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Evaluation of total load sediment transport formulas using ANN
International Journal of Sediment Research 24 (2009) 274-286
Evaluation of total load sediment transport formulas using ANN Chih Ted YANG1, Reza MARSOOLI2, and Mohammad Taghi AALAMI3
Abstract The calculated results from various sediment transport formulas often differ from each other and from measured data. Some parameters in the sediment transport formulas are more effective than others to estimate total sediment load. In this study, an Artificial Neural Network (ANN) model is trained using four dominant parameters of sediment transport formulas. ANN models are able to reveal hidden laws of natural phenomena such as sediment transport process. The results of ANN and some total bed material load sediment transport formulas have been compared to indicate the importance of variables which can be used in developing sediment transport formulas. To train ANN, average flow velocity, water surface slopes, average flow depth, and median particle diameter are used as dominant parameters to estimate total bed material load. Two hundreds and fifty samples are used to train the ANN model. Twenty-four sets of field data not used in the training nor calibration of ANN are used to compare or verify the accuracy of ANN and some well-known total bed material load formulas. The test results show that the ANN model developed in this study using minimum number of dominant factors is a reliable and uncomplicated method to predict total sediment transport rate or total bed material load transport rate. Results show that the accuracy of formulas in descending order are those by Yang (1973), Laursen (1958), Engelund and Hansen (1972), Ackers and White (1973), and Toffaleti (1969). These results are similar to those made by ASCE (1982) based on laboratory and field data not used in this paper. Study results also show that the formulas based on physical laws of sediment transport, like those formulas that were developed based on power concept, are more accurate than other formulas for estimating total bed material sediment load in rivers. Key Words: Artificial Neural Network (ANN), Total bed material load, Sediment transport formulas
1 Introduction Determination of sediment load or transport rate is important to a wide range of water resources projects, such as the design of dams and reservoirs, sediment and pollution transport in rivers and lakes, channel design and maintenance. There are continuing interests in developing new sediment transport formula based mainly on laboratory investigations. At least 100 published transport formulas can be found in the literature. Verification of the accuracies of these formulas is mainly based on laboratory data and limited field data under some carefully designed data collection programs. It is desirable to use routine data collected from river gauging stations to determine the accuracies of some commonly used sediment transport formulas. The Artificial Neural Network (ANN) model has been used in recent years for the assessment of the accuracy of sediment transport formulas and other hydraulic and hydrologic phenomena. ANN will be used in this paper to estimate the total bed material load in rivers using four dominant parameters. According to data used in this study collected by the U.S. Geological Survey from river gauging stations, 1
Borland Prof. of Water Resources, Colorado State University, Fort Collins, CO 80523, U.S.A. E-mail: [email protected] 2 Graduate student of Civil-Hydraulics Engineering, University of Tabriz, Iran 3 Assist. Prof. of Civil-Hydraulics Engineering, University of Tabriz, Iran Note: The original manuscript of this paper was received in Dec. 2008. The revised version was received in Feb. 2009. Discussion open until Sept. 2010. - 274 -
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five well-known formulas including Yang (1973), Laursen (1958), Engelund and Hansen (1972), Ackers and White (1973), and Toffaleti (1969) are chosen for comparison. 2 Artificial Neural Network model (ANN) 2.1 Overview of ANN An ANN consists of a number of data processing elements called neurons or nodes that are grouped in layers. The input layer neurons receive input data or information and transmit the values to the next layer of processing elements across connections. This process is continued until the output layer is reached. This type of network in which data flows in one direction (forward) is known as a feed-forward network. The application of ANN models has been the topic of a large number of recent literatures, such as the book by Lingireddy and Brion (2005).
Fig.1
Basic concepts of artificial neuron
Figure 1 shows the basic concept of an artificial neuron. In this figure, various inputs to the network are represented by the symbol x n . Another input in the figure is called a bias, θ , and it is always equal to one. Each of these inputs is multiplied by a connection weight. These weights are represented by wn . In the simplest case, these products are simply summed, fed through a transfer function, σ , to generate a result, and then output. In mathematical form, a neuron with n number of inputs is shown in Eq. (1). n output = σ ⎛⎜ ∑ wi × xi + θ ⎞⎟ (1) ⎝ i =1 ⎠ 2.2 Training of ANN The process of determining ANN weights is called learning or training, and is similar to the calibration of a mathematical model. The ANNs are trained with a training set of input data and known output data. At the beginning of training, the weights are initialized either with a set of random values or based on previous experiences. Next, the weights are systematically changed by the learning algorithm such that for a given input the difference between the ANN output and actual output is minimized. Many learning examples are repeatedly presented to the network, and the process is terminated when output and observed value difference is less than a specified value. This process requires that the neural network computes the error derivative of the weights (EW). In other words, it must calculate how the error changes as each weight is increased or decreased slightly. This back propagation algorithm is the most widely used method for determining the EW. The algorithm involves calculating the derivatives of the network training error with respect to the weights by the application of chain rule and gradient descent optimization to adjust the weights to minimize the error. The equations related to this algorithm are available in ANN text books such as the book by Lingireddy and Brion (2005). 2.3 Application of ANN in hydraulics and hydrology Motivated by successful applications in modeling nonlinear system behavior in a wide range of areas, International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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ANN models have been applied to hydraulics and hydrology studies such as sediment transport, rainfallrunoff modeling, flow prediction, etc. Nourani et al. (2006) used ANN as a black box to assess the effect of flow discharge and temperature in suspended sediment load. In their study, Lighvan Chay watershed in Iran was used for the case study. They concluded that ANN can perform better than classic statistics models for sediment transport studies. Nagy et al. (2002) estimated the total sediment discharge concentration using a neural network approach. They used a number of effective parameters on the dynamics of sediments, such as tractive shear stress, velocity ratio, suspension parameters, Froude number, Reynolds number, stream width-depth ratio, and so on, as inputs for training the ANN. They showed that ANN can be successfully applied to sediment transport studies. Cigizoglu (2002) made a comparison between Artificial Neural Networks (ANNs) and sediment rating curve for suspended sediment estimation. Agarwalet et al. (2006) simulated the runoff and sediment yield using artificial neural networks as daily, weekly, ten-daily, and monthly monsoon runoff and sediment yield from an Indian catchment using Back Propagation Artificial Neural Network (BPANN) technique, and compared the results with observed values obtained from using single- and multi-input linear transfer function models. Raghuwanshi et al. (2006) made five ANN models developed for predicting runoff and sediment yield. 3 Total bed material load sediment transport formulas 3.1 Overview Sediment transport is a complex phenomenon while most theoretical treatments are based on some idealized and simplified assumptions. These assumptions have been used by researchers to develop their equations using one or two dominant factors such as water discharge, average flow velocity, energy slope and shear stress. Due to different assumptions used, various formulas often have different computed results from each other and from measured data. Analytical comparisons of some of the basic transport formulas are made to determine the accuracy and the interrelationships among them. Alonso and co-workers in 1980 made a comparison of sediment transport equations. The results of their comparison showed that the Yang (1973), Ackers-White (1973), Engelund-Hansen (1972), and Laursen (1958) formulas have the best agreement with measured data while the Yalin (1963), and Bagnold (1966) formulas gave unsatisfactory results. Also, the MPME method, which estimates the total load by adding the bed load predicted by Meyer-Peter and Müller’s (1948) formula to the suspended load computed by Einstein’s (1950) procedure, always worked poorly. In 1982, the Yang (1973), Laursen (1958), Ackers-White (1973), Engelund-Hansen (1972), Bagnold (1966), MPME, Meyer-Peter and Müller’s (1948), and Yalin (1963) formulas were ranked in descending order of accuracy by ASCE Sedimentation Committee (1982) based on the results obtained by Alonso (1980). Wu et al. (2008) evaluated the applicability of some sediment transport methods to the Yellow River in China. The best predictions were obtained by the Yang 1996 method. On the other hand, some methods such as Ackers and White, Engelund and Hansen were not applicable to the Yellow River. Some researches developed some methods to predict sediment transport rate. Mekonnen and Dargahi (2007) used a 3D numerical model, ECOMSED (open source code), to simulate flow and sediment transport in rivers to account for the dynamics of the mobile bed boundary, a model for the bed load transport was included in the code. The model successfully predicted the general flow patterns and sediment transport characteristics of their case study. Yang et al. (2007) re-examined Bagnold method of sediment transport. They showed that the total load of sediment-laden flow is related to near bed energy dissipation rate. 3.2 Formulas based on Einstein’s bed load function Einstein (1950) developed two ideas that broke with the concept used previously. He believed that the critical criterion for incipient motion should be avoided, because it is difficult to define. According to his belief the bed load transport is related to the turbulent flow fluctuations rather than to the average values of forces exerted by the flow on sediment particles. Einstein developed a probabilistic approach that the beginning and ceasing of sediment motion is expressed in terms of probability. Toffaleti (1969) developed his transport functions based on Einstein theory with some modifications. In Tofalleti approach, the total depth of flow is divided into four zones including upper, middle, lower, and bed zones. The sediment - 276 -
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discharge is calculated for each zone and the total sediment load is:
qt = q B + q su + q sm + q sl
(2)
where qt is total sediment load discharge per unit width, q B is bed load discharge per unit width, and
qsu , qsm , q sl are suspended load discharge per unit width in the upper, middle, and lower zones, respectively. The input required for Toffaleti’s method consist of the average velocity, stream depth, stream width, water temperature, and d 65 size. Details of calculation process can be found in most sediment transport books, such as the one by Yang (1996, 2003). 3.3 Formulas based on the power concept Some total load sediment transport formulas such as Yang, Engelund-Hansen, and Ackers-White were developed based on the power concept. According to the power concept, the sediment transport rate should be related to the rate of energy dissipation used in transport sediment. Bagnold (1966) considered the relationship between the rate of energy available to an alluvial rivers and the rate of work being done by the river in transporting sediment. Engelund and Hansen (1972) developed a sediment transport function using Bagnold’s stream power concept (Yang, 2002) as shown below. 2
0.5
0.05γ s (γ DS )1.5 ⎛ V ⎞ ⎛ γ ⎞ ⎟ ⎜ (3) ⎜ γ − γ ⎟ ⎜⎜ g ⎟⎟ d 50 ⎠⎝ ⎠ ⎝ s where qt is total sediment discharge by weight per unit width, V is average flow velocity, S is energy slope, γ s and γ are specific weights of sediment and water, respectively, d50 is median particle diameter, and D is flow depth. Also, based on Bagnold’s stream power concept (Yang, 2002), Ackers and White (1973) developed the following formula.
qt =
⎛ γDS ⎞ ⎟ ⎜ γ XD ⎜ ρ ⎟ Ggr = ⎟ ⎜ dγ S ⎜ V ⎟ ⎟ ⎜ ⎠ ⎝
n
(4)
where X is the rate of sediment transport in terms of mass flow per unit mass flow rate, Ggr and n are a dimensionless function and transition exponent, respectively. Yang (1972, 1973) stated that the rate of work being done by a unit weight of water in transporting sediment should be directly related to the rate of work available to a unit weight of water. The rate of energy per unit weight of water available for transporting water and sediment in an open channel with reach length x and total drop of Y is dY dx dY (5) = = VS dt dt dx where VS is defined as the unit stream power. Yang (1972, 1973) considered the following relation based on unit stream power concept to determine total sediment concentration. φ (Ct ,VS ,U * ,υ , ω , d ) = 0 (6) where Ct is total sediment concentration, U * is shear velocity sediment.
(gDS )
0.5
, and ω is fall velocity of
3.4 Other transport functions Laursen (1958) developed a functional relationship between the flow condition and the resulting sediment discharge. 76 n ⎞ ⎛U ⎞ ⎛ d ⎞ ⎛ τ′ (7) Ct = 0.01γ ∑ pi ⎜ i ⎟ ⎜⎜ − 1⎟⎟ f ⎜⎜ * ⎟⎟ i =1 D τ ⎝ ⎠ ⎝ ci ⎠ ⎝ ωi ⎠ International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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where pi is percentage of materials available in size fraction i , τ ci is critical tractive stress for sediment size d i , and τ ′ is bed shear stress 1/ 3
τ′ =
ρV 2 ⎛ d 50 ⎞
(8) ⎜ ⎟ 58 ⎝ D ⎠ As shown in equations Eqs. (3), (4), (6), and (8), variables V, S, D, and d 50 are important factors needed for the computation of shear velocity, shear stress, unit stream power, and eventually total bed material load concentration.
4 Regression approach Regression approach is an alternative to estimate the sediment load when there is no sediment data, such as particle size, to use a predictive sediment transport formula. Sediment-discharge rating curve is a regression approach that connects the river hydrology to sediment transport. The rating curve estimate sediment load using flow discharge in the following form Qs = aQwb (9) where Qs is suspended sediment discharge, Qw is flow discharge, and a and b are the coefficients determined by regression analysis. Flow velocity and flow depth are two required data for using rating curve. These two parameters, which are measured in gauging stations, are the most important factors to determine the flow conditions. Water surface slope is another important parameter of flow regime. Yang (1975) defines the unit stream power as the V and S product. The rate of work being done by a unit weight of water in transporting sediment must be directly related to the rate of work available to a unit weight of water. Thus, total sediment concentration or total bed material load must be directly related to unit stream power (V × S ) . While rating curve can simulate a linear relation between flow characteristics, it seems necessary to use artificial neural networks, which are suited to complex nonlinear models, for the analysis of relation between hydrology and sediment transport. In this research, a nonlinear regression is developed using ANN to reveal the relation among flow velocity, flow depth; water surface slope, and median sediment particle size to develop a nonlinear regression among flow characteristics and sediment transport using ANN. 5 Application of ANN for forecasting total bed material sediment discharge 5.1 Parameters and data used in training the ANN model Important parameters in most sediment transport formulas are average flow velocity V, water surface slope S, average water depth D, and median particle diameter d50 . In this study, field data published by Williams and Rosgen of the U.S. Geological Survey in 1986 are used. These data were collected from the following river stations in the United States: 1- Susitna River near Talkeetna, Alaska (1982-1985) 2- Chulitnariver below Canyon near Talkeetna, Alaska (1982-1985) 3- South Fork of Salmon River near Cascade, Idaho (1985-1986) 4- Clearwater River at Spalding, Idaho (1972-1979) 5- Wisconsin River at Muscoda, Wisconsin (1976-1979) 6- Chippewa River near Carville, Wisconsin (1976-1979) 7- Muddy Creek near Pinedale, Wyoming (1975-1976) 8- Snake River near Anatone, Washington (1972-1979) 9- Black River near Galesville, Wisconsin (1977-1979) 10- Tanana River at Fairbank, Alaska (1977-1981) Measured V, S, D, and d 50 values are used as input data to the model and total bed material sediment discharge in tons per day as output of model. A total of 250 samples were used for training the model and 76 samples, or about 25 percent of total measured data randomly selected to test or verify the model. Number of samples used for training and testing of ANN model are shown in Table 1 for each river. Table - 278 -
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2 shows the statistic parameters of samples used for training and testing the ANN. 5.2 QNET for training ANNs To test the ANN models by trial-and-error, the public domain QNET Software is used. QNET back propagation neural networks are multi-layered and feed-forward (connections must connect to the next layer) in design. QNET can perform the data normalization automatically. If this option is selected during training setup, all data for the nodes in the input layer and/or training targets in the output layer will be normalized between the limits of 0.15 and 0.85. Table 1
Number of samples used for training and testing the ANN model Number of samples Location Total Training Testing Susitna River near Talkeetna, Alaska 29 24 5 Chulitna River below Canyon near Talkeetna, Alaska 36 26 10 South Fork of Salmon River near Cascade, Idaho 52 42 10 Clearwater River at Spalding, Idaho 44 34 10 Wisconsin River at Muscoda, Wisconsin 37 27 10 Chippewa River near Caryville, Wisconsin 15 10 5 Muddy creek near Pinedale, Wyoming 15 10 5 Snake River near Anatone, Washington 45 35 10 Black River near Galesville, Wisconsin 22 17 5 Tanana River at Fairbank, Alaska 31 25 6 Total 326 250 76 Table 2
Statistic parameters of samples used for ANN model
Data
Training or calibration data
Testing or verification data
Input data
Average velocity (m/s) Average depth (m) Water surface slope (m/m)
d50 Output data Input data
Total sediment load (ton/day) Average velocity (m/s) Average depth (m) Water surface slope (m/m)
d50 Output data
Total sediment load (ton/day)
Average
Max
Min
1.48 2.47 0.00072 0.68
3.40 5.80 0.0083 2.00
0.23 0.16 0.00011 0.19
Standard deviation 0.78 1.55 0.00063 0.40
14145.93
554152.30
0.65
41391.42
1.44 2.43 0.00068 0.71
2.80 5.30 0.0015 1.84
0.23 0.22 0.00009 0.27
0.77 1.55 0.00042 0.37
12621.71
144443.52
3.05
27572.38
The learning rate coefficient (η ) can be controlled manually during training or QNET can control it automatically using its Learn Rate Control (LRC) feature. The learning rate coefficient determines the size of the node weights adjustments during training. The learning rate coefficient Eta’s valid range is between 0.0 and 1.0. LRC will drive η higher or lower in a systematic fashion depending on the current learning activity. If the network appears to be learning at a relatively slow rate, η is driven up quickly. Conversely, if the network is learning at a fast pace, QNET will hold η constant or even lower it to avoid instabilities. QNET gives the option of selecting four transfer functions: the sigmoid, Gaussian, hyperbolic tangent and hyperbolic secant. These functions set the output signal strength between 0.0 and 1.0. As it is seen in Eq. (1), the input signal to the transfer function is the dot product of all the node’s input signals and the node’s weight vector plus bias factor. As it is explained above, QNET has a high and intelligent ability to simulate complex networks easily while preparing a code for these networks could be excruciating. For practical problems, using an easy International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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method, which is usable for different cases, is more acceptable than sophisticated methods. Also, using low cost public domain software would be more satisfactory for researchers, especially students, instead of costly software like MATLAM. In summary, QNET is a professional user friendly software and that is why it was used to make a lot of simple and complex networks in this research. 5.3 Statistic analysis tools QNET uses two statistic parameters to determine the accuracy of the models. These parameters are the correlation coefficient and root mean square error and their values are monitored during training process. Correction Coefficient = Root Mea Square Estimation =
∑( x − x)( y − y )
(10)
∑ ( x − x) 2 ( y − y ) 2 1 n
∑ (y n
i =1
actual
i
− y forecast
)
2
i
(11)
5.4 ANN model development For choosing the best network for forecasting, different models were built with various nodes and middle layers. In Learn Rate Control (LRC) feature, 0.001 and 0.15 were used as the minimum and maximum limit for auto control of learning rate. In this study, 0.8 was used for momentum factor α , which damps high frequency weight changes and can be set in the 0.8 to 0.9 range for the majority of networks. The input combinations to estimate total sediment load are 1) V, S, D, and d50 ; 2) V, S, and D; and 3) V and D; 4) V and S; 5) S and D. The first combination includes hydrological and sediment parameters required for the majority of sediment transport formulas. Other four combinations include hydrological dominant parameters of flow conditions. After try-and-error tests, the best network found in this study is the one with 3 layers including one input layer, one middle layer, and one output layer. The best model consists of 4 inputs including V, S, D, and d50, 7 middle, and one output nodes which should have a Sigmoid transfer function. The Sigmoid function represented by 1/(1+e − x ) and set the output signal strength between 0 and 1. The correlation coefficient and RMS error of this ANN are 0.97 and 0.013 for the training data set and 0.95 and 0.011 for the test data set, respectively. The final architecture of the ANN model is given in Table 3.
Model inputs V, S, D, d 50
Model output Total sediment discharge
Table 3 Characteristics of the final ANN model ANN Transfer Training set structure function Correlation RMSE (4,7,1)
Sigmoid
0.97
0.013
Test set Correlation RMSE 0.95
0.011
Accuracy of different ANN models developed in this study, is summarized in Table 4. The ANN model (4-7-1) which has been trained using four important parameters of sediment transport formulas has slightly more accurate results than other models with four inputs. The accuracy of the chosen ANN model (4-7-1) in comparison with other models of 4 inputs, specially those ones with more than one hidden layer, has insignificant difference. According to the intricacy and longer time needed for training of models with more than one hidden layer, the selection of ANN model (4-7-1) is reasonable. It can be concluded from Table 4 that those models which use only hydrological data as input, especially the ANN (3-4-1) with three dominant parameters of flow conditions, are as accurate as the ANN (4-7-1), which uses both hydrological and sediment parameters as input. In other words, the total sediment load is estimable using hydrological data accurately. Using hydrological data, which are measured in gauging stations, is easier than using both hydrological and sediment data. This would be more useful in some cases that there is only measured hydrological data and no sediment data, such as many rivers in Iran. However, it should be noted that most sediment transport formulas include a parameter for sediment particle size. - 280 -
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Table 4 Input data
Hydrological and sediment
Combination Dominant parameters of sediment transport formulas Dominant parameters of flow conditions
Hydrological
Parameters of rating curve Unit stream power
.
Other combinations
Sediment (kg/s)
Accuracy of some ANN models
Model inputs V, S, D, d50 V, S, D, d50 V, S, D, d50 V, S, D, d50 V, S, D V, S, D V, S, D V, S, D V, D V, D V, D V, S V, S V, S S, D S, D S, D
10000.00 1000.00 100.00 10.00 1.00 0.10 0 0.01 0.00
Training set
ANN structure (4-2-1) (4-7-1) (4-3-7-1) (4-5-4-7-1) (3-1-1) (3-4-1) (3-3-3-1) (3-3-3-2-1) (2-5-1) (2-6-3-1) (2-7-6-2-1) (2-4-1) (2-4-3-1) (2-6-4-2-1) (2-7-1) (2-9-6-1) (2-9-5-3-1)
Correlation 0.92 0.97 0.97 0.97 0.75 0.96 0.96 0.97 0.89 0.95 0.94 0.89 0.94 0.93 0.94 0.95 0.95
RMSE 0.020 0.013 0.013 0.013 0.035 0.014 0.012 0.012 0.023 0.015 0.017 0.023 0.017 0.019 0.017 0.014 0.014
Test set Correlation 0.87 0.95 0.95 0.94 0.62 0.95 0.94 0.94 0.91 0.91 0.93 0.79 0.74 0.79 0.93 0.94 0.95
RMSE 0.020 0.011 0.011 0.012 0.034 0.011 0.012 0.012 0.015 0.015 0.014 0.023 0.023 0.022 0.012 0.012 0.011
Measured ANN
50
100
150
200
250
300
.
Pattern sequence ANN (3-4-1)
Measured ANN
10000.00
Sediment (kg/s)
1000.00 100.00 10.00 1.00 0.10 0
50
100
150
200
250
300
0.01 0.00
Pattern sequence ANN (4-7-1) Fig. 2 Comparison of measured data and ANN results (training output)
The comparison of the measured data and estimated total sediment load using ANN for training and testing procedures are shown in Figs. 2 and 3, respectively. It can be seen that the estimates of ANN model are close to the measured data and they are in good agreement. It can be concluded that the ANN model can be used to detect hidden relationship between sediment transport rate and some important parameters of transport formulas or flow conditions such as average velocity, water surface slope, average water depth, and median particle diameter.
International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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10000
Measured ANN
Sediment (Kg/s)
1000 100 10 1 0.1 0
20
0.01
40
60
80
Pattern sequence
ANN (3-4-1) 10000
Measured ANN
Sediment (Kg/s)
1000 100 10 1 0.1 0 0.01
20
40
60
80
Pattern sequence
ANN (4-7-1) Fig. 3 Comparison of measured data and ANN results (testing output)
6 Comparison of ANN and sediment transport equations results 6.1 Field data 24 sets of field data with accurate hydraulic information, not used in the training and verification of ANN in previous section, are used to compare the results of ANN model and sediment transport formulas. Field data include average flow velocity, average flow depth, channel width, water surface slope, sediment particle size distribution, water temperature and total bed-material load. These data were measured from: Chippewa River at Durand, Wisconsin (1975-1979), Chippewa River near Pepin, Wisconsin (1976-1979), and Wisconsin River at Muscoda, Wisconsin (1976-1979). The range of data used to test the ANN model results and five sediment transport formulas are summarized in Table 5. The five formulas tested are Yang (1973), Engelund-Hansen (1972), AckersWhite (1973), Laursen (1958) and Toffaleti (1969). The first four formulas were considered by ASCE (1982) as the four most accurate formulas for total bed material load in the sand size range. Toffaleti formula is included in the test because it was developed for sand transport in rivers similar to those used in this paper. Table 5 Data Average velocity (m/s) Average depth (m) Water surface slope (m/m) Water surface width (m) Water temp (Co) d35 (mm) d50 (mm) d65 (mm) d90 (mm) - 282 -
Statistic parameters of samples used for comparison Average Max Min Standard deviation 0.78 2.4 0.48 0.38 1.33 3.2 0.61 0.57 0.00047 0.0037 0.00017 0.00069 236.63 299 59 53.12 13.2 25.5 3.5 7.45 0.46 0.92 0.35 0.12 0.63 1.57 0.42 0.27 1.08 4.89 0.48 0.97 4.94 24.73 0.92 6.35 International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 274–286
The Yang, Engelund-Hansen and Ackers-White formulas are all based on the power concept. EngelundHansen and Ackers-White formula are based on Bagnold’s stream power concept indirectly (Yang, 2002), and Yang’s formula is based on his unit stream power concept directly. Toffaleti approach is based on the concept of Einstein (1950) and Einstein and Chien (1953). Laursen formula is developed using shear stress concept. 6.2 Results Figure 4 shows the comparisons between computed results from five sand transport formulas and the computed results from the ANN models. It shows that ANN (3-4-1) has a trend to overestimate small loads and underestimate large sediment loads while ANN (4-7-1) estimates sediment load with a more stable trend.
Computed total sediment discharge (kg/s) .
10000
Yang Laursen ANN(4-7-1) Engelund-Hansen Ackers-White Toffaleti ANN(3-4-1)
1000 100 10 1 1
10
100
1000
10000
Measured total sediment discharge (kg/s) Fig. 4 Comparison of ANN and the formulas results with measured data
The accuracy of each formula and ANN are shown in Table 6. This table shows that the ANN model has accuracy similar to the well-known total bed material load formulas. It shows that the ANN, Yang, and Laursen approaches are more accurate than other methods for estimating total sand transport rates in rivers compared in this study. The ANN (3-4-1), which uses three dominant parameters of flow conditions; V, S, and D, as input data; has about the same accuracy as the ANN (4-7-1) using V, S, D, and d50 four parameters as input. This reveals that the increasing number of parameters of input data set for training ANN, which is a regression approach, may not increase the accuracy of estimation due to complexity of nonlinear regression calculation process. Table 6 Rank 1 2 3 4 5 6 7
Comparison of correlation coefficients of six formulas and the ANN model Equation Correlation ANN (3-4-1) 0.998 Yang 0.996 ANN (4-7-1) 0.996 Laursen 0.995 Engelund-Hansen 0.979 Ackers-White (d50) 0.935 Toffaleti 0.833
The Yang and Laursen formulas are based on physical rules of sediment transport, but ANN is based on trial-and-error interpolation method of the data used in this study. The ANN model in this study has been developed to simulate the inner relation among four important parameters in sediment transport formulas. As shown in Table 6, the Yang formula based on the unit stream power concept has accuracy similar to ANN. Also, the Engelund-Hansen and Ackers-White formulas, which have been developed based indirectly on Bagnold’s stream power concept, also have good agreement with measured data. The International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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Laursen formula, which is not based on power concept, shows the accuracy close to that of Yang and ANN. The Toffaleti formula shows the least agreement with measured data. The Toffaleti formula has been developed based on a probabilistic approach introduced by Einstein (1950). The high correlations shown in Table 6 are due to the inclusion of a set of data collected during a flood with measured total sediment discharge of 1,714 kg/s, which is significantly higher than other used set data in this section. From statistical point of view, if one set of data has the value outside of other data range, the inclusion of this particular data set can have significant impact on the overall correlation coefficient. Table 7 shows the comparison of correlation coefficient if the data set collected during the flood is excluded. In this case, the correlation coefficients are more in line with other published results in the literature. Table 7
Comparison of correlation coefficients of six formulas and the ANN model without flood set data Rank Equation Correlation 1 ANN (3-4-1) 0.852 2 Yang 0.845 3 Laursen 0.819 4 ANN (4-7-1) 0.815 5 Engelund-Hansen 0.796 6 Ackers-White (d50) 0.784 7 Toffaleti 0.771
Comparisons between rankings of selected sediment transport formulas by ASCE and this study are shown in Table 8. It shows that the accuracies of selected sediment transport formulas are similar to those made by ASCE (1982) based on laboratory and field data not used in this paper. In both studies, Yang (1973) and Laursen (1958) methods are more accurate than others. Table 8 Summary of rating of selected sediment transport equations Equation Ranking Ranking by ASCE (1982) by this research Ackers-White 3 4 Engelund-Hansen 4 3 Laursen 2 2 Yang 1 1
7 Summary and conclusions ANN model is used for estimating total sediment load in rivers in this study. 250 samples were used for training the ANN model and 76 samples were used to verify the model. Inputs for training the model include average flow velocity V, water surface slope S, average water depth D, and median particle diameter d50. The results show that the ANN model with four input nodes V, S, D, and d50 is able to estimate the sediment transport rate accurately but it is worthy of mention that the ANN with three input data estimates the sediment load as accurate as some well known formulas. This ANN model has been developed using the minimum number of three flow effective parameters including V, S, and D. The minimum required input data make the ANN model in this paper is easy to use for estimating the total bed material load when there is no sediment data to use the well known sediment transport formulas such as Yang and Laursen. On the other hand, this ANN model might not have the stable trend to estimate sediment load like those ANN with four input and some reliable formulas such as Yang. Comparisons of accuracy of the ANN model against sediment transport formulas were made using 24 sets of measured samples not used in the training and testing of the ANN model. The comparison shows that the unit stream power approach used by Yang is most reliable for the estimation of sediment transport rate. The Laursen, Engelund-Hansen, and Ackers-White formulas are also fairly accurate. The Toffaleti formula based on probabilistic approach is the least accurate one. This ranking is similar to the ranking of accuracy of sediment transport formulas by the ASCE Sedimentation Committee (1982) without using the - 284 -
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ANN approach. In summary, the ANN model trained using three important parameters V, S, and D can be used to estimate total bed material load in lack of sediment data. However, it should be pointed out that the accuracy of an ANN model is data sensitive, and should not be applied to the conditions outside of the data range used in training the ANN model without verification. In other words, for other cases with data range out of this study, an appropriate ANN should be trained using data set of that case study to estimate total sediment load. References Ackers P, and White W. R. 1973, Sediment transport: new approach and analysis. Journal of the Hydraulics Division, ASCE Vol. 99, No. HY11, pp. 2041–2060. Agarwal A. h., Mishra S. K., and Singh J. K. 2006, Simulation of runoff and sediment yield using Artifitial Neural Networks. Biosystems Engineering, Vol. 97, No. 4, pp. 597–613, Published by Elsevier Ltd. ASCE Task Committee on Relations between Morphology of Small Streams and Sediment Yield of the Committee on Sedimentation of the Hydraulics Division 1982. Relationships between morphology of small stream and sediment yields,” Journal of the Hydraulics Divisions, ASCE, Vol. 108, No. HY11, pp. 1328–1365. Bagnold R, A. 1966, An approach to the sediment transport problems from general physics. U.S. Geological Survey Professional Paper 422-J. Cigizoglu H. 2002, Suspended sediment estimation for rivers using artificial neural networks and sediment rating curves. Turkish Journal of Engineering Environmental Sciences, Vol. 26, No. 1, pp.27–36. Einstein H. A. 1950, The bed-load function for sediment transportation in open channel flows. U.S. Department of Agriculture, Soil Conservation service, Technical Bulletin No. 1026. Einstein H. A. and Chen N. 1953, Transport of sediment mixtures with large ranges of grain sizes. Missouri River Division Sediment Series No. 2, U.S. Army Corps of the Engineers Missouri River Division, University of California Institute of Engineering Research, Berkley. Engelund F. and Hansen E. 1972, A Monograph on Sediment Transport in Alluvial Streams. Teknisk Forlag, Copenhagen. Laursen E. M. 1958, The total sediment load of streams. Journal of the Hydraulics Division, ASCE Vol. 84, No. 1, pp. 1530–1–1530–36. Lingireddy S. and Brion G. M. 2005, Artificial Neural Networks In Water Supply Engineering. Published by the American Society of Civil Engineers. Mekonnen, M. A., Dargahi, B. 2007, Three dimensional numerical modeling of flow and sediment transport in rivers. International Journal of Sediment Research, Vol. 22, No. 3, pp. 188-198. Meyer-Peter E. and Müller R. 1948, Formula for bed-load transport. Proceedings of International Association for Hydraulic Research, 2nd meeting, Stockholm. Nagy H. M., Watanabe K., and Hirano M. 2002, Prediction of sediment load concentration in rivers using artificial neural network model. Journal of Hydraulics Engineering, ASCE Vol. 128, No. 6, pp. 588–595. Nourani V., Aalami M. T., Nazmara H., Hosseinzadeh H. 2006, Estimation of suspended sediment load in Talkhehrood using artificial neural network. 3rd National Comgress on Civil Engineering, University of Tabriz, Tabriz, Iran, 1-3 May 2007, Water Engineering, Vol., Paper No. 442. Nourani V., Aalami M. T., Aminfar M. H., Nourpour A. 2006, Application of artificial neural network in sensitivity analysis of effective parameters of suspended sediment load (Case Study: Lighvan Chay, 3rd National Congress on Civil Engineering, University of Tabriz, Tabriz, Iran, 1-3 May 2007, Water Eng. Vol., Paper No. 57. Raghuwanshi N. S., Singh R., and Reddy L. S. 2006, Runoff and sediment yield modeling using artificial neural networks: Upper Siwane River. India. Journal of Hydrologic Engineering, Vol. 11, No. 1. Rumelhart D. E., Hinton G. E., and Williams R. J. 1986, Learning internal representation by error propagation. MIT Press, Cambridge, Massachusetts, USA. Toffaleti F. B. 1969, Definitive computations of sand discharge in rivers. Journal of the Hydraulics Division, ASCE Vol 95, No. HY1, pp.225–246. Wiliams P. and Rosgen L. 1986, Measured total sediment load for 93 United States streams. U.S. Geological Survey. Wu, B., Maren, D. S., and Li, L., 2008, Predictibility of sediment transport in the Yellow River using selected transport formulas. International Journal of Sediment Research, Vol. 23, No. 4, pp. 283-298. Yalin, M. S., 1963, An expression for bed load transportation. Journal Hydraulic Division, ASCE Vol. 89, pp. 221–250. Yang C. T. 1972, Unit stream power and sediment transport. Journal of the Hydraulics Division, ASCE Vol. 98, No HY 10, pp. 1805–1826. Yang C. T. 1973, Incipient motion and sediment transport. Journal of the Hydraulics Division, ASCE Vol. 99, No. HY10, pp. 1679–1704. Yang C. T. 1996, Sediment Transport Theory and Practice. McGraw-Hill Companies, Inc. (reprint by Krieger P International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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Company, 2003). Yang C. T. 2002, Sediment transport and stream power. International Journal of Sediment Research, Vol. 17, No. 1, pp. 31–38. Yang C. T., Molinas, A., and Wu B. 1996, Sediment transport in the Yellow River. Journal of Hydraulic Engineering, ASCE Vol. 122, No. 5, pp. 237–244. Yang, S. Q., Koh, S. C., Kim, I. S., Song, Y. C. 2007, Sediment transport capacity - An improved Bagnold formula. International Journal of Sediment Research, Vol. 22, No. 1, pp. 27-38.
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Evaluation of total load sediment transport formulas using ANN Chih Ted YANG1, Reza MARSOOLI2, and Mohammad Taghi AALAMI3
Abstract The calculated results from various sediment transport formulas often differ from each other and from measured data. Some parameters in the sediment transport formulas are more effective than others to estimate total sediment load. In this study, an Artificial Neural Network (ANN) model is trained using four dominant parameters of sediment transport formulas. ANN models are able to reveal hidden laws of natural phenomena such as sediment transport process. The results of ANN and some total bed material load sediment transport formulas have been compared to indicate the importance of variables which can be used in developing sediment transport formulas. To train ANN, average flow velocity, water surface slopes, average flow depth, and median particle diameter are used as dominant parameters to estimate total bed material load. Two hundreds and fifty samples are used to train the ANN model. Twenty-four sets of field data not used in the training nor calibration of ANN are used to compare or verify the accuracy of ANN and some well-known total bed material load formulas. The test results show that the ANN model developed in this study using minimum number of dominant factors is a reliable and uncomplicated method to predict total sediment transport rate or total bed material load transport rate. Results show that the accuracy of formulas in descending order are those by Yang (1973), Laursen (1958), Engelund and Hansen (1972), Ackers and White (1973), and Toffaleti (1969). These results are similar to those made by ASCE (1982) based on laboratory and field data not used in this paper. Study results also show that the formulas based on physical laws of sediment transport, like those formulas that were developed based on power concept, are more accurate than other formulas for estimating total bed material sediment load in rivers. Key Words: Artificial Neural Network (ANN), Total bed material load, Sediment transport formulas
1 Introduction Determination of sediment load or transport rate is important to a wide range of water resources projects, such as the design of dams and reservoirs, sediment and pollution transport in rivers and lakes, channel design and maintenance. There are continuing interests in developing new sediment transport formula based mainly on laboratory investigations. At least 100 published transport formulas can be found in the literature. Verification of the accuracies of these formulas is mainly based on laboratory data and limited field data under some carefully designed data collection programs. It is desirable to use routine data collected from river gauging stations to determine the accuracies of some commonly used sediment transport formulas. The Artificial Neural Network (ANN) model has been used in recent years for the assessment of the accuracy of sediment transport formulas and other hydraulic and hydrologic phenomena. ANN will be used in this paper to estimate the total bed material load in rivers using four dominant parameters. According to data used in this study collected by the U.S. Geological Survey from river gauging stations, 1
Borland Prof. of Water Resources, Colorado State University, Fort Collins, CO 80523, U.S.A. E-mail: [email protected] 2 Graduate student of Civil-Hydraulics Engineering, University of Tabriz, Iran 3 Assist. Prof. of Civil-Hydraulics Engineering, University of Tabriz, Iran Note: The original manuscript of this paper was received in Dec. 2008. The revised version was received in Feb. 2009. Discussion open until Sept. 2010. - 274 -
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five well-known formulas including Yang (1973), Laursen (1958), Engelund and Hansen (1972), Ackers and White (1973), and Toffaleti (1969) are chosen for comparison. 2 Artificial Neural Network model (ANN) 2.1 Overview of ANN An ANN consists of a number of data processing elements called neurons or nodes that are grouped in layers. The input layer neurons receive input data or information and transmit the values to the next layer of processing elements across connections. This process is continued until the output layer is reached. This type of network in which data flows in one direction (forward) is known as a feed-forward network. The application of ANN models has been the topic of a large number of recent literatures, such as the book by Lingireddy and Brion (2005).
Fig.1
Basic concepts of artificial neuron
Figure 1 shows the basic concept of an artificial neuron. In this figure, various inputs to the network are represented by the symbol x n . Another input in the figure is called a bias, θ , and it is always equal to one. Each of these inputs is multiplied by a connection weight. These weights are represented by wn . In the simplest case, these products are simply summed, fed through a transfer function, σ , to generate a result, and then output. In mathematical form, a neuron with n number of inputs is shown in Eq. (1). n output = σ ⎛⎜ ∑ wi × xi + θ ⎞⎟ (1) ⎝ i =1 ⎠ 2.2 Training of ANN The process of determining ANN weights is called learning or training, and is similar to the calibration of a mathematical model. The ANNs are trained with a training set of input data and known output data. At the beginning of training, the weights are initialized either with a set of random values or based on previous experiences. Next, the weights are systematically changed by the learning algorithm such that for a given input the difference between the ANN output and actual output is minimized. Many learning examples are repeatedly presented to the network, and the process is terminated when output and observed value difference is less than a specified value. This process requires that the neural network computes the error derivative of the weights (EW). In other words, it must calculate how the error changes as each weight is increased or decreased slightly. This back propagation algorithm is the most widely used method for determining the EW. The algorithm involves calculating the derivatives of the network training error with respect to the weights by the application of chain rule and gradient descent optimization to adjust the weights to minimize the error. The equations related to this algorithm are available in ANN text books such as the book by Lingireddy and Brion (2005). 2.3 Application of ANN in hydraulics and hydrology Motivated by successful applications in modeling nonlinear system behavior in a wide range of areas, International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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ANN models have been applied to hydraulics and hydrology studies such as sediment transport, rainfallrunoff modeling, flow prediction, etc. Nourani et al. (2006) used ANN as a black box to assess the effect of flow discharge and temperature in suspended sediment load. In their study, Lighvan Chay watershed in Iran was used for the case study. They concluded that ANN can perform better than classic statistics models for sediment transport studies. Nagy et al. (2002) estimated the total sediment discharge concentration using a neural network approach. They used a number of effective parameters on the dynamics of sediments, such as tractive shear stress, velocity ratio, suspension parameters, Froude number, Reynolds number, stream width-depth ratio, and so on, as inputs for training the ANN. They showed that ANN can be successfully applied to sediment transport studies. Cigizoglu (2002) made a comparison between Artificial Neural Networks (ANNs) and sediment rating curve for suspended sediment estimation. Agarwalet et al. (2006) simulated the runoff and sediment yield using artificial neural networks as daily, weekly, ten-daily, and monthly monsoon runoff and sediment yield from an Indian catchment using Back Propagation Artificial Neural Network (BPANN) technique, and compared the results with observed values obtained from using single- and multi-input linear transfer function models. Raghuwanshi et al. (2006) made five ANN models developed for predicting runoff and sediment yield. 3 Total bed material load sediment transport formulas 3.1 Overview Sediment transport is a complex phenomenon while most theoretical treatments are based on some idealized and simplified assumptions. These assumptions have been used by researchers to develop their equations using one or two dominant factors such as water discharge, average flow velocity, energy slope and shear stress. Due to different assumptions used, various formulas often have different computed results from each other and from measured data. Analytical comparisons of some of the basic transport formulas are made to determine the accuracy and the interrelationships among them. Alonso and co-workers in 1980 made a comparison of sediment transport equations. The results of their comparison showed that the Yang (1973), Ackers-White (1973), Engelund-Hansen (1972), and Laursen (1958) formulas have the best agreement with measured data while the Yalin (1963), and Bagnold (1966) formulas gave unsatisfactory results. Also, the MPME method, which estimates the total load by adding the bed load predicted by Meyer-Peter and Müller’s (1948) formula to the suspended load computed by Einstein’s (1950) procedure, always worked poorly. In 1982, the Yang (1973), Laursen (1958), Ackers-White (1973), Engelund-Hansen (1972), Bagnold (1966), MPME, Meyer-Peter and Müller’s (1948), and Yalin (1963) formulas were ranked in descending order of accuracy by ASCE Sedimentation Committee (1982) based on the results obtained by Alonso (1980). Wu et al. (2008) evaluated the applicability of some sediment transport methods to the Yellow River in China. The best predictions were obtained by the Yang 1996 method. On the other hand, some methods such as Ackers and White, Engelund and Hansen were not applicable to the Yellow River. Some researches developed some methods to predict sediment transport rate. Mekonnen and Dargahi (2007) used a 3D numerical model, ECOMSED (open source code), to simulate flow and sediment transport in rivers to account for the dynamics of the mobile bed boundary, a model for the bed load transport was included in the code. The model successfully predicted the general flow patterns and sediment transport characteristics of their case study. Yang et al. (2007) re-examined Bagnold method of sediment transport. They showed that the total load of sediment-laden flow is related to near bed energy dissipation rate. 3.2 Formulas based on Einstein’s bed load function Einstein (1950) developed two ideas that broke with the concept used previously. He believed that the critical criterion for incipient motion should be avoided, because it is difficult to define. According to his belief the bed load transport is related to the turbulent flow fluctuations rather than to the average values of forces exerted by the flow on sediment particles. Einstein developed a probabilistic approach that the beginning and ceasing of sediment motion is expressed in terms of probability. Toffaleti (1969) developed his transport functions based on Einstein theory with some modifications. In Tofalleti approach, the total depth of flow is divided into four zones including upper, middle, lower, and bed zones. The sediment - 276 -
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discharge is calculated for each zone and the total sediment load is:
qt = q B + q su + q sm + q sl
(2)
where qt is total sediment load discharge per unit width, q B is bed load discharge per unit width, and
qsu , qsm , q sl are suspended load discharge per unit width in the upper, middle, and lower zones, respectively. The input required for Toffaleti’s method consist of the average velocity, stream depth, stream width, water temperature, and d 65 size. Details of calculation process can be found in most sediment transport books, such as the one by Yang (1996, 2003). 3.3 Formulas based on the power concept Some total load sediment transport formulas such as Yang, Engelund-Hansen, and Ackers-White were developed based on the power concept. According to the power concept, the sediment transport rate should be related to the rate of energy dissipation used in transport sediment. Bagnold (1966) considered the relationship between the rate of energy available to an alluvial rivers and the rate of work being done by the river in transporting sediment. Engelund and Hansen (1972) developed a sediment transport function using Bagnold’s stream power concept (Yang, 2002) as shown below. 2
0.5
0.05γ s (γ DS )1.5 ⎛ V ⎞ ⎛ γ ⎞ ⎟ ⎜ (3) ⎜ γ − γ ⎟ ⎜⎜ g ⎟⎟ d 50 ⎠⎝ ⎠ ⎝ s where qt is total sediment discharge by weight per unit width, V is average flow velocity, S is energy slope, γ s and γ are specific weights of sediment and water, respectively, d50 is median particle diameter, and D is flow depth. Also, based on Bagnold’s stream power concept (Yang, 2002), Ackers and White (1973) developed the following formula.
qt =
⎛ γDS ⎞ ⎟ ⎜ γ XD ⎜ ρ ⎟ Ggr = ⎟ ⎜ dγ S ⎜ V ⎟ ⎟ ⎜ ⎠ ⎝
n
(4)
where X is the rate of sediment transport in terms of mass flow per unit mass flow rate, Ggr and n are a dimensionless function and transition exponent, respectively. Yang (1972, 1973) stated that the rate of work being done by a unit weight of water in transporting sediment should be directly related to the rate of work available to a unit weight of water. The rate of energy per unit weight of water available for transporting water and sediment in an open channel with reach length x and total drop of Y is dY dx dY (5) = = VS dt dt dx where VS is defined as the unit stream power. Yang (1972, 1973) considered the following relation based on unit stream power concept to determine total sediment concentration. φ (Ct ,VS ,U * ,υ , ω , d ) = 0 (6) where Ct is total sediment concentration, U * is shear velocity sediment.
(gDS )
0.5
, and ω is fall velocity of
3.4 Other transport functions Laursen (1958) developed a functional relationship between the flow condition and the resulting sediment discharge. 76 n ⎞ ⎛U ⎞ ⎛ d ⎞ ⎛ τ′ (7) Ct = 0.01γ ∑ pi ⎜ i ⎟ ⎜⎜ − 1⎟⎟ f ⎜⎜ * ⎟⎟ i =1 D τ ⎝ ⎠ ⎝ ci ⎠ ⎝ ωi ⎠ International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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where pi is percentage of materials available in size fraction i , τ ci is critical tractive stress for sediment size d i , and τ ′ is bed shear stress 1/ 3
τ′ =
ρV 2 ⎛ d 50 ⎞
(8) ⎜ ⎟ 58 ⎝ D ⎠ As shown in equations Eqs. (3), (4), (6), and (8), variables V, S, D, and d 50 are important factors needed for the computation of shear velocity, shear stress, unit stream power, and eventually total bed material load concentration.
4 Regression approach Regression approach is an alternative to estimate the sediment load when there is no sediment data, such as particle size, to use a predictive sediment transport formula. Sediment-discharge rating curve is a regression approach that connects the river hydrology to sediment transport. The rating curve estimate sediment load using flow discharge in the following form Qs = aQwb (9) where Qs is suspended sediment discharge, Qw is flow discharge, and a and b are the coefficients determined by regression analysis. Flow velocity and flow depth are two required data for using rating curve. These two parameters, which are measured in gauging stations, are the most important factors to determine the flow conditions. Water surface slope is another important parameter of flow regime. Yang (1975) defines the unit stream power as the V and S product. The rate of work being done by a unit weight of water in transporting sediment must be directly related to the rate of work available to a unit weight of water. Thus, total sediment concentration or total bed material load must be directly related to unit stream power (V × S ) . While rating curve can simulate a linear relation between flow characteristics, it seems necessary to use artificial neural networks, which are suited to complex nonlinear models, for the analysis of relation between hydrology and sediment transport. In this research, a nonlinear regression is developed using ANN to reveal the relation among flow velocity, flow depth; water surface slope, and median sediment particle size to develop a nonlinear regression among flow characteristics and sediment transport using ANN. 5 Application of ANN for forecasting total bed material sediment discharge 5.1 Parameters and data used in training the ANN model Important parameters in most sediment transport formulas are average flow velocity V, water surface slope S, average water depth D, and median particle diameter d50 . In this study, field data published by Williams and Rosgen of the U.S. Geological Survey in 1986 are used. These data were collected from the following river stations in the United States: 1- Susitna River near Talkeetna, Alaska (1982-1985) 2- Chulitnariver below Canyon near Talkeetna, Alaska (1982-1985) 3- South Fork of Salmon River near Cascade, Idaho (1985-1986) 4- Clearwater River at Spalding, Idaho (1972-1979) 5- Wisconsin River at Muscoda, Wisconsin (1976-1979) 6- Chippewa River near Carville, Wisconsin (1976-1979) 7- Muddy Creek near Pinedale, Wyoming (1975-1976) 8- Snake River near Anatone, Washington (1972-1979) 9- Black River near Galesville, Wisconsin (1977-1979) 10- Tanana River at Fairbank, Alaska (1977-1981) Measured V, S, D, and d 50 values are used as input data to the model and total bed material sediment discharge in tons per day as output of model. A total of 250 samples were used for training the model and 76 samples, or about 25 percent of total measured data randomly selected to test or verify the model. Number of samples used for training and testing of ANN model are shown in Table 1 for each river. Table - 278 -
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2 shows the statistic parameters of samples used for training and testing the ANN. 5.2 QNET for training ANNs To test the ANN models by trial-and-error, the public domain QNET Software is used. QNET back propagation neural networks are multi-layered and feed-forward (connections must connect to the next layer) in design. QNET can perform the data normalization automatically. If this option is selected during training setup, all data for the nodes in the input layer and/or training targets in the output layer will be normalized between the limits of 0.15 and 0.85. Table 1
Number of samples used for training and testing the ANN model Number of samples Location Total Training Testing Susitna River near Talkeetna, Alaska 29 24 5 Chulitna River below Canyon near Talkeetna, Alaska 36 26 10 South Fork of Salmon River near Cascade, Idaho 52 42 10 Clearwater River at Spalding, Idaho 44 34 10 Wisconsin River at Muscoda, Wisconsin 37 27 10 Chippewa River near Caryville, Wisconsin 15 10 5 Muddy creek near Pinedale, Wyoming 15 10 5 Snake River near Anatone, Washington 45 35 10 Black River near Galesville, Wisconsin 22 17 5 Tanana River at Fairbank, Alaska 31 25 6 Total 326 250 76 Table 2
Statistic parameters of samples used for ANN model
Data
Training or calibration data
Testing or verification data
Input data
Average velocity (m/s) Average depth (m) Water surface slope (m/m)
d50 Output data Input data
Total sediment load (ton/day) Average velocity (m/s) Average depth (m) Water surface slope (m/m)
d50 Output data
Total sediment load (ton/day)
Average
Max
Min
1.48 2.47 0.00072 0.68
3.40 5.80 0.0083 2.00
0.23 0.16 0.00011 0.19
Standard deviation 0.78 1.55 0.00063 0.40
14145.93
554152.30
0.65
41391.42
1.44 2.43 0.00068 0.71
2.80 5.30 0.0015 1.84
0.23 0.22 0.00009 0.27
0.77 1.55 0.00042 0.37
12621.71
144443.52
3.05
27572.38
The learning rate coefficient (η ) can be controlled manually during training or QNET can control it automatically using its Learn Rate Control (LRC) feature. The learning rate coefficient determines the size of the node weights adjustments during training. The learning rate coefficient Eta’s valid range is between 0.0 and 1.0. LRC will drive η higher or lower in a systematic fashion depending on the current learning activity. If the network appears to be learning at a relatively slow rate, η is driven up quickly. Conversely, if the network is learning at a fast pace, QNET will hold η constant or even lower it to avoid instabilities. QNET gives the option of selecting four transfer functions: the sigmoid, Gaussian, hyperbolic tangent and hyperbolic secant. These functions set the output signal strength between 0.0 and 1.0. As it is seen in Eq. (1), the input signal to the transfer function is the dot product of all the node’s input signals and the node’s weight vector plus bias factor. As it is explained above, QNET has a high and intelligent ability to simulate complex networks easily while preparing a code for these networks could be excruciating. For practical problems, using an easy International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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method, which is usable for different cases, is more acceptable than sophisticated methods. Also, using low cost public domain software would be more satisfactory for researchers, especially students, instead of costly software like MATLAM. In summary, QNET is a professional user friendly software and that is why it was used to make a lot of simple and complex networks in this research. 5.3 Statistic analysis tools QNET uses two statistic parameters to determine the accuracy of the models. These parameters are the correlation coefficient and root mean square error and their values are monitored during training process. Correction Coefficient = Root Mea Square Estimation =
∑( x − x)( y − y )
(10)
∑ ( x − x) 2 ( y − y ) 2 1 n
∑ (y n
i =1
actual
i
− y forecast
)
2
i
(11)
5.4 ANN model development For choosing the best network for forecasting, different models were built with various nodes and middle layers. In Learn Rate Control (LRC) feature, 0.001 and 0.15 were used as the minimum and maximum limit for auto control of learning rate. In this study, 0.8 was used for momentum factor α , which damps high frequency weight changes and can be set in the 0.8 to 0.9 range for the majority of networks. The input combinations to estimate total sediment load are 1) V, S, D, and d50 ; 2) V, S, and D; and 3) V and D; 4) V and S; 5) S and D. The first combination includes hydrological and sediment parameters required for the majority of sediment transport formulas. Other four combinations include hydrological dominant parameters of flow conditions. After try-and-error tests, the best network found in this study is the one with 3 layers including one input layer, one middle layer, and one output layer. The best model consists of 4 inputs including V, S, D, and d50, 7 middle, and one output nodes which should have a Sigmoid transfer function. The Sigmoid function represented by 1/(1+e − x ) and set the output signal strength between 0 and 1. The correlation coefficient and RMS error of this ANN are 0.97 and 0.013 for the training data set and 0.95 and 0.011 for the test data set, respectively. The final architecture of the ANN model is given in Table 3.
Model inputs V, S, D, d 50
Model output Total sediment discharge
Table 3 Characteristics of the final ANN model ANN Transfer Training set structure function Correlation RMSE (4,7,1)
Sigmoid
0.97
0.013
Test set Correlation RMSE 0.95
0.011
Accuracy of different ANN models developed in this study, is summarized in Table 4. The ANN model (4-7-1) which has been trained using four important parameters of sediment transport formulas has slightly more accurate results than other models with four inputs. The accuracy of the chosen ANN model (4-7-1) in comparison with other models of 4 inputs, specially those ones with more than one hidden layer, has insignificant difference. According to the intricacy and longer time needed for training of models with more than one hidden layer, the selection of ANN model (4-7-1) is reasonable. It can be concluded from Table 4 that those models which use only hydrological data as input, especially the ANN (3-4-1) with three dominant parameters of flow conditions, are as accurate as the ANN (4-7-1), which uses both hydrological and sediment parameters as input. In other words, the total sediment load is estimable using hydrological data accurately. Using hydrological data, which are measured in gauging stations, is easier than using both hydrological and sediment data. This would be more useful in some cases that there is only measured hydrological data and no sediment data, such as many rivers in Iran. However, it should be noted that most sediment transport formulas include a parameter for sediment particle size. - 280 -
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Table 4 Input data
Hydrological and sediment
Combination Dominant parameters of sediment transport formulas Dominant parameters of flow conditions
Hydrological
Parameters of rating curve Unit stream power
.
Other combinations
Sediment (kg/s)
Accuracy of some ANN models
Model inputs V, S, D, d50 V, S, D, d50 V, S, D, d50 V, S, D, d50 V, S, D V, S, D V, S, D V, S, D V, D V, D V, D V, S V, S V, S S, D S, D S, D
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Correlation 0.92 0.97 0.97 0.97 0.75 0.96 0.96 0.97 0.89 0.95 0.94 0.89 0.94 0.93 0.94 0.95 0.95
RMSE 0.020 0.013 0.013 0.013 0.035 0.014 0.012 0.012 0.023 0.015 0.017 0.023 0.017 0.019 0.017 0.014 0.014
Test set Correlation 0.87 0.95 0.95 0.94 0.62 0.95 0.94 0.94 0.91 0.91 0.93 0.79 0.74 0.79 0.93 0.94 0.95
RMSE 0.020 0.011 0.011 0.012 0.034 0.011 0.012 0.012 0.015 0.015 0.014 0.023 0.023 0.022 0.012 0.012 0.011
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The comparison of the measured data and estimated total sediment load using ANN for training and testing procedures are shown in Figs. 2 and 3, respectively. It can be seen that the estimates of ANN model are close to the measured data and they are in good agreement. It can be concluded that the ANN model can be used to detect hidden relationship between sediment transport rate and some important parameters of transport formulas or flow conditions such as average velocity, water surface slope, average water depth, and median particle diameter.
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ANN (4-7-1) Fig. 3 Comparison of measured data and ANN results (testing output)
6 Comparison of ANN and sediment transport equations results 6.1 Field data 24 sets of field data with accurate hydraulic information, not used in the training and verification of ANN in previous section, are used to compare the results of ANN model and sediment transport formulas. Field data include average flow velocity, average flow depth, channel width, water surface slope, sediment particle size distribution, water temperature and total bed-material load. These data were measured from: Chippewa River at Durand, Wisconsin (1975-1979), Chippewa River near Pepin, Wisconsin (1976-1979), and Wisconsin River at Muscoda, Wisconsin (1976-1979). The range of data used to test the ANN model results and five sediment transport formulas are summarized in Table 5. The five formulas tested are Yang (1973), Engelund-Hansen (1972), AckersWhite (1973), Laursen (1958) and Toffaleti (1969). The first four formulas were considered by ASCE (1982) as the four most accurate formulas for total bed material load in the sand size range. Toffaleti formula is included in the test because it was developed for sand transport in rivers similar to those used in this paper. Table 5 Data Average velocity (m/s) Average depth (m) Water surface slope (m/m) Water surface width (m) Water temp (Co) d35 (mm) d50 (mm) d65 (mm) d90 (mm) - 282 -
Statistic parameters of samples used for comparison Average Max Min Standard deviation 0.78 2.4 0.48 0.38 1.33 3.2 0.61 0.57 0.00047 0.0037 0.00017 0.00069 236.63 299 59 53.12 13.2 25.5 3.5 7.45 0.46 0.92 0.35 0.12 0.63 1.57 0.42 0.27 1.08 4.89 0.48 0.97 4.94 24.73 0.92 6.35 International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 274–286
The Yang, Engelund-Hansen and Ackers-White formulas are all based on the power concept. EngelundHansen and Ackers-White formula are based on Bagnold’s stream power concept indirectly (Yang, 2002), and Yang’s formula is based on his unit stream power concept directly. Toffaleti approach is based on the concept of Einstein (1950) and Einstein and Chien (1953). Laursen formula is developed using shear stress concept. 6.2 Results Figure 4 shows the comparisons between computed results from five sand transport formulas and the computed results from the ANN models. It shows that ANN (3-4-1) has a trend to overestimate small loads and underestimate large sediment loads while ANN (4-7-1) estimates sediment load with a more stable trend.
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The accuracy of each formula and ANN are shown in Table 6. This table shows that the ANN model has accuracy similar to the well-known total bed material load formulas. It shows that the ANN, Yang, and Laursen approaches are more accurate than other methods for estimating total sand transport rates in rivers compared in this study. The ANN (3-4-1), which uses three dominant parameters of flow conditions; V, S, and D, as input data; has about the same accuracy as the ANN (4-7-1) using V, S, D, and d50 four parameters as input. This reveals that the increasing number of parameters of input data set for training ANN, which is a regression approach, may not increase the accuracy of estimation due to complexity of nonlinear regression calculation process. Table 6 Rank 1 2 3 4 5 6 7
Comparison of correlation coefficients of six formulas and the ANN model Equation Correlation ANN (3-4-1) 0.998 Yang 0.996 ANN (4-7-1) 0.996 Laursen 0.995 Engelund-Hansen 0.979 Ackers-White (d50) 0.935 Toffaleti 0.833
The Yang and Laursen formulas are based on physical rules of sediment transport, but ANN is based on trial-and-error interpolation method of the data used in this study. The ANN model in this study has been developed to simulate the inner relation among four important parameters in sediment transport formulas. As shown in Table 6, the Yang formula based on the unit stream power concept has accuracy similar to ANN. Also, the Engelund-Hansen and Ackers-White formulas, which have been developed based indirectly on Bagnold’s stream power concept, also have good agreement with measured data. The International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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Laursen formula, which is not based on power concept, shows the accuracy close to that of Yang and ANN. The Toffaleti formula shows the least agreement with measured data. The Toffaleti formula has been developed based on a probabilistic approach introduced by Einstein (1950). The high correlations shown in Table 6 are due to the inclusion of a set of data collected during a flood with measured total sediment discharge of 1,714 kg/s, which is significantly higher than other used set data in this section. From statistical point of view, if one set of data has the value outside of other data range, the inclusion of this particular data set can have significant impact on the overall correlation coefficient. Table 7 shows the comparison of correlation coefficient if the data set collected during the flood is excluded. In this case, the correlation coefficients are more in line with other published results in the literature. Table 7
Comparison of correlation coefficients of six formulas and the ANN model without flood set data Rank Equation Correlation 1 ANN (3-4-1) 0.852 2 Yang 0.845 3 Laursen 0.819 4 ANN (4-7-1) 0.815 5 Engelund-Hansen 0.796 6 Ackers-White (d50) 0.784 7 Toffaleti 0.771
Comparisons between rankings of selected sediment transport formulas by ASCE and this study are shown in Table 8. It shows that the accuracies of selected sediment transport formulas are similar to those made by ASCE (1982) based on laboratory and field data not used in this paper. In both studies, Yang (1973) and Laursen (1958) methods are more accurate than others. Table 8 Summary of rating of selected sediment transport equations Equation Ranking Ranking by ASCE (1982) by this research Ackers-White 3 4 Engelund-Hansen 4 3 Laursen 2 2 Yang 1 1
7 Summary and conclusions ANN model is used for estimating total sediment load in rivers in this study. 250 samples were used for training the ANN model and 76 samples were used to verify the model. Inputs for training the model include average flow velocity V, water surface slope S, average water depth D, and median particle diameter d50. The results show that the ANN model with four input nodes V, S, D, and d50 is able to estimate the sediment transport rate accurately but it is worthy of mention that the ANN with three input data estimates the sediment load as accurate as some well known formulas. This ANN model has been developed using the minimum number of three flow effective parameters including V, S, and D. The minimum required input data make the ANN model in this paper is easy to use for estimating the total bed material load when there is no sediment data to use the well known sediment transport formulas such as Yang and Laursen. On the other hand, this ANN model might not have the stable trend to estimate sediment load like those ANN with four input and some reliable formulas such as Yang. Comparisons of accuracy of the ANN model against sediment transport formulas were made using 24 sets of measured samples not used in the training and testing of the ANN model. The comparison shows that the unit stream power approach used by Yang is most reliable for the estimation of sediment transport rate. The Laursen, Engelund-Hansen, and Ackers-White formulas are also fairly accurate. The Toffaleti formula based on probabilistic approach is the least accurate one. This ranking is similar to the ranking of accuracy of sediment transport formulas by the ASCE Sedimentation Committee (1982) without using the - 284 -
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ANN approach. In summary, the ANN model trained using three important parameters V, S, and D can be used to estimate total bed material load in lack of sediment data. However, it should be pointed out that the accuracy of an ANN model is data sensitive, and should not be applied to the conditions outside of the data range used in training the ANN model without verification. In other words, for other cases with data range out of this study, an appropriate ANN should be trained using data set of that case study to estimate total sediment load. References Ackers P, and White W. R. 1973, Sediment transport: new approach and analysis. Journal of the Hydraulics Division, ASCE Vol. 99, No. HY11, pp. 2041–2060. Agarwal A. h., Mishra S. K., and Singh J. K. 2006, Simulation of runoff and sediment yield using Artifitial Neural Networks. Biosystems Engineering, Vol. 97, No. 4, pp. 597–613, Published by Elsevier Ltd. ASCE Task Committee on Relations between Morphology of Small Streams and Sediment Yield of the Committee on Sedimentation of the Hydraulics Division 1982. Relationships between morphology of small stream and sediment yields,” Journal of the Hydraulics Divisions, ASCE, Vol. 108, No. HY11, pp. 1328–1365. Bagnold R, A. 1966, An approach to the sediment transport problems from general physics. U.S. Geological Survey Professional Paper 422-J. Cigizoglu H. 2002, Suspended sediment estimation for rivers using artificial neural networks and sediment rating curves. Turkish Journal of Engineering Environmental Sciences, Vol. 26, No. 1, pp.27–36. Einstein H. A. 1950, The bed-load function for sediment transportation in open channel flows. U.S. Department of Agriculture, Soil Conservation service, Technical Bulletin No. 1026. Einstein H. A. and Chen N. 1953, Transport of sediment mixtures with large ranges of grain sizes. Missouri River Division Sediment Series No. 2, U.S. Army Corps of the Engineers Missouri River Division, University of California Institute of Engineering Research, Berkley. Engelund F. and Hansen E. 1972, A Monograph on Sediment Transport in Alluvial Streams. Teknisk Forlag, Copenhagen. Laursen E. M. 1958, The total sediment load of streams. Journal of the Hydraulics Division, ASCE Vol. 84, No. 1, pp. 1530–1–1530–36. Lingireddy S. and Brion G. M. 2005, Artificial Neural Networks In Water Supply Engineering. Published by the American Society of Civil Engineers. Mekonnen, M. A., Dargahi, B. 2007, Three dimensional numerical modeling of flow and sediment transport in rivers. International Journal of Sediment Research, Vol. 22, No. 3, pp. 188-198. Meyer-Peter E. and Müller R. 1948, Formula for bed-load transport. Proceedings of International Association for Hydraulic Research, 2nd meeting, Stockholm. Nagy H. M., Watanabe K., and Hirano M. 2002, Prediction of sediment load concentration in rivers using artificial neural network model. Journal of Hydraulics Engineering, ASCE Vol. 128, No. 6, pp. 588–595. Nourani V., Aalami M. T., Nazmara H., Hosseinzadeh H. 2006, Estimation of suspended sediment load in Talkhehrood using artificial neural network. 3rd National Comgress on Civil Engineering, University of Tabriz, Tabriz, Iran, 1-3 May 2007, Water Engineering, Vol., Paper No. 442. Nourani V., Aalami M. T., Aminfar M. H., Nourpour A. 2006, Application of artificial neural network in sensitivity analysis of effective parameters of suspended sediment load (Case Study: Lighvan Chay, 3rd National Congress on Civil Engineering, University of Tabriz, Tabriz, Iran, 1-3 May 2007, Water Eng. Vol., Paper No. 57. Raghuwanshi N. S., Singh R., and Reddy L. S. 2006, Runoff and sediment yield modeling using artificial neural networks: Upper Siwane River. India. Journal of Hydrologic Engineering, Vol. 11, No. 1. Rumelhart D. E., Hinton G. E., and Williams R. J. 1986, Learning internal representation by error propagation. MIT Press, Cambridge, Massachusetts, USA. Toffaleti F. B. 1969, Definitive computations of sand discharge in rivers. Journal of the Hydraulics Division, ASCE Vol 95, No. HY1, pp.225–246. Wiliams P. and Rosgen L. 1986, Measured total sediment load for 93 United States streams. U.S. Geological Survey. Wu, B., Maren, D. S., and Li, L., 2008, Predictibility of sediment transport in the Yellow River using selected transport formulas. International Journal of Sediment Research, Vol. 23, No. 4, pp. 283-298. Yalin, M. S., 1963, An expression for bed load transportation. Journal Hydraulic Division, ASCE Vol. 89, pp. 221–250. Yang C. T. 1972, Unit stream power and sediment transport. Journal of the Hydraulics Division, ASCE Vol. 98, No HY 10, pp. 1805–1826. Yang C. T. 1973, Incipient motion and sediment transport. Journal of the Hydraulics Division, ASCE Vol. 99, No. HY10, pp. 1679–1704. Yang C. T. 1996, Sediment Transport Theory and Practice. McGraw-Hill Companies, Inc. (reprint by Krieger P International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 274–286
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