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Advances in Computational Design, Vol. 6, No. 2 (2021) 135-151
DOI: https://doi.org/10.12989/acd.2021.6.2.135 135
Copyright © 2021 Techno-Press, Ltd.
http://www.techno-press.org/?journal=acd&subpage=7 ISSN: 2383-8477 (Print), 2466-0523 (Online)
Discharge coefficient estimation for rectangular side weir
using GEP and GMDH methods
Ajmal Hussain1a, Ali Shariq1, Mohd Danish2b and Mujib A. Ansari1c
1Department of Civil Engineering, Zakir Hussain College of Engineering & Technology,
Aligarh Muslim University, Aligarh-202002, India
2Civil Engineering Section, University Polytechnic, Aligarh Muslim University, Aligarh-202002, India
(Received June 18, 2020, Revised December 22, 2020, Accepted December 24, 2020)
Abstract. Flow through the rectangular side weir is a spatially varied type flow with decreasing discharge and
used as a flow diversion structure. They are mainly used in the field of hydraulic, irrigation, and environmental
engineering for diverting and controlling the flow of water in irrigation–drainage systems, drainage canal systems,
and wastewater channels. In this study, gene expression programming and group method of data handling were used
to estimate the coefficient of discharge for rectangular side weir under subcritical flow condition. Based on
dimensional analysis, the coefficient of the discharge depends on the ratio of the crest height to length, ratio of the
width of channel to crest length, ratio of the upstream depth in the channel to crest length and the approach Froude
number. The performance of the proposed GMDH and GEP model is based on the coefficient of correlation (0.91),
mean absolute percentage error (3.54), average absolute deviation (3.3), root mean square error (0.027) and the
coefficient of correlation (0.905), mean absolute percentage error (4.12) average absolute deviation (3.9), root mean
square error (0.029), respectively. Finally, the results reveal that GMDH model could provide more satisfactorily
estimations as compared to those obtained by traditional regression and GEP models.
Keywords: rectangular side weir; coefficient of discharge; froude number; GMDH; GEP
1. Introduction
The side weirs may be of different shapes such as triangular, trapezoidal, rectangular or their
combination according to application. They are generally used in river-control structures,
reservoirs, dams, river-intake facilities, irrigation canals, and wastewater-treatment plants. The
study of diversion of flow from the primary channel to the secondary channel, the main river to
another river, or the main canal to sub-canal is important aspects for hydraulic engineering. The
various hydraulic structures used to divert flow are weirs, spillway, sluice gate, and orifice.
(Hussain et al. 2014, Hussain et al. 2016, Shariq et al. 2018, Ansari et al. 2019, Shariq et al.
2020). Spatially varied flow with decreasing discharge are observed in side weirs and side orifices
Corresponding author, Ph.D. Student, E-mail: shariq.ali792@gmail.com
aAssistant Professor, E-mail: ajmalamin.iitr@gmail.com
bAssistant Professor, E-mail: mohd.danish999@gmail.com
cProfessor, E-mail: mujibansari68@gmail.com
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
that are used for diverting water from irrigation or drainage systems, for controlling the water
depth in a canal, and in flood schemes relief on the river.
In past studies, the extensive literature on side weirs is available due to its wide range of
applications in environmental and hydraulic engineering. De Marchi (1934) provides the first
theoretical approach on the hydraulics of rectangular side weir in a rectangular channel.
Hydraulics and flow characteristics of rectangular side weir have been widely studied
experimentally, theoretically and numerically for different shapes (rectangular, triangular,
trapezoidal, and circular) of the channels by many researchers (De Marchi 1934, Emiroglu et al.
2011, Ranga Raju et al. 1979, Shariq et al. 2018, Shariq 2016, Vatankhah 2012, Hager 1987,
Mohammed et al. 2013, Mohammed and Golijanek-Jędrzejczyk 2020).
Many Researcher's studies have formulated discharge coefficients equation for side weirs. The
flow through the side weir in a rectangular channel has been the subject of many investigations
(Subramanya and Awasthy 1972, Ranga Raju et al. 1979, Hager 1987). De Marchi (1934) provides
the first theoretical approach for the discharge passed through the rectangular side weir in a
rectangular channel. For developing a general expression, it is assumed that specific energy along
the rectangular side weir is constant, uniform flow is maintained in the primary channel, and the
edges of the rectangular side weir are sharp. One of the most common and fundamental bases for
designing of side weirs is De Marchi’s approach. Dominguez (1999) reported the following
discharge equation for the rectangular side weir.
)(
2
15
4
12
5.2
1
5.2
2hh hh
gLCQ d
(1)
Where, Q is discharge passed through the rectangular side weir, g is the acceleration due to
gravity, L is the crest length of the rectangular side weir, Cd is coefficient of discharge, and h is the
head over the crest of rectangular side weir. The upstream and downstream sections of side weir
are referred by the subscript 1 and 2, respectively. For developing a general expression, it is
assumed that specific energy along the rectangular side weir is constant, uniform flow is
maintained in the primary channel, and the edges of the rectangular side weir are sharp.
Kaveh et al. (2018a) adopted four soft computing-based techniques for Analysis of slope
stability failures, Patient Rule-Induction Method (PRIM), M5 algorithm, Group Method of Data
Handling (GMDH) and Multivariate Adaptive Regression Splines (MARS). Kaveh et al. (2018b)
predicted shear strength of both FRP-reinforced concrete members with and without stirrups using
the Group Method of Data Handling (GMDH) technique. Alkroosh and Sarker (2019) used gene
expression programming (GEP) for predicting the compressive strength of fly ash geopolymer
concrete. Kose and Kayadelen (2010) predicted the effects of infill walls on-base reactions and
roof drift of reinforced concrete frames using adaptive neuro-fuzzy inference system (ANFIS) and
gene expression programming (GEP). Khorrami and Derakhshani (2019) predict the ultimate
bearing capacity of the shallow foundations using a combination of the M5-GP approach.
Mohammed and Sharifi (2020) also provided the coefficient of discharge equation for obliged side
weir using GEP method.
In recent past, various artificial intelligence techniques such as artificial neural networks
(ANNs), adaptive neuro-fuzzy inference system (ANFIS), genetic programming, support vector
machines (SVMs) were used extensively for solving various problems in different fields of civil
engineering (Azmathulla et al. 2010, Ansari and Atthar 2013, Ansari et al. 2019, Ayaz and
Mansoor 2018, Dutta et al. 2018, Alam et al. 2017, Ansari et al. 2018, Shao et al. 2014, Li et al.
136
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
2016, Saridemir 2016). Recently, the GMDH network is used in many fields to forecast and model
the behaviours of unknown or complex systems based on different sets of multi-input-single-
output data pairs (Amanifard et al. 2008). Moreover, in various researches such as energy
conservation, economics and engineering geology, control engineering system identification, the
GMDH approach is applied (Srinivasan 2008, Najafzadeh et al. 2013, Ansari 2014, Faisal et al.
2020, Rizvi et al. 2020).
The Gene Expression Programming technique is an extended form of genetic programming
(GP), and it is an evolutionary artificial intelligence technique introduced by Ferreira. Gene
Expression Programming evolves computer programs with various lengths and shapes encoded in
linear chromosomes with a fixed size.
The present study aims to re-analyze the databases and to develop a GMDH and GEP model for
the prediction of the coefficient of discharge of rectangular side weir. Few studies available in
literature related to application of GMDH on side weir, an attempt has been made to developed a
model to estimate a coefficient of discharge of side rectangular weir, which provide satisfactory
results. The proposed equation obtained through the GMDH and GEP model is also compared with
existing regression equations available in literature. Among all computational intelligence
methods, the Group Method of Data Handling (GMDH) is known as a self-organized system with
the capability of solving extremely complex nonlinear problems (Amanifard et al. 2008). This
specific approach has been used because several studies related to application of GMDH methods
have reported that it is one of the best approaches in dealing with problems related to water
resources engineering.
2. Dimensional analysis
Dimensional analysis was performed to estimate the functional relationship for the coefficient
of discharge for rectangular side weir. Coefficient of discharge of rectangular side weir can be
expressed as a function of the upstream depth of flow (y1), acceleration due to gravity (𝑔), average
flow velocity over the cross-section of the channel (𝑉), the dynamic viscosity of water (μ), the
density of water (ρ), a crest length of side weir (𝐿), the width of the main channel (𝐵), and crest
height of side weir (𝑃).
1
, , , , , , ,
d
C f P L B g V y
(2)
111
, , ,
dyV
PB
C f F
L L L
gy
(3)
3. Data collection
The data sets presented by Shariq et al. (2018), Azza and Al-Talib (2012), and Bagheri et al.
(2014) have been used in this study. The experimental set-up of Shariq et al. (2018) consisted of a
primary flume of length, width, and depth of 12.8 m, 0.29 m, and 0.39 m, respectively. A
rectangular side weir was constructed on the right wall from the upstream end of the primary
137
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
Table 1 Range of experimental data for the present study
Parameters
Unit
Range of data
Q1
l/s
7.1 – 44.6
Q2
l/s
0.4 – 29.07
B
cm
29 & 40
y1
cm
9 – 32.1
L
cm
15 – 60.5
F1
-
0.11-0.77
Fig. 1 Variation of Cd with Froude number
Fig. 2 Variation of Cd with y1/L
Fig. 3 Variation of Cd with P/L
138
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
Fig. 4 Variation of Cd with B/L
Table 2 Available equation of Cd in literature
S.No.
Source
Discharge coefficient equations for rectangular side weirs
1.
Ghodsian (1997)
]
)(
075.0611.0)[63.01( 1
33.0
1P
Py
FCd
2.
Shariq et al. (2018)
0.2322
3.6295
0.0394 0.0357
0.8292 1
1
1.1308 1.5396 0.1492 0.0105 0.487
dy
PB
CF
L L L
3
Borghei et al. (1999)
1
47.055.0 FCd
channel at 8.20 m distance. Discharge over the rectangular side weir was passed into a secondary
channel consisted of 4.18 m length, 0.2 m width, and 0.35 m depth and, then, moved to a return
channel. The set-up of Bagheri et al. (2014) consisted of rectangular channels of length, height,
and width are 8 m, 0.4 m, and 0.6 m, respectively. All the experiments conducted under subcritical
flow conditions. The range of experimental data collected for the present study is shown in Table
1.
4. Analysis of data, results, and discussions
4.1 Effect of the dimensionless parameter on Cd
The effect of the dimensionless parameters y1/L, F1, P/L, and B/L on the observed coefficient of
discharge, Cd was conducted. Thorough data analysis indicates that B/L, F1, P/L, and y1/L are the
affecting dimensionless parameters for Cd. To show the variation of Cd against upstream Froude
number, F1 by keeping the other affecting parameters y1/L, B/L, and P/L as constant, is shown in
Fig. 1. It indicates that Cd decrease with the increase of F1. In Fig. 2, the variation of Cd against
y1/L while keeping the affecting parameters F1, B/L, and P/L as constant, indicates that Cd
increases with the increase of y1/L. Similarly, in Fig. 3 the variation of Cd against P/L, shows that
Cd decreases with the increase in P/L when other affecting parameters such as y1/L, B/L, and F1
remain constant. The variation of Cd against B/L indicates that Cd increases with the increase of
139
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
Fig. 5 Comparison between observed and predicted Cd for Bhorghei et al. (1999) model for all data
sets
Fig. 6 Comparison between observed and predicted Cd for Ghodsian (1997) model for all data sets
B/L when other affecting parameters such as y1/L, P/L, and F1 remains constant, as shown in Fig.
4.
4.2 Accuracy of existing relationships for Cd
Extensive literature is available for the estimation of the coefficient of discharge. In order to
verify the accuracy of the existing models, the entire available range of data was used. Table 1
shows the range of data for all the parameters used in the present investigation and Table 2 shows
the models proposed by Borghei et al. (1999), Ghodsian (1997), and Shariq et al. (2018). These
models were selected for comparison in the present study. The comparison between the observed
Cd of rectangular side weir and those computed by the proposed available models are shown in
Figs. 5-7, and the qualitative performance parameters are presented in Table 4. A close study of
Figs. 5-7 reveals that none of the existing models was able to estimate the values of Cd of
rectangular side weir for the range of data used in the present study.
140
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
Fig. 7 Comparison between observed and predicted Cd for Shariq et al. (2018) model for all data
sets
Fig. 8 Network Architecture of the GMDH model for predicting the coefficient of discharge
4.3 Proposed GMDH model for the coefficient of discharge of rectangular side weir
Group Method of Data Handling (GMDH) traditionally uses quadratic two-variable polynomial
while developing the network. A modified form of GMDH network can be obtained by introducing
several other types of polynomials and functions to enhance the performance of the model. In the
present study, the GMDH network was modified by using two variable quadratic polynomial and
one variable logarithmic function, as shown in Eqs. (4)-(5).
Quadratic: 2 variables
22
0 1 2 3 4 5
ˆ()
i j i j i j i j
y G x x a a x a x a x x a x a x
(4)
Log: 1 variable
0 1 2
ˆ( ) log( )
i j i
y G x x a a x a
(5)
Besides, the results obtained by the GMDH model were compared with the regression models
proposed by Borghei et al. (1999), Ghodsian (1997) and Shariq et al. (2018). The proposed
GMDH network under consideration yielded a correlation coefficient of 0.91.
One of the critical properties of GMDH networks is that it provides analytical equations, which
was obtained using a logarithmic function and quadratic polynomial. Analytical Eqs. (A1)-(A13)
141
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
(a)
(b)
Fig. 9 Comparison between predicted and observed Cd using present GMDH model for training
data sets
obtained by GMDH network for predicting Cd of rectangular side weir are presented in the
Appendix.
In Eqs. (A1)-(A13), the subscript and superscript of each parameter represent the number of
pertaining layers and neurons, respectively. The proposed structure of the GMDH network
containing five selective neurons in the first layer, four selective neurons in the second layer, two
selective neurons in the third and one selective neuron in the fourth respectively and a selective
neuron in the output layer (5-4-2-1) for predicting the coefficient of discharge is presented in Fig.
8. The predicted values of Cd have been plotted against its observed values for training and
validation data sets, as shown in Fig. 9 for the GMDH model. It can be observed from Fig. 9 that
most of the data lie within ±7% error band. Therefore, the GMDH model, along with
corresponding logarithmic function with one variable and quadratic function with two variable
polynomials (Eqs. (A1)-(A13)) is recommended for general use to predict Cd of rectangular side
weir.
4.4 Proposed Gene Expression Programming model for the coefficient of discharge of
rectangular side weir
Gene Expression Programming (GEP) is a procedure that mimics biological evolution to create
a computer program to model some phenomena (Ferreira 2001, Azamathulla et al. 2011,
Mohammed and Sharifi 2020). It is a system for encoding articulation that allows fast operation of
an extensive range of mutations and cross-breeding methods while ensuring that the resulting
expression will always be acceptable (Ferreira 2001, Ferreira 2006). It is associated with the
principle of natural selection that is fit; healthier individuals should breed and yields generation at
a rapid rate than unfit, sick individuals. Through this alternative process, each offspring becomes
fitter and healthier.
The healthier individuals in each breed are unconditionally reproduced unchanged into the next
breed. An expression tree is a better way to describe expression in a system because the tree can be
complicated, and expression trees can be evaluated immediately (Ferreira 2001).
To identify the best combination of the model building parameter of GEP and determining the
most favourable value of population size, gene head length, gene per chromosome, maximum
142
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
Table 3 GEP model parameters
Parameter
Setting
Population size
55
Number of genes per chromosome
05
Gene head length
12
Number of generations
10000
Generation without improvement
10000
Linking function
+
Fitness function
RRSE
Function set
+, -, ×, ÷, logistic 4
Chromosome length
66
Mutation rate
0.044
Inversion rate
0.1
Fig. 10 Sub expression trees corresponding to each gene for the Eq. (6)
generation, and generations without improvement (GWI) was found by minimizing the variation
between the estimated values and the desired output of GEP model. The GEP method has also
been used for determining the Cd of the rectangular side weir. The performance of GEP models
was deduced based on Mean Absolute Percentage Error (MAPE), Root Mean Squared Error
(RMSE), Efficiency coefficient (E) & Average Absolute Deviation (AAD) and coefficient of
correlation (R). The training of the GEP models was stopped when it achieved a satisfactory
precision, or the maximum generation reached the recommended limit. Table 3 shows the
parameters used in developing the GEP model.
143
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
Fig. 11 The expression for the logistic function
(a)
(b)
Fig. 12(a) Comparison between predicted and observed Cd using present GEP model for training
data sets
The explicit formulation of the GEP model for Cd of rectangular side weir has been optimized
as Eq. (7):
1
1
11
1
342.77 0.123
1 exp 99.47 / 1.327
/
1 exp 142.8 /
1 exp 4.75 4.75 9.5
/
1.673 (
4.78
/
1 exp 3.736 ( 0.204)
1 exp / 1.68
d
CBL
yL PL
yL
FF
PL
B L F
1
0.134 / )yL
(6)
From Eq. (6), it has been observed that there is sub-expression corresponding to each gene in
the equation. The sub-expression trees of the gene are shown in Fig. 10. Logistic4 (a,b,c,d) is
shown in Fig. 11 can be represented as Eq. (7).
The observed and predicted values of the Cd of rectangular side weir using a GEP model for the
training and validation data are compared graphically, as shown in Fig. 12. It shows that the
predicted Cd lies within ±7% of the observed values for training data as well as validation data,
which is a better estimation of Cd for side rectangular sharp-crested weir. The qualitative
performance of the present GEP model for all data sets has a mean absolute percentage error of
4.12 and the average absolute deviation of 3.9 with a coefficient of correlation of 0.905.
144
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
Table 5 Comparison between existing relations, GEP and GMDH model
Source
Percentage of data having error less than
±4%
±8%
±12%
±16%
Ghodsian (1997)
0.55
3.29
3.38
9.34
Borghei et al. (1999)
1.64
3.83
9.87
18.11
Shariq et al. (2018)
53.84
78.56
92.29
99.98
GEP Model (Eq. (6))
63.18
87.45
96.24
100
GMDH Model (Eq. (A1)-(A13))
73.62
91.75
97.24
100
4.5 Comparison between GMDH, GEP model and available equations in literature
Tables 4 and 5 show the comparison between performance parameters and percentage error of
GMDH, GEP model, and available equation of coefficient of discharge of rectangular side weir in
literature. Both GMDH and GEP models predicted results satisfactorily as compared to the
available equations of Cd for rectangular side weir. The qualitative performance of the present GEP
has lowest MAPE (4.12), AAD (3.9), RMSE (0.029), E (0.820), highest R (0.905) and GMDH
model has lowest MAPE (3.45), AAD (3.33), RMSE (0.027), E (0.832), highest R (0.91),
respectively, which indicates that it has better performance as compared to other existing
predictors. The percentage of data having error less than ±8% for Ghodsian (1997), Borghei et al.
1999, and Shariq et al. 2018 have been found 3.29%, 3.83%, and 78.56%, respectively, which
were lesser as compared to present GEP and GMDH model. The proposed GEP and GMDH
models provided results with a maximum error of ±12% for about 96.24% and 97.24 % of the total
data, respectively, that shows the favourable performance of the present GEP and GMDH models.
Table 4 Performance parameters of existing, GEP and GMDH models
R
MAPE
AAD
RMSE
E
Ghodsian (1997)
Training
0.26
29.203
30.11
0.1792
-5.951
Testing
0.28
29.403
30.15
0.1816
-6.640
All
0.27
29.244
30.12
0.1797
-6.056
Shariq et al. (2018)
Training
0.87
4.95
4.79
0.0340
0.754
Testing
0.85
4.65
4.65
0.0324
0.717
All
0.87
4.89
4.73
0.0337
0.748
Borghei et al. (1999)
Training
0.076
27.847
28.876
0.1801
-6.538
Testing
0.086
27.834
28.792
0.1804
-6.538
All
0.081
27.844
28.859
0.1801
-6.090
GEP Model (Eq. (6))
Training
0.928
3.621
3.470
0.025
0.861
Testing
0.832
6.260
5.815
0.042
0.689
All
0.905
4.120
3.912
0.029
0.820
GMDH Model (Eq. (A13))
Training
0.912
3.158
3.071
0.028
0.827
Testing
0.847
6.368
5.755
0.042
0.685
All
0.91
3.454
3.301
0.027
0.832
145
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
5. Conclusions
In this study, the Group method of data handling (GMDH) and Gene expression programming
(GEP) model have been used to estimate the coefficient of discharge for rectangular side weir.
• The variation of Cd with the upstream Froude number shows that Cd decreases with the
increase of Froude number.
• The variation of Cd with P/L indicates that Cd decreases with the increase of P/L. The
variation of Cd with y1/L indicates that Cd is directly proportioned to y1/L.
• Observed and calculated values of Cd of rectangular side weir using GMDH model for the test
data are compared graphically. It shows that the computed Cd lies within ±7% of the observed
values, which may be considered as a satisfactory estimation of the coefficient of discharge for
rectangular side weir.
• The qualitative performance of the present GEP model for all data sets has Mean absolute
percentage error (4.12) & average absolute deviation (3.9), root mean square error (0.029),
efficiency coefficient (0.820), and coefficient of correlation (0.905).
• The qualitative performance of the present GMDH model indicates that it has the lowest
MAPE (3.4), AAD (3.33), RMSE (0.027), E (0.832) and highest R (0.91) as compared to other
existing predictors.
• Proposed GEP and GMDH model provides much better results as compared to the available
models in the literature (Shariq et al. 2018, Bhorghei et al. 1999, Ghodsian 1997).
• The proposed GEP and GMDH models produced results with a maximum error of ±12% for
about 96.24% and 97.24% of the total data, respectively, that shows the excellent performance
of both the models.
References
Ahmad, F., Ansari, M.A., Hussain, A. and Jahangeer, J. (2020), “Model development for estimation of
sediment removal efficiency of settling basins using group methods of data handling”, J. Irrigation
Drainage Eng., 147(2), 04020043. https://doi.org/10.1061/(ASCE)IR.1943-4774.0001532.
Alam, J., Kim, D. and Choi, D. (2017), “Seismic probabilistic risk assessment of weir structures considering
the earthquake hazard in the Korean Peninsula”, Earthq. Struct., 4(13), 421-427.
https://doi.org/10.12989/eas.2018.13.4.421.
Alkroosh, I.S. and Sarker, P.K. (2019), “Prediction of the compressive strength of fly ash geopolymer
concrete using gene expression programming”, Comput. Concrete, 24(4), 295-302.
https://doi.org/10.12989/cac.2019.24.4.295.
Amanifard, N., Nariman-Zadeh, N., Farahani, M.H. and Khalkhali, A. (2008), “Modeling of multiple short-
length-scale stall cells in an axial compressor using evolved gmdh neural networks”, J Energy Convers
Manage, 49(10), 2588-2594. https://doi.org/10.1016/j.enconman.2008.05.025.
Ansari, M.A. (2014), “Sediment removal efficiency computation in vortex settling chamber using artificial
neural networks”, Water Energy Int., 71(1), 54-67.
Ansari, M.A. and Athar M. (2013), “Artificial neural networks approach for estimation of sediment removal
efficiency of vortex settling basins”, ISH J. Hydraulic Eng., 19(1), 38-48.
doi:10.1080/09715010.2012.758415.
Ansari, M.A., Ansari, S.A. and Alam, S. (2018), “Computation of scour depth below pipelines using
artificial neural networks”, Water Energy Int., 61(6), 55-62.
Ansari, M.A., Hussain, A., Shariq, A. and Alam, F. (2019), “Experimental and numerical study for the
estimation of coefficient of discharge for side compound weir”, Canadian J. Civil Eng., 46(10), 887-895.
146
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
https://doi.org/10.1139/cjce-2017-0689.
Ayaz, M. and Mansoor, T. (2018), “Discharge coefficient of oblique sharp crested weir for free and
submerged flow using trained ann model”, Water Sci, 32(2), 192-212.
https://doi.org/10.1016/j.wsj.2018.10.002.
Azamathulla, H.M., Ghani, A.A., Leo, C.S., Chang, C.K. and Zakaria, N.A. (2011), “Gene-expression
programming for the development of a stage-discharge curve of the pahang river”, Water Resources
Management, 25, 2901-2916.
Azamathulla, H.M., Ghani, A.A., Zakaria, N.A. and Guven, A. (2010), “Genetic programming to predict
bridge pier scour”, J. Hydraulic Eng., 136(3),165-169. https://doi.org/10.1061/(ASCE)HY.1943-
7900.0000133.
Azza, N. and Al-Talib, (2012), “Flow over oblique side weir”, J. Damascus Univ., 28(1), 15-22.
Bagheri, S., Kabiri-Samani, A.R. and Heidarpour, M. (2014), “Discharge coefficient of rectangular sharp-
crested side weirs part i: traditional weir equation”, Flow Measure. Instrumentation, 35, 109-115.
https://doi.org/10.1016/j.flowmeasinst.2013.11.005.
Borghei, S.M., Jalili, M.R. and Ghodsian, M., (1999), “Discharge coefficient for sharp-crested side weirs in
subcritical flow”, J. Hydraul Eng., 125(10), 1051-1056. https://doi.org/10.1061/(ASCE)0733-
9429(1999)125:10(1051).
Dutta, S., Samui, P. and Kim, D. (2018), “Comparison of machine learning techniques to predict
compressive strength of concrete”, Comput. Concrete, 21(4), 463-470.
https://doi.org/10.12989/cac.2018.21.4.463.
Emiroglu, M.E., Agaccioglu, H. and Kaya, N. (2011), “Discharging capacity of rectangular side weirs in
straight open channels”, Flow Measure. Instrumentation, 22, 319-330.
https://doi.org/10.1016/j.flowmeasinst.2011.04.003.
F.J. Domínguez, (1999) Editorial Universitaria, Santiago, Chile, (in Spanish).
Ferreira, C. (2001), “Gene expression programming: a new adaptive algorithm for solving problems”, J.
Complex Syst., 13(2), 87-129.
Ferreira, C. (2006), “Gene expression programming; mathematical modelling by an artificial intelligence”,
Springer, Heidelberg.
Ghodsian, M. (1997), “Elementary discharge coefficient for rectangular side weir”, Proceeding of 4th Int.
Conf. on Civil Engineering, Tehran.
Hager, W.H. (1987), “Lateral outflow of side weirs”, J. Hydraul. Eng., 113(4), 491-504.
https://doi.org/10.1061/(ASCE)0733-9429(1987)113:4(491).
Hussain, A., Ahmad, Z. and Ojha, C.S.P. (2016), “Flow through lateral circular orifice under free and
submerged flow conditions”, Flow Measure. Instrumentation, 36(10), 32-35.
https://doi.org/10.1016/j.flowmeasinst.2016.09.007.
Hussain, S., Hussain, A. and Ahmad, Z. (2014), “Discharge characteristics of orifice spillway under oblique
approach flow”, Flow Measure. Instrumentation, 39, 9-18.
https://doi.org/10.1016/j.flowmeasinst.2014.05.022.
Kaveh, A., Bakhshpoori, T. and Hamze-Ziabari, S.M. (2018b), “Gmdh-based prediction of shear strength of
frp-rc beams with and without stirrups”, Comput. Concrete, 22(2), 197-207.
https://doi.org/10.12989/cac.2018.22.2.197.
Kaveh, A., Hamze-Ziabari, S.M. and Bakhshpoori, T. (2018a) “Soft computing-based slope stability
assessment: A comparative study”, Geomech. Eng., 14(3), 257-269.
http://dx.doi.org/10.12989/gae.2018.14.3.257.
Khorrami, R. and Derakhshani, A. (2019), “Estimation of ultimate bearing capacity of shallow foundations
resting on cohesionless soils using a new hybrid m5\gp model”, Geomech. Eng., 19(2), 127-139.
https://doi.org/10.12989/gae.2019.19.2.127.
Kose, M.M. and Kayadelen, C. (2010), “Effects of infill walls on RC buildings under time history loading
using genetic programming and neuro-fuzzy”, Struct. Eng. Mech., 47(3), 401-419.
https://doi.org/10.12989/sem.2013.47.3.401.
Li, S., Yu, S., Shangguan, Z. and Wang, Z. (2016), “Estimating model parameters of rockfill materials based
147
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
on genetic algorithm and strain measurements”, Geomech. Eng., 10(1), 37-48.
http://dx.doi.org/10.12989/gae.2016.10.1.037.
Marchi, D. (1934), “G. Essay on the performance of lateral weirs”, L Energia Electrica Milano, 11(11), 849-
860.
Mohammed, A.Y. and Golijanek-Jędrzejczyk, A. (2020), “Estimating the uncertainty of discharge coefficient
predicted for oblique side weir using monte carlo method”, Flow Measure. Instrumentation, 73(4),1-6.
https://doi.org/10.1016/j.flowmeasinst.2020.101727.
Mohammed, A.Y. and Sharifi, A. (2020), “Gene expression programming (gep) to predict coefficient of
discharge for oblique side weir”, Appl. Water Sci., 10, 145. https://doi.org/10.1007/s13201-020-01211-5.
Mohammed, A.Y., Al-Talib, A.N. and Basheer, T.A. (2013), “Simulation of flow over the side weir using
simulink. Scientiairanica”, 20(4), 1094-1100.
Najafzadeh, M., Barani, G.A. and Hessami Kermani, M.R. (2013), “Abutment scour in clear-water and live-
bed conditions by GMDH network”, Water Sci. Technol., 67(5), 1121-1128.
https://doi.org/10.2166/wst.2013.670.
Ranga Raju, K.G., Prasad, B. and Gupta, S.K. (1979), “Side weirs in rectangular channels”, J. Hydraul Div.
105(5) 547-554.
Rizvi, Z.H., Baqir Husain, S.M., Haider, H. and Wuttke, F. (2020), “Effective thermal conductivity of sands
estimated by group method of data handling (gmdh), Proc. of Materials Today, 26(2), 2103-2107.
Saridemir, M. (2016), “Empirical modeling of flexural and splitting tensile strengths of concrete containing
fly ash by gep”, Comput. Concrete, 17(4), 489-498. https://doi.org/10.12989/cac.2016.17.4.489.
Shao, G., Jiang, L. and Chouw, N. (2014), “Experimental investigations of the seismic performance of
bridge piers with rounded rectangular cross-sections”, Earthq. Struct., 7(4), 463-484.
https://doi.org/10.12989/eas.2014.7.4.463.
Shariq, A. (2016), Flow characteristics of side weirs in open channel, Master’s Thesis, Aligarh Muslim
University, Aligarh, India.
Shariq, A., Hussain, A. and Ahmad, Z. (2020), “Discharge equation for the gabion weir under through flow
condition”, Flow Measurement Instrumentation, 74, 101769.
Shariq, A., Hussain, A. and Ansari, M.A. (2018), “Lateral flow through the sharp crested side rectangular
weirs in open channels”, Flow Measure. Instrumentation, 59, 8-17.
https://doi.org/10.1016/j.flowmeasinst.2017.11.007.
Srinivasan, D. (2008), “Energy demand prediction using gmdh networks”, Neuro Computing, 72(1-3), 625-
629. https://doi.org/10.1016/j.neucom.2008.08.006.
Subramanya, K. and Awasthy, S.C. (1972), “Spatially varied flow over side weirs”, J. Hydraul Div. (ASCE),
98(1), 1-10. https://doi.org/10.1061/JYCEAJ.0003188.
Vatankhah, A. (2012), “Analytical solution for water surface profile along a side weir in a triangular
channel”, Flow Measure. Instrument., 23(1), 76-79. https://doi.org/10.1016/j.flowmeasinst.2011.10.001.
CC
148
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
Appendix
122
1 1 1
40.635 0.228* / 0.127 / 0.713 0.289 0.526* / *
d
C B L B L F F B L F
(A1)
122
1 1 1
70.707 0.178* / 0.093* / 0.38* / 0.079 / 0.140* / * /
d
C y L y L P L P L y L P L
(A2)
22
2 1 1 1 1 1 1
3 4 4 7 7 4 7
3.157 8.822 0.703 3.135 4.306 13.053
d d d d d d d
C C C C C C C
(A3)
122
11 0.9 0.406* / 0.1328* / 0.452* / 0.1317* / 0.166* / */
d
C B L B L P L P L B L P L
(A4)
2
2 1 1 1
2
1 1 1
10 11 11 11
4.119 0.172 / 0.103 / 15.558 12.62 0.621 /
d d d d
C y L y L C C y L C
(A5)
22
3 2 2 2 2 2 2
2 3 3 10 10 10 10
0.267 7.907 5.628 5.975 17.703 24.149
d d d d d d d
C C C C C C C
(A6)
1
14 0.534 0.0282*log / 0.615
d
C B L
(A7)
2
2 1 1
2
13 14 14
1
14
1665.113 1107.707* / 18.856 / 5929.096 5627.62
2138.981 /
d d d
d
C B L B L C C
B L C
(A8)
222
1 1 1
16 0.694 0.112 0.085 0.192 / 0.195 / 0.588* * /
d
C F F P L P L F P L
(A9)
22
2 1 1 1 1 1 1
15 16 16 11 11 16 11
1.278 3.489 1.57 1.997 0.078 2.26
d d d d d d d
C C C C C C C
(A10)
22
3 2 2 2 2 2 2
12 13 13 15 15 13 15
0.904 0.826 5.795 4.986 0.237 8.784
d d d d d d d
C C C C C C C
(A11)
22
4 3 3 3 3 3 3
1 2 2 12 12 2 12
0.694 5.414 1.837 7.025 13.133 12.558
d d d d d d d
C C C C C C C
(A12)
4
1
0.548 0.983*log 0.4535
dd
CC
(A13)
149
Ajmal Hussain, Ali Shariq, Mohd Danish and Mujib A. Ansari
Appendix II: Performance indices
The qualitative performances of the available equations in terms of coefficient of correlation
(R), Root mean square error (RMSE), Mean absolute percentage error (MAPE), and Average
Absolute Deviation (AAD) are also calculated and defined below.
The coefficient of correlation describes the degree of co-linearity between simulated and
measured data, which ranges from -1 to +1, and is an index of the degree of the linear relationship
between observed and simulated data. If R = 0, no linear relationship exists. If R = ±1, a perfect
positive or negative linear relationship exists. Its equation is
n
i
df
df
n
i
d
d
n
i
df
df
d
d
CiC
n
CiC
n
CiCCiC
n
R
1
2
1
2
0
0
1
0
0
11
1
(8)
R and R2 have widely been used for model evaluation, though they are oversensitive to high
extreme values (outlier) and insensitive to additive and proportional differences between model
predictions and measure data.
Mean Absolute Percentage Error (MAPE) is a measure of the accuracy in a fitted time series
value in statistics and has been used for discharge prediction evaluation. It expresses the accuracy
as a percentage and is defined as
eMean
iC
iCiC
n
MAPE n
ido
dodf
100
1
1
(9)
where Cdo(i) and Cdf(i) are observed and predicted discharge, respectively.
Cdo
&
Cdf
denote their
mean observed and predicted discharge respectively, and n is a number of data considered.
The average absolute deviation (AAD) or simply deviation of a data set is the average of an
absolute deviation from a central point. In the general form, the central point can be the mean,
median, mode, or the result of another measure of central tendency.
n
iiXnX
n
AAD 1)(
1
(10)
Root Mean Squared Error (RMSE) is often used to measure the difference between values predicted by
a model and those actually observed from the thing being modeled. RMSE is one of the commonly
used error-index statistics and is defined as
n
iCiC
RMSE
n
idodf
1
2
(11)
The Nash-Sutcliffe model efficiency coefficient is used to assess the predictive power of
hydrological models. It is the normalized statistic that determines the relative magnitude of the
residual variance (“noise”) compared to the measured data variance and indicates how well the
plot of observed versus predicted data fits the 1:1 line. It is defined as
150
Discharge coefficient estimation for rectangular side weir using GEP and GMDH methods
n
idodo
n
idfdo
CiC
iCiC
E
1
1
1
(12)
Nash-Sutcliffe efficiencies ranges between (-∞, 1]: E=1 correspond to a perfect match of
predicted coefficient of discharge to the observed data; E=0 shows that the model are as accurate
as the mean of the observed data; and -∞<E<0 occurs when the observed mean is a better than the
model, which indicates unacceptable performance.
151
