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ELSEVIER
MATHEMATICAL
AND
COMPUTER MODELLING
D I FI E! (~'11""
Mathematical and Computer Modelling 40 (2004) 839-846 www.elsevier.com/locate/mcm
Rainfall-Runoff Model Using an Artificial N e u r a l N e t w o r k A p p r o a c h S. RIAD AND J. MANIA Universit4 des Sciences et Technologies de Lille-USTL-LML-EPUL. UMR CNRS 8107 Ecole Polytechnique Universitaire de Lille, D6partement de g~otechnique et g6nie civil Cit~ scientifique, Avenue Paul Langevin, 59655 Villeneuve d'Ascq Cedex, France ©polyt ech-lille, fr rsouad_2OO4©yahoo, fr
L. BOUCHAOU Universit6 Ibn Zohr, Facult~ des sciences, D6partement de G6ologie Laboratoire de G~ologie Appliqu6e et G~oenvironnement (G.A.G.E) Equipe d'Hydrog~ologie, BP 28/S, 80000 Agadir, Maroc Bouchaoul©cara~ail. com
Y. NAJJAR Kansas State University, Department of Civil Engineering Manhattan, KS 66505, U.S.A. ea4146©ksu, edu
(Received March 2003; revised and accepted October 2003) A b s t r a c t - - T h e use of artificial neural networks (ANNs) is becoming increasingly common in the analysis of hydrology and water resources problems. In this research, an ANN was developed and used to model the rainfall-runoff relationship, in a catchment located in a semiarid climate in Morocco. The multilayer perceptron (MLP) neural network was chosen for use in the current study. The results and comparative study indicate that the artificial neural network method is more suitable to predict river runoff than classical regression model. (~) 2004 Elsevier Ltd. All rights reserved. Keywords--Rainfall-runoff, Catchment, Semiarid climate, MLP, Modelling, Artificial neural network, Multiple regression, Morocco.
1. I N T R O D U C T I O N T h e A N N s models are powerful prediction tools for the relation between rainfall and runoff parameters. T h e results will s u p p o r t decision making in the area of water resources planning and m a n a g e m e n t . Besides, t h e y assist u r b a n planners and m a n a g e r s u n d e r t a k e t h e necessary measures to face the b a d predictions. Thus, t h e y help avoid losses in public and private properties, and health and ecological hazards t h a t are likely to occur due t o flooding. Moreover, the A N N models have been used increasingly in various aspects of science and engineering because of its ability to model b o t h linear and nonlinear systems w i t h o u t the need to make a n y a s s u m p t i o n s as are implicit in most traditional statistical approaches. In some of the hydrologic problems, A N N s have already been successfully used for river flow prediction [1-8],
0895-7177/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.10.012
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for rainfall-runoff process [9-11], for the prediction of water quality parameters [12] and for characterization of soil pollution [13]. In addition, ANNs are applied for prediction of evaporation [14], for rainfall-runoff forecasting [15-19], for prediction of flood disaster [20], and for river flow time series prediction [21]. In these hydrological applications, a multilayer feed-forward backpropagation algorithm is used [22]. It usually is composed of a large number of interconnected nodes, arranged in an input layer, an output layer, and one or more hidden layers. The transfer function selected for the network was the sigmoid function. The aim of this paper is to model the rainfall-runoff relationship in the Ourika catchment located in semiarid climate in Morocco using a black box type model based on ANN methodology.
2. T H E A R T I F I C I A L N E U R A L N E T W O R K S
APPROACH
2.1. O v e r v i e w of A N N The ANN technology is an alternate computational approach inspired by studies of the brain and nervous systems [23]. It is based on theories of the massive interconnection and parallel processing architecture of biological neural systems. The main theme of ANN research focuses on modelling of the brain as a parallel computational device for various computational tasks that were performed poorly by traditional serial computers. ANNs have a number of interconnected processing elements (PEs) that usually operate in parallel and are configured in regular architectures. The collective behavior of ANN, like a human brain, demonstrates the ability to learn, recall, and generalize from training patterns or data. The advantage of neural networks is they are capable of modelling linear and nonlinear systems. In the present study, we use an MLP trained with a backpropagation algorithm to predict the drainage basin runoff. The MLP consists of an input layer consisting of node(s) representing various input variable(s), the hidden layer consisting of many hidden nodes, and an output layer consisting of output variable(s). The input nodes pass on the input signal values to the nodes in the hidden layer unprocessed. The values are distributed to all the nodes in the hidden layer depending on the connection weights W~j and Wjk [24-26] between the input node and the hidden nodes. Connection weights are the interconnecting links between the neurons in successive layers. Each neuron in a certain layer is connected to every single neuron in the next layer by links having an appropriate and an adjustable connection weight.
ot
AWeights Wi~
Weights
i Pt
6,
1
~
Input layer Input
Xt ~
Hidden layer Neuron
)
Output layer Output
~ S
X. Figure 1. Architecture of the neural network model used in this study.
Rainfall-Runoff Model
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The architecture of the neural network used in this study and the schematic representation of a neuron are shown in Figure 1. Each node j receives incoming signals from every node i in the previous layer. Associated with each incoming signal (Xi) is a weight (Wij). The effective incoming signal (Sj) to node j is the weighted sum of all the incoming signals and bj is the neuron threshold value. n
sj =
xiwq + b.
(1)
i=l
The effective incoming signal, Sy, is passed through a nonlinear activation function to produce the outgoing signal (yj) of the node. The most commonly used in this type of networks is the logistic sigmoid function. This transfer function is continuously differentiable, monotonic, symmetric, bounded between 0 and 1 [27]. It is expressed mathematically as: 1
f (Sj)
-
1 + e-sj
(2)
2.].. A N N P e r f o r m a n c e In this study, both statistical and graphical criteria were adopted to select the desired optimal network model. The statistical criteria consist of average squared of error (ASE), coefficient of determination (R 2) and the mean absolute relative error (MARE). They are given by N
2
ASE = 4=1
Y N(Q
E R 2 =
1 -
'
(3)
~)2
t~ - Qt~
i=l
N
,
(4)
E (Qti-Qt0 2 i=1 N
MARE = ~=1
N
x
loo,
(5)
where Qti and Qti are respectively, the actual and predicted value of flow (normalized between 0 and 1), (~ti is the mean of Qti values and N is the total number of data sets. The R 2 statistic measures the linear correlation between the actual and predicted flows values. The ASE and MARE statistic measures are used to quantify the error between observed and predicted values. The optimal value for R 2 is equal to 1.0 and for ASE and MARE is equal to 0.0. The graphical performance indicator gives better results when the data pairs are closing to 45 ° line and the good superposition between the desired and calculated flow values in the training and testing phases. For the data set considered in the present study, the input variables as well as the target variables are first normalized linearly in the range of 0 and 1. This range is selected because of the use of the logistic function (which is bounded between 0.0 and 1.0) as the activation function for the output layer, i.e., equation (2). The normalization is done using the following equation. .,~ =
X -- X m i n . X m a x -- X m i n '
(6)
where .~ is the standardized value of the input, X m i n and Xmax are respectively, the minimum and maximum of the actual values, in all observations and X is the original data set. The main reason for standardizing the data matrix is that the variables are usually measured in different units. By standardizing the variables and recasting them in dimensionless units, the arbitrary effect of similarity between objects is removed.
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3. T H E
STUDY
CATCHMENT
AND
DATABASE
In the present study, the flow and rainfall series observed in Ourika basin at Aghbalou station in Morocco is analyzed using the ANN model. The Ourika basin is the most important subcatchment of Tensift basin drainage located in semi-arid region of Marrakech, which is draining an area of about 503 km 2. (See Figure 2.) Tensift drainage basin
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Figure 2. Location of Ourika Wadi in t h e Tensift basin.
The Rainfall and Runoff daily data at the Aghbalou station was used for model investigation. The data contains information for a period of seven years (1990 to 1996). The entire database is represented by 2550 daily values of rainfall and runoff pairs. The ANN model was trained using the resulting runoff and rainfall daily data. The database was collected by the Rabat hydraulic administration. The input vector is represented by rainfall and runoff values for the preceding seven days, (i.e., t - 1, t - 2, t - 3, t - 4, t - 5, t - 6, t - 7) as well as the rainfall value expected for day t. Accordingly, the output vector represents the expected runoff value for day t (Qt).
4. R E S U L T S
AND
DISCUSSIONS
The database compiled represents seven years daily sets of rainfail-runoff values for the Ourika Wadi basin. In this paper, we used the data for the last year (1996) for model testing, while the other remaining data (1990 to 1995) was used for model training/calibration. The training phase of ANN model was terminated when the average squared error (ASE) on the testing databases was minimal. The goal of the training process is to reach an optimal solution based on some performance measurements such as ASE, coefficient of determination known as R-square value (R2), and the MARE. Therefore, required ANN model was developed in two phases: training (calibration) phase, and testing (generalization or validation) phase. In the training phase, a larger part for database (six years) was used to train the network and the remaining part of the database (one year) is used in the testing phase. Testing sets are
Rainfall-Runoff Model
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Table 1. Statistical accuracy measures of this network model at testing and training phases. ASE R2 MARE Training Phase
0.000076
0.948
1.029%
Testing Phase
0.000007
0.917
1.524%
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.,~ 14o ~ 120 ,~ 100
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Actual flow (m3/s) Figure 3. Comparison between the actual and ANN predicted flow values.
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Actual flow (m~/s) Figure 3. (cont.) Table 2. Statistical parameters of the predicted and actual flow at training and testing phases. (b) Testing Phase
(a) Training Phase Statistical Parameters
Actual Flow (ma/s)
Predicted Flow (ma/s)
Statistical Parameters
Actual Flow (ma/s)
Predicted Flow (ma/s)
Average
9.79
9.70
Average
2,71
3.25
Standard of Deviation
18.93
17.81
Standard of Deviation
4.33
3.96
Minimum
0
1.37
Minimum
0.10
1.38
Maximum
184
187.17
Maximum
31.90
31.16
Coefficient of Variation
1.60
1.22
Coefficient of Variation
1.93
1.84
u s u a l l y used to select the best p e r f o r m i n g network model. I n this research, t h e A N N was o p t i m a l at 600 i t e r a t i o n s w i t h 12 h i d d e n nodes. T h e c o r r e s p o n d i n g a c c u r a c y m e a s u r e s of this n e t w o r k m o d e l on t e s t i n g a n d t r a i n i n g d a t a are given i n t h e following t a b l e (Table 1). Generally, accuracy m e a s u r e s o n t r a i n i n g d a t a are b e t t e r t h a n those o n t e s t i n g data.
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160
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40
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80
100
120
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160
180
Actual flow (m~/s) Figure 4. Comparison between the a~tual and predicted flow values by multiple linear regression (MLR).
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15
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,
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0.948 I 0.924 0.917 i 0.888
The comparison between the predicted and actual flow values at training and testing phases show excellent agreement with the R 2 are respectively 0,948 and 0,917 (see Figure 3). Note that, data pairs closer to the 45 ° line represent better prediction cases. The good performance and convergence of the model are illustrated in Figure 3. The statistical parameters of the predicted and actual values of flow for the entire database are practically identical (see Table 2). ]In order to evaluate the performance of the ANN, the multiple linear regression (MLR) technique was applied with the same data sets used in the ANN model. Figure 4 shows the comparative results obtained by MLR technique. The R 2 values for MLR and ANN models are presented in Table 3. Apparently, the ANN approach gives much better prediction than the traditional method (MLR). 5. C O N C L U S I O N The artificial neural network (ANN) models show good capability to model hydrological process. T h e y are useful and powerful tools to handle complex problems compared with the other traditional models. In this study, the results obtained show clearly t h a t the artificial neural networks are capable of model rainfall-runoff relationship in the arid and semiarid regions in which the rainfall and runoff are very irregular, thus, confirming the general enhancement achieved by using neural networks in m a n y other hydrological fields. T h e results and comparative s t u d y indicate t h a t the artificial neural network method is more suitable to predict river runoff than classical regression model. The ANN approach could provide a very useful and accurate tool to solve problems in water resources studies and management.
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