The discharge estimation in rivers is crucial in implementing flood management techniques and essential flood defence and drainage systems. During the normal flood season, water flows solely in the main channel. During a flood, rivers comprise a main channel and floodplains, collectively called a compound channel. Computing the discharge is challenging in non-prismatic compound channels where the floodplains converge or diverge in a longitudinal direction. Various soft computing techniques have nowadays become popular in the field of water resource engineering to solve these complex problems. This paper uses a hybrid soft computing technique – artificial neural network and particle swarm optimization (ANN–PSO) and multivariate adaptive regression splines (MARS) to model the discharge in non-prismatic compound open channels. The analysis considers nine non-dimensional parameters – bed slope, relative flow depth, relative longitudinal distance, hydraulic radius ratio, angle of convergence or divergence, flow aspect ratio, relative friction factor, and area ratio – as influencing factors. A gamma test is carried out to determine the optimal combination of input variables. The developed MARS model has produced satisfactory results, with a mean absolute percentage error (MAPE) of less than 7% and an R2 value of more than 0.90.

  • Using traditional methods to estimate discharge in non-prismatic compound channels provides unsatisfactory results.

  • Discharge is estimated in non-prismatic compound channels using two soft computing techniques ANN–PSO and MARS.

  • Influencing parameters for the prediction of discharge are identified using the Gamma test.

  • Different model performances have been carried out for different ranges of width ratio and relative flow depth.

Qfp

discharges carried by the floodplain.

Q

measured discharge

Qmc

discharges carried by the main channel

Rfp

hydraulic radius of the floodplain

Rmc

hydraulic radius of the main channel

S0

bed slope of the channel

n

Manning's roughness coefficient

H

total flow depth over the main channel

h

bankfull depth of the main channel

P

wetted perimeter

R

hydraulic radius

A

area of the compound channel

fr

relative friction factor

Ar

area ratio

Rr

relative hydraulic radius

Xr

relative longitudinal distance

α

width ratio

β

relative flow depth

δ*

flow aspect ratio of the main channel

θ

diverging or converging angle

R2

correlation coefficient

ANFIS

adaptive neuro-fuzzy interface systems

ANNs

artificial neural networks

DCM

divided channel method

GEP

gene expression programming

MAPE

mean absolute percentage error

MARS

multivariate adaptive regression splines

PSO

particle swarm optimization

RMSE

root mean square error

SCM

single-channel method

Rivers are an important domestic, industrial, and agricultural water source. For a long time, easy access to water has attracted the establishment of civilizations and industries in their immediate floodplains. Accurate prediction of the discharge of a flooded river is a key challenge in river engineering due to changes in river geometry and hydraulic properties. Due to numerous settlements on the riverbank, the floodplain width has been reduced in some places and extended in others, resulting in a converging or diverging shape known as a non-prismatic floodplain. Moreover, due to natural geological formations, the rivers often converge and diverge in planforms (Karmaker et al. 2020). In such cases, the flow pattern might change from uniform to non-uniform because of variations in cross-sectional area. When the watercourse overflows its banks, it causes floods, which can have catastrophic impacts and result in a loss of livelihood. Among many natural disasters floods are the ones which have the most devastating effect near river sections and can range in frequency from high to low flow flood. This can adversely affect urban and agricultural areas, and commercial activities, making it essential to estimate discharges during floods precisely. By doing so, the severity of the flood can be predicted and used to develop flood prevention strategies.

Generally, the flow velocity of floodplains is comparatively lower than that of the velocity of the main channel. The interaction that occurs between the fast-moving water in the main channel and the slow-moving water in the floodplain creates a large hindrance in flow and makes discharge prediction difficult. This interaction also causes the transfer of mass and momentum at the transition site, where momentum is consumed. Different authors have researched the hydraulics of prismatic sections and presented shear force equations for apparent shear force and correlations between flow depth, discharge, and velocity. (Knight & Demetriou 1983; Khuntia et al. 2018; Khuntia et al. 2019; Kumar et al. 2022; Choudhary et al. 2023). In natural rivers, variations in the cross-sectional area can cause the flow state to change from uniform to non-uniform, making hydraulic analysis more difficult (Bousmar et al. 2006; Rezaei 2006; Proust et al. 2010). Bousmar & Zech (2004) developed a lateral distribution model (LDM) for uniform flow and modified it for non-uniform flow. They also created an expression for non-prismatic sections using the Shiono-Knight Method (SKM) model. Mehrabani et al. (2020) analyzed the vegetation effect on mean flow and large-scale turbulence for converging floodplains through laboratory experiments. Few studies have been conducted on determining discharge in a compound channel with converging and diverging floodplains. Different authors have used analytical and soft computing techniques to predict discharge (Das et al. 2020, 2021; Yonesi et al. 2022).

Over the past three decades, various artificial intelligence (AI) algorithms have been used to calculate the discharge capacity of channels. Thirumalaiah & Deo (1998) used artificial neural networks (ANNs) to predict water levels at different stations along the river Indravathi. Srinivasulu & Jain (2006) used ANNs to develop a rainfall–runoff model for the Kentucky River basin. Kumar et al. (2015) analyzed the flood frequency for the lower Godavari basin using L-moments, ANN, and fuzzy interface systems. Abraham et al. (2001) developed a rainfall time series model for Kerala state in India employing multivariate adaptive regression splines (MARS) and other regression techniques. Deo et al. (2017) developed MARS, support vector machine (SVM), and M5tree models to predict drought in eastern Australia. Shaghaghi et al. (2019) predicted the geometry of regime rivers using MARS, M5Tree, and SVM models using 85 cross-section data of the Gamasiab River, Kaab River, and Behesht Abad River of Iran. Gaur et al. (2013) proposed the management of groundwater resources of the Dore River basin using ANN–PSO. Mazandaranizadeh & Motahari (2017) developed an ANN–PSO model for a rainfall–runoff response for the Karaj basin. Balavalikar et al. (2018) used the particle swarm optimization-based artificial neural network model (ANN–PSO) to predict the groundwater level in the Udupi district, India. Jahanpanah et al. (2019) estimated discharge with free overfall in rectangular channels using ANN, GEP, M5 tree, and MARS model and found that the ANN model is a better predictive model for their collected datasets. Das et al. (2020, 2021) used GEP and ANFIS, and Yonesi et al. (2022) used GMDH (Group method of data handling) and MARS soft computing techniques to estimate discharge in compound channels with non-prismatic floodplains. Generally, the prediction of discharge in converging and diverging compound channels using numerical methods is complex and often leads to poor estimated values at different sections.

Novelty of the work

The novelty of the present study is as follows:

  • Traditional models of discharge computation fail to provide the better discharge predictive value for converging and diverging compound channels, therefore two artificial intelligence-machine learning (AI-ML) techniques, ANN -PSO and MARS, are adopted to develop the discharge predictive model using experimental datasets of different researchers.

  • A wide range of datasets is used in the present study by considering six different input parameters including width ratio, relative flow depth, flow aspect ratio, hydraulic radius ratio, converging and diverging angles, and bed slope.

  • Width ratio and relative flow depth are the two most influencing input parameters which are used by different researchers to predict the discharge of prismatic compound channels. Therefore, in the present study, performances of existing discharge predictive models have been carried out for different ranges of these input parameters.

The aim of this study is to develop a discharge predictive model for non-prismatic compound channels. To attain this objective, a comprehensive dataset was collected from different literature works, and its details are addressed in the sources of dataset section. The methodology section describes the approaches, which include the use of the Gamma test (GT) as well as two AI-ML techniques: ANN–PSO and MARS. The outcomes of these approaches are extensively examined in the Results and Discussion section. The findings drawn from these results converge in the Conclusions section, thereby providing a comprehensive culmination to the study.

Data were collected from experiments performed by different researchers on non-prismatic compound channels. A total of 290 datasets of non-prismatic compound channels are collected from the experiments performed by different researchers and the maximum, minimum, standard deviation, mean, and median of these datasets are provided in Table 1. Of these, 218 data, i.e., 75% of the total dataset were randomly selected for the training of the ANN–PSO and MARS model and the rest of the data are used to validate (test) the performance of the model. From the findings of Das et al. (2016, 2020), Khuntia et al. (2023) and Yonesi et al. (2022), the most influential parameters for predicting the discharge of non-prismatic compound channels are selected as follows: Ar is the area ratio, ratio of area of main channel to the floodplain, Rr is the relative hydraulic radius, hydraulic radius of main channel to the hydraulic radius of the flood plain, δ* is the flow aspect ratio, ratio of width (b) of the main channel to flow depth (H), α is the width ratio, width of the floodplain to the width of the main channel, β is the relative flow depth, (Hh)/H, where h is the main channel depth, Xr is the relative longitudinal distance, the ratio of the distance (l) of section in longitudinal direction of the channel to the total length (L) of the non-prismatic channel, θ is the converging or diverging angle of floodplain, S0 is the bed slope of the channel, FF is the friction factor ratio, which is the ratio of main channel friction factor fmc to the floodplain ffp.

Table 1

Summary of the dataset collected from the literature for diverging and converging compound channels

Authors’ datasetRangeffArRrβS0δ*αXrθQ/Qmc
Bousmar (2002)/Cv3.81 Max 0.837 10.72 3.70 0.5380 0.0010 5.770 3.000 0.833 −3.81 2.908 
Min 0.646 0.93 1.70 0.2780 0.0010 3.690 1.340 0.000 −3.81 1.745 
Avg 0.754 3.19 2.41 0.4256 0.0010 4.595 2.168 0.417 −3.81 2.326 
Std.Dev 0.057 2.73 0.57 0.0967 0.774 0.567 0.285 0.582 
Bousmar (2002)/Cv11.31 Max 0.835 9.71 4.40 0.5310 0.0010 6.360 3.000 0.250 −11.31 2.326 
Min 0.610 0.94 1.72 0.2050 0.0010 3.750 1.500 0.000 −11.31 1.454 
Avg 0.749 2.86 2.48 0.4110 0.0010 4.711 2.250 0.125 −11.31 1.841 
Std.Dev 0.063 2.14 0.67 0.1118 0.894 0.559 0.093 0.363 
Rezaei (2006)/Cv 1.91 Max 0.824 9.76 4.46 0.5090 0.0020 6.360 3.020 0.750 −3.81 2.177 
Min 0.607 0.98 1.79 0.2020 0.0020 3.910 1.510 0.000 −3.81 1.215 
Avg 0.718 3.28 2.85 0.3528 0.0020 5.151 2.263 0.375 −3.81 1.639 
Std.Dev 0.068 2.30 0.84 0.1100 0.876 0.562 0.280 0.360 
Rezaei (2006)/Cv 3.81 Max 0.830 4.05 4.59 0.5220 0.0020 6.540 3.020 1.000 −1.91 3.195 
Min 0.602 0.96 1.75 0.1790 0.0020 3.800 2.010 0.000 −1.91 1.210 
Avg 0.754 1.97 2.41 0.3912 0.0020 4.844 2.481 0.531 −1.91 2.206 
Std.Dev 0.071 0.85 0.86 0.1127 0.898 0.356 0.354 0.772 
Rezaei (2006)/Cv 11.31 Max 0.825 4.92 4.22 0.5060 0.0020 6.380 3.020 0.833 −11.31 2.046 
Min 0.619 0.98 1.78 0.1990 0.0020 3.930 2.010 0.667 −11.31 1.136 
Avg 0.724 2.39 2.81 0.3504 0.0020 5.173 2.515 0.750 −11.31 1.517 
Std.Dev 0.071 1.19 0.88 0.1127 0.899 0.505 0.083 0.343 
Bousmar et al. (2006)/Dv3.81 Max 0.832 12.80 4.20 0.5250 0.0010 6.290 3.000 1.000 3.81 2.908 
Min 0.620 0.95 1.73 0.2140 0.0010 3.800 1.330 0.167 3.81 1.745 
Avg 0.724 3.71 2.75 0.3697 0.0010 5.043 2.165 0.583 3.81 2.326 
Std.Dev 0.062 3.12 0.71 0.1066 0.853 0.568 0.285 0.475 
Bousmar et al. (2006)/Dv5.71 Max 0.821 11.38 3.85 0.5390 0.0010 5.890 2.330 1.000 5.71 2.908 
Min 0.638 1.39 1.81 0.2640 0.0010 3.690 1.330 0.250 5.71 1.745 
Avg 0.728 4.25 2.67 0.3977 0.0010 4.819 1.833 0.625 5.71 2.326 
Std.Dev 0.051 2.80 0.55 0.1002 0.802 0.372 0.280 0.475 
Yonesi et al. (2013)/Dv3.81 Max 0.806 20.60 35.09 0.3640 0.0009 1.900 3.000 1.000 3.81 12.319 
Min 0.305 1.37 1.91 0.1450 0.0009 1.410 1.330 0.167 3.81 8.213 
Avg 0.554 5.84 9.37 0.2543 0.0009 1.657 2.165 0.583 3.81 10.266 
Std.Dev 0.145 5.21 8.96 0.0983 0.219 0.568 0.285 2.053 
Yonesi et al. (2013)/Dv11.31 Max 0.801 11.41 19.43 0.3590 0.0009 1.900 3.000 0.333 11.31 12.319 
Min 0.372 1.39 1.94 0.1460 0.0009 1.420 1.600 0.100 11.31 8.213 
Avg 0.579 4.48 7.19 0.2530 0.0009 1.659 2.300 0.217 11.31 10.266 
Std.Dev 0.133 3.04 5.34 0.0987 0.222 0.539 0.090 2.053 
Naik & Khatua (2016)/Cv5 Max 0.716 21.08 6.85 0.3250 0.0011 4.410 1.800 0.500 −5.00 1.734 
Min 0.527 3.85 2.73 0.1180 0.0011 3.370 1.400 0.000 −5.00 1.426 
Avg 0.641 8.13 4.01 0.2356 0.0011 3.823 1.657 0.181 −5.00 1.589 
Std.Dev 0.059 4.67 1.27 0.0671 0.336 0.182 0.226 0.117 
Naik & Khatua (2016)/Cv9 Max 0.712 15.59 5.31 0.3190 0.0011 4.200 1.800 0.500 −9.00 1.580 
Min 0.573 3.94 2.77 0.1600 0.0011 3.400 1.400 0.000 −9.00 1.233 
Avg 0.647 7.58 3.80 0.2403 0.0011 3.798 1.647 0.193 −9.00 1.406 
Std.Dev 0.045 3.39 0.81 0.0560 0.280 0.176 0.219 0.129 
Naik & Khatua (2016)/Cv12.3 Max 0.715 22.59 7.26 0.3240 0.0011 4.450 1.800 0.595 −13.38 1.541 
Min 0.516 3.86 2.73 0.1110 0.0011 3.380 1.400 0.000 −13.38 1.194 
Avg 0.634 8.64 4.12 0.2268 0.0011 3.867 1.633 0.238 −13.38 1.368 
Std.Dev 0.055 4.99 1.21 0.0670 0.335 0.170 0.257 0.129 
Das & Khatua (2018)/Dv5.93 Max 2.608 3.327 4.386 0.5130 0.001 2.588 5.824 1.000 5.93 5.146 
Min 0.481 0.404 1.341 0.1400 0.001 1.466 2.765 0.000 5.93 1.490 
Avg 1.113 1.325 2.542 0.3010 0.001 2.103 4.294 0.500 5.93 2.509 
Std.Dev 0.541 0.688 0.897 0.1186 0.357 1.081 0.354 1.023 
Das & Khatua (2018)/Dv9.83 Max 2.363 3.363 4.386 0.2369 0.001 2.588 5.824 1.000 9.83 5.151 
Min 0.435 0.396 1.322 0.1314 0.001 1.435 2.765 0.000 9.83 1.483 
Avg 1.021 1.325 2.538 0.1675 0.001 2.099 4.294 0.500 9.83 2.472 
Std.Dev 0.516 0.698 0.908 0.0325 0.362 1.081 0.354 0.996 
Das & Khatua (2018)/Dv14.57 Max 2.218 3.375 4.327 0.5190 0.001 2.582 5.824 1.000 14.57 5.064 
Min 0.395 0.399 1.330 0.1420 0.001 1.447 2.765 0.000 14.57 1.472 
Avg 0.927 1.326 2.539 0.3024 0.001 2.099 4.294 0.500 14.57 2.428 
Std.Dev 0.463 0.702 0.907 0.1200 0.361 1.081 0.354 0.969 
Mehrabani et al. (2020)/Dv7.25–11.3 Max 0.042 1.521 2.816 0.3103 0.001 4.402 3.333 0.500 11.30 2.291 
Min 0.009 1.222 2.456 0.2663 0.001 4.138 3.167 0.000 7.25 2.068 
Avg 0.026 1.378 2.650 0.2857 0.001 4.286 3.250 0.250 9.28 2.153 
Std.Dev 0.014 0.109 0.128 0.0159 0.095 0.083 0.250 2.025 0.085 
Authors’ datasetRangeffArRrβS0δ*αXrθQ/Qmc
Bousmar (2002)/Cv3.81 Max 0.837 10.72 3.70 0.5380 0.0010 5.770 3.000 0.833 −3.81 2.908 
Min 0.646 0.93 1.70 0.2780 0.0010 3.690 1.340 0.000 −3.81 1.745 
Avg 0.754 3.19 2.41 0.4256 0.0010 4.595 2.168 0.417 −3.81 2.326 
Std.Dev 0.057 2.73 0.57 0.0967 0.774 0.567 0.285 0.582 
Bousmar (2002)/Cv11.31 Max 0.835 9.71 4.40 0.5310 0.0010 6.360 3.000 0.250 −11.31 2.326 
Min 0.610 0.94 1.72 0.2050 0.0010 3.750 1.500 0.000 −11.31 1.454 
Avg 0.749 2.86 2.48 0.4110 0.0010 4.711 2.250 0.125 −11.31 1.841 
Std.Dev 0.063 2.14 0.67 0.1118 0.894 0.559 0.093 0.363 
Rezaei (2006)/Cv 1.91 Max 0.824 9.76 4.46 0.5090 0.0020 6.360 3.020 0.750 −3.81 2.177 
Min 0.607 0.98 1.79 0.2020 0.0020 3.910 1.510 0.000 −3.81 1.215 
Avg 0.718 3.28 2.85 0.3528 0.0020 5.151 2.263 0.375 −3.81 1.639 
Std.Dev 0.068 2.30 0.84 0.1100 0.876 0.562 0.280 0.360 
Rezaei (2006)/Cv 3.81 Max 0.830 4.05 4.59 0.5220 0.0020 6.540 3.020 1.000 −1.91 3.195 
Min 0.602 0.96 1.75 0.1790 0.0020 3.800 2.010 0.000 −1.91 1.210 
Avg 0.754 1.97 2.41 0.3912 0.0020 4.844 2.481 0.531 −1.91 2.206 
Std.Dev 0.071 0.85 0.86 0.1127 0.898 0.356 0.354 0.772 
Rezaei (2006)/Cv 11.31 Max 0.825 4.92 4.22 0.5060 0.0020 6.380 3.020 0.833 −11.31 2.046 
Min 0.619 0.98 1.78 0.1990 0.0020 3.930 2.010 0.667 −11.31 1.136 
Avg 0.724 2.39 2.81 0.3504 0.0020 5.173 2.515 0.750 −11.31 1.517 
Std.Dev 0.071 1.19 0.88 0.1127 0.899 0.505 0.083 0.343 
Bousmar et al. (2006)/Dv3.81 Max 0.832 12.80 4.20 0.5250 0.0010 6.290 3.000 1.000 3.81 2.908 
Min 0.620 0.95 1.73 0.2140 0.0010 3.800 1.330 0.167 3.81 1.745 
Avg 0.724 3.71 2.75 0.3697 0.0010 5.043 2.165 0.583 3.81 2.326 
Std.Dev 0.062 3.12 0.71 0.1066 0.853 0.568 0.285 0.475 
Bousmar et al. (2006)/Dv5.71 Max 0.821 11.38 3.85 0.5390 0.0010 5.890 2.330 1.000 5.71 2.908 
Min 0.638 1.39 1.81 0.2640 0.0010 3.690 1.330 0.250 5.71 1.745 
Avg 0.728 4.25 2.67 0.3977 0.0010 4.819 1.833 0.625 5.71 2.326 
Std.Dev 0.051 2.80 0.55 0.1002 0.802 0.372 0.280 0.475 
Yonesi et al. (2013)/Dv3.81 Max 0.806 20.60 35.09 0.3640 0.0009 1.900 3.000 1.000 3.81 12.319 
Min 0.305 1.37 1.91 0.1450 0.0009 1.410 1.330 0.167 3.81 8.213 
Avg 0.554 5.84 9.37 0.2543 0.0009 1.657 2.165 0.583 3.81 10.266 
Std.Dev 0.145 5.21 8.96 0.0983 0.219 0.568 0.285 2.053 
Yonesi et al. (2013)/Dv11.31 Max 0.801 11.41 19.43 0.3590 0.0009 1.900 3.000 0.333 11.31 12.319 
Min 0.372 1.39 1.94 0.1460 0.0009 1.420 1.600 0.100 11.31 8.213 
Avg 0.579 4.48 7.19 0.2530 0.0009 1.659 2.300 0.217 11.31 10.266 
Std.Dev 0.133 3.04 5.34 0.0987 0.222 0.539 0.090 2.053 
Naik & Khatua (2016)/Cv5 Max 0.716 21.08 6.85 0.3250 0.0011 4.410 1.800 0.500 −5.00 1.734 
Min 0.527 3.85 2.73 0.1180 0.0011 3.370 1.400 0.000 −5.00 1.426 
Avg 0.641 8.13 4.01 0.2356 0.0011 3.823 1.657 0.181 −5.00 1.589 
Std.Dev 0.059 4.67 1.27 0.0671 0.336 0.182 0.226 0.117 
Naik & Khatua (2016)/Cv9 Max 0.712 15.59 5.31 0.3190 0.0011 4.200 1.800 0.500 −9.00 1.580 
Min 0.573 3.94 2.77 0.1600 0.0011 3.400 1.400 0.000 −9.00 1.233 
Avg 0.647 7.58 3.80 0.2403 0.0011 3.798 1.647 0.193 −9.00 1.406 
Std.Dev 0.045 3.39 0.81 0.0560 0.280 0.176 0.219 0.129 
Naik & Khatua (2016)/Cv12.3 Max 0.715 22.59 7.26 0.3240 0.0011 4.450 1.800 0.595 −13.38 1.541 
Min 0.516 3.86 2.73 0.1110 0.0011 3.380 1.400 0.000 −13.38 1.194 
Avg 0.634 8.64 4.12 0.2268 0.0011 3.867 1.633 0.238 −13.38 1.368 
Std.Dev 0.055 4.99 1.21 0.0670 0.335 0.170 0.257 0.129 
Das & Khatua (2018)/Dv5.93 Max 2.608 3.327 4.386 0.5130 0.001 2.588 5.824 1.000 5.93 5.146 
Min 0.481 0.404 1.341 0.1400 0.001 1.466 2.765 0.000 5.93 1.490 
Avg 1.113 1.325 2.542 0.3010 0.001 2.103 4.294 0.500 5.93 2.509 
Std.Dev 0.541 0.688 0.897 0.1186 0.357 1.081 0.354 1.023 
Das & Khatua (2018)/Dv9.83 Max 2.363 3.363 4.386 0.2369 0.001 2.588 5.824 1.000 9.83 5.151 
Min 0.435 0.396 1.322 0.1314 0.001 1.435 2.765 0.000 9.83 1.483 
Avg 1.021 1.325 2.538 0.1675 0.001 2.099 4.294 0.500 9.83 2.472 
Std.Dev 0.516 0.698 0.908 0.0325 0.362 1.081 0.354 0.996 
Das & Khatua (2018)/Dv14.57 Max 2.218 3.375 4.327 0.5190 0.001 2.582 5.824 1.000 14.57 5.064 
Min 0.395 0.399 1.330 0.1420 0.001 1.447 2.765 0.000 14.57 1.472 
Avg 0.927 1.326 2.539 0.3024 0.001 2.099 4.294 0.500 14.57 2.428 
Std.Dev 0.463 0.702 0.907 0.1200 0.361 1.081 0.354 0.969 
Mehrabani et al. (2020)/Dv7.25–11.3 Max 0.042 1.521 2.816 0.3103 0.001 4.402 3.333 0.500 11.30 2.291 
Min 0.009 1.222 2.456 0.2663 0.001 4.138 3.167 0.000 7.25 2.068 
Avg 0.026 1.378 2.650 0.2857 0.001 4.286 3.250 0.250 9.28 2.153 
Std.Dev 0.014 0.109 0.128 0.0159 0.095 0.083 0.250 2.025 0.085 

Two traditional approaches such as the single-channel method (SCM) and the divided channel method (DCM) are used to calculate the flow of non-prismatic compound channels. The GT is performed to determine the most influencing parameters which affect the discharge in a compound channel section and then the discharge prediction models are developed using ANN–PSO and MARS soft computing techniques. Details about the different methods used in the present study are described as below.

Single-channel method

In the SCM, both the main channel and floodplain are considered as a single element, and there is no difference between a compound channel and a normal channel in terms of calculation of discharge using Equation (1). The shortcoming of this method is that it provides unsatisfactory results when employed for discharge computation in the compound channel. During the flood, the water level increases and the floodplain gets covered by the floodwater, leading to a drastic increase in the wetted perimeter compared to the depth. Thus, the SCM underestimates the discharge; however, with an increased floodplain flow depth, the SCM gives more accurate results.
(1)
where Q is the discharge in the compound channel, n is the equivalent roughness coefficients, R is the hydraulic radius (A/P), A is the cross-sectional area, P is the wetted perimeter, S0 is the energy slope.

Divided channel method

The DCM was developed by Lotter (1933) in which the compound channel is divided into subsections for determining the discharge of the compound channel. There are different ways of dividing the compound channel, one of which is dividing the compound channel by introducing imaginary vertical lines between the main channel and left/right floodplains, and the total discharge is calculated by the summation of discharge of all three subparts. The imaginary line that divides the compound channel can be vertical, horizontal, or diagonal, drawn from the intersection of the main channel and floodplain, as demonstrated in Figure 1.
Figure 1

Types of subsection dividing lines between floodplains and main channel in the DCM.

Figure 1

Types of subsection dividing lines between floodplains and main channel in the DCM.

Close modal
Manning's formula can be used to calculate discharge in each subsection using Equation (2)
(2)

Here, n is the Manning's roughness coefficient, Ai is the the area of the ith subsection, Ri is the hydraulic radius of the ith subsection, S0 is the energy slope of the channel.

Further modification in this method is carried out as ‘H, V, and D’ representing the planes of the division of the main channel and the floodplain; ‘i’ and ‘e’ represent separating lines if included or excluded, respectively, for the calculation of the wetted perimeter. Numerous commercial software were developed based on the DCM, e.g., HEC-RAS, MIKE 11, and ISIS (Atabay & Knight 2006). It has been possible to assess the accuracy of the traditional approaches, empirical formula approach, ANN, and ANN–PSO model by calculating statistical error indices like the coefficient of determination (R2), mean absolute percentage error (MAPE), mean absolute error (MAE), root mean square error (RMSE), and Nash–Sutcliffe coefficient (E).

None of the traditional methods provided satisfactory results for non-prismatic compound channels, as shown in Table 2. In the traditional methods listed above, QDCM(V−i) gives the least MAPE and RMSE, 62.75% and 22.48, respectively, whereas QDCM(V−e) provides R2 equal to 0.19. Thus, there is a need for study in this domain to get better discharge results. Figure 2 depicts the observed and predicted %Qmc obtained from different traditional methods such as QSCM,QDCM(D−i),QDCM(D−e),QDCM(V−i),QDCM(V−e),QDCM(H−i), and QDCM(H−e). The coefficient of determination (R2) value for all the traditional approaches is found to be less than 0.1 except for the QDCM(V−e) where the R2 value is obtained as 0.19. Overall, all the traditional models provide a poor estimation of discharge value for non-prismatic compound channels.
Table 2

Statistical indices for the outcomes obtained from the conventional techniques

MethodMAPERMSER2EId
QSCM 70.64 25.24 0.05 −0.83 0.33 
QDCM(D−I) 66.41 22.96 0.06 −0.51 0.39 
QDCM(D−E) 64.97 23.21 0.06 −0.55 0.27 
QDCM(V−I) 62.75 22.49 0.06 −0.45 0.23 
QDCM(V−E) 65.74 37.54 0.19 −3.04 −1.69 
QDCM(H−I) 73.09 24.88 0.07 −0.78 0.45 
QDCM(H−E) 66.81 25.91 0.07 −0.93 0.02 
MethodMAPERMSER2EId
QSCM 70.64 25.24 0.05 −0.83 0.33 
QDCM(D−I) 66.41 22.96 0.06 −0.51 0.39 
QDCM(D−E) 64.97 23.21 0.06 −0.55 0.27 
QDCM(V−I) 62.75 22.49 0.06 −0.45 0.23 
QDCM(V−E) 65.74 37.54 0.19 −3.04 −1.69 
QDCM(H−I) 73.09 24.88 0.07 −0.78 0.45 
QDCM(H−E) 66.81 25.91 0.07 −0.93 0.02 

The bolded values indicates the best input combinations among other input combinations.

Figure 2

Comparison plots of observed vs. predicted %Qmc from SCM and different DCMs: (a) QSCM, (b) QDCM(D−i), (c) QDCM(D−e), (d) QDCM(V−i), (e) QDCM(V−e), (f) QDCM(H−i), and (g) QDCM(H−e).

Figure 2

Comparison plots of observed vs. predicted %Qmc from SCM and different DCMs: (a) QSCM, (b) QDCM(D−i), (c) QDCM(D−e), (d) QDCM(V−i), (e) QDCM(V−e), (f) QDCM(H−i), and (g) QDCM(H−e).

Close modal
Figure 3

Flow chart for the methodology of the ANN–PSO algorithm.

Figure 3

Flow chart for the methodology of the ANN–PSO algorithm.

Close modal

Gamma test

The Gamma test (GT) was first proposed by Agalbjorn et al. (1997) and later many discussions and improvisations were made to it by different researchers (Tsui 1999; Durrant 2001; Tsui et al. 2002). Using this method, estimation of the best mean-squared error can be achieved by a smooth model with an unseen set of data for a particular combination of inputs before the development of the model. The GT results can be organized by considering another term V-ratio, which restores a scaled invariant clamour evaluation near 0 and 1. The V-ratio equation is shown in Equation (3).
(3)

y’ is the output and σ2(y) is the variance of output. If V-ratio is near 0, it indicates a high degree of predictive accuracy of the output in the model (Evans & Jones 2002).

Artificial neural network-particle swarm optimization

ANN–PSO, an Artificial Neural Network Particle Swarm Optimization Algorithm, is a well-known technique for optimizing neural network weights. ANNs have some limitations like a slow learning rate and getting stuck in the local minima. These limitations can be eradicated by using some optimization algorithm in conjunction with ANNs. In this method, the PSO algorithm is used in addition to the ANN.

The hybrid ANN–PSO model achieves higher accuracy in less time and, therefore, is widely employed in a variety of fields. The ANN–PSO model approach begins with the initialization of a collection of random particles. The population of particles is termed a swarm. The positions of particles that reflect the ANN connection factors (e.g., biases and weights) are specified in this step. Normally, particle selection will be done randomly. The system begins with a random population of solutions and attempts to discover the optimal solution in a specific search space by improving these solutions in successive iterations. The hybrid PSO model network is then trained using the initial positions of the particles (i.e., with its initial biases and weights components). The fitness of the trained model can be calculated based on the error between the actual and observed output. With each successive iteration, the solution directs the swarm toward the ideal objective by utilizing each particle's capacity to draw on others' expertise. Each subsequent iteration characterizes two values, local best (pid) and global best (pgd) (Kiran et al. 2006; Kwok et al. 2006). The position that is best among all previously earned individual best positions is known as the ‘global best’, while the best position that a particle has attained so far is termed as ‘local best’. Weights are introduced (Mohandes 2012) so particles can attain balance during global and local exploration.
(4)
(5)
where R1 and R2 are random values in the range between zero and unity, acceleration constant, i.e., C1 and C2 usually vary in the range of 1 and 3, pid and pgd indicate the individual and global best value and W1 is the inertia weight. C1 is the cognitive learning factor; C2 is the social learning factor and both are acceleration coefficients which affect the local best solution and global best solution during the process of optimization. C1 and C2 signify weights provided to the best historical position and global best position, respectively. The flow chart of the ANN-PSO model is depicted in Figure 3.

Multivariate adaptive regression splines

MARS technique was developed by Friedman (1991). The MARS model can accurately predict the nonlinear relationships between input and output variables. Since the structure of the model was unknown before modelling it can be said that the MARS model is a non-parametric one. Furthermore, instead of using all data together the MARS model splits data into sub-categories and executes modelling procedures for each sub-category. These sub-categories are mathematical models and are termed local. The MARS model can detect a hidden nonlinear pattern in a dataset with many variables. The MARS model evaluation function can be defined without the need to combine several statistical functions. The MARS model is based on a few basis functions specified for each variable.
(6)
where t is the node and is an explanatory variable in practice. The basis functions are spline functions mirrored in the node t-pair. The MARS model takes the following general form.
The following equation represents MARS:
(7)
is the response predicted, denotes the estimated error, and denotes the linear combination of the basis function and coefficient represented as:
(8)
is the intercept, Ci is the coefficient corresponding to their basis function, . The total number of basis functions is denoted by N. Input variables should not be highly correlated without any missing data; these are the basic requirements for obtaining good results with the MARS model. MARS-based designed equations predict the discharge ratio in converging–diverging floodplains. The flowchart of the MARS model is shown in Figure 4.
Figure 4

Flow chart for the methodology of the MARS model.

Figure 4

Flow chart for the methodology of the MARS model.

Close modal

Results obtained from the GT

In the GT, 50 experiments of different input combinations are performed using Win-Gamma software. The best input combinations from the GT are presented in Table 3. It can be seen that the combination of six parameters (Rr, β, So, δ*, α, and θ) with mask [001111101] gives a Gamma value close to zero, along with the least gradient as compared to other combinations chosen for developing the model.

Table 3

Results obtained from the GT

Serial No.Combination of the input parametersGammaGradientStd. errorV-ratioMasking
1. Ar,δ*,α, Xr,θ −0.000580 0.2447 0.00120 −0.0170 [010001111] 
2. Ar,β,So,δ*,θ −0.000370 0.0746 0.00060 −0.0110 [010111001] 
3. Rr,β,So,δ*, α,θ − 0.000272 0.0844 0.00050 − 0.0080 [001111101] 
4. Ar,Rr,β,θ −0.000100 1.2974 0.00290 −0.0030 [011100001] 
5. Ar,Rr,β,So,θ 0.000170 0.5232 0.00300 0.0050 [011110001] 
6. All-β 0.000298 0.1015 0.00110 0.0080 [111011111] 
7. δ*,α,Xr,θ 0.000533 0.2552 0.00150 0.0160 [000001111] 
8. Ar,Rr,So,δ*,α 0.000569 0.2236 0.00050 0.0170 [011011100] 
9. Ff,Rr,β,So,θ 0.000713 0.6449 0.00340 0.0210 [101110001] 
10. All nine inputs −0.000013 0.0000 0.00001 −0.0623 [111111111] 
Serial No.Combination of the input parametersGammaGradientStd. errorV-ratioMasking
1. Ar,δ*,α, Xr,θ −0.000580 0.2447 0.00120 −0.0170 [010001111] 
2. Ar,β,So,δ*,θ −0.000370 0.0746 0.00060 −0.0110 [010111001] 
3. Rr,β,So,δ*, α,θ − 0.000272 0.0844 0.00050 − 0.0080 [001111101] 
4. Ar,Rr,β,θ −0.000100 1.2974 0.00290 −0.0030 [011100001] 
5. Ar,Rr,β,So,θ 0.000170 0.5232 0.00300 0.0050 [011110001] 
6. All-β 0.000298 0.1015 0.00110 0.0080 [111011111] 
7. δ*,α,Xr,θ 0.000533 0.2552 0.00150 0.0160 [000001111] 
8. Ar,Rr,So,δ*,α 0.000569 0.2236 0.00050 0.0170 [011011100] 
9. Ff,Rr,β,So,θ 0.000713 0.6449 0.00340 0.0210 [101110001] 
10. All nine inputs −0.000013 0.0000 0.00001 −0.0623 [111111111] 

The bolded values indicate the best PSO-ANN model performance.

Results of the ANN–PSO model

In ANN–PSO, the result obtained by best-fitting examination for the different swarm size, number of neurons, and acceleration factors (C1 and C2) with their evaluating indices (such as MAPE, RMSE, and R2) are depicted in Table 4. During the training and testing stages of the modelling, it was noticed that with changes in the acceleration factors, swarm size, and the number of neurons, RMSE, and R2 values also change. It can be easily observed from Table 4 that in trials with multiple nodes equal to 8 and swarm size equal to 35, acceleration factors C1 and C2 as 1.5 and 2.5, respectively, provide high R2 values. The model is selected on the basis of the least RMSE and MAPE values and a high R2 value. The training and testing stage plots of the best ANN–PSO model are depicted in Figure 5. During the training stage, the RMSE, R2, and MAPE were found to be 0.026, 0.981, and 14.78%, and for the testing stage, the values were 0.030, 0.965 and 14.92% respectively. Therefore, the most optimized model obtained by the fitting test is 8–35–1.5–2.5. During optimization with change in the acceleration factors, it was observed that the best output came from 1.5 and 2.5 as the values of C1 and C2, respectively (Rukhaiyar et al. 2018). The quality of output deteriorates with trials for different combinations of acceleration factors. Work done by the earlier researchers has not focused on the best-fitting experiments for the best-optimized ANN–PSO model. Keeping the parameter constant can be considered as a primitive approach, and more study for optimizing the model is needed.
Table 4

Parametric study of the hybrid PSO-ANN model

Number of neuronSwarm sizeBest performing acceleration factor
Training stage
Testing stage
C1C2MAPERMSER2MAPERMSER2
10 1.5 2.5 19.44 0.035 0.966 22.00 0.039 0.955 
15 1.5 2.5 19.44 0.035 0.966 29.77 0.074 0.839 
20 1.5 2.5 47.19 0.089 0.779 41.71 0.079 0.806 
30 1.5 2.5 31.79 0.049 0.926 32.72 0.062 0.898 
35 1.5 2.5 16.02 0.029 0.975 21.62 0.041 0.958 
40 1.5 2.5 25.88 0.048 0.935 25.20 0.048 0.927 
50 1.5 2.5 22.98 0.036 0.959 25.45 0.049 0.935 
10 1.5 2.5 26.77 0.047 0.932 31.13 0.061 0.911 
15 1.5 2.5 19.10 0.038 0.955 29.77 0.051 0.951 
20 1.5 2.5 24.71 0.042 0.947 30.24 0.051 0.941 
30 1.5 2.5 25.46 0.047 0.934 30.44 0.056 0.934 
8 35 1.5 2.5 14.78 0.026 0.981 14.92 0.030 0.965 
40 1.5 2.5 28.39 0.059 0.894 30.44 0.056 0.934 
50 1.5 2.5 17.07 0.031 0.971 20.41 0.047 0.941 
10 1.5 2.5 16.90 0.029 0.975 22.29 0.042 0.951 
20 1.5 2.5 23.35 0.047 0.932 30.88 0.059 0.917 
30 1.5 2.5 21.15 0.045 0.939 26.77 0.045 0.954 
40 1.5 2.5 17.27 0.033 0.966 26.18 0.051 0.927 
50 1.5 2.5 24.42 0.041 0.949 31.82 0.060 0.915 
10 10 1.5 2.5 43.52 0.075 0.837 52.42 0.089 0.878 
10 20 1.5 2.5 26.12 0.043 0.944 31.82 0.051 0.939 
10 30 1.5 2.5 18.31 0.027 0.977 23.73 0.044 0.947 
10 40 1.5 2.5 17.01 0.027 0.977 23.28 0.040 0.959 
10 50 1.5 2.5 17.70 0.030 0.972 19.89 0.036 0.966 
Number of neuronSwarm sizeBest performing acceleration factor
Training stage
Testing stage
C1C2MAPERMSER2MAPERMSER2
10 1.5 2.5 19.44 0.035 0.966 22.00 0.039 0.955 
15 1.5 2.5 19.44 0.035 0.966 29.77 0.074 0.839 
20 1.5 2.5 47.19 0.089 0.779 41.71 0.079 0.806 
30 1.5 2.5 31.79 0.049 0.926 32.72 0.062 0.898 
35 1.5 2.5 16.02 0.029 0.975 21.62 0.041 0.958 
40 1.5 2.5 25.88 0.048 0.935 25.20 0.048 0.927 
50 1.5 2.5 22.98 0.036 0.959 25.45 0.049 0.935 
10 1.5 2.5 26.77 0.047 0.932 31.13 0.061 0.911 
15 1.5 2.5 19.10 0.038 0.955 29.77 0.051 0.951 
20 1.5 2.5 24.71 0.042 0.947 30.24 0.051 0.941 
30 1.5 2.5 25.46 0.047 0.934 30.44 0.056 0.934 
8 35 1.5 2.5 14.78 0.026 0.981 14.92 0.030 0.965 
40 1.5 2.5 28.39 0.059 0.894 30.44 0.056 0.934 
50 1.5 2.5 17.07 0.031 0.971 20.41 0.047 0.941 
10 1.5 2.5 16.90 0.029 0.975 22.29 0.042 0.951 
20 1.5 2.5 23.35 0.047 0.932 30.88 0.059 0.917 
30 1.5 2.5 21.15 0.045 0.939 26.77 0.045 0.954 
40 1.5 2.5 17.27 0.033 0.966 26.18 0.051 0.927 
50 1.5 2.5 24.42 0.041 0.949 31.82 0.060 0.915 
10 10 1.5 2.5 43.52 0.075 0.837 52.42 0.089 0.878 
10 20 1.5 2.5 26.12 0.043 0.944 31.82 0.051 0.939 
10 30 1.5 2.5 18.31 0.027 0.977 23.73 0.044 0.947 
10 40 1.5 2.5 17.01 0.027 0.977 23.28 0.040 0.959 
10 50 1.5 2.5 17.70 0.030 0.972 19.89 0.036 0.966 

The bolded values indicates the best MARS model performance.

Figure 5

Selected ANN–PSO model for the present study: (a) training stage and (b) testing stage.

Figure 5

Selected ANN–PSO model for the present study: (a) training stage and (b) testing stage.

Close modal

Results of the MARS model

The results obtained from the MARS model after trying different numbers of iterations are presented in Table 5. The model with iterations equal to 2 and maximum functions 20 came out as best fitting as it has the high R2 value, and low values of RMSE and MAPE. The training and testing stage plots of the best MARS model are depicted in Figure 6. The statistical indices MAPE, RMSE, and R2 for the training stage were found as 9.11%, 0.016 and 0.993 and for the testing stage, 12.34%, 0.020 and 0.984, respectively. The MARS-based model gives sightly more accurate results than the ANN–PSO model. The developed MARS model produced 20 different basis functions as given in Table 6. The developed MARS model output is represented in the form of an equation as shown in Equation (9).
(9)
Table 5

Statistical indices of the MARS model performance during training and testing stages

Number of iterations triedTraining
Testing
MAPERMSER2MAPERMSER2
2 9.106 0.0164 0.9925 12.344 0.0201 0.9836 
9.128 0.0164 0.9920 13.864 0.0224 0.9800 
8.545 0.0153 0.9930 12.903 0.0236 0.9780 
Number of iterations triedTraining
Testing
MAPERMSER2MAPERMSER2
2 9.106 0.0164 0.9925 12.344 0.0201 0.9836 
9.128 0.0164 0.9920 13.864 0.0224 0.9800 
8.545 0.0153 0.9930 12.903 0.0236 0.9780 
Table 6

Basis functions and their coefficient of the developed MARS model

Basis functionCoefficient
BF1 = max(0,0.13816 -S0) 5.874 
BF2 = max(0,0.13246 -δ*) 3.745 
BF3 = max(0, S0 −0.13816) * max(0, δ* −0.11941) −0.075 
BF4 = max(0, S0 −0.13816) * max(0,0.11941 -δ*) −10.724 
BF5 = BF2 * max(0,0.065669 -Rr) 124.242 
BF6 = max(0, S0 −0.13816) * max(0,0.65771 -β0.111 
BF7 = max(0, θ −0.41934) * max(0, β −0.60093) −3.379 
BF8 = max(0, θ −0.41934) * max(0,0.60093 -β2.712 
BF9 = max(0, θ −0.31984) * max(0, β −0.33178) 1.916 
BF10 = max(0, θ −0.31984) * max(0,0.33178 -β−1.689 
BF11 = max(0,0.10671 -Rr) * max(0,0.17731 -δ*) −6.381 
BF12 = max(0,0.28634 -α0.064 
BF13 = max(0, β −0.92056) 0.985 
BF14 = max(0,0.92056 -β−0.095 
'BF15 = max(0, θ −0.41934) * max(0, δ* −0.2518) −1.258 
'BF16 = max(0, θ −0.41934) * max(0,0.2518 -δ*) 2.268 
BF17 = max(0,0.10671 -Rr) * max(0, α −0.18419) 1.928 
'BF18 = BF14 * max(0, θ −0.60352) −0.759 
'BF19 = max(0, θ −0.31984) * max(0, δ* −0.55451) 0.558 
'BF20 = max(0, θ −0.31984) * max(0,0.55451 -δ*) −1.265 
Basis functionCoefficient
BF1 = max(0,0.13816 -S0) 5.874 
BF2 = max(0,0.13246 -δ*) 3.745 
BF3 = max(0, S0 −0.13816) * max(0, δ* −0.11941) −0.075 
BF4 = max(0, S0 −0.13816) * max(0,0.11941 -δ*) −10.724 
BF5 = BF2 * max(0,0.065669 -Rr) 124.242 
BF6 = max(0, S0 −0.13816) * max(0,0.65771 -β0.111 
BF7 = max(0, θ −0.41934) * max(0, β −0.60093) −3.379 
BF8 = max(0, θ −0.41934) * max(0,0.60093 -β2.712 
BF9 = max(0, θ −0.31984) * max(0, β −0.33178) 1.916 
BF10 = max(0, θ −0.31984) * max(0,0.33178 -β−1.689 
BF11 = max(0,0.10671 -Rr) * max(0,0.17731 -δ*) −6.381 
BF12 = max(0,0.28634 -α0.064 
BF13 = max(0, β −0.92056) 0.985 
BF14 = max(0,0.92056 -β−0.095 
'BF15 = max(0, θ −0.41934) * max(0, δ* −0.2518) −1.258 
'BF16 = max(0, θ −0.41934) * max(0,0.2518 -δ*) 2.268 
BF17 = max(0,0.10671 -Rr) * max(0, α −0.18419) 1.928 
'BF18 = BF14 * max(0, θ −0.60352) −0.759 
'BF19 = max(0, θ −0.31984) * max(0, δ* −0.55451) 0.558 
'BF20 = max(0, θ −0.31984) * max(0,0.55451 -δ*) −1.265 
Figure 6

Selected MARS model for the present study: (a) training stage and (b) testing stage.

Figure 6

Selected MARS model for the present study: (a) training stage and (b) testing stage.

Close modal
Data sets of different researchers have been utilized to predict %Qmc value using the present approaches, i.e., ANN–PSO and MARS model. Percentage main channel discharge %Qmc is calculated by using Equation (10).
(10)
It can be seen that the MARS model provides a slightly better prediction of the Q/Qmc compared to ANN–PSO. Therefore, the developed MARS model was found to be appropriate for predicting the discharge in a non-prismatic compound channel. Two AI methods, i.e., ANN–PSO and MARS are used to compute the discharge of the main channel %Qmc, and a graph has been plotted between observed and predicted discharge (%Qmc) as shown in Figure 7. The correlation diagram shown in Figure 8 indicates that the present MARS model predicts discharge very accurately for the datasets of Rezaei (2006)/Cv3.81, Das (2018), and Yonesi et al. (2013) and for Rezaei (2006)/Cv1.91, ANN–PSO was a better method than MARS. The present AI model has been successfully applied to the individual authors' dataset for the computation of %Qmc.
Table 7

Performance of present models (ANN–PSO and MARS) in predicting %Qmc for different author datasets

S.NoDataset of different authorsANN–PSO
MARS
MAPERMSER2MAPERMSER2
Bousmar (2002)/Cv3.81 10.39 4.69 0.89 7.96 4.28 0.90 
Bousmar (2002)/Cv11.31 12.77 9.02 0.61 9.57 6.43 0.66 
Rezaei (2006)/Cv 1.91 9.94 5.10 0.96 8.38 7.04 0.89 
Rezaei (2006)/Cv 3.81 9.10 5.94 0.98 3.69 2.87 0.98 
Rezaei (2006)/Cv 11.31 9.10 9.43 0.96 6.48 4.67 0.96 
Bousmar et al. (2006)/Dv3.81 18.95 9.66 0.25 14.13 7.97 0.35 
Bousmar et al. (2006)/Dv5.71 16.04 8.38 0.29 13.11 7.17 0.47 
Yonesi et al. (2013)/Dv3.81 5.15 0.73 0.93 1.07 0.13 0.99 
Yonesi et al. (2013)/Dv 11.31 7.14 1.12 0.95 1.08 0.12 0.99 
10 Naik & Khatua (2016)/Cv5 10.06 7.44 0.49 9.65 7.24 0.91 
11 Naik & Khatua (2016)/Cv 9 10.86 8.79 0.78 4.21 3.3 0.97 
12 Naik & Khatua (2016)/Cv 12.3 4.90 4.90 0.77 4.81 3.7 0.98 
13 Das & Khatua (2018)/Dv14.57 8.29 6.02 0.88 3.48 1.86 0.99 
14 Das & Khatua (2018)/Dv5.93 11.77 11.53 0.79 3.08 2.4 0.99 
15 Das & Khatua (2018)/Dv9.83 9.31 9.55 0.84 4.25 2.42 0.98 
16 Mehrabani et al. (2020)/Dv (7.25–11.3) 8.61 5.44 0.72 5.56 2.86 0.67 
S.NoDataset of different authorsANN–PSO
MARS
MAPERMSER2MAPERMSER2
Bousmar (2002)/Cv3.81 10.39 4.69 0.89 7.96 4.28 0.90 
Bousmar (2002)/Cv11.31 12.77 9.02 0.61 9.57 6.43 0.66 
Rezaei (2006)/Cv 1.91 9.94 5.10 0.96 8.38 7.04 0.89 
Rezaei (2006)/Cv 3.81 9.10 5.94 0.98 3.69 2.87 0.98 
Rezaei (2006)/Cv 11.31 9.10 9.43 0.96 6.48 4.67 0.96 
Bousmar et al. (2006)/Dv3.81 18.95 9.66 0.25 14.13 7.97 0.35 
Bousmar et al. (2006)/Dv5.71 16.04 8.38 0.29 13.11 7.17 0.47 
Yonesi et al. (2013)/Dv3.81 5.15 0.73 0.93 1.07 0.13 0.99 
Yonesi et al. (2013)/Dv 11.31 7.14 1.12 0.95 1.08 0.12 0.99 
10 Naik & Khatua (2016)/Cv5 10.06 7.44 0.49 9.65 7.24 0.91 
11 Naik & Khatua (2016)/Cv 9 10.86 8.79 0.78 4.21 3.3 0.97 
12 Naik & Khatua (2016)/Cv 12.3 4.90 4.90 0.77 4.81 3.7 0.98 
13 Das & Khatua (2018)/Dv14.57 8.29 6.02 0.88 3.48 1.86 0.99 
14 Das & Khatua (2018)/Dv5.93 11.77 11.53 0.79 3.08 2.4 0.99 
15 Das & Khatua (2018)/Dv9.83 9.31 9.55 0.84 4.25 2.42 0.98 
16 Mehrabani et al. (2020)/Dv (7.25–11.3) 8.61 5.44 0.72 5.56 2.86 0.67 
Figure 7

Comparison of observed and predicted %Qmc of present ANN–PSO and MARS model.

Figure 7

Comparison of observed and predicted %Qmc of present ANN–PSO and MARS model.

Close modal
Figure 8

(a–p) Comparison of observed and predicted %Qmc computed from present ANN–PSO and MARS model for individual author datasets.

Figure 8

(a–p) Comparison of observed and predicted %Qmc computed from present ANN–PSO and MARS model for individual author datasets.

Close modal

Comparison of the present model with the existing %Qmc model

Four empirical equations developed by Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016), and Das et al. (2022) have been used for determining the Qmc (%). Knight & Demetriou (1983) proposed an empirical relation for determining the %Qmc as depicted in Equation (11):
(11)
Khatua & Patra (2007) provided Equation (12) for computation of %Qmc,
(12)
Devi et al. (2016) developed Equation (13) for the %Qmc model using regression analysis
(13)
Das et al. (2022) performed a regression analysis and developed Equation (14) for determining %Qmc.
(14)
The observed vs. predicted %Qmc by four empirical formulae developed by Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016), and Das et al. (2022) are shown in Figure 9. The present models ANN-PSO and MARS are applied to different channel dataset and the statistical error indices are presented in Table 7. Table 8 depicts the error analysis results in predicting discharges using present models (i.e., ANN–PSO and MARS) and four empirical approaches by Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016) and Das et al. (2022). It is found that the four empirical methods by Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016) and Das et al. (2022) provide R2 values of 0.47, 0.41, 0.39, and 0.18, respectively, whereas the present model ANN–PSO model and MARS model show a high R2 value of 0.84 and 0.93, respectively, for %Qmc values.
Table 8

Statistical error indices in predicting %Qmc from the present models and existing empirical methods for the collected dataset

Different modelsMAPERMSER2EId
(Present Model) ANN–PSO 11.17 8.03 0.839 0.81 0.96 
(Present Model) MARS 6.93 5.05 0.928 0.93 0.98 
Knight & Demetriou (1983)  44.72 23.15 0.467 −1.16 0.63 
Khatua & Patra (2007)  44.02 22.95 0.405 −1.12 0.62 
Devi et al. (2016)  53.10 26.72 0.389 −1.83 0.57 
Das et al. (2022)  43.52 22.44 0.183 −1.01 0.60 
Different modelsMAPERMSER2EId
(Present Model) ANN–PSO 11.17 8.03 0.839 0.81 0.96 
(Present Model) MARS 6.93 5.05 0.928 0.93 0.98 
Knight & Demetriou (1983)  44.72 23.15 0.467 −1.16 0.63 
Khatua & Patra (2007)  44.02 22.95 0.405 −1.12 0.62 
Devi et al. (2016)  53.10 26.72 0.389 −1.83 0.57 
Das et al. (2022)  43.52 22.44 0.183 −1.01 0.60 
Figure 9

Comparison of observed and predicted %Qmc by existing empirical methods and present models.

Figure 9

Comparison of observed and predicted %Qmc by existing empirical methods and present models.

Close modal

For different ranges of width ratio and relative flow depth, the performance of all six models, i.e., present models, ANN–PSO and MARS, and four empirical methods, i.e., Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016), and Das et al. (2022) are shown in Tables 9 and 10. This comparison clearly indicates the suitability of different models for different input ranges of width ratios and relative flow depths.

Table 9

Performance of models with different ranges of width ratio (α)

Different modelsΑ < 1.51.5 < α < 2.02.0 < α < 2.52.5 < α < 3.03.0 < α < 5.8
Present model (ANN–PSO) 6.16 5.67 5.85 5.35 2.96 
11.56 11.31 11.44 11.16 6.22 
7.28 6.93 7.22 7.28 4.34 
0.84 0.89 0.88 0.89 0.93 
0.83 0.88 0.88 0.87 0.91 
0.96 0.92 0.93 0.97 0.98 
Present model (MARS) 4.8 4.56 4.44 3.26 1.72 
8.98 8.68 8.7 6.1 3.66 
6.05 5.98 5.72 4.97 2.2 
0.88 0.92 0.9 0.94 0.98 
0.88 0.91 0.9 0.94 0.98 
0.97 0.98 0.97 0.98 0.99 
Knight & Demetriou (1983)  34.91 24.28 16.32 15.55 17.22 
71.99 51.09 39.23 35.45 45.55 
37.19 26.8 19.6 18.99 20.03 
0.25 0.48 0.52 0.31 0.59 
−5.6 −2.05 −0.65 −0.63 −0.84 
0.39 0.52 0.63 0.61 0.68 
Khatua & Patra (2007)  36.22 24.94 17.33 15.67 13.81 
74.54 52.75 41.79 36.09 40.23 
38.44 27.58 20.6 18.63 16.92 
0.25 0.45 0.49 0.28 0.52 
−6.05 −2.23 −0.82 −0.57 −0.28 
0.38 0.5 0.6 0.59 0.68 
Devi et al. (2016)  41.57 29.06 21.36 18.63 17.73 
88.02 60.55 50.4 43.1 50.48 
43.28 31.37 24.3 21.97 20.64 
0.17 0.45 0.49 0.29 0.51 
−9.93 −3.18 −1.53 −1.18 −0.9 
0.34 0.91 0.55 0.55 0.63 
Das et al. (2022)  23.75 22.11 20.48 20.32 14.94 
51.11 47.4 48.01 47.59 37.99 
28.36 24.96 23.23 23.5 17.86 
0.08 0.21 0.29 0.17 0.04 
−3.01 −1.64 −1.31 −1.5 −0.42 
0.37 0.49 0.53 0.51 0.53 
Different modelsΑ < 1.51.5 < α < 2.02.0 < α < 2.52.5 < α < 3.03.0 < α < 5.8
Present model (ANN–PSO) 6.16 5.67 5.85 5.35 2.96 
11.56 11.31 11.44 11.16 6.22 
7.28 6.93 7.22 7.28 4.34 
0.84 0.89 0.88 0.89 0.93 
0.83 0.88 0.88 0.87 0.91 
0.96 0.92 0.93 0.97 0.98 
Present model (MARS) 4.8 4.56 4.44 3.26 1.72 
8.98 8.68 8.7 6.1 3.66 
6.05 5.98 5.72 4.97 2.2 
0.88 0.92 0.9 0.94 0.98 
0.88 0.91 0.9 0.94 0.98 
0.97 0.98 0.97 0.98 0.99 
Knight & Demetriou (1983)  34.91 24.28 16.32 15.55 17.22 
71.99 51.09 39.23 35.45 45.55 
37.19 26.8 19.6 18.99 20.03 
0.25 0.48 0.52 0.31 0.59 
−5.6 −2.05 −0.65 −0.63 −0.84 
0.39 0.52 0.63 0.61 0.68 
Khatua & Patra (2007)  36.22 24.94 17.33 15.67 13.81 
74.54 52.75 41.79 36.09 40.23 
38.44 27.58 20.6 18.63 16.92 
0.25 0.45 0.49 0.28 0.52 
−6.05 −2.23 −0.82 −0.57 −0.28 
0.38 0.5 0.6 0.59 0.68 
Devi et al. (2016)  41.57 29.06 21.36 18.63 17.73 
88.02 60.55 50.4 43.1 50.48 
43.28 31.37 24.3 21.97 20.64 
0.17 0.45 0.49 0.29 0.51 
−9.93 −3.18 −1.53 −1.18 −0.9 
0.34 0.91 0.55 0.55 0.63 
Das et al. (2022)  23.75 22.11 20.48 20.32 14.94 
51.11 47.4 48.01 47.59 37.99 
28.36 24.96 23.23 23.5 17.86 
0.08 0.21 0.29 0.17 0.04 
−3.01 −1.64 −1.31 −1.5 −0.42 
0.37 0.49 0.53 0.51 0.53 

Note: Six values presented in each cell represent MAE, MAPE, RMSE, R2, E, and Id, respectively.

Table 10

Performance of models with different ranges of relative flow depth (β)

Different models0.1 < β< 0.20.2 < β< 0.30.3 < β< 0.40.4 < β< 0.5
Present model (ANN–PSO) 5.9 5.14 5.81 4.37 
10.33 8.96 12.83 10.62 
9.33 6.61 7.71 5.45 
0.86 0.82 0.83 0.72 
0.84 0.79 0.83 0.71 
0.96 0.95 0.95 0.91 
Present model (MARS) 2.9 3.27 4.71 3.16 
4.78 5.72 9.22 7.56 
4.2 4.3 6.69 4.16 
0.97 0.92 0.89 0.83 
0.97 0.92 0.87 0.83 
0.99 0.98 0.96 0.95 
Knight & Demetriou (1983)  18.45 20.72 21.42 21.33 
33.49 44.02 46.25 59.32 
21.15 24.68 24.71 25.04 
0.3 0.23 0.2 0.22 
−0.52 −1.75 −3.34 −5.16 
0.79 0.5 0.49 0.43 
Khatua & Patra (2007)  14.25 19.48 23.17 23.04 
27.13 39.94 63.55 63.61 
17.62 23.72 27.82 26.43 
0.3 0.21 0.07 0.21 
−0.09 −1.72 −3.58 −5.98 
0.82 0.52 0.45 0.41 
Devi et al. (2016)  18 21.2 26.16 26.91 
36.08 45.03 57.04 74.93 
21.55 24.71 29.5 30.02 
0.35 0.3 0.14 0.24 
−0.57 −1.72 −5.01 −7.75 
0.77 0.51 0.44 0.38 
Das et al. (2022)  13.21 16.12 21.2 27.88 
23.56 32.34 61.75 76.15 
16.53 19.07 24.67 30.16 
0.3 0.25 0.25 0.4 
0.11 −0.7 −2.13 −7.83 
0.79 0.63 0.5 0.42 
Different models0.1 < β< 0.20.2 < β< 0.30.3 < β< 0.40.4 < β< 0.5
Present model (ANN–PSO) 5.9 5.14 5.81 4.37 
10.33 8.96 12.83 10.62 
9.33 6.61 7.71 5.45 
0.86 0.82 0.83 0.72 
0.84 0.79 0.83 0.71 
0.96 0.95 0.95 0.91 
Present model (MARS) 2.9 3.27 4.71 3.16 
4.78 5.72 9.22 7.56 
4.2 4.3 6.69 4.16 
0.97 0.92 0.89 0.83 
0.97 0.92 0.87 0.83 
0.99 0.98 0.96 0.95 
Knight & Demetriou (1983)  18.45 20.72 21.42 21.33 
33.49 44.02 46.25 59.32 
21.15 24.68 24.71 25.04 
0.3 0.23 0.2 0.22 
−0.52 −1.75 −3.34 −5.16 
0.79 0.5 0.49 0.43 
Khatua & Patra (2007)  14.25 19.48 23.17 23.04 
27.13 39.94 63.55 63.61 
17.62 23.72 27.82 26.43 
0.3 0.21 0.07 0.21 
−0.09 −1.72 −3.58 −5.98 
0.82 0.52 0.45 0.41 
Devi et al. (2016)  18 21.2 26.16 26.91 
36.08 45.03 57.04 74.93 
21.55 24.71 29.5 30.02 
0.35 0.3 0.14 0.24 
−0.57 −1.72 −5.01 −7.75 
0.77 0.51 0.44 0.38 
Das et al. (2022)  13.21 16.12 21.2 27.88 
23.56 32.34 61.75 76.15 
16.53 19.07 24.67 30.16 
0.3 0.25 0.25 0.4 
0.11 −0.7 −2.13 −7.83 
0.79 0.63 0.5 0.42 

Note: Six values presented in each cell represent MAE, MAPE, RMSE, R2, E, and Id, respectively.

Tables 9 and 10 provide the MAPE, RMSE, and R2 values of two present models and three predictive discharge equations for selected ranges of width ratio (α) and relative flow depth (β), respectively. In Table 9, for all the range of α (i.e., α < 1.5, 1.5 < α < 2.0, 2.0 < α < 2.5, 2.5 < α < 3.0 and 3.0 < α < 5.8) MAPE is found high for methods by Devi et al. (2016), Knight & Demetriou (1983), Khatua & Patra (2007) and ANN–PSO (the present model). In addition, RMSE is found to be high in models by Devi et al. (2016), Knight & Demetriou (1983), Khatua & Patra (2007) and ANN–PSO (the present model) and R2 is low for models by Devi et al. (2016), Knight & Demetriou (1983), Khatua & Patra (2007) and ANN–PSO (present model). For the ranges of α < 1.5, 1.5 < α < 2.0 and 2.0 < α < 2.5, Knight & Demetriou (1983) show a good result (low MAE and RMSE and higher R2) as compared to Khatua & Patra (2007). The previous models provide unsatisfactory results with high errors and low R2 values as they have not considered the converging and diverging compound channels for model development. However, for all the ranges of α, MARS (present model) performs well with minimum MAPE and RMSE and high R2 value.

In Table 10, for the range of β (i.e., 0.1 < β < 0.2), MAE is found to be high for models by Knight & Demetriou (1983), Devi et al. (2016), Khatua & Patra (2007), Das et al. (2022) and ANN–PSO (the present model). For the range of 0.2 < β < 0.3 and 0.3 < β < 0.4, a high value of MAE is found by Devi et al. (2016), whereas for the range of 0.4 < β < 0.5, the model by Das et al. (2022) provides a high MAE value. Further, a greater MAPE is found for the model by Devi et al. (2016) for the range of 0.1 < β < 0.2 and 0.2 < β < 0.3, whereas Khatua & Patra (2007) and Das et al. (2022) show a high MAPE value for the range 0.3 < β < 0.4 and 0.4 < β < 0.5, respectively. In addition to that, RMSE is found to be high for the models by Devi et al. (2016), Knight & Demetriou (1983), Khatua & Patra (2007) and ANN–PSO (i.e., the present model) for the range of 0.1 < β < 0.4, whereas the model by Das et al. (2022) demonstrates a large RMSE value for the range of 0.4 < β < 0.5. The coefficient of determination, R2 is found to be high for Devi et al. (2016) in the range of 0.1 < β < 0.2 and 0.2 < β < 0.3 and the model by Das et al. (2022) depicts a high R2 value for the range of 0.3 < β < 0.5. The model by Das et al. (2022) is found to be satisfactory with high values of E as compared to other previous models for the range of 0.1 < β < 0.2, 0.2 < β < 0.3, and 0.3 < β < 0.4, whereas Knight & Demetriou (1983) indicates the good prediction in the range of 0.4 < β < 0.5. For the range of 0.1 < β < 0.2 and 0.4 < β < 0.5, the Id values are found to be high for Khatua & Patra (2007) and Knight & Demetriou (1983), respectively, whereas the model by Das et al. (2022) is found to be satisfactory for both ranges of β (i.e., 0.2 < β < 0.3 and 0.3 < β < 0.4). The models developed by previous researchers provide unsatisfactory results with high errors and low R2, E, and Id values as they have not included the convergent and divergent compound channels in model development. However, for all the ranges of β, MARS (present model) performs well with minimum MAE, MAPE, and RMSE and high values of R2, E, and Id as compared to previous models including ANN–PSO (another present model).

The traditional approaches, i.e., SCM and DCM, are employed to determine the discharge of compound channels with non-prismatic floodplains. QDCM (V−i) gives the lowest MAPE and RMSE of 62.75% and 22.48, respectively. These traditional methods provide less satisfactory results for determining the discharge of compound channels. Four empirical methods, i.e., Knight & Demetriou (1983), Khatua & Patra (2007), Devi et al. (2016), and Das et al. (2022) are also used for determining the discharge, which provides the R2 value of less than 0.50. The GT indicates that relative hydraulic radius, relative flow depth, bed slope of the channel, flow aspect ratio, and converging and diverging angle are the most influential parameters for estimating the discharge of the non-prismatic compound channel. In the present study, ANN–PSO and MARS soft computing approaches successfully predicted the discharge in the converging and diverging compound channels with a high R2 value of more than 0.80 and 0.90, respectively, and an MAPE value of less than 12 and 7%, respectively. The performances of the developed models and four existing %Qmc models have been checked for different ranges of width ratio, and relative flow depth, and the present MARS model was found to provide better results for all the different ranges of α and β. Overall, it was observed that the MARS model is a more efficient method as compared to the ANN–PSO model in predicting the discharges in non-prismatic compound channels for the present ranges of datasets.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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