−1ÞD50
,
RMSE
0.6829 0.4596 0.1210
Re, Hb, Re,
R D50 H R , b D50
DC
Bedform characteristics Re,h/L,L/H
0.1263 0.5908 0.2995 0.0210 Best PSO-LSSVM model
V :H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 gðs−1ÞD50
Re, Hb, Re,
R D50
H R , b D50
V , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Rb gðs−1ÞD50
,
V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
V :H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 gðs−1ÞD50
, Rb ,
0.6681 0.4138
0.1260 0.6228 0.3334 0.0204
0.6728
0.1261 0.5998 0.3295 0.0208
0.4136
R2 0.5959 0.7571 R2
Verification DC 0.3415 0.5403 DC
RMSE 0.0054 0.0045 RMSE
Re L/H h/L Eliminated variable
0.5428 0.8179 0.8248 R2
0.2845 0.6541 0.6758 DC
0.0058 0.0045 0.0044 RMSE
Re R/D50 Rb Eliminated variable
0.7796 0.8239 0.8398 R2
0.5987 0.6780 0.7050 DC
0.0042 0.0038 0.0036 RMSE
Rb v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.60180 0.3258 0.57305 0.2562
0.0209 0.1119
H/b Re R/D50 Eliminated variable
0.60123 0.5942 0.6023 R2
0.3168 0.3191 0.3210 DC
0.0234 0.0211 0.0207 RMSE
v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.5929
0.3304
0.1346
H/b Re
0.6288 0.6684
0.3589 0.4042
0.0201 0.0193
Re H/b Eliminated variable
gðs−1ÞD50
0.0211
Relative dune height h/H v ffi Re, H/b, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
Performance criteria
gðs−1ÞD50
K. Roushangar et al. / International Journal of Sediment Research 32 (2017) 515–526
523
Table 9 Some of semi-empirical equations for computing Manning’s roughness coefficient. Reference
Expression
R2
Bruschin (1985)
D50 w 1=7:3 ½RñS n ¼ 12:8 D50
0.1109
0.1182
0.0121
D50in millimeter
0.6840
0.7123
0.0038
D50in meter
0.2760
0.3152
0.0098
Fr o 0.4
Karim (1995)
n¼
Camacho and Yen (1992)
0:175 R6ffiffi n¼p =Re0:19 g 3:78T
½ff 0:465 0:037D0:126 50 0 1
1 R6
DC
RMSE
0.4 o Fr o 0.7
n ¼ pffiffig 0:0081ðDR50 Þ0:125 Re Fr 0:88 n¼ Strickler (1923)
1
R6ffiffi p g
0:05
0.7 o Fr o 1
0:078ðDR50 Þ0:166 =Fr 0:444
n ¼ 0:0474D50
0.2311
1=6
Condition
0.3087
0.0102
D50in meter
Note: R: hydraulic radius, D50:median grain diameter, Sw: energy slope, f 0 ,f: grain and total Darcy-Weisbach friction factor, T : Camacho parameter V H/b, R/D50, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , and Rb for prediction the ratio of dune height gðs−1ÞD50
3.6. Sensitivity analysis
2
to its length led to the best outcome (R ¼ 0.6228, DC¼0.3334 and RMSE ¼ 0.0204). Likewise, for predicting of h/H, the model with V parameters Re, H/b, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi were the dominant parameters gðs−1ÞD50
(R2 ¼ 0.6829, DC ¼0.4596 and RMSE¼ 0.121). By comparing the various input combinations, it can be concluded that adding R/D50 to other parameters could cause an increase in the accuracy of the results. Further, according to the V results, it can be inferred from Table 7 that the effect of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is gðs−1ÞD50
rational. Finally, according to the results, it is almost impossible to make a sensible relation between h/H and each one of the input parameters solely. So, it can be deduced that the relative dune height has a suitable correlation with a combination of Re, H=b, V and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . The scatterplot of predicted relative dune height vs. gðs−1ÞD50
observed values (for the verification dataset) is shown in Fig. 7.
To investigate the impacts of different parameters of the best PSO-LSSVM models on the prediction of Manning's roughness coefficient, sensitivity analysis was done. The significance of each parameter was evaluated by eliminating the parameter. Table 8 reveals that in prediction of Manning's roughness coefficient with flow characteristics, Reynolds number is the most important parameter, and for bedform and sediment characteristics, Reynolds number has the most significant impact on Manning's roughness coefficient. Finally, for prediction relative dune height, v is the dominant parameter. for both objective functions, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs−1ÞD50
3.7. Comparison with previous studies To evaluate the accuracy and capability of the proposed models obtained using the PSO-LSSVM method in prediction of Manning's
Fig. 8. Comparison between the models obtained using the PSO-LSSVM method and semi-empirical equations for prediction of Manning's roughness coefficient.
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K. Roushangar et al. / International Journal of Sediment Research 32 (2017) 515–526
Table 10 Some of the data used in this study from Guy et al. (1966), Roushangar (2008), U. S. Army Corps of Engineers (1935), and Williams (1970). Fr
Re
D50
H/D50
H/b
h/H
h/L
n
0.369657 0.39 0.372286 0.345435 0.378988 0.46345 0.486625 0.3 0.518009 0.37 0.497534 0.526518 0.372286 0.428234 0.45268 0.565729 0.440286 0.478581 0.406684 0.354876 0.493791 0.41 0.552517 0.460552 0.415476 0.385559 0.33 0.414997 0.380171 0.617886 0.417059 0.363652 0.393771 0.365077 0.34 0.675378 0.403609 0.480999 0.490137 0.34319 0.32 0.440567 0.39 0.44 0.412364 0.643489 0.426351 0.395141 0.501371 0.527824 0.428909 0.485984 0.44 0.596941 0.521621 0.475066 0.403362 0.431068 0.649278 0.41 0.836536 0.527375 0.652066 0.43 0.500517 0.484267 0.457655 0.609749 0.442899 0.490176 0.35 0.745421 0.667081 0.423369 0.746437
17,356.84 80,500 17,447.97 28,066.5 23,270.46 11,932.3 12,496.33 17,1700 13,197.7 80,000 47,936.42 13,200.44 30,314.01 15,985.7 16,921.84 14,565.64 25,086.1 18,422.78 19271.29 45,413.74 19,812.91 73,600 22,306.72 43,947.18 48,974.61 31,836.14 185,900 37,510.19 35,641.7 15,867.05 25,564.61 47,454.2 24,464.47 87,342.55 66,400 17,479.14 71,133 17,930.29 30,868.79 61,553.21 149,900 53,466.01 89,200 81,500 54,155.75 25,337.19 35,054.7 51,032.57 32,420.5 19,703.31 41,324.5 30,193.49 79,900 22,314.51 42,473.86 44,449.75 74,508.67 27,048.79 24,135.4 54,500 21,313.05 28,980.2 30,814.69 92,300 26,487.9 25,627.9 28,528.15 50,721.05 54,519.68 23,479.85 19,1900 28,223.18 41,789.45 76,890.93 28,019.3
0.00135 0.00093 0.00135 0.00135 0.00135 0.00135 0.00135 0.00093 0.00135 0.00093 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00019 0.00135 0.00135 0.00135 0.00135 0.00093 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00033 0.00135 0.00135 0.00135 0.00135 0.00135 0.00019 0.00135 0.00028 0.00093 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00033 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00028 0.00135 0.00135 0.00135 0.00093 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00093 0.00135 0.00135 0.00135 0.00135
158.9476 176.9806 158.496 158.0444 111.3084 68.63644 68.41067 340.8516 67.73333 190.0903 116.7271 66.37867 158.496 113.1147 113.3404 68.63644 70.89422 66.15289 161.2053 157.1413 69.088 834.1895 69.53956 115.5982 107.9218 161.4311 344.129 109.5022 113.5662 68.41067 111.0827 160.528 112.8889 164.8178 480.2909 69.088 153.7547 112.8889 64.79822 156.2382 1524 110.6311 674.9143 173.7032 161.6569 67.73333 160.528 158.7218 66.15289 113.1147 116.7271 112.8889 452.5818 113.3404 158.496 113.3404 160.528 114.2436 112.4373 478.9714 67.73333 68.63644 160.528 183.5355 66.37867 66.37867 113.3404 163.0116 111.9858 163.9147 340.8516 65.024 114.0178 158.0444 114.0178
2.816 0.0675 2.808 1.4 0.986 1.216 1.212 0.13 1.2 0.0725 0.517 1.176 1.404 2.004 2.008 1.216 0.314 0.586 2.856 0.696 0.612 0.065 0.616 0.512 0.239 1.43 0.13125 0.485 0.503 1.212 0.984 0.711 1 0.187179 0.26 1.224 0.3405 2 0.1435 0.346 0.11875 0.245 0.0775 0.06625 0.716 0.6 1.422 0.703 0.1465 2.004 0.517 1 0.245 2.008 1.404 0.502 0.3555 1.012 1.992 0.055 1.2 0.304 2.844 0.07 0.294 0.294 1.004 1.444 0.248 2.904 0.13 0.576 1.01 0.35 2.02
0.034091 0.037037 0.037037 0.038571 0.042596 0.046053 0.046205 0.048077 0.05 0.051724 0.056093 0.057823 0.058405 0.05988 0.065737 0.065789 0.066879 0.068259 0.070028 0.071839 0.071895 0.076923 0.081169 0.082031 0.083682 0.083916 0.085714 0.086598 0.087475 0.089109 0.089431 0.090014 0.092 0.09589 0.096154 0.098039 0.099853 0.1 0.101045 0.102601 0.105263 0.110204 0.112903 0.113208 0.114525 0.116667 0.116737 0.118065 0.119454 0.11976 0.121857 0.122 0.122449 0.123506 0.123932 0.125498 0.126582 0.128458 0.134538 0.136364 0.136667 0.138158 0.140647 0.142857 0.142857 0.142857 0.143426 0.144044 0.145161 0.14876 0.153846 0.15625 0.156436 0.157143 0.158416
0.005322 0.006897 0.006132 0.006429 0.002917 0.00231 0.00231 0.013514 0.002703 0.010714 0.004677 0.003864 0.006949 0.004491 0.009244 0.007435 0.008898 0.003017 0.013055 0.006329 0.006395 0.023529 0.009804 0.014483 0.006897 0.010508 0.034615 0.010769 0.004112 0.01015 0.013665 0.012075 0.013218 0.019444 0.021739 0.014019 0.019429 0.013369 0.004677 0.018933 0.025 0.014876 0.046667 0.02069 0.031538 0.018919 0.028041 0.022432 0.010671 0.017751 0.0225 0.024206 0.021429 0.027434 0.037179 0.0252 0.029032 0.019118 0.032057 0.033333 0.035652 0.011667 0.03367 0.024242 0.007241 0.004719 0.022642 0.044068 0.023226 0.032432 0.044444 0.03 0.036239 0.030556 0.025
0.0092 0.017 0.00959 0.01 0.0104 0.0107 0.0099 0.017 0.0099 0.017 0.0147 0.0101 0.0098 0.00971 0.00946 0.0102 0.0123 0.0106 0.0093 0.0111 0.011 0.02 0.0114 0.0136 0.0141 0.01 0.019 0.0129 0.0127 0.0101 0.0119 0.0106 0.0115 0.0171 0.022 0.0108 0.0122 0.01 0.0127 0.0124 0.021 0.0126 0.021 0.02 0.0141 0.012 0.0109 0.0126 0.0141 0.0105 0.0141 0.0127 0.024 0.0108 0.0111 0.0143 0.0165 0.0124 0.0108 0.021 0.011 0.0135 0.0095 0.021 0.0133 0.013 0.0125 0.0111 0.0149 0.0103 0.025 0.0128 0.0134 0.017 0.0108
K. Roushangar et al. / International Journal of Sediment Research 32 (2017) 515–526
525
Table 10 (continued ) Fr
Re
D50
H/D50
H/b
h/H
h/L
n
0.410035 0.579776 0.37 0.57667 0.740893 0.826197 0.505139 0.41 0.56456 0.37 0.58502 0.621766 0.33 0.402121 0.381987 0.538448 0.36 0.34 0.426535 0.41 0.525043 0.4 0.435772
38,898.66 35,599.18 10,3900 36,157.05 34,691.3 38,360.88 27,348.33 68,800 29,991.91 190,900 30,959.88 32,398.47 150,600 92,649.71 87,550.88 98,095.13 163,600 100,100 40,147.05 147,500 28,745.57 188,400 97,955.71
0.00135 0.00135 0.00032 0.00135 0.00135 0.00135 0.00135 0.00032 0.00135 0.00093 0.00135 0.00135 0.00027 0.00135 0.00135 0.00135 0.00019 0.00032 0.00135 0.00032 0.00135 0.00054 0.00135
114.9209 111.3084 561.975 64.57244 158.2702 156.2382 67.73333 533.4 66.60444 327.7419 66.37867 65.47556 1049.867 159.8507 159.1733 158.496 1491.916 533.4 114.0178 600.075 68.41067 496.7111 156.6898
0.509 0.986 0.295 0.143 2.804 2.768 0.3 0.28 0.295 0.125 0.294 0.29 0.11625 0.181538 0.180769 0.351 0.11625 0.28 0.505 0.315 0.303 0.44 0.177949
0.163065 0.166329 0.169492 0.171329 0.174037 0.176301 0.176667 0.178571 0.179661 0.18 0.180272 0.182759 0.182796 0.183616 0.184397 0.185185 0.193548 0.196429 0.205941 0.206349 0.207921 0.215909 0.216138
0.015962 0.037788 0.0625 0.020588 0.047656 0.091729 0.011042 0.1 0.015143 0.051429 0.023043 0.022083 0.033333 0.030952 0.030233 0.041935 0.033333 0.033333 0.028108 0.028889 0.014651 0.052778 0.06
0.0132 0.0132 0.022 0.0143 0.0095 0.0104 0.0138 0.022 0.0134 0.026 0.0139 0.0142 0.023 0.017 0.016 0.017 0.019 0.023 0.013 0.027 0.0133 0.032 0.0202
roughness coefficient, a comparison between the proposed methods and some of the conventional semi-empirical equations was undertaken for the verification data sets. A variety of the analytical and semi-empirical approaches has been developed to predict the roughness coefficient in which some have simple and some have complex structure. The considered approaches are listed in Table 9. There are different concepts and approaches that are used in the derivation and extraction process of the roughness coefficient. The approach developed by Strickler (1923) considers only the median grain diameter (D50). The formula developed by Karim (1995), was designed by considering the grain and total Darcy-Weisbach friction factors (f0 and f) and the median grain diameter. The formula developed by Bruschin (1985) considered the interaction among the median grain diameter, energy slope (Sw), and hydraulic radius (R). For Froude values below 0.4 and higher Camacho and Yen (1992) developed different formulas. The approach developed by Camacho and Yen (1992) took advantage of Reynolds (Re) and Froude (Fr) values interaction along with R and D50. The formula varies for Froude values between 0.4 and 0.7, and between 0.7 and 1. All of the approaches main conditions consider D50 as a main parameter. According to Table 9, among the Classic approaches Karim's formula predicted the RC better in comparison with the other approaches. Also Fig. 8 shows the slightly better performance of the prediction equation obtained using the PSO-LSSVM method than the empirical equations for Manning's roughness coefficient prediction. As shown in Fig. 8, almost all the predicted values using the PSO-LSSVM approach are within a 10% margin of the measured values.
conclusions can be stated. Results proved that the PSO-LSSVM models for Manning's roughness coefficient demonstrate better performance than for the Darcy-Weisbach roughness coefficient for all cases. According to the outcome, in the prediction of Manning's roughness coefficient using flow characteristics, adding the ratio of flow depth to channel width (H/b) to Reynolds number (Re) improved the results. Also, it was deduced that prediction of RC for Reynolds number values less than 80,000 had more accuracy than for higher Reynolds number values. In prediction of Manning's roughness coefficient using flow and bedform characteristics, the single input consisting of the Reynolds number had the highest influence and has important role in predicting Manning's roughness coefficient. Regarding Manning's roughness coefficient with sediment characteristics, Reynolds number, the ratio of hydraulic radius to median grain diameter, and grain Reynolds number (Re, R/D50, and Rb) yielded the best results. In estimation of the relative dune height (h/H), using the Re, H/b, and v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yielded the best results. As an overall consequence, the gðs−1ÞD50
obtained results confirmed that the Reynolds number (Re) is the most important and influential parameter for prediction Manning's roughness coefficient. It should, however, be noted that the SVM is a data-driven model and the SVM-based models are data sensitive, so further studies using data ranges beyond this study and field data should be done to determine the merits of the models to estimate roughness coefficients in real flow conditions (Table 10).
References 4. Conclusions In the present study, the performance of the PSO-LSSVM approach was evaluated in prediction of the Manning and DarcyWeisbach roughness coefficients by applying different modeling scenarios. The outputs from the proposed PSO-LSSVM method were compared with conventional approaches of semi-empirical equations. The outcome revealed the accurate prediction obtained using the PSO-LSSVM approach in comparison to the classical equations. According to the results, the following prominent
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