Neural Comput & Applic (2013) 23:873–880 DOI 10.1007/s00521-012-1011-5
ORIGINAL ARTICLE
A practical approach to formulate stage–discharge relationship in natural rivers Aytac Guven • Ali Aytek • H. Md. Azamathulla
Received: 9 May 2012 / Accepted: 21 June 2012 / Published online: 7 July 2012 Ó Springer-Verlag London Limited 2012
Abstract This study proposes a new formulation technique for modeling stage–discharge relationship, as an alternative approach to standard regression techniques. An explicit neural network formulation (ENNF) is derived by using data obtained from United States Geological Survey data base. The neural network model is trained and tested using time series of daily stage and discharge data from two stations in Pennsylvania, USA. The model is compared with the standard rating curve (SRC) technique. Statistical parameters such as average, standard deviation, minimum, and maximum values, as well as criteria such as root mean square error, the efficiency coefficient (E), and determination coefficient (R2) are used to measure the performance of the ENNF. Considerably, well performance is achieved in modeling streamflow by using ENNF. The comparison results reveal that the suggested formulations perform better than the conventional SRC. Keywords Stage discharge Neural networks Rating curve Modeling
A. Guven (&) Department of Civil Engineering, University of Gaziantep, 27310 Gaziantep, Turkey e-mail:
[email protected] A. Aytek Sahinbey Municipality, Gaziantep, Turkey e-mail:
[email protected] H. Md. Azamathulla River Engineering and Urban Drainage Research Centre (REDAC), Universiti Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Pulau Pinang, Malaysia e-mail:
[email protected]
1 Introduction Both the stage and the discharge of a stream vary most of the time and, in order to obtain a continuous record of discharge, the stage is recorded and the discharge computed from a correlation of stage and discharge. This correlation, or calibration, is known as the stage–discharge relationship. Accurate information about rate of flow in rivers is important for a variety of hydrologic applications such as water resources planning and operation, hydraulic and hydrologic modeling, water and sediment budget analyses, and design of storage and conveyance structures. However, the logistics of collecting direct measurement of discharge on a continuous basis is costly, especially during large flood events. Therefore, it is a common practice to convert records of water stages into discharges by using a relationship between stage and discharge [1]. Two distinct approaches for modeling stage–discharge relationships are available. The first approach is based on numerical solutions of the dynamic, unsteady, non-uniform flow equations [9, 10, 31]. This approach can be successfully used to model the observed stage–discharge relationships if accurate information on boundary conditions and channel geometry are available. The second approach is data driven and is based on nonlinear techniques. An alternative approach to standard regression methods is based on data-driven, nonparametric, nonlinear models where no defined functional form of the rating curve is required. A popular example of such alternative models is the artificial neural networks (ANNs). The use of ANN models has received much attention in several water resources and hydrologic applications [18, 19]. More details on NNs can be found in [3–6, 11–17, 21, 27]. The ANN models have been already converted into tractable design equations (ENNF) for solving civil engineering problems.
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Tawfik et al. [35] used a neural network to model stage– discharge rating curves for two locations on the White Nile River and found NN to be more accurate than three traditional techniques. This work is based on a simple fixed network structure, and the results were validated using records that were included in training the network. The problem of setting stage–discharge relationships using NNs has also been addressed by Jain and Chalisgaonkar [22], and Sudheer and Jain [33]. These analyses are based on idealized hypothetical looping rating curves rather than actual measurements. Supharatid [34] developed a neural network model to forecast tidal-level variations at the mouth of a river in Thailand. The neural network model was then used to construct stage–discharge rating curves, Bhattacharya and Solomatine [8] and Thirumalaiah and Deo [36] also applied ANN to model stage–discharge relationships, Liong et al. [25] used NN for river stage forecasting in Bangladesh, Torsten et al. [37] extrapolated stage–discharge relationships by numerical modeling, Sivapragasam and Mutill [32] proposed a new approach to discharge rating curve extension, Overleir [28] modeled stage–discharge relationships affected by hysteresis using the Jones formula and nonlinear regression, Overleir [29] developed a new rating curve model for estimating stage–discharge relationships in reservoir-like situations, Liao and Knight [24] studied analytic stage–discharge formulae for flow in straight trapezoidal open channels, Lohani et al. [26] used fuzzy logic to derive stage–discharge–sediment concentration relationships. Most recently, a new research by Azamathulla et al. [7] and Guven and Aytek [20], which presents a relatively new soft computing method (Gene-Expression Programming), was presented as an alternative tool for modeling stage–discharge relationship. The purpose of this paper is to develop a new formulation for stage–discharge relationship based on ENNF. The study applies two modeling techniques: an explicit neural network formulation (ENNF) and the standard rating curve (RC). To the best of the authors’ knowledge, the former technique (ENNF) has not been previously used to model stage–discharge relationships. The study uses concurrent stage and discharge measurements for two USGS station Pennsylvania, USA.
2 Methodology 2.1 Explicit neural networks formulation (ENNF) The term ENNF is not very common in ANNs studies, which has been first pronounced by Guven et al. [18]. ENNF utilizes the optimal weights obtained from welltrained ANN model and the nonlinear transfer function (e.g., log-sigmoid and tangent-hyperbolic) used in determining the total information (Eq. 2) a neuron receives (Eq. 3) and the scalar output (Eqs. 11, 12). A schematic
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representation of ENNF algorithm is given following section. ENNF translates the hidden neural network internal mechanisms into a transparent equation, which can be further utilized by researchers or practitioners, who want to re-evaluate the performance of corresponding ANN model by applying it on their own data set. In this study, the multilayer perceptron (MLP) neural network with one single hidden layer was considered. MLPs are layered feed forward networks typically trained with static backpropagation. These networks have found their way into countless applications requiring static pattern classification. Their main advantage is that they are easy to use and that they can approximate any input/output map. The key disadvantages are that they train slowly and require lots of training data (typically three times more training samples than network weights). The basic element of a ANN is an artificial neuron, which consists of three main components; weights, bias, and an activation function. Each neuron receives inputs xi (i ¼ 1; 2; . . .; n) attached with a weight wij (j C 1), which shows the connection strength for a particular input for each connection. Every input is then multiplied by the corresponding weight of the neuron connection and summed as n X Wi ¼ wij xj ð1Þ j¼1
A bias bi, is a type of correction weight with a constant nonzero value, is added to the summation in Eq. 1 as n X U i ¼ W i þ bi ¼ wij xj þ bi ð2Þ j¼1
In the architecture, logistic transfer function is utilized as 1 yi ¼ f ðUi Þ ¼ ð3Þ 1 þ eUi The Levenberg–Marquardt backpropagation algorithm was employed to minimize the MSE of the network in this study. The representation of ENNF in matrix form can be illustrated as: 2
w11 6w 6 21 6 6 .. 6 . 6 Ui ¼ 6 . 6 .. 6 6 . 6 . 4 .
w12
w22 .. . .. . .. .
wi1 wi2 wij 3 2 3 f ðU1 Þ a1 6 f ðU Þ 7 6 7 a 6 2 7 7 6 27 6 6 .. 7 6 .. 7 7 6 . 7 6 . 7 7 6 6 W ¼6 . 76 .. 7 7 þ ½B 6 .. 7 6 . 7 7 6 6 6 6 . 7 6 . 7 6 . 7 4 .. 7 5 4 . 5 2
f ðUj Þ y ¼ ½f ðWÞ
aj
3
2x 3 2 3 b1 1 7 7 6 7 w2j 7 6 x 6 b2 7 7 6 27 6 . 7 .. 7 .. 7 6 6 7 6 7 . 7 6 .. 7 . 7 7 6 6 . 7þ6 . 7 7 .. 7 6 . 7 6 . 7 . 7 6 . 7 . 7 6 6 7 6 7 . 7 6 . 7 .. 7 7 6 . 5 4 .. 5 4 .. 5 w1j
xj
bi
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where ai are weights used in hidden layer and B is bias, f represents the transfer function, y is the output of neural network. 2.2 Stage–discharge rating curve The functional relationship between stage and discharge can be established by field measurement of stage and discharge and thereafter can be expressed as a rating curve. Ideally, a rating curve describes a unique functional relationship between stage and discharge; therefore, it is obtained as a smooth and continuous curve with reasonable degree of sensitivity. The relation can be estimated by a sufficient number of measurements suitably distributed throughout the range in stage, taking into account the shape of the stage–discharge relationship. The number and spacing of the observations are made to conform to the relative frequency of flow at the various stages, that is, the number of observations at various sub-ranges is in proportion to the probable occurrence of discharge at these same ranges, covering the whole range of discharge for which the relation is plotted. The stage–discharge relationship may be expressed by an equation of the form Q ¼ Cðh þ aÞn
ð4Þ
which is the equation of a parabola where Q is the discharge, h is the gauge height, C and n are constants, and a is the stage at zero flow (datum correction). This equation may be transformed by logarithms to log Q ¼ log C þ n logðh þ aÞ
ð5Þ
which is in the form of the equation of a straight line y ¼ n0 x þ C 0
ð6Þ
where n0 is the gradient and C is the intersection of the line on the y axis. By plotting Q against (h ? a), therefore, on double logarithmic graph paper, a straight line is obtained. Often two or more straight lines may be required to fit the data, and it is usually possible, initially, to decide on the approximate location of the break points of each range by a careful investigation of the controls. The actual break points may be determined by solving the two equations concerned for Q and h or by purely graphical means. For very irregular channels, or for non-uniform flow, Eq. 6 cannot be expected to apply throughout the whole range of stage.
3 Study area and data used The data set used in this study was obtained from US Geological Survey (USGS). The time series of daily stage and discharge data from two stations; 01470500 (Lat
40°310 2100 , long 75°590 5500 ) Schuylkill River at Berne, PA, USA, and 01474500 (Lat 39°580 0400 , long 75°110 2000 ) Philadelphia County, PA, are used. The drainage areas at these sites are 919.45 km2 for the upstream station and 4,902.85 km2 for the downstream station. Information on the daily time series for these stations can be acquired from the USGS web server (http://webserver.cr.usgs.gov/sediment). The data of July 01, 2000–September 30, 2005 (84 % of the whole data i.e 1825) were chosen for the training of proposed ANN model, and data of October 1, 2005–September 30, 2006 (16 % of the whole data i.e 365) were chosen for testing of the model. In the upstream data set, the discharge values range between 2.12 and 971.03 m3/s, while in the downstream data set discharge values range between 0.76 and 125.41 m3/s.
4 Performance measures It is important to define the criteria by which the performance of the model and its prediction accuracy will be evaluated in model development process. Various statistical measures have been developed and used to assess the model performance. The determination coefficient (R2), the root mean square error (RMSE), the coefficient of efficiency (E), and the adjusted coefficient of efficiency (E1), defined as follows. Legates and McCabe [23] are used in the current study. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 1X 3 RMSEðm =sÞ ¼ ð7aÞ ðQe Þi ðQo Þi n i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Pn 1 i¼1 ðQe Þi ðQo Þi n RMSEð%Þ ¼ 100 ð7bÞ Qm 2 Pn i¼1 ðQe Þi ðQo Þi Eð%Þ ¼ 1 Pn ð8aÞ 2 i¼1 ðQe Þi Qm Pn i¼1 ðQe Þi Qo Þi E1 ð%Þ ¼ 1 Pn ð8bÞ i¼1 ððQe Þi Qm Þ where Qm, (Qo)i and (Qe)i are mean, observed, and estimated discharges, respectively; and n is the number observation data. The RMSE describes the average difference between model results and observations in units of the discharge (Eq. 7a) and can be normalized to provide a relative measure with respect to the mean discharge (Eq. 7b). Physically, the coefficient of efficiency, E, measures the differences between the observations and predictions relative to the variability in the observed data itself. According to Eq. 8a, E may range from -? to 1.0, where A value of 90 % and above indicates very satisfactory performance, whereas a value below 80 % indicates an unsatisfactory performance.
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Legates and McCabe [23] raised the issue of oversensitivity of E to extreme values (caused by squaring the difference terms), and therefore, introduced a modified coefficient of efficiency, E1, which uses the absolute differences, rather than their squares. The combined use of RMSE, E, and E1, will provide a sufficient assessment of each model’s performance and will allow comparison of the accuracy of the three modeling approaches used in this study.
5 Results and discussion The optimal network architecture which is related to the number of neurons in the hidden layer is one of the most important tasks in NN studies. Generally, the trial-anderror approach is used. In this study, the best architecture of the network was obtained by trying different number of neurons. The trial started from two, and the performance of each network was checked by employing Akaike information criterion (AIC) defined as [2]; AIC ¼ N lnðMSEÞ þ 2k
ð9Þ
where N is the number of exemplars in the training set, k is the number of network weights, and MSE is the mean squared error. AIC is used to measure the exchange between training performance and network size. The goal is to minimize AIC to obtain a network with the best generalization. The relationship between the number of neurons ranging from 2 to 5 and the corresponding AIC values obtained is presented in Fig. 1. As it is seen in Fig. 1a, AIC values decrease with increasing number of neurons in the training stage, which means that the
5.1 Derivation of ENNF for stage–discharge relationship Input parameters and weights of the trained NN were extracted to form an explicit expression in the following manner. Each input was multiplied by a connection weight (Eq. 1) and then biases were simply added to this multiplication (Eq. 2), and finally, the sum was transformed
(a) 0.980
(b)
-7600
1
-15200
R2 (Test) 0.996
-8400
0.992
0.960 -8800 0.950 0.940 2
3
4
-9200
0.984
-9600
0.98
5
-16400 -16800 1
No. of neurons
(d)
-8400
3
4
5
0.998
R2 (Train) AIC (Train)
-13200
0.996
-13600
0.994
-14000
0.992
-14400
-8800
0.994
R2
R2 (Test) AIC (Test)
AIC
R2
2
No. of neurons
(c)0.998 0.996
-16000
0.988
-9200 0.992 0.990
-9600 1
2
3
4
No. of neurons
123
5
0.99
-14800 1
2
3
4
No. of neurons
5
AIC
1
-15600
R2 (Train) AIC (Train)
AIC
-8000
R2
AIC (Test)
AIC
0.970
R2
Fig. 1 The effect of number of hidden neurons on the NN performance in a testing, b training sets for upstream (01470500) station and c testing, d training sets for downstream (01474500) station
architecture of the network improves in the learning process with increasing number of neurons. In the testing process, however, AIC values reduce with increasing number of neurons until the number of neurons reaches two and then the AIC values start to increase, which implies that the network becomes more generalized with increasing number of neurons until an optimum value is obtained. Beyond this optimum point, the network turns out to be specialized only on the training set and it deviates from producing reasonable results in the testing stage. This procedure is a common experience in NN studies. The determination coefficient, R2, is also shown in Fig. 1. The correlation coefficient seems to be slightly affected by increasing number of neurons in the training stage (Fig. 1a) up to two neurons beyond which no change was noticed. However, Fig. 1b shows that R2 starts to decrease with increase in the number of neurons after the two neurons. These findings are in agreement with previous studies on the AIC. Based on these analyses, the optimal architecture of the NN was constructed as 1–2–1, representing the number of inputs, neurons, and outputs, respectively.
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through a transfer function (sigmoid) (Eq. 3) to generate an output. In order to acquire accurate results from the ENNF, prior to the execution of the training process of the ANN, input and output parameters were normalized in the range of (-0.95; 0.95) by Cnormalized ¼ cC þ d
ð10Þ
where C represents parameters used in the NN training process, c and d are normalization coefficients of that particular parameter [4]. Taking these independent parameters into account, the stage–discharge relationship can functionally be expressed as 1076:5340
Q¼ 1þe
8:8664 1þe0:9931hþ4:6440
þ
1:1365 1þe3:2250h5:9326
2:3563
51:6736
R2 = 0.990 for upstream and RMSE = 0.378 and R2 = 0.995 for downstream station. Comparing the NN predictions with the observed data for the test stage demonstrates a high generalization capacity of the proposed model with relatively low error and high correlation (RMSE = 9.594 and R2 = 0.985 for upstream and (RMSE = 8.284 and R2 = 0.998 for downstream), which exhibits a successful performance of the NN model for estimating stage–discharge relationship both in training and testing stages (Table 1). Observed and estimated discharge for upstream (1470500) station in testing period (2006 water year) is given in Fig. 3. As it is seen from the Fig. 3, a flood is occurred from the date June 26 to June 29, 2006 with a magnitude of 275.17, 469.95, 971.03, and 336.89 m3/s,
ð11Þ Table 1 The daily statistical parameters of data set for the stations
for upstream station and, Q¼
1453:9072 35:7221
1 þ e1þe1:2426h1:3711
3:8206 1þe14:7307hþ14:2248
70:6852 þ 7:5859
Training period
Testing period
Upstream
Upstream
Downstream
Downstream
ð12Þ
Qm
21.56
94.86
24.29
109.31
for downstream station. Figure 2 compares the NN estimates to the observed data via scatter plots of training (Fig. 2a) and testing sets (Fig. 2b) for upstream station and; training (Fig. 2c) and testing sets (Fig. 2d) for downstream station. It is clearly noted in Fig. 2 that the proposed NN model has impressively well learned the nonlinear relationship between the input and the output variables with RMSE = 6.785 and
Sx Csx
26.27 5.29
109.11 3.96
61.82 11.55
147.72 5.43
2.12
2.24
2.52
5.68
xmax
399.17
1,310.75
971.03
1,483.44
1.22
1.15
2.54
1.35
Cv
Qm mean observed discharge, Sx standard deviation, Csx skewness, xmin minimum observed discharge, xmax maximum observed discharge, Cv coefficient of variation
(a)
(b) 1200
y = 1.005x + 0.031 R2 = 0.990
400 350 300
1000
Observed
Observed Discharge
450
250 200 150 100
y = 0.917x + 0.671 R2 = 0.985
800 600 400 200
50 0
0 0
50 100 150 200 250 300 350 400 450
0
Estimated Discharge(0147050_train)
(c)
200
400
600
800
1000 1200
Estimated Dischage (0147050_test)
(d) 1400
1600
y = 1.000x + 0.257 R2 = 0.998
1200
y = 1.034x - 2.752 R2 = 0.998
1400 1200
1000
Observed
Observed Discharge
Fig. 2 Scatter plots comparing simulated (ENNF) model and observed flows for a training, b testing sets for upstream station and, c training, d testing sets for downstream station
xmin
800 600 400
1000 800 600 400
200
200
0
0 0
365
730
1095
1460
Estimated Discharge(01474500_train)
0
500
1000
1500
Estimated Dischage (01474500_test)
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1000 750 500 250 0
1000 800 600
25.6 26.6 27.6 28.6 29.6 30.6 1.7
Discharge (m3/s)
1200
400
Eq.13 Observed
200 0 0
30
60
90 120 150 180 210 240 270 300 330 360
Time (day)
Fig. 3 Daily hydrograph simulated by ENNF (Eq. 11) model and observed flows for upstream (1470500) station in testing period
respectively. The proposed model (Eq. 11) estimates these events successfully as 301.90, 629.33, 972.28, and 411.85 m3/s, respectively. The maximum flow is observed in this period and the model evaluates with a very low error percent. The daily hydrographs and scatter plots of high and low flows are best represented by the ENNF model. Observed and estimated flows for downstream (1474500) station in testing period (2006 water year) are given in Fig. 4. The same flood starts from June 26 to July 6 as 258.67, 925.42, 1,356.83, 1,323.73, 681.23, 390.41 m3/s, and 312.60, 286.34, 258.67, 254.17, 229.95 m3/s, respectively. These floods estimated successfully as 291.59, 925.74, 1483.44, 1,341.89, 682.27, 387.25 m3/s, and 311.41, 283.10, 256.77, 252.81, 227.90 m3/s, respectively by ENNF (Eq. 12). It can be deduced from these results that the ENNF model can simulate high flows and low flows quite well as stated also by Panagoulia [30]. 5.2 ENNF versus SRC The most commonly used technique in modeling stage– discharge modeling, SRC, was also taken into consideration in this study. The performance of SRC was analyzed by computing the coefficient of efficiency E, the determination coefficient R2, and the RMSE of daily observation flows between the ENNF and other methods. RMSE and R2
were calculated to measure the deviation from and approximation to observed flows obtained from two stations in PA, USA. Overall, statistical characteristics of the three models were listed in Table 2. The lower RMSE implies the better performance of the applied method. The estimated average RMSE on the daily basis for ENNF model are 9.594 and 8.284 m3/s in upstream and downstream, respectively. However, RMSE values of SRC are roundly as 1,530 and 79 for upstream and downstream stations, respectively. The estimated determination coefficient of ENNF for both station (R2 = 0.985 and 0.998) are much higher than SRC. The average ratio above and below 1.0 for different methods indicates over- and underestimation of flows relative to observed data. As shown in Table 2, the average ratio defined as the ratio of arithmetic mean of estimated discharge to mean observed discharge gave the values of 1.06 and 0.99 for ENNF, whereas 5.16 and 0.66 for SRC. As an overall performance, SRC resulted in the overestimation of discharge. Based upon the RMSE, R2, and E criterions, comparison made to the results of the observed data, presented in Table 2 for both stations, shows that the ENNF model has the closest approximation.
6 Further applications One of the important problem faced using NN applications is the estimation period. Forecasting of hydrologic time series is one of the most complicated tasks due to the wide range of data, the uncertainties in the parameters, and the reliable availability of adequate data. The models proposed in this study (Eqs. 11, 12) are tested against another data set. Stage discharge of both stations for October 1, 2006– August 25, 2007 has been taken into consideration for Table 2 Statistical performance of ANN-based model versus SRC in testing period
Eq. 11
Eq. 12
SRC
SRC
25.77
125.32
108.38
Av. ratio
1.06
5.16
0.99
0.66
66.92 10.62
1,583.30 18.10
142.72 5.51
168.60 14.15
71.781
xmin (m3/s)
2.13
4.83
13.57
24.31
xmax (m3/s)
972.28
29,693.11
1,356.83
2,215.37
12.63
1.32
2.35
Cv
2.60
R2
0.985
0.792
0.998
0.833
RMSE (m3/s)
9.594
1,529.798
8.284
78.579
6,297.42
7.58
71.89
RMSE (%)
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Downstream (1474500)
Qm (m3/s) Sx Csx
Fig. 4 Daily hydrograph simulated by ENNF (Eq. 12) model and observed flows for downstream (1470960) station in testing period
Upstream (147500)
39.50
E (%)
0.979
0.07
0.997
0.793
E1 (%)
0.90
0.15
0.972
0.405
Neural Comput & Applic (2013) 23:873–880 400
300 250
Eq.13
200
Observed
150 100 50
Observed Discharge (m3/s)
400
350
Discharge (m3/s)
Fig. 5 Observed and estimated discharge for upstream (1470500) station in forecasting period
879
y = 0.900x - 1.192
350
R2 = 0.994
300 250 200 150 100 50 0
0 0
30 60 90 120 150 180 210 240 270 300 330
Time (day)
0
50 100 150 200 250 300 350 400
Estimated Discharge (m3/s)
Fig. 6 Observed and estimated discharge for downstream (1470960) station in forecasting period
Table 3 Observed and estimated flood values for downstream station in prediction period Date
Observed discharge (m3/s)
Estimated discharge (Eq. 12) (m3/s)
Error (%)
22.06.2006
21.77
22.03
23.06.2006
20.82
21.06
1.19 1.14
24.06.2006 25.06.2006
47.90 161.66
55.77 175.81
16.43 8.75
26.06.2006
258.67
291.59
12.73
27.06.2006
925.42
925.74
0.03
28.06.2006
1,356.83
1,483.44
9.33
29.06.2006
1,323.73
1,341.89
1.37
30.06.2006
681.23
682.27
0.15
01.07.2006
390.41
387.85
-0.66
02.07.2006
312.60
311.41
-0.38
03.07.2006
286.34
283.10
-1.13
04.07.2006
258.67
256.77
-0.73
05.07.2006
254.17
252.81
-0.53
06.07.2006
229.95
227.90
-0.89
07.07.2006
171.05
170.71
-0.20
08.07.2006
126.47
127.40
0.73
09.07.2006
105.62
106.45
0.78
estimation period. It is emphasized that the data of this period has neither been used in training nor used in testing period. The observed and estimated flows for upstream (1470500) and downstream (01474500) stations are given
in Figs. 5 and 6, respectively. As it is seen, the daily hydrographs and scatter plots are best represented by the ENNF model. The model has a high determination coefficient, R2 = 0.994 for upstream and 0.998 for downstream stations. A flood has occurred at November 17, 2006 with a magnitude of 305.75 m3/s for upstream station. The ENNF model computes it successfully as 355.70 m3/s. As stated before, the ENNF model evaluates peak values quite well. Table 3 shows observed and estimated flood values occurred at various periods for downstream station. The forecasted hydrographs generally agree well with the observed hydrographs included all storms.
7 Conclusions Traditionally, NNs are used as black-box models in which no one is interested in the fundamental hidden formulation. The question to be answered is that, how anyone can apply this kind of models in any other study, while the model has not been formulated. The main aim of this study was to determine an explicit neural network formulation (ENNF) for modeling stage–discharge relationship in rivers. ENNF was developed to estimate the river flow from measured gauge stage. The data of two gauging stations were used to compare the performance of ENNF versus SRC approach. ENNF was found to be considerably better than the conventional SRC. ENNF is also quite successful, especially,
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in estimating peak discharge values during flood events and completing missing data. The results of this study highly promise and suggest that ENNF modeling is a more versatile and improved alternative to the conventional approaches for developing stage–discharge relationship. Acknowledgments The data used in this study were downloaded from the web server of the USGS. The authors wish to thank the staff of the USGS who are associated with data observation, processing, and management of USGS Web sites.
References 1. Ab. Ghani A, Chang CK, Leow CS, Zakaria NA (2012) Sungai Pahang digital flood mapping: 2007 flood. Int J River Basin Manag 10(2):139–148 2. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control AC-19:716–723 3. Alavi AH, Gandomi AH, Mollahasani A, Heshmati AAR, Rashed A (2010) Modeling of maximum dry density and optimum moisture content of stabilized soil using artificial neural networks. J Plant Nutr Soil Sci 173(3):368–379 4. Alavi AH, Gandomi AH (2011) Prediction of principal groundmotion parameters using a hybrid method coupling artificial neural networks and simulated annealing. Comput Struct 89(23–24): 2176–2194 5. ASCE Task Committee (2000) Artificial neural networks in hydrology. I: preliminary concepts. J Hydrol Eng 5(2):115–123 6. ASCE Task Committee (2000) Artificial neural networks in hydrology. II: hydrologic applications. J Hydrol Eng 5(2): 124–137 7. Azamathulla HM, Ghani A, Leow CS, Chang CK, Zakaria NA (2011) Gene-expression programming for the development of a stage-discharge curve of the Pahang River. Water Resour Manag 25(11):2901–2916 8. Bhattacharya B, Solomatine DP (2000) Application of artificial neural network in stage-discharge relationship. In: Proceedings of 4th international conference on hydroinformatics. IAHR, Iowa City 9. Fread DL (1973) A dynamic model of stage-discharge relations affected by changing discharge. NOAA Tech. Memo. NWS HYDRO-16, National Weather Service, Silver Spring 10. Fread DL (1975) Computation of stage-discharge relationships affected by unsteady flow. Water Resour Bull 11(2):213–228 11. Gandomi AH, Alavi AH, Mirzahosseini MR, Moqhadas Nejad F (2011) Nonlinear genetic-based models for prediction of flow number of asphalt mixtures. J Mater Civ Eng ASCE 23(3): 248–263 12. Gandomi AH, Tabatabaie SM, Moradian MH, Radfar A, Alavi AH (2011) A new prediction model for load capacity of castellated steel beams. J Constr Steel Res 67(7):1096–1105 13. Gandomi AH, Babanajad SK, Alavi AH, Farnam Y (2012) A novel approach to strength modeling of concrete under triaxial compression. J Mater Civ Eng ASCE (in press). doi:10. 1061/(ASCE)MT.1943-5533.0000494 14. Gandomi AH, Alavi AH (2011) Applications of computational ıntelligence in behavior simulation of concrete materials. In: Yang XS, Koziel S (eds) Chapter 9 in computational optimization and applications in engineering and industry, vol 359. Springer SCI, pp 221–243
123
Neural Comput & Applic (2013) 23:873–880 15. Gandomi AH, Alavi AH (2011) Multi-stage genetic programming: a new strategy to nonlinear system modeling. Inf Sci 181(23):5227–5239 16. Gandomi AH, Alavi AH (2012) A new multi-gene genetic programming approach to nonlinear system modeling. Part II: geotechnical and earthquake engineering problems. Neural Comput Appl 21(1):189–201 17. Goh ATC, Kulhawy FH, Chua CG (2005) Bayesian neural network analysis of undrained side resistance of drilled shafts. J Geotech Geoenviron Eng 131(1):84–93 18. Guven A, Gunal M, Cevik AK (2006) Prediction of pressure fluctuations on stilling basins. Can J Civ Eng 33(11):1379–1388 19. Guven A, Aytek A, Yuce MI, Aksoy H (2007) Genetic programming-based empirical model for daily reference evapotranspiration estimation. Clean-Soil Air Water 36(10–11):905–912 20. Guven A, Aytek A (2009) A new approach for stage-discharge relationship: gene-expression programming. J Hydrol Eng 14(8): 812–820 21. Haykin S (1999) Neural networks: a comprehensive foundation. Pearson Education Inc., New Jersey 22. Jain SK, Chalisgaonkar D (2000) Setting up stage-discharge relations using ANN. J Hydrol Eng 5(4):428–433 23. Legates DR, McCabe GJ (1999) Evaluating the use of goodnessof-fit measures in hydrologic and hydroclimatic model validation. Water Resour Res 35(1):233–241 24. Liao H, Knight DW (2007) Analytic stage–discharge formulae for flow in straight trapezoidal open channels. Adv Water Resour. doi:10.1016/j.advwatres.2007.05.002 25. Liong SY, Lim W, Paudyal GN (2000) River stage forecasting in Bangladesh: neural network approach. J Comput Civ Eng 14(1): 1–18 26. Lohani AK, Goel NK, Bhatia KKS (2007) Deriving stage–discharge–sediment concentration relationships using fuzzy logic. Hydrol Sci 52(4):793–807 27. Maier HR, Dandy GC (2000) Neural networks for the prediction and forecasting of water resources variables: a review of modeling issues and applications. Environ Model Softw 15(1):101–124 28. Overleir P (2006) Modelling stage–discharge relationships affected by hysteresis using the Jones formula and nonlinear regression. Hydrol Sci 51(3):365–388 29. Overleir P (2006) A robust stage-discharge rating curve model based on critical flow from a reservoir. Hydrol Res 37(3):217–233 30. Panagoulia D (2006) Artificial neural networks and high and low flows in various climate regimes. Hydrol Sci 51(4):563–587 31. Schmidt AR, Yen BC (2002) Stage-discharge ratings revisited. In: Wahl TL, Pugh CA, Oberg KA, Vermeyen TB (eds) Hydraulic measurements and experimental methods, Proceedings of EWRI and IAHR joint conference, Estes Park 32. Sivapragasam C, Mutill N (2005) Discharge rating curve extension—a new approach. Water Resour Manag 19:505–520 33. Sudheer KP, Jain SK (2003) Radial basis function neural network for modeling rating curves. J Hydrol Eng 8(3):161–164 34. Supharatid S (2003) Application of a neural network model in establishing a stage-discharge relationship for a tidal river. Hydrol Process 17(15):3085–3099 35. Tawfik M, Ibrahim A, Fahmy H (1997) Hysteresis sensitive neural network for modeling rating curves. J Comput Civ Eng 11(3):206–211 36. Thirumalaiah K, Deo MC (1998) River stage forecasting using artificial neural networks. J Hydrol Eng 3(1):26–32 37. Torsten D, Gerd M, Torsten S (2002) Extrapolating stage-discharge relationships by numerical modeling. In: International conference on hydraulic engineering, Warshaw, pp 1–8
