Artificial neural network and regression models for flow velocity at sediment incipient deposition

Accepted Manuscript Research papers Artificial neural network and regression models for flow velocity at sediment incipient deposition Mir-Jafar-Sadegh Safari, Hafzullah Aksoy, Mirali Mohammadi PII: DOI: Reference:

S0022-1694(16)30536-4 http://dx.doi.org/10.1016/j.jhydrol.2016.08.045 HYDROL 21480

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

27 January 2016 31 July 2016 25 August 2016

Please cite this article as: Safari, M-J., Aksoy, H., Mohammadi, M., Artificial neural network and regression models for flow velocity at sediment incipient deposition, Journal of Hydrology (2016), doi: http://dx.doi.org/10.1016/ j.jhydrol.2016.08.045

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Artificial neural network and regression models for flow velocity at sediment incipient deposition Mir-Jafar-Sadegh Safari1, Hafzullah Aksoy*2 & Mirali Mohammadi3 1

Istanbul Technical University, Department of Civil Engineering, Istanbul, Turkey. e-mail: [email protected] Istanbul Technical University, Department of Civil Engineering, Istanbul, Turkey. e-mail: [email protected] 3 Urmia University, Department of Civil Engineering, Urmia, Iran. e-mail: [email protected] *Corresponding author 2

Abstract: A set of experiments for the determination of flow characteristics at sediment incipient deposition has been carried out in a trapezoidal cross-section channel. Using experimental data, a regression model is developed for computing velocity of flow in a trapezoidal cross-section channel at the incipient deposition condition and is presented together with already available regression models of rectangular, circular, and U-shape channels. A generalized regression model is also provided by combining the available data of any cross-section. For comparison of the models, a powerful tool, the artificial neural network (ANN) is used for modelling incipient deposition of sediment in rigid boundary channels. Three different ANN techniques, namely, the feed-forward back propagation (FFBP), generalized regression (GR), and radial basis function (RBF), are applied using six input variables; flow discharge, flow depth, channel bed slope, hydraulic radius, relative specific mass of sediment and median size of sediment particles; all taken from laboratory experiments. Hydrodynamic forces acting on sediment particles in the flow are considered in the regression models indirectly for deriving particle Froude number and relative particle size, both being dimensionless. The accuracy of the models is studied by the root mean square error (RMSE), the mean absolute percentage error (MAPE), the discrepancy ratio (Dr) and the concordance coefficient (CC). Evaluation of the models finds ANN models superior and some regression models with an acceptable performance. Therefore, it is concluded that appropriately constructed ANN and regression models can be developed and used for the rigid boundary channel design. KEY WORDS: artificial neural network (ANN); critical velocity; incipient deposition; open channel; regression model; sediment transport 1

1. INTRODUCTION

Sediment transport is an important issue due to its tremendous impact on the design of sediment carrying channels such as urban drainage, sewer, and irrigation systems as well as power plant intakes; rigid boundary channels in general. Deposition of sediment in such channels decreases the channel cross-section area and changes the hydraulic resistance, velocity and wall shear stress distribution of the channel (Ackers et al., 1996; Ota and Nalluri, 2003; De Sutter et al., 2003). Consequently, it substantially impacts on the carrying capacity of a channel, and therefore additional costs arise for cleaning the channel under the effect of sediment deposition. The subject of sediment transport in rigid boundary channels received interest also because of its environmental effects such as the contamination of sediment with toxic substances due to deposition of sediment in urban drainage channels and sewer systems (Ashley et al., 1992). As an urban hydrology practice, the drainage systems are designed to remove the runoff from urbanized areas to prevent flooding. However, in order to design a sustainable drainage system with reliable efficiency, the sediment transport process should be taken into account (Mays, 2001; Butler and Davies, 2004). Therefore, sediment transport modelling is a challenging task in the fields of urban hydrology and hydraulic structures.

Linked to the sediment transport, sediment threshold is a concept that can be evaluated by incipient motion and incipient deposition (ASCE Task Force Committee, 1966; Loveless, 1992; Safari et al., 2015). Incipient motion is the flow condition that is just adequate to initiate sediment particles to move. Incipient deposition is defined as the flow condition when sediment particles in suspension begin to deposition in which sediment particles either continue their motion in the channel as bed load or they are accumulated at the channel bed partially without making a deposited bed (Safari et al., 2015).

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Two common approaches, critical velocity and critical shear stress, are applied for evaluating the sediment threshold condition. As examples to mention, incipient motion of sediment in rigid boundary channels was investigated by Novak and Nalluri (1984), El-Zaemey (1991), Ab Ghani et al. (1999), Mohammadi (2005), Bong et al. (2013) and Safari et al. (2013a) who used the critical velocity or critical shear stress approaches for developing regression models by considering different sediment particle sizes in various channels with particular cross-sections. Safari et al. (2014, 2015) extended the experimental work of Loveless (1992) to develop regression models for computing the critical flow velocity at the incipient deposition condition, and found higher critical flow velocities for the incipient deposition than the incipient motion under the same flow conditions when non-cohesive sediment is concerned.

Regression models are widely used because of their relatively simple structure and their ability to work with limited input data, however they are not able to represent the channel cross-section and sediment particle size that influence the critical velocity of flow. Such models succeeded in defining the essential factors of the problem in one hand. To get a discrete formula, on the other hand, some important variables were disregarded for simplicity, dummy constants were added for consistency, and some boundary conditions were considered for applicability. Consequently, it is questionable whether any formula can be applied successfully or not to the diversity of channel cross-sections and sediment sizes.

Nowadays, the artificial neural network (ANN) approach has been applied to many branches of engineering. The ANN approach is becoming a strong tool for providing hydraulic, hydrology and environmental engineers with sufficient details for the design purposes and management practices (Nagy et al., 2002; Tsai et al., 2015; Chang and Tsai, 2016; Chang et al., 2016). ANNs have many distinct advantages. For example, they can approximate any arbitrary continuous function and simulate a nonlinear system without a priori assumption of processes involved (ASCE Task 3

Committee, 2000a, b). Although ANNs have been widely used for modelling sediment transport in alluvial channels (Nagy et al., 2002; Zhu et al., 2007; Wang et al., 2008; Yang et al., 2009; Kakaei Lafdani et al., 2013; Atieh et al., 2015; Thompson et al., 2016), there is no study on incipient deposition of sediment using ANN techniques other than Safari et al. (2013b) who did a preliminary analysis. Also Ab Ghani and Azamathulla (2010), Azamathulla et al. (2012) and Ebtehaj and Bonakdari (2013, 2014) can be mentioned as examples for use of artificial intelligence techniques, such as gene-expression programming and adaptive neural fuzzy inference system in the sediment transport problem without considering incipient deposition in sewer systems.

This study is an attempt for providing incipient deposition model for trapezoidal cross-section channel. Moreover, no study exists in the literature yet to model the incipient deposition using ANN techniques. Experimentally investigating sediment incipient deposition in rigid boundary channels this study aims to estimate velocity of flow at the incipient deposition condition by comparing different ANN and regression models. The architecture of ANN models, decided after several trials, are trained with experimental data composed of fluid, flow and sediment characteristics while dimensionless variables based on the hydrodynamics of flow at the sediment incipient deposition is considered in the regression models.

2. EXPERIMENTAL DATA

Data compiled from the literature and data from an experimental study performed within this study are used jointly. Following sub-sections explain details of the data.

2.1. Experimental data from the literature Sediment transport in rigid boundary channels was studied experimentally by Loveless (1992) at the incipient deposition condition where sediment is about to deposit. Experiments were performed at University of London, UK, in circular, rectangular and U-shape cross-section channels. It is seen 4

from the experimental characteristics given in Table 1 that the rectangular and circular channels have a cross-section area of approximately 60 cm2 each while the cross-section of the U-shape channel is more than one order of magnitude bigger. Although the cross-sections are not comparable in terms of their size, granular sand with the same size is used as sediment in the three channels. Experiments were designed by Loveless (1992) as follows:

At the beginning of each experiment, channel slope and flow discharge were set to provide the nondeposition flow condition. Next, the slope was gradually reduced until local deposition of sediment began to occur. This procedure was applied to identify the incipient deposition condition which occurred in 77 experiments out of 208.

2.2. Experimental data of this study An experimental apparatus was constructed in Hydraulics Laboratory of Istanbul Technical University, Turkey to conduct experiments in this study (Aksoy and Safari, 2014). The apparatus is composed of a support structure, channel, fluctuation prevention tank, sediment feeder, tailgate and sand trap at the downstream section of the channel (Figure 1). Mounted on the support structure the channel is 12 m-long and has a trapezoidal cross-section with 30 cm-bottom width and 60 degreeouter angle for the side walls each 300 mm-long. Water is supplied from the constant head-storage of the laboratory through a pipe into the fluctuation prevention tank before it enters the channel. Flow discharge into the channel was adjusted by a valve on the pipe. Sediment was poured into the channel using the vibrant sediment feeder. Uniform flow was established in the channel by adjusting the tailgate at the channel downstream outlet. The utilized sediment was accumulated in the sand trap tank at the end of the channel; water was cleaned from the sediment before it returned back to the laboratory water storage tank.

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In order to identify incipient deposition, 132 experiments were carried out for certain sizes of sand and channel bed slopes. The bed slope was adjusted using a hydraulic jack placed under the support structure, determined using a mapping camera, and checked by a digital inclinometer. In the experiments, channel bed was given slopes changing from 0.1% to 1%. For each experiment, different flow characteristics such as flow depth, discharge and sediment concentration were measured. The flow discharge was measured by an ultrasonic flow meter. Volumetric concentration of sediment poured into the channel was measured using a glass cylinder. Flow depth was measured at several points along the 4 m-long observation section located 4 m from the upstream and 4 m from the downstream sections of the channel (Figure 1). Sediment motion was observed optically. Observations and measurements were done in uniform flow conditions adjusted with the tailgate.

Each experiment started with a discharge high enough to prevent the deposition of sediment within flow; i.e., the non-deposition mode was observed. Incipient deposition was achieved by reducing flow velocity, hence discharge. Gradual decrease in the discharge allows flow to switch from the non-deposition mode to the incipient deposition. Flow condition in which sediment particles begin to deposit is the incipient deposition. As used by Loveless (1992), in the experiments, incipient deposition was identified as the sediment transport mode in which sediment particles were clustered visibly in certain areas at the bottom of the channel. The incipient deposition condition was observed in 15 experiments out of 132.

The fine and coarse non-cohesive uniform sands with median diameter of d50 = 0.15 mm and d50 = 0.83 mm, and specific mass of 2600 kg/m3 and 2680 kg/m3, respectively, were used in the experiments. For a sediment be considered uniform, (1)

6

should be satisfied in which and

is geometric standard deviation of the particle size distribution,

are the 84 and 16% finer sediment sizes, respectively. The geometric standard deviation of

fine and coarse sands was computed as 1.36 and 1.21, respectively, demonstrating that the sediment is uniform.

Figure 1

Table 1

3. HYDRODYNAMICS OF INCIPIENT DEPOSITION AND REGRESSION MODELS

3.1. Hydrodynamics of incipient deposition Sediment motion is linked to the concepts of incipient motion and incipient deposition (ASCE Task Force Committee, 1966; Loveless, 1992; Safari et al., 2014, 2015), both computed in terms of either critical flow velocity or critical bed shear stress. In this study, the critical velocity approach is considered by analysing hydrodynamic forces acting on sediment particles which are the drag force, the lift force, the buoyed weight of the sediment particles and the resisting force against motion. The drag force

in the flow direction is given by (2)

where

is the drag coefficient,

the fluid specific mass,

near bed flow velocity. The lift force

the sediment median size, and

the

acts vertically upward. It can be expressed as (3)

in which

is the lift coefficient. The buoyed weight of sediment

is written as (4)

7

where

is the specific mass of the sediment and

the gravity acceleration. The resistance force

in terms of normal force components can be expressed as (5) in which

is the Coulomb friction coefficient.

The drag force becomes equal to the resistance force

under the threshold condition of

sedimentation for which (6) is obtained when Equations (2-5) are considered. In Equation (6), the sediment to the fluid (

is the relative specific mass of

). The left hand side of Equation (6) is known as the particle

Froude number, while the right hand side depends on the channel and sediment particle geometry.

3.2. Development of regression models The above hydrodynamics was taken into account in developing regression models in the literature. For instance, Novak and Nalluri (1984) used the dimensionless relative particle size (

where

is the hydraulic radius) as the geometry parameter of the channel and the sediment particle, and considered the critical flow mean velocity of the near bed flow velocity (

needed for the sediment threshold condition instead

) to simplify Equation (6) as (7)

in which

and

, the coefficient and the exponent, are constants to be determined based on

experimental data. Using this methodology Safari et al. (2014) studied the incipient deposition of sediment particles in rigid boundary channels for different cross-sections, and proposed the incipient deposition equations as

(8) 8

(9)

(10)

for rectangular, circular and U-shape cross-section channels, respectively. The trapezoidal crosssection channel was additionally studied as an experimental work in this study to obtain (11) for computing the incipient deposition critical velocity in the trapezoidal cross-section channel. As another output of this study, Equations (8-11) were generalized to any cross-section as (12) by simply combining data available in the literature for the rectangular, circular, U-shape channels with data for the trapezoidal channels in this study. Regression models (Equations 8-12) were all developed using the SPSS statistical package (Landau and Everitt, 2004).

3.3. Comparison of regression models Equations (8-12) based on experimental data are compared in Figure 2 from which it is seen that the rectangular, circular and U-shape channel data remained above the trapezoidal channel. It indicates, in the trapezoidal channel, that sediment particles begin to deposit at a lower flow mean velocity in comparison with the rectangular, circular and U-shape channels for indication can be extended to the range of

. This

when the trapezoidal channel is

compared to the U-shape channel. Models give similar results regardless of the channel crosssection as the relative particle size,

, increases. It should be noted that the smallest particle size

used in the rectangular, circular and U-shape channels had the median size of 0.45 mm (see Table 1). In the experiments of the trapezoidal channel, however, fine sand with median size of 0.15 mm was used as the smallest particle size. Sediment particle characteristics such as the particle size and 9

the specific mass are essential factors which affect the flow mean velocity in the incipient deposition condition. Higher flow velocity is required for the incipient deposition of sediment with increasing particle size when hydrodynamic forces acting on the sediment particles are considered. Performance of the models in Figure 2 could possibly be affected by the channel cross-section each with its own specific geometry, boundary shear stress, secondary flows, etc. These are all important points not considered in this study but needed to be discussed through detailed experiments.

Figure 2

4. PERFORMANCE CRITERIA Evaluation of the model accuracy provides the most important means for the credibility of the established models. To this extent, the performance of the models developed in this study are evaluated in terms of four different criteria; the root mean square error ( percentage error ( these,

), the discrepancy ratio (

), the mean absolute

) and the concordance coefficient (

). Among

indicates the difference between the calculated and measured flow velocities by (13)

in which

and

, respectively, are the measured and calculated critical velocities of flow in the

incipient deposition condition, and

the number of data.

is calculated through a term-by-

term error comparison of the measured and calculated outputs. It gives the accuracy as in percentage as defined in Wang et al. (2009) by .

(14)

In order to check the model robustness, the distribution of

calculated as (15)

is used.

is clustered in two ranges of 0.9 <

< 1.1 and 0.75 <

< 1.25 to demonstrate the

soundness of the models with 10% and 25% unreliability, respectively, meaning that the model 10

calculates outputs with less than 10% and 25% errors. The model underestimates when while it overestimates for

> 1.

< 1

is the concordance between the measured and calculated

outputs and has a range from -1 to 1, with a perfect agreement at 1. It is computed by (16) in which

is the correlation coefficient,

and

the standard deviation and

and

the

average of the measured and calculated critical flow velocities, respectively.

5. ANN TECHNIQUES

Selection of the input variables is an essential issue in the ANN modelling. Critical flow velocity ( ), water discharge ( ), flow depth ( ), longitudinal slope ( ), particle median diameter ( ), hydraulic radius ( ), relative specific mass of sediment to the fluid ( ), kinematic viscosity ( ), and acceleration due to gravity ( ) are important variables in sediment transport when the critical velocity approach is considered for the determination of the threshold condition. Among the variables, kinematic viscosity ( ) and acceleration gravity ( ) are constant; they are, therefore, eliminated to achieve the final expression for the critical velocity of flow as .

(17)

The independent variables in Equation (17) describe the inputs of the network while the dependent variable corresponds to the output.

Application of the ANNs consisted of three steps of training, validation and testing; the first two steps are used in the establishing ANNs. The network was trained with the training data set. The training step includes determination of the exact hidden relationship between the independent and dependent variables; i.e., inputs and outputs. The accuracy of the results obtained from the network are evaluated by comparing their outputs with the validation data set. In other words, the validation data are used to check the generalization capability of the models until the best model is found by 11

changing the ANN structure. Finally, a test data set is used to verify the model and to estimate its expected performance for an unknown set of data. Among the data of 92 experiments [15 experiments performed in this study + 77 experiments by Loveless (1992)], 70 randomly selected experiments were used for training the models, 11 for validation and 11 for testing. In order to ensure that each variable was treated equally in the model, data were rescaled to the interval [0, 1] (Dawson and Wilby, 2001). MATLAB codes were used for the application (Araghinejad, 2014).

5.1. Feed-Forward Back Propagation Neural Network The most common ANN approach used in engineering problems is the feed forward back propagation (FFBP) neural network. The FFBP neural network has one or more hidden layers whose computation nodes are correspondingly called hidden neurons. The function of the hidden neurons intervenes between the external inputs and the network output in a useful manner. With each additional hidden layer, the network is able to extract higher order statistics although one hidden layer is found precise enough for water resources engineering problems (Hornik et al., 1989; Zhu et al., 2007). The source nodes in the input layers of the network supply respective elements of the activation pattern, which constitutes the input signals applied to the neurons in the second layer. The output signals of the second layer are used as inputs to the third layer, and so on for the rest of the network (Cigizoglu, 2008). The external input information at the input nodes is propagated forward to compute the output information signal at the output layer using the LevenbergMarquardt (LM) optimization technique. In this study, a tangent sigmoid function was used as the neuron transfer function. Also, linear transfer functions at the output layer nodes were employed. The structure and number of neurons in the hidden layer of the FFBP neural network model affect its performance.

In order to find the optimal structure of the FFBP and to fix the number of neurons in the hidden layer, a trial-and-error procedure was carried out. Different networks were examined with number 12

of neurons between 1-20. Figure 3 shows the performance of the FFBP neural network with different number of neurons in the hidden layer based on

and

of the validation data

set. The optimum number of the neurons in the hidden layer was found to be three or four; with and

of 0.029 and 4.69%, for three neurons, and 0.028 and 4.36%, for four neurons,

respectively. Four neurons in the hidden layer give slightly better performance than three neurons; therefore, four neurons were chosen as the number of neurons in the hidden layer of the FFBP ANN. Figure 4 compares the critical flow velocity measured in the experiments and calculated by the FFBP neural network with four neurons in the hidden layer on the validation data. It is seen that all validation data are close to the bisector line. Consequently, the FFBP network (6, 4, 1) composed of one input layer with 6 neurons and one hidden layer with 4 neurons provided the best performance criteria, i.e. the lowest

and

were obtained on the validation data.

Figure 3

Figure 4

5.2. Generalized Regression Neural Network The generalized regression (GR) neural network approximates any arbitrary function between the input and output vectors by estimating a function directly from the training data without using an iterative training procedure (Specht, 1991; Tsoukalas and Uhrig, 1997). The GR neural network consists of four layers: input, pattern, summation and output. The number of neurons in the input layer is equal to the total number of input variables. Each layer is fully connected to the adjacent layers by a set of weights between the neurons. The GR neural network was used in this study to regress the critical flow velocity ( ) on the input variables listed in Equation (17) as a non-linear function captured by the training data. The pattern layer has one neuron for each training pattern. Each neuron in the pattern layer is connected to the neurons in the summation layers in which the 13

sum of the weighted outputs of the pattern layer is computed. Once the weights are set, the GR neural network calculates outputs (Besaw et al., 2010).

Different spreads in the range of 0.001-4 were examined in order to find the best value that gives the minimum

and

spreads in terms of

. Figure 5 shows the performance of the GR model with different and

on the validation data. It is seen from Figure 5 that the GR

model has an acceptable performance in the spreads between 0.001-0.09 for the incipient deposition, the optimum spread being 0.05 with

and

of 0.059 and 6.85%,

respectively. Figure 6 shows how the critical flow velocity measured in the experiments is close to the corresponding value calculated by the GR neural network with spread of 0.05 on the validation data.

Figure 5

Figure 6

5.3. Radial Basis Function Neural Network The radial basis function (RBF) introduced by Lowe and Broomhead (1988) consists of input, hidden and output layers. The basis functions produce a significant non-zero response to the input stimulus only when the input falls within a small localized region of the input space. The transformation has a nonlinear type from the input layer to the hidden layer while it is linear between the hidden and output layers. There are different types of functions, such as the Cauchy, Gaussian, multiquadratic and inverse multiquadratic, used for the transformation. The spread constant and the number of hidden layers were decided with a trial-and-error approach before the RBF simulation starts.

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In this study, the maximum number of neurons in the hidden layer was set as 10, the optimum number of neurons in the hidden layer for all spreads was found two. The Gaussian function was used for the transformation because of its popularity (Caiti and Parisini, 1994; Wang et al., 2010; Hadzima-Nyarko et al., 2014).

The spreads between 0.1-5 were examined in order to find the best spread. The performance of the RBF neural network model on the validation data with different spreads is shown in Figure 7. Spreads in the range of 0.7-1.1 were found good; 0.9 was selected as the best spread with and

of 0.085 and 12.05%, respectively. Figure 8 compares the measured and calculated

critical flow velocities with the selected spread on the validation data.

It should be mentioned that optimum spread parameters in the GR and the RBF neural networks differ from each other although both use the Gaussian function in their structures. Also, the GR neural network uses the same functions as the RBF neural network in the input layer; however, they use different output functions. The spread parameter is different in the output functions of both networks. Therefore, the best spreads vary in each case.

Figure 7

Figure 8

6. DISCUSSION Comparison was made between the ANN and regression models based on the test data by examining the goodness-of-fit with the performance criteria; models (Tables 2 and 3). The mean values of

,

,

and

of the

for the FFBP, GR and RBF ANN models are 0.98,

1.01, 0.93, respectively (Table 2), indicating that, the three ANN models show an almost similar 15

performance under the same data requirement or the input combination. However, when the percentage of the data in the range of 0.9 <

< 1.1 is considered, it is seen that the FFBP gives a

more accurate prediction in comparison with other regression and ANN models. When a wider range (0.75 <

< 1.25) is taken into account, ANN models become more superior to the

regression models. Equations (8-9) among the regression models give better results. Comparing results in Table 3 based on

and

demonstrates that the FFBP has the best performance

in predicting the critical flow velocity at the incipient deposition condition. Another comparison by the concordance coefficient (

) indicates that the FFBP is again superior to other models. The

performances of the GR and RBF are also good with low

and

, and high

. Also

scatter plots of the critical flow velocities measured in the experiments and calculated by the models at the incipient deposition condition are given in Figure 9. A quick look at the scatter diagrams shows the prominent similarity between the measured and calculated velocities in ANN models in general and the FFBP ANN in particular. As a result, the FFBP among the ANN models outperforms all others.

Table 2

Table 3

Figure 9

Performance criteria in Tables 2-3 and scatter diagrams in Figure 9 show that regression models have different performances. Each regression model has been developed for a specific channel cross-section that is considered the main reason for the difference in their performances. Therefore, the channel cross-section is one of the main parameters to be considered in the process of sediment transport in rigid boundary channels. Regression models use the relative particle size ( 16

) as the

input variable changing with sediment, channel and flow characteristics. In other words, regression models consider not only channel characteristics but also sediment size and hydraulic features.

This study indicates that the ANN techniques are powerful tools to model the critical flow velocity at the incipient deposition. They can make high-performance predictions. The best performance was obtained by the ANN models in terms of different evaluation criteria at the testing phases of the models. On the other hand, regression models have an acceptable ability for the prediction of the critical flow velocity. In regression models, the sediment incipient deposition variables are lumped into two dimensionless parameters, the critical particle Froude number and the relative particle size. In the ANN models, the sediment incipient deposition variables are considered individually. This gives an opportunity to ANN models to have higher number of variables than regression models. At the same time, it can be considered an advantage given to the ANN models to better perform over regression models. Through developing ANN models, the best models were obtained by calibration to adjust parameters in the models. However, regression models are best-fit relationships adopted on experimental data composed of two dimensionless parameters, the critical particle Froude number and the relative particle size. This seems to be a reason why ANNs outperform over regression models.

Indeed, over-fitting is the aptitude of ANN models to fit training events too exactly due to large number of weights with the use of higher number of variables in the input layer with which ANN models of irrelevant inputs may behave poorly. This reminds the concept of curse of dimensionality (Bishop, 1995; Bowden et al., 2005; Giustolisi and Laucelli, 2005). As the number of input variables increases, additional weights become necessary at the expense of model parsimony. Therefore, a restricted number of weights (model parameters) are recommended in order to avoid over-fitting and the curse of dimensionality (Bellman, 1957; Haykin, 1999; Carcano et al., 2008) as there could always be a possibility that the network design would be encountered with these 17

problems (Bishop, 1995; Parchami-Araghi et al., 2013). However, the ANN models in this study were not expected to have such a problem as the input variables used in developing ANN models were decided based on the physics of sediment transport. On the other hand, it is worthy to mention that the FFBP model is much less susceptible to the curse of dimensionality (Bishop, 1995).

A further general discussion can be made due to the fact that sediment transport in the flow is a complicated process which has a stochastic nature because of the turbulence characteristics of the flow. Therefore, an analytical model that can perform well in any case is very difficult to develop. It should also be mentioned that most of the sediment transport models existing in the literature ignore some of the effective variables although they consider the physics behind the sediment transport processes. However, it is seen in this study that a well-established ANN model can be used as a tool for constructing a relationship for the flow velocity at the incipient motion mode of sediment particles within flow.

Finally, the importance of the issue should be mentioned. Sedimentation and storm water management are considered crucial in urban hydrology. Although hydrological processes in urban areas occur at smaller scales compared to the rural regions, they are both similar (Mays, 2001). The rigid boundary drainage systems are designed to remove the runoff from urban areas as fast as possible for flood prevention. As it is well discussed by Delleur (2003), another main objective of a sustainable urban drainage system is the protection of safety and health of communities by removing the sediment generated by the catchment. This goal is achieved by designing urban drainage systems from the hydraulic engineering viewpoints as it has been considered in this study.

7. CONCLUSIONS In this study, a series of ANN and regression models are developed and applied to the incipient deposition of sediment within flow in rigid boundary channels. The models are based on the critical 18

velocity approach. A cross-section-specific regression model is proposed for the trapezoidal channel. A regression model generalized over different cross-sections is also suggested utilizing all available incipient deposition experimental data. The ANN models use hidden relationships between the critical flow velocity and the characteristics of flow and sediment; namely flow mean velocity, channel bed slope, flow depth, discharge, hydraulic radius, sediment relative specific mass and median particle diameter. Regression models are based on a relation between the particle Froude number and relative particle size, both being dimensionless and providing the model with flow, sediment and channel characteristics. Among the developed ANN models, the FFBP is found generally superior to other ANN and all regression models; however regression models may compete in some cases. The better performance of the ANN models can be related to the higher number of individually taken input variables that are lumped into the critical particle Froude number and the relative particle size in regression models. As a conclusion, it is found that appropriately constructed ANN and regression models can be successfully applied for the estimation of flow velocity at the incipient deposition condition and be used for such practical problems of urban hydrology as sedimentation, storm water management and drainage system design.

ACKNOWLEDGEMENTS Experimental data in this study were taken from the research project no. 113M062, entitled “Incipient deposition of sediment particles in rigid boundary channels”, under coordination of Prof. H. Aksoy, funded by the Scientific and Technological Research Council of Turkey (TUBITAK). Authors used additional data available in the literature for which they wish to express their sincere gratitude and appreciation to Dr. J.H. Loveless. Continuous, constructive and courageous comments from the reviewers and the editor improved the content and presentation of this paper substantially. These efforts are all greatly appreciated. REFERENCES

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Table captions Table 1. Channel and sediment characteristics of experimental data Table 2. Performance of the ANN and regression models for the critical flow velocity based on Table 3. Performance of the ANN and regression models for the critical flow velocity based on ,

and

Figure captions Figure 1. Experimental apparatus and channel cross-section Figure 2. Comparison of the regression models (Lines show best-fit regression models of each set of data. Solid line (Eq. 12) is the regression model generalized to the four cross-section data) Figure 3. Performance of the FFBP ANN model with different number of neurons in the hidden layer on validation data Figure 4. Comparison of velocities measured in the experiments ( ANN model (

) and calculated by the FFBP

) on validation data

Figure 5. Performance of the GR ANN model with different spreads on validation data Figure 6. Comparison of velocities measured in the experiments ( GR ANN model (

) and calculated by the

) on validation data

Figure 7. Performance of the RBF ANN model with different spreads on validation data Figure 8. Comparison of velocities measured in the experiments ( ANN model (

) on validation data

Figure 9. Comparison of velocities measured in the experiments ( (

) and calculated by the RBF

) on testing data

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) and calculated by the models

Table 1. Channel and sediment characteristics of experimental data Data source Channel characteristics Sediment (mm) Cross-section Geometry width = 5.9 cm height = 10 cm length = 7.2 m 0.45 Rectangular 1.30 width = 10 cm height = 5.9 cm length = 7.2 m Loveless (1992) diameter = 8.8 cm 0.45 Circular length = 7.2 m 1.30 width = 22 cm 0.45 U-shape height = 40 cm 1.30 length = 7 m 6.00 bottom width = 30 cm inclined length = 30 cm 0.15 Present study Trapezoidal 0 outer angle = 60 0.83 length = 12 m

Table 2. Performance of the ANN and regression models for the critical flow velocity based on Method ANN Models FFBP GR RBF Regression Models Rectangular (Equation 8) Circular (Equation 9) U-shaped (Equation 10) Trapezoidal (Equation 11) General (Equation 12)

Max

Min

Mean

Percentage of data in range 0.9-1.1 0.75-1.25

1.09 1.16 1.06

0.86 0.72 0.80

0.98 1.01 0.93

81.81 63.63 54.54

100.00 90.90 100.00

1.34 1.40 1.51 1.07 1.26

0.65 0.68 0.72 0.51 0.60

1.03 1.05 1.10 0.74 0.90

63.63 63.63 45.45 9.09 36.36

72.72 72.72 72.72 54.54 81.81

Table 3. Performance of the ANN and regression models for the critical flow velocity based on , and Method (%) ANN models FFBP 0.037 5.48 0.97 GR 0.065 8.06 0.90 RBF 0.072 9.52 0.86 Regression models Rectangular (Equation 8) 0.124 13.77 0.64 Circular (Equation 9) 0.119 14.12 0.64 U-shaped (Equation 10) 0.119 15.78 0.62 Trapezoidal (Equation 11) 0.203 27.15 0.20 General (Equation 12) 0.123 14.54 0.53

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27

28

29

30

31

32

33

34

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HIGHLIGHTS 1. Experimental data of sediment incipient deposition were used in modelling. 2. Artificial neural networks and regression models were developed. 3. Models were established using flow, fluid, sediment and channel characteristics. 4. Feed forward back propagation artificial neural network model was found superior.

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