Abstract
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.
HIGHLIGHTS
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.
NOMENCLATURE
- 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
ABBREVIATIONS
INTRODUCTION
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.
SOURCES OF DATASET
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, (H–h)/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.
Summary of the dataset collected from the literature for diverging and converging compound channels
Authors’ dataset . | Range . | ff . | Ar . | Rr . | β . | S0 . | δ* . | α . | Xr . | θ . | Q/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 | 0.774 | 0.567 | 0.285 | 0 | 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 | 0.894 | 0.559 | 0.093 | 0 | 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 | 0.876 | 0.562 | 0.280 | 0 | 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 | 0.898 | 0.356 | 0.354 | 0 | 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 | 0.899 | 0.505 | 0.083 | 0 | 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 | 0.853 | 0.568 | 0.285 | 0 | 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 | 0.802 | 0.372 | 0.280 | 0 | 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 | 0.219 | 0.568 | 0.285 | 0 | 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 | 0.222 | 0.539 | 0.090 | 0 | 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 | 0.336 | 0.182 | 0.226 | 0 | 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 | 0.280 | 0.176 | 0.219 | 0 | 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 | 0.335 | 0.170 | 0.257 | 0 | 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 | 0.357 | 1.081 | 0.354 | 0 | 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 | 0.362 | 1.081 | 0.354 | 0 | 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 | 0.361 | 1.081 | 0.354 | 0 | 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 | 0.095 | 0.083 | 0.250 | 2.025 | 0.085 |
Authors’ dataset . | Range . | ff . | Ar . | Rr . | β . | S0 . | δ* . | α . | Xr . | θ . | Q/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 | 0.774 | 0.567 | 0.285 | 0 | 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 | 0.894 | 0.559 | 0.093 | 0 | 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 | 0.876 | 0.562 | 0.280 | 0 | 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 | 0.898 | 0.356 | 0.354 | 0 | 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 | 0.899 | 0.505 | 0.083 | 0 | 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 | 0.853 | 0.568 | 0.285 | 0 | 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 | 0.802 | 0.372 | 0.280 | 0 | 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 | 0.219 | 0.568 | 0.285 | 0 | 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 | 0.222 | 0.539 | 0.090 | 0 | 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 | 0.336 | 0.182 | 0.226 | 0 | 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 | 0.280 | 0.176 | 0.219 | 0 | 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 | 0.335 | 0.170 | 0.257 | 0 | 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 | 0.357 | 1.081 | 0.354 | 0 | 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 | 0.362 | 1.081 | 0.354 | 0 | 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 | 0.361 | 1.081 | 0.354 | 0 | 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 | 0.095 | 0.083 | 0.250 | 2.025 | 0.085 |
METHODOLOGY
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
Divided channel method
Types of subsection dividing lines between floodplains and main channel in the DCM.
Types of subsection dividing lines between floodplains and main channel in the DCM.
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).
Statistical indices for the outcomes obtained from the conventional techniques
Method . | MAPE . | RMSE . | R2 . | E . | Id . |
---|---|---|---|---|---|
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 |
Method . | MAPE . | RMSE . | R2 . | E . | Id . |
---|---|---|---|---|---|
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.
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).
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).
Gamma test
‘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.
Multivariate adaptive regression splines


RESULTS AND DISCUSSIONS
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.
Results obtained from the GT
Serial No. . | Combination of the input parameters . | Gamma . | Gradient . | Std. error . | V-ratio . | Masking . |
---|---|---|---|---|---|---|
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 parameters . | Gamma . | Gradient . | Std. error . | V-ratio . | Masking . |
---|---|---|---|---|---|---|
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
Parametric study of the hybrid PSO-ANN model
Number of neuron . | Swarm size . | Best performing acceleration factor . | Training stage . | Testing stage . | |||||
---|---|---|---|---|---|---|---|---|---|
C1 . | C2 . | MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | ||
5 | 10 | 1.5 | 2.5 | 19.44 | 0.035 | 0.966 | 22.00 | 0.039 | 0.955 |
5 | 15 | 1.5 | 2.5 | 19.44 | 0.035 | 0.966 | 29.77 | 0.074 | 0.839 |
5 | 20 | 1.5 | 2.5 | 47.19 | 0.089 | 0.779 | 41.71 | 0.079 | 0.806 |
5 | 30 | 1.5 | 2.5 | 31.79 | 0.049 | 0.926 | 32.72 | 0.062 | 0.898 |
5 | 35 | 1.5 | 2.5 | 16.02 | 0.029 | 0.975 | 21.62 | 0.041 | 0.958 |
5 | 40 | 1.5 | 2.5 | 25.88 | 0.048 | 0.935 | 25.20 | 0.048 | 0.927 |
5 | 50 | 1.5 | 2.5 | 22.98 | 0.036 | 0.959 | 25.45 | 0.049 | 0.935 |
8 | 10 | 1.5 | 2.5 | 26.77 | 0.047 | 0.932 | 31.13 | 0.061 | 0.911 |
8 | 15 | 1.5 | 2.5 | 19.10 | 0.038 | 0.955 | 29.77 | 0.051 | 0.951 |
8 | 20 | 1.5 | 2.5 | 24.71 | 0.042 | 0.947 | 30.24 | 0.051 | 0.941 |
8 | 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 |
8 | 40 | 1.5 | 2.5 | 28.39 | 0.059 | 0.894 | 30.44 | 0.056 | 0.934 |
8 | 50 | 1.5 | 2.5 | 17.07 | 0.031 | 0.971 | 20.41 | 0.047 | 0.941 |
9 | 10 | 1.5 | 2.5 | 16.90 | 0.029 | 0.975 | 22.29 | 0.042 | 0.951 |
9 | 20 | 1.5 | 2.5 | 23.35 | 0.047 | 0.932 | 30.88 | 0.059 | 0.917 |
9 | 30 | 1.5 | 2.5 | 21.15 | 0.045 | 0.939 | 26.77 | 0.045 | 0.954 |
9 | 40 | 1.5 | 2.5 | 17.27 | 0.033 | 0.966 | 26.18 | 0.051 | 0.927 |
9 | 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 neuron . | Swarm size . | Best performing acceleration factor . | Training stage . | Testing stage . | |||||
---|---|---|---|---|---|---|---|---|---|
C1 . | C2 . | MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | ||
5 | 10 | 1.5 | 2.5 | 19.44 | 0.035 | 0.966 | 22.00 | 0.039 | 0.955 |
5 | 15 | 1.5 | 2.5 | 19.44 | 0.035 | 0.966 | 29.77 | 0.074 | 0.839 |
5 | 20 | 1.5 | 2.5 | 47.19 | 0.089 | 0.779 | 41.71 | 0.079 | 0.806 |
5 | 30 | 1.5 | 2.5 | 31.79 | 0.049 | 0.926 | 32.72 | 0.062 | 0.898 |
5 | 35 | 1.5 | 2.5 | 16.02 | 0.029 | 0.975 | 21.62 | 0.041 | 0.958 |
5 | 40 | 1.5 | 2.5 | 25.88 | 0.048 | 0.935 | 25.20 | 0.048 | 0.927 |
5 | 50 | 1.5 | 2.5 | 22.98 | 0.036 | 0.959 | 25.45 | 0.049 | 0.935 |
8 | 10 | 1.5 | 2.5 | 26.77 | 0.047 | 0.932 | 31.13 | 0.061 | 0.911 |
8 | 15 | 1.5 | 2.5 | 19.10 | 0.038 | 0.955 | 29.77 | 0.051 | 0.951 |
8 | 20 | 1.5 | 2.5 | 24.71 | 0.042 | 0.947 | 30.24 | 0.051 | 0.941 |
8 | 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 |
8 | 40 | 1.5 | 2.5 | 28.39 | 0.059 | 0.894 | 30.44 | 0.056 | 0.934 |
8 | 50 | 1.5 | 2.5 | 17.07 | 0.031 | 0.971 | 20.41 | 0.047 | 0.941 |
9 | 10 | 1.5 | 2.5 | 16.90 | 0.029 | 0.975 | 22.29 | 0.042 | 0.951 |
9 | 20 | 1.5 | 2.5 | 23.35 | 0.047 | 0.932 | 30.88 | 0.059 | 0.917 |
9 | 30 | 1.5 | 2.5 | 21.15 | 0.045 | 0.939 | 26.77 | 0.045 | 0.954 |
9 | 40 | 1.5 | 2.5 | 17.27 | 0.033 | 0.966 | 26.18 | 0.051 | 0.927 |
9 | 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.
Selected ANN–PSO model for the present study: (a) training stage and (b) testing stage.
Selected ANN–PSO model for the present study: (a) training stage and (b) testing stage.
Results of the MARS model
Statistical indices of the MARS model performance during training and testing stages
Number of iterations tried . | Training . | Testing . | ||||
---|---|---|---|---|---|---|
MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | |
2 | 9.106 | 0.0164 | 0.9925 | 12.344 | 0.0201 | 0.9836 |
3 | 9.128 | 0.0164 | 0.9920 | 13.864 | 0.0224 | 0.9800 |
4 | 8.545 | 0.0153 | 0.9930 | 12.903 | 0.0236 | 0.9780 |
Number of iterations tried . | Training . | Testing . | ||||
---|---|---|---|---|---|---|
MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | |
2 | 9.106 | 0.0164 | 0.9925 | 12.344 | 0.0201 | 0.9836 |
3 | 9.128 | 0.0164 | 0.9920 | 13.864 | 0.0224 | 0.9800 |
4 | 8.545 | 0.0153 | 0.9930 | 12.903 | 0.0236 | 0.9780 |
Basis functions and their coefficient of the developed MARS model
Basis function . | Coefficient . |
---|---|
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 -β) | 0.111 |
BF7 = max(0, θ −0.41934) * max(0, β −0.60093) | −3.379 |
BF8 = max(0, θ −0.41934) * max(0,0.60093 -β) | 2.712 |
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 -α) | 0.064 |
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 function . | Coefficient . |
---|---|
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 -β) | 0.111 |
BF7 = max(0, θ −0.41934) * max(0, β −0.60093) | −3.379 |
BF8 = max(0, θ −0.41934) * max(0,0.60093 -β) | 2.712 |
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 -α) | 0.064 |
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 |
Selected MARS model for the present study: (a) training stage and (b) testing stage.
Selected MARS model for the present study: (a) training stage and (b) testing stage.
Performance of present models (ANN–PSO and MARS) in predicting %Qmc for different author datasets
S.No . | Dataset of different authors . | ANN–PSO . | MARS . | ||||
---|---|---|---|---|---|---|---|
MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | ||
1 | Bousmar (2002)/Cv3.81 | 10.39 | 4.69 | 0.89 | 7.96 | 4.28 | 0.90 |
2 | Bousmar (2002)/Cv11.31 | 12.77 | 9.02 | 0.61 | 9.57 | 6.43 | 0.66 |
3 | Rezaei (2006)/Cv 1.91 | 9.94 | 5.10 | 0.96 | 8.38 | 7.04 | 0.89 |
4 | Rezaei (2006)/Cv 3.81 | 9.10 | 5.94 | 0.98 | 3.69 | 2.87 | 0.98 |
5 | Rezaei (2006)/Cv 11.31 | 9.10 | 9.43 | 0.96 | 6.48 | 4.67 | 0.96 |
6 | Bousmar et al. (2006)/Dv3.81 | 18.95 | 9.66 | 0.25 | 14.13 | 7.97 | 0.35 |
7 | Bousmar et al. (2006)/Dv5.71 | 16.04 | 8.38 | 0.29 | 13.11 | 7.17 | 0.47 |
8 | Yonesi et al. (2013)/Dv3.81 | 5.15 | 0.73 | 0.93 | 1.07 | 0.13 | 0.99 |
9 | 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.No . | Dataset of different authors . | ANN–PSO . | MARS . | ||||
---|---|---|---|---|---|---|---|
MAPE . | RMSE . | R2 . | MAPE . | RMSE . | R2 . | ||
1 | Bousmar (2002)/Cv3.81 | 10.39 | 4.69 | 0.89 | 7.96 | 4.28 | 0.90 |
2 | Bousmar (2002)/Cv11.31 | 12.77 | 9.02 | 0.61 | 9.57 | 6.43 | 0.66 |
3 | Rezaei (2006)/Cv 1.91 | 9.94 | 5.10 | 0.96 | 8.38 | 7.04 | 0.89 |
4 | Rezaei (2006)/Cv 3.81 | 9.10 | 5.94 | 0.98 | 3.69 | 2.87 | 0.98 |
5 | Rezaei (2006)/Cv 11.31 | 9.10 | 9.43 | 0.96 | 6.48 | 4.67 | 0.96 |
6 | Bousmar et al. (2006)/Dv3.81 | 18.95 | 9.66 | 0.25 | 14.13 | 7.97 | 0.35 |
7 | Bousmar et al. (2006)/Dv5.71 | 16.04 | 8.38 | 0.29 | 13.11 | 7.17 | 0.47 |
8 | Yonesi et al. (2013)/Dv3.81 | 5.15 | 0.73 | 0.93 | 1.07 | 0.13 | 0.99 |
9 | 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 |
Comparison of observed and predicted %Qmc of present ANN–PSO and MARS model.
(a–p) Comparison of observed and predicted %Qmc computed from present ANN–PSO and MARS model for individual author datasets.
(a–p) Comparison of observed and predicted %Qmc computed from present ANN–PSO and MARS model for individual author datasets.
Comparison of the present model with the existing %Qmc model
Statistical error indices in predicting %Qmc from the present models and existing empirical methods for the collected dataset
Different models . | MAPE . | RMSE . | R2 . | E . | Id . |
---|---|---|---|---|---|
(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 models . | MAPE . | RMSE . | R2 . | E . | Id . |
---|---|---|---|---|---|
(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 |
Comparison of observed and predicted %Qmc by existing empirical methods and present models.
Comparison of observed and predicted %Qmc by existing empirical methods and present models.
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.
Performance of models with different ranges of width ratio (α)
Different models . | Α < 1.5 . | 1.5 < α < 2.0 . | 2.0 < α < 2.5 . | 2.5 < α < 3.0 . | 3.0 < 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 models . | Α < 1.5 . | 1.5 < α < 2.0 . | 2.0 < α < 2.5 . | 2.5 < α < 3.0 . | 3.0 < 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.
Performance of models with different ranges of relative flow depth (β)
Different models . | 0.1 < β< 0.2 . | 0.2 < β< 0.3 . | 0.3 < β< 0.4 . | 0.4 < 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 models . | 0.1 < β< 0.2 . | 0.2 < β< 0.3 . | 0.3 < β< 0.4 . | 0.4 < 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).
CONCLUSIONS
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.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.