## Abstract

This study is trying to develop an alternative approach to the issues of sediment transport simulation. A machine learning method, named least square support vector regression (LSSVR) and a meta-heuristic approach, called particle swarm optimization (PSO) algorithm are used to estimate bed material load transport rate. PSO algorithm is utilized to calibrate the parameters involved in the model to facilitate a desirable simulation by LSSVR. Implementing on a set of laboratory and field data, the model is capable of performing more satisfactorily in comparison to candidate traditional methods. Similarly, the proposed method has a better performance than a specific version of decision tree method. To enhance the model, the variables are scaled in logarithmic form, leading to an improvement in the results. Thus, the proposed model can be an efficient alternative to conventional approaches for the simulation of bed material load transport rates providing comparable accuracy.

## NOTATION

*C*_{t}Concentration (ppm by weight)

*D*Channel depth (m)

Dimensionless particle diameter (–)

*d*_{50}Median particle diameter (m)

- Fr
Densimetric Froude number (–)

*G*Gravity acceleration (m·s

^{−2})*Q*_{w}Water discharge (m

^{3}·s^{−1})Shear Reynolds number (–)

- Re
Reynolds number (–)

*S*_{f}Channel slope (–)

- SG
Specific gravity (–)

*T*Temperature (°C)

Shear velocity (m·s

^{−1})*V*Flow velocity (m·s

^{−1})Critical velocity (m·s

^{−1})*W*Channel width (m)

Bed shear stress (N·m

^{−2})Critical shear stress (N·m

^{−2})Water specific weight (N·m

^{−3})Sediment fall velocity (m·s

^{−1})

## INTRODUCTION

Sediment transport science deals with the interactions between flow and sediment particles. Sediment transport and sedimentation lead to the development of sedimentary bars, resulting in a decline in flood flow transportation, channel sedimentation, dam lifetime and reservoir storage capacity, corrosion in river structure installations, and contaminant transport. Sediment transport is a highly complex phenomenon often subjected to theoretical, empirical, and semi-empirical treatment. Most theory-driven approaches, such as those by Meyer-Peter & Muller (1948), Bagnold (1966), and Einstein (1942) are based on ideal and simplified assumptions in which sediment transport rate is determined according to a small number of dominant factors. However, the obtained results from various approaches based on statistical analyses differ dramatically from each other as well as from field observations. Due to the nonlinearity and complexity of sediment transport and the variability of particle diameters, as well as the various forms the channel bed can take, data-driven methods may provide a better response to such problems.

Computational intelligence techniques, such as artificial neural networks (ANNs), fuzzy logic, genetic programming (GP), or a combination of these techniques, have been successfully used to solve complex problems associated with water resources management. Among them, ANN is the most widely used learning method. This method has two disadvantages. In the first place, its structure needs to be defined prior to training. Second, the regularization techniques, such as early termination and training with noise are somewhat restricted. Furthermore, ANNs can also be trapped in local minima in the process of model training (Smola 1996). These drawbacks do not exist in a support vector machine (SVM) approach. Therefore, in this study, a combination of least square support vector regression (LSSVR) and particle swarm optimization (PSO) algorithm is applied to the most important variables in sediment transport problems including channel geometry and hydraulic properties. In general, the calibration of LSSVR and kernel parameters is a challenging process. As such, PSO algorithm is used for the calibration of these parameters.

SVM is a newly developed learning technique that has gained enormous popularity in the areas of classification, pattern recognition, and regression. SVM works based on the structural risk minimization principle offering greater generalizability. SVM was initially introduced by Vapnik & Lerner (1963) and extended to nonlinear conditions by Cortes & Vapnik (1995). A great deal of previous research into SVM application in water resources has focused on flood forecasting, some of which are mentioned as follows. Having investigated a case study application of SVM in flood prediction, Yu *et al.* (2006) reported the model accuracy in predicting water surface levels 1 to 6 hours after the flood. Sivapragasam *et al.* (2001) introduced a useful prediction technique based on singular spectrum analysis coupled with SVM. Compared with nonlinear prediction methods, their hybrid model enjoyed greater accuracy when it came to predicting hydrologic parameters. Yilin *et al.* (2006) used a combination of SVM and the shuffled complex evolution (SCE) optimization algorithm in forecasting long-term discharge and concluded that SVM satisfactorily predicts long-term discharge. Similarly, Asefa *et al.* (2006) offered an SVM-based method for hourly and seasonal flow prediction. In the same vein, Shuquan & Lijun (2007) used least square support vector machine (LSSVM) to predict both mid-term and long-term runoff. They concluded that LSSVM functions more effectively in comparison to ANN. Han *et al.* (2007) conducted research on a case study flood flow analysis and found that SVM excels any other benchmarking models that are being used for flood forecasting. Huang *et al.* (2010) used SVM to comprehensively assess flood disaster loss. Noori *et al.* (2011) investigated the impact of three input selection techniques on SVM efficiency for monthly flow prediction, showing that the pre-processing of input variables through principal component analysis and Gamma tests can improve SVM performance. In another investigation into the application of SVM and regression tree methods over a case study research on outflow forecasting, Sahraei *et al.* (2015) found that support vector regression (SVR) surpasses the regression tree in terms of outlet discharge prediction.

There is also a large volume of published studies describing the role of machine learning methods in estimating sediment discharge, one of which is the research done by Jain (2001). He developed an ANN-based sediment rating curve for two gauging sites on the Mississippi River and demonstrated that his model is more precise than the conventional sediment rating curve techniques in terms of sediment load calculation. Nagy *et al.* (2002) applied the ANN model to datasets measured in different rivers to develop an estimator for natural sediment discharge. Their results indicated that the new approach yielded better results compared to several commonly used formulas. Likewise, Bhattacharya *et al.* (2007), in a comparison with the existing approaches, asserted that adopting two machine learning methods, namely, ANN and MT, for estimating bed load and total load transport rate can enhance the sediment model accuracy to some extent. However, the existing literature on the application of SVR over sediment load is extensive and focuses particularly on either meteorological data or streamflow as input variables for their model (Kisi *et al.* 2012; Afan *et al.* 2016). Having utilized SVR to forecast the suspended sediment concentration of rivers in the USA in 2006, Çimen showed that their method performs better than fuzzy logic and gradual evolution approaches (Çimen 2006). In the same way, Azamathulla *et al.* (2010) found that SVR technique enjoys superior performance in comparison to existing sediment transport methods in three rivers in Malaysia. Misra *et al.* (2009) demonstrated that simulated daily, weekly, and monthly runoff and sediment yield by SVR was significantly better than those yielded by ANN at model training, calibration, and verification.

Although some research has been carried out on SVR, none considered estimation of bed material load as an approach applicable to a wide range of flow and sediment characteristics. The main goal of this paper is to present a reliable approach for the estimation of bed material load if there is no concern about the data availability. The remainder of this paper is organized as follows. The next section details underlying methodology of this research and describes how LSSVR and PSO methods work. This is followed by a section presenting a discussion regarding the method's suitability to estimation of the sediment transport rate by comparing with the traditional methods. The final section presents the conclusions of the study.

## MATERIALS AND METHODS

### Data description and dimensional analysis

*Q*), channel width (

_{w}*W*), channel depth (

*D*), water surface slope (

*S*), water temperature (

_{f}*T*), median particle diameter (

*d*

_{50}), specific gravity (

*SG*), and bed material concentration by weight (

*C*). The source of data as well as their averages and standard deviations are presented in Table 1. It is preferred to use kinematic viscosity for two reasons (Yang 1996): (1) most of the traditional methods consider viscosity as decision variable rather than temperature and (2) variables are converted to dimensionless variables prior to modeling. As a result, it substitutes the temperature using Equation (1): where

_{t}*T*is the water temperature in centigrade and

*v*is the kinematic viscosity in square meters per second.

Experimental data^{a} | Number of data | Index | Discharge (10^{−4} c·m/s) | Width (m) | Depth (m) | Surface slope (10^{−5} m/m) | Kinematic viscosity (10^{−8} m^{2}/s) | Median particle diameter (mm) | Specific gravity |
---|---|---|---|---|---|---|---|---|---|

Barton & Lin (1955) | 30 | Average | 1,340 | 1.219 | 0.192 | 119 | 95.7 | 0.18 | 2.65 |

Standard deviation | 754 | 0 | 0.065 | 44 | 6.74 | 0 | 0 | ||

Vanoni & Brooks (1957) | 21 | Average | 91 | 0.267 | 0.071 | 245 | 91.4 | 0.145 | 2.65 |

Standard deviation | 28 | 0 | 0.012 | 52 | 8.31 | 0 | 0 | ||

Kennedy & Brooks (1963) | 9 | Average | 396 | 0.851 | 0.102 | 181.1 | 94.8 | 0.142 | 2.65 |

Standard deviation | 1.37 | 0 | 0.034 | 55.7 | 7.55 | 0 | 0 | ||

Nomicos (1957) | 26 | Average | 88.3 | 0.267 | 0.073 | 236 | 90.9 | 0.149 | 2.65 |

Standard deviation | 32.5 | 0 | 1.18E-03 | 40 | 8.92 | 0.005 | 0 | ||

Soni (1980) | 23 | Average | 55.9 | 0.2 | 0.065 | 406 | 83.3 | 0.32 | 2.65 |

Standard deviation | 20 | 0 | 0.021 | 175 | 1.97 | 0 | 0 | ||

Stein (1965) | 57 | Average | 2,685 | 1.219 | 0.218 | 434 | 93.1 | 0.4 | 2.65 |

Standard deviation | 1,064 | 0 | 0.072 | 288 | 4.75 | 0 | 0 | ||

Taylor & Vanoni (1972) | 6 | Average | 479 | 0.657 | 0.090 | 198 | 76.1 | 0.198 | 2.65 |

Standard deviation | 321 | 0.302 | 0.018 | 8 | 14.5 | 0.046 | 0 | ||

Vanoni & Brooks (1957) | 15 | Average | 409 | 0.851 | 0.110 | 177 | 94.2 | 0.137 | 2.65 |

Standard deviation | 252 | 0 | 0.047 | 83.7 | 5.42 | 0 | 0 | ||

Willis et al. (1972) | 96 | Average | 2,579 | 1.219 | 0.243 | 97.8 | 93.3 | 0.1 | 2.65 |

Standard deviation | 1,045 | 0 | 0.062 | 38.9 | 4.90 | 0 | 0 | ||

Willis (1979) | 32 | Average | 307 | 0.360 | 0.128 | 394 | 96.4 | 0.54 | 2.65 |

Standard deviation | 109 | 0 | 0.015 | 221 | 17.2 | 0 | 0 | ||

Field data | |||||||||

Brownlie (1981) | 289 | Average | 488,790 | 58.264 | 0.649 | 119 | 114 | 0.384 | 2.65 |

Standard deviation | 667,180 | 25.879 | 0.486 | 33.7 | 21.4 | 0.184 | 0 | ||

Toffaleti (1968) | 38 | Average | 1,433,290 | 120.64 | 0.825 | 80.5 | 103 | 0.304 | 2.65 |

Standard deviation | 719,450 | 46.119 | 0.280 | 3.5 | 7.49 | 0.031 | 0 | ||

Sum | 642 | Average | 305,600 | 33.799 | 0.424 | 177 | 103 | 0.305 | 2.65 |

Standard deviation | 604,850 | 40.906 | 0.422 | 159 | 18.8 | 0.179 | 0 |

Experimental data^{a} | Number of data | Index | Discharge (10^{−4} c·m/s) | Width (m) | Depth (m) | Surface slope (10^{−5} m/m) | Kinematic viscosity (10^{−8} m^{2}/s) | Median particle diameter (mm) | Specific gravity |
---|---|---|---|---|---|---|---|---|---|

Barton & Lin (1955) | 30 | Average | 1,340 | 1.219 | 0.192 | 119 | 95.7 | 0.18 | 2.65 |

Standard deviation | 754 | 0 | 0.065 | 44 | 6.74 | 0 | 0 | ||

Vanoni & Brooks (1957) | 21 | Average | 91 | 0.267 | 0.071 | 245 | 91.4 | 0.145 | 2.65 |

Standard deviation | 28 | 0 | 0.012 | 52 | 8.31 | 0 | 0 | ||

Kennedy & Brooks (1963) | 9 | Average | 396 | 0.851 | 0.102 | 181.1 | 94.8 | 0.142 | 2.65 |

Standard deviation | 1.37 | 0 | 0.034 | 55.7 | 7.55 | 0 | 0 | ||

Nomicos (1957) | 26 | Average | 88.3 | 0.267 | 0.073 | 236 | 90.9 | 0.149 | 2.65 |

Standard deviation | 32.5 | 0 | 1.18E-03 | 40 | 8.92 | 0.005 | 0 | ||

Soni (1980) | 23 | Average | 55.9 | 0.2 | 0.065 | 406 | 83.3 | 0.32 | 2.65 |

Standard deviation | 20 | 0 | 0.021 | 175 | 1.97 | 0 | 0 | ||

Stein (1965) | 57 | Average | 2,685 | 1.219 | 0.218 | 434 | 93.1 | 0.4 | 2.65 |

Standard deviation | 1,064 | 0 | 0.072 | 288 | 4.75 | 0 | 0 | ||

Taylor & Vanoni (1972) | 6 | Average | 479 | 0.657 | 0.090 | 198 | 76.1 | 0.198 | 2.65 |

Standard deviation | 321 | 0.302 | 0.018 | 8 | 14.5 | 0.046 | 0 | ||

Vanoni & Brooks (1957) | 15 | Average | 409 | 0.851 | 0.110 | 177 | 94.2 | 0.137 | 2.65 |

Standard deviation | 252 | 0 | 0.047 | 83.7 | 5.42 | 0 | 0 | ||

Willis et al. (1972) | 96 | Average | 2,579 | 1.219 | 0.243 | 97.8 | 93.3 | 0.1 | 2.65 |

Standard deviation | 1,045 | 0 | 0.062 | 38.9 | 4.90 | 0 | 0 | ||

Willis (1979) | 32 | Average | 307 | 0.360 | 0.128 | 394 | 96.4 | 0.54 | 2.65 |

Standard deviation | 109 | 0 | 0.015 | 221 | 17.2 | 0 | 0 | ||

Field data | |||||||||

Brownlie (1981) | 289 | Average | 488,790 | 58.264 | 0.649 | 119 | 114 | 0.384 | 2.65 |

Standard deviation | 667,180 | 25.879 | 0.486 | 33.7 | 21.4 | 0.184 | 0 | ||

Toffaleti (1968) | 38 | Average | 1,433,290 | 120.64 | 0.825 | 80.5 | 103 | 0.304 | 2.65 |

Standard deviation | 719,450 | 46.119 | 0.280 | 3.5 | 7.49 | 0.031 | 0 | ||

Sum | 642 | Average | 305,600 | 33.799 | 0.424 | 177 | 103 | 0.305 | 2.65 |

Standard deviation | 604,850 | 40.906 | 0.422 | 159 | 18.8 | 0.179 | 0 |

^{a}Experimental data are provided in Brownlie (1981).

*W*/

*D*

*>*4, the bed shear stress can be calculated using flow depth. Williams (1970) focused on smooth wall correction to obtain the bed shear stress for measured data with an aspect ratio of

*W/D*

*>*4, which has been converted to SI unit system as follows: where is the bed shear stress and

*S*is surface slope. Critical shear stress is the bed shear stress in the initiation of motion condition and Rao & Sreenivasulu (2006) have developed the following equation to approximately compute Shields' stress: where is shear Reynolds' number and denotes shear velocity.

_{f}*Fr*is the densimetric Froude number, is surface slope, is relative bed roughness, is aspect ratio,

*Re*is Reynolds' number, and is equivalent to the dimensionless Shields' parameter (Shields 1936). As the term (

*SG*−1) is nearly constant, it is excluded from the inputs. Details of the average and standard deviation (SD) of the dimensionless input and output variables can be observed in Table 2.

Variables | Densimetric Froude | ||||||
---|---|---|---|---|---|---|---|

Average | 2,580.6 | 16.96 | 0.00177 | 0.00131 | 65.78 | 467,941 | 2.654 |

Standard deviation | 3,361.1 | 9.60 | 0.00159 | 0.00139 | 83.4 | 785,637 | 2.927 |

Variables | Densimetric Froude | ||||||
---|---|---|---|---|---|---|---|

Average | 2,580.6 | 16.96 | 0.00177 | 0.00131 | 65.78 | 467,941 | 2.654 |

Standard deviation | 3,361.1 | 9.60 | 0.00159 | 0.00139 | 83.4 | 785,637 | 2.927 |

### Support vector machine

*f*(

*X*) function for training patterns (

*X*) so that it has maximum margin from the training target values (

*Y*). In other words, SVR is a model that fits a tube to the data with a radius of so that minimum error occurs in the test dataset. Let us first assume that a training set (

*T*) is considered as follows to compare with its prediction values: where is an

*m*-dimensional vector, each of which belongs to a specific decision variable.

*y*is the associated output variable and

_{i}*N*is the number of samples. Second, Vapnik asserts that the term representing the complexity of set of functions must be minimized in order to minimize the test error according to Equation (7). Therefore, the SVR method uses a set of linear functions in the form of ( is the weight vector and

*b*is a bias value) for prediction. Trivially, the predictions are different from the measurements and, in some cases, it is impossible to consider error values lower than . As such, a deviation from , known as slack variable , is defined. The error value is then minimized using Equation (8). According to Equation (8), maximum margin requires minimizing the norm of weight vector: where determines the estimated tube range and

*C*controls the deviation error more than , both of which are above zero.

*et al.*2002): where is the regularization parameter that creates a trade-off between the uniformity of fitness curve and fitness error minimization. The Lagrangian function of the constrained optimization problem can be written as follows:

By derivation of Equation (12) with respect to , *b*, *e*, and , thereby substituting the terms in the equation, the optimization problem is converted to a linear programming and multipliers are obtained.

### PSO algorithm

The original algorithm of PSO is inspired by the social behavior of birds trying to find their food in nature. This algorithm works based on the interaction and co-operation of the birds searching for food within an area (Kennedy & Eberhart 1995). The birds follow the one that is nearest to the food while simultaneously taking advantage of their earlier experiences finding the food (Kennedy & Eberhart 2001). The food can be found only in one point in the search space that the birds are not aware of. Each solution acts like a bird, which is called a particle in this paper. Each particle enjoys a merit value by means of optimizing the objective function. PSO first generates an initial random solution each of which has an *n*-dimensional position where *n* is the number of decision variables. As well, it assigns a velocity vector between the maximum and minimum allowable velocities to each particle. In order to produce a new population for the next generation and have progress in terms of convergence, PSO requires specifying some parameters, including inertia weight (*w*), inertia weight reduction factor , minimum inertia weight (*w _{min}*), personal and global acceleration values (

*c*

_{1}and

*c*

_{2}), and maximum velocity reduction factor . In every generation, the personal (, the best position which every single particle has ever had since the beginning to the current generation) and global (, the best ever position among all particles) best particles are found to update the population for the next iteration.

*k*is the iteration number. The velocity values are limited to to guarantee the convergence of PSO to the optimal objective and either when there is no significant change in the value of objective function (Equation (15)) or after a specified number of generation the algorithm terminates:

### Model tree (M5P)

*T*denotes the subset that reached the node.

*T*represents the new subset being created due to division based on the selected attribute and

_{i}*S*is the standard deviation (Wang & Witten 1997). Solomatine and Dulal were among the first to demonstrate the model trees as a substitute for rainfall-runoff modeling. Additional details can be found in the works of Solomatine & Dulal (2003).

_{d}### Traditional methods

In spite of the complexity of sediment transport phenomenon, most conventional methods work on some idealized or simplified assumptions (Yang 1996). Currently, some of the most commonly used methods for measuring total bed material load are by Engelund & Hansen (1972), Ackers & White (1973), Graf (1971), and Yang (1972). Due to its inclusion in the modeling, Yang's (1972) methodology is briefly explained in the following. However, Yang's (1996) textbook gives complete details about the rationale for development of the aforementioned traditional methods.

*V*is flow velocity.

### Cross-validation

Cross-validation is a model validation technique for generalization of the results of statistical indices to an independent dataset. Cross-validation is used for random subsampling. The *K*-fold cross-validation has a general form in which the dataset is divided by *K* roughly equal parts. In this method, each sample is used (*K*−1) times in the training process and only once in the testing process. For instance, suppose that the initial dataset is divided by two subsets. First, one of the subsets is used in the training and the other is treated as the test dataset. Then, the roles of the two datasets are reversed and the modeling process is repeated. That is to say, the part that was used for testing in the first stage is used for training in the second and vice versa. The result is calculated by averaging the outputs of both runs. This approach is referred to as two-fold cross-validation. As a result, for a general form, the process is repeated *K* times and final output values are then computed by taking average of the *K* output values. There has been a wide range of research in the field of water resources and the study by Kazeminezhad *et al.* (2010) is just one example that applied this technique in an ANN method for scour around pipelines.

### Experimental setup

In this study, the estimation and parameter calibration processes were codified in MATLAB^{®} environment. LSSVMlab toolbox (Least squares support vector machine Lab (LS-SVMlab) toolbox 2011), which is available for free, was partly used for the purpose of estimation and PSO algorithm was codified and combined with LSSVR. The ten-fold cross-validation technique was then applied to the data, and is the prevalent technique compared to other models and has been considered as the model of choice in recent studies. In summary, data were shuffled at the beginning, thereby dividing by ten approximately equal parts. Each time, one subcategory of the dataset was set aside for testing the model. The other nine folds were used for model training and calibration of LSSVR parameters. These parameters include the regularization parameter (*γ*) and the one related to the kernel parameter, which differs depending on the type of kernel. After pre-processing and normalization of the data, the learning process was carried out and the parameters were calibrated using the validation dataset. As previously stated, PSO algorithm was adopted for the purpose of calibration to facilitate the identification of the magnitude of LSSVR parameters. When various kernel functions were considered, it became clear that the radial basis function (RBF) had the best generalization among all other available functions. The search space embraced a range from 0 to 50,000 for both regularization and RBF parameters. A flowchart representing the combined model is shown in Figure 1. In general, the hybrid model of PSO-LSSVR requires a larger computational budget than when each of the methods are used individually for simulation purposes. However, the combined model may be more reliable due to the enhanced accuracy it offers. PSO has predetermined parameters, which must be entered prior to execution. Table 3 shows the pre-set magnitudes of PSO parameters, including inertia weight and its corresponding minimum amount, inertia weight reduction factor, personal and swarm learning rate, maximum velocity reduction factor, critical probability, and termination condition. The PSO parameter selection is based on the recommended values and ranges in the literature (Shi & Eberhart 1998; Rini *et al.* 2011). The learning rate of the model was then analyzed via the testing dataset (the last fold). This process was repeated ten times in total and, finally, the average of the results was compared with the measurements and the statistical indices were provided.

Parameter | w | w _{min} | C_{1} | C_{2} | Critical probability (P_{cr}) | Termination condition (ε) | |||
---|---|---|---|---|---|---|---|---|---|

Value | 0.9 | 0.9 | 0.4 | 1 | 0.4 | 2 | 2 | 0.1 | 0.01 |

Parameter | w | w _{min} | C_{1} | C_{2} | Critical probability (P_{cr}) | Termination condition (ε) | |||
---|---|---|---|---|---|---|---|---|---|

Value | 0.9 | 0.9 | 0.4 | 1 | 0.4 | 2 | 2 | 0.1 | 0.01 |

### Performance metrics

*E*), correlation coefficient (

*R*),

*RMSE*, scatter index (

*SI*), and relative error (

*RE*).

*E*is a well-known index in water resources modeling and it has a range of . An efficiency of one pertains to an excellent match between the predictions and observations, while a negative value corresponds to a model with a predictive power of worse than the observed mean.

*R*is a measure of linear correlation between the predictions and observations.

*RE*represents the absolute bias.

*RMSE*shows the model's error of estimation and when it is normalized by the mean measured value is referred to as the scatter index (

*SI*): where and

*y*, respectively, denote the estimated and measured sediment concentration and

*n*is the number of data. These metrics can generally give the analyst an insight into the model performance and robustness.

## RESULTS AND DISCUSSION

Three models were basically developed in this research. In the first model, the dimensionless variables in Equation (5) were normalized and then entered in the hybrid estimator. The associated results appear in Figure 2 representing the error histogram of data. The abscissa axis in Figure 2 shows the relative error (*RE*) and the ordinate axis depicts the data frequency. It is apparent from the figure that the simulation error proved to be high. However, it was still more satisfactory than traditional methods (see Table 4) since the majority of data lie in the range of %−200 and %200 and this is convincing for the phenomenon of sediment transport. The reason why there still occurs great errors in a few number of data points is the existence of considerable diversity in sediment concentration (ranging from 0 to approximately 40,000 mg/liter). Due to this, LSSVR attributes more error to the samples having lower sediment concentration as it is focusing upon the minimization of the sum of squared errors rather than relative errors in the optimization problem.

R | E | RMSE | RE | SI | RE <0.4 | ||
---|---|---|---|---|---|---|---|

1st type of PSO-LSSVR | All data | 0.9605 | 0.9217 | 940.38 | 23.30% | 36.44% | 62.46% |

2nd type of PSO-LSSVR | All data | 0.9662 | 0.9321 | 876.07 | 21.62% | 33.95% | 64.64% |

3rd type of PSO-LSSVR | All data | 0.9558 | 0.9100 | 1,008.5 | 21.80% | 39.08% | 71.50% |

M5P | 0.9059 | 0.8182 | 1,428.1 | 34.07% | 55.34% | 53.20% | |

Ackers & White (1973) | 0.744 | 0.2342 | 2,941.4 | 52.48% | 113.98% | 38.63% | |

Engelund & Hansen (1972) | 0.676 | −73.232 | 28,958 | 515.99% | 1122.1% | 2.96% | |

Graf (1971) | 0.397 | −0.3041 | 3,838.2 | 89.35% | 148.73% | 8.41% | |

Yang (1972) | 0.869 | 0.7157 | 1,792.1 | 41.34% | 69.44% | 43.93% |

R | E | RMSE | RE | SI | RE <0.4 | ||
---|---|---|---|---|---|---|---|

1st type of PSO-LSSVR | All data | 0.9605 | 0.9217 | 940.38 | 23.30% | 36.44% | 62.46% |

2nd type of PSO-LSSVR | All data | 0.9662 | 0.9321 | 876.07 | 21.62% | 33.95% | 64.64% |

3rd type of PSO-LSSVR | All data | 0.9558 | 0.9100 | 1,008.5 | 21.80% | 39.08% | 71.50% |

M5P | 0.9059 | 0.8182 | 1,428.1 | 34.07% | 55.34% | 53.20% | |

Ackers & White (1973) | 0.744 | 0.2342 | 2,941.4 | 52.48% | 113.98% | 38.63% | |

Engelund & Hansen (1972) | 0.676 | −73.232 | 28,958 | 515.99% | 1122.1% | 2.96% | |

Graf (1971) | 0.397 | −0.3041 | 3,838.2 | 89.35% | 148.73% | 8.41% | |

Yang (1972) | 0.869 | 0.7157 | 1,792.1 | 41.34% | 69.44% | 43.93% |

At the first stage of simulation, it is attempted to portray the functionality of the hybrid PSO-LSSVR model versus M5P. The scatter diagram in Figure 3 compares the proposed model with the M5P method and depicts the simulated bed material load concentration versus the measured concentrations. As the figure illustrates, the PSO-LSSVR model fits the data to a relatively high degree and confirms its superiority over the M5P model (see the statistical indices in Table 4). Therefore, we refrained from further simulations by M5P for the next models. It is noteworthy that a handful of data with low concentrations were predicted as zero by PSO-LSSVR and M5P. Hence, they do not appear in the logarithmic scale.

*d*

_{50}> 0.15 mm and

*Fr*

*<*0.8. On the other hand, Yang's unit stream power approach has the best performance among all the traditional approaches, enjoying better correlation with the observed data. The PSO-LSSVR model demonstrates greater density around the bisector of the first quadrant of the Cartesian system in Figure 4 which means it estimates bed material load transport rate more accurately than traditional methods. However, Yang's unit stream power method is comparable with the PSO-LSSVR model in terms of estimating concentrations less than 50 ppm. Equation (27) shows a single execution of the model without utilizing cross-validation technique. In this equation,

*X*denotes the vector of decision variables for the training and validation datasets.

_{i}*λ*is the Lagrangian coefficient belonging to the

_{i}*i*-th sample.

*X*is the vector of decision variables for the test sample. ,

*n*, and

*b*are the RBF parameter, the total number of training datasets, and bias term, respectively. It is meaningless to present the optimized values for these parameters because one may have a different dataset and use a specific shuffling procedure on their data, thus producing values specific to their model.

*RMSE*and correlation coefficient/NSE cannot solely explain the intricacies of the model. In other words, Nash–Sutcliffe, correlation coefficient, and root mean squared errors are all biased towards the high values of sediment concentration when they are considered as the objective function for tuning LSSVR parameters. As a result, there were still noticeable oscillations in the estimation of the samples with low concentration values. By exploring the theoretical background of sediment hydraulics, and reviewing the related literature, it is observed that most equations have been presented logarithmically. Yang's equation is a clear-cut example to illustrate the point. By doing so, the difference between the maximum and minimum sediment concentrations decreases and the model can emphasize low concentrations as well. Thus, the predictor and response variables were logarithmically scaled and ten further simulations were applied. The outcomes were converted to their initial scales at the end of the simulation process. Finally, the results were compared with that of the secondary model in Figure 5. The resulting function of a trained PSO-LSSVR following the procedures of the third model can be seen in Equation (28). Note that the logarithms are at base 10: As shown in Table 4, all three simulated models performed far better than traditional approaches; this is evident in the high correlation between the predicted and observed values. Moreover, the resulting

*RMSE*,

*SI*, and

*RE*for the combined model are less than those for the traditional methods. The statistical indices attest to the superiority of the PSO-LSSVR method over other approaches introduced in this study, thus making this model a viable alternative for calculation of sediment loads.

As mentioned before, adding the four dimensionless variables of Yang's method to the PSO-LSSVR model partially improved the results of the model. When the dependent and independent variables were logarithmically inserted in the model, *RMSE* and *SI* were slightly increased and the correlation and *NSE* approximately remained constant. By contrast, the percentage of data whose single *REs* were lower than 0.4 increased, which led to a decline in error dispersion and better distribution of errors in the histogram. Therefore, the *RE* of each sample must be taken into consideration in conjunction with the statistical performance metrics and the conclusion should be drawn by observing the histogram of errors. In the second model, the estimation accuracy of almost all the samples with high concentration values was satisfactory, but samples containing low concentrations were highly fault tolerant. In the final model, the histogram of relative errors was compressed such that 71.5% of the dataset had a *RE* of less than 0.4, whereas 64.64% of the data fell within this range in the second model. According to Figure 6, a large percentage of samples with great errors has interestingly declined and resulted in a denser error distribution in the final model (see Table 4 and Figures 6 and 7). Figures 8 and 9 present the error histograms of Engelund & Hansen (1972) and Yang (1972) methods. It can be observed that the errors' magnitude was dramatically greater than the final version of PSO-LSSVR. By looking at the figures, it is seen that Engelund and Hansen's method yielded outlier predictions during the estimation of bed sediment concentrations. Yang's method can be a relatively good way to estimate the concentration of bed sediments although it is not comparable with the hybrid model of PSO-LSSVR.

## CONCLUSION

This study presented an alternative LSSVR method for estimating sediment transport rate in channels. This method was coupled with PSO algorithm to calibrate the regularization and kernel parameters in LSSVR. The data were converted to a dimensionless form and then normalized. The cross-validation technique was applied to the data and a primary PSO-LSSVR model was created. It was found that the hybrid model had the best performance compared with those obtained from M5P and traditional methods. However, the Yang approach was still relatively superior for samples with low concentrations. As a result, its dimensionless variables were added to the hybrid model leading to an improvement in the model training. Due to the wide range of concentration, all the data were converted into a logarithmic scale. When trained, it was found that a higher percentage of data had a *RE* of less than 0.4 and estimation of samples with low concentration became less volatile resulting in a denser error histogram. This study demonstrates a rigorous methodology for bed material load estimation and highlights key performance issues that may influence modeling in sediment transport. Overall, PSO-LSSVR, developed as part of this study, shows great potential as an efficient, reliable, and easy-to-use machine learning approach for sediment transport rate estimation.

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*Geological Survey Professional Paper*No. 422-I

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