1 Introduction

With growing population across the world, increasingly more people are living in and exploring the areas along rivers, leading to serious problems related to natural flooding, which often results in heavy life losses, in addition to economic damage. Trend analyses have shown that the proportion of both global losses and deaths attributable to flood disasters continue to rise rapidly (Berz, 2000). Given that flooding occurs when the volume of water flowing through a channel is greater than the channel’s capacity, predicting the flow rate of natural streams is crucial for flood prevention. As a means to reduce casualties and salvage assets as much as possible, accurate flow parameter prediction in flood circumstances, therefore, has aroused the attention of academics and engineers over recent years, with various techniques and methodologies devised to precisely gauge and anticipate river discharge, velocity distribution, shear stress distribution, and water surface level in the event of overbank floods. Compound channels are a prevalent river feature during the period of overbank flows. The hydrodynamics of a river’s discharge can induce alterations to the morphology of floodplains, leading to the development of a two-stage channel system that exhibits either converging or diverging characteristics. The transfer of energy from main channels to flood zones is comparatively higher in a composite waterway with narrowing flood zones, and thereby rendering the replication of flows in such a channel is a challenging task. Although the flow models of straight and meandering prismatic two-stage channels have been investigated by some researchers, including Sellin (1964), Myers and Elsawy (1975), Knight et al. (2010), and Khatua et al. (2012), very little is known about non-prismatic compound channels. When a channel is shaped like a converging or diverging path, the floodplain flow would increase and the floodplain flow would decrease (James & Brown, 1977). After investigating compound channels with symmetrically descending floodplains, Bousmar and Zech (2002), Bousmar et al. (2004), Rezaei (2006), and Rezaei and Knight (2009) discovered additional losses of head as well as the transfer of momentum from main channels to floodplains. Proust et al., (2006) have surveyed asymmetric geometry at a faster convergence rate, finding that there were more mass transfers and more head losses at a large convergence angle of 22 degrees. Flows in curved, two-stage converging and diverging channels were investigated by Chlebek et al. (2010), who found that when floodplains expand, the velocity gradient would rise and the head losses would increase, owing to mass and momentum transfer. There are discernible variations in the flow distribution between main channels and floodplains as a result of varying flow forces in each segment. Rezaei and Knight (2011) have explored the depth-averaged velocity, local velocity distributions, and boundary shear stress distributions in compound channels with nonprismatic floodplains in a range of convergence degrees, discovering an increase in velocity near the main channel walls, especially during the second half of the convergence reach; in addition, the effects of the lateral flows that enter main channels were observed at a high relative depth, as shown with the depth-averaged velocities. Overbank flows in compound channels with non-prismatic floodplains were studied by Yonesi et al. (2013), who found that when the depth ratio is increased or the roughness ratio is decreased, the velocity gradient between main channels and floodplains may be reduced at the divergence stretch’s middle and end sections. As the divergence angle becomes greater, so does the velocity gradient. The proportion of the split discharge is affected by the roughness ratio and the depth ratio. Furthermore, the shear stress gradient would increase with higher roughness of floodplains. Naik and Khatua (2016) have developed a multivariable regression model to forecast the water surface profile for compound channels with non-prismatic floodplains. Their proposed model matches actual facts and other researchers’ conclusions well. In non-prismatic compound channels, high-flow features have received little attention, and more data are needed for prediction of water surface profiles. To recreate precise profiles, more tests shall be conducted on non-prismatic compound channels with converging floodplains.

Building profile models for water surface by using different approaches makes it difficult to analyze dependencies among components, and these models would get complicated and time-consuming. This cuts in half the time needed for experiments and labour-intensive calculations. Support vector machine (SVM), gene expression programming (GEP), artificial neural networks (ANN), fuzzy neural networks (ANFIS), and M5 tree decision models are all often used to estimate flows in composite channels (Parsaie et al., 2017b). Unlike other soft computing algorithms like ANN and SVM, GEP can really produce mathematical relationships (Karimi et al., 2015), although GEP-based river engineering has been neglected (Azamathulla et al., 2013; Pradhan & Khatua, 2017). Parsaie et al. (2015) have used the support vector machine (SVM) method to forecast discharges in a compound open channel, finding that SVM can deliver the maximum accuracy, as illustrated by the error indices. By employing various analytical techniques, including the single channel method (SCM), coherence method (COHM), and divided channel method (DCM), Parsaie et al. (2016) have forecasted the discharges of a compound open channel. To enhance the precision of the flow discharge prediction, they have also devised a radian basis neural network (RBF). Finally, they found that the DCM featuring a horizontally separated boundary between subsections exhibited a correlation of determination (R2) of 0.76, with more precise results acquired in analytical methods. The evaluation of the MLP model’s outcomes exhibited a higher degree of precision (R2 = 0.95) than the RBF (R2 = 0.85) and other analytical methodologies. In another study, Parsaie and Haghiabi (2017) employed certain analytical techniques, specifically artificial neural network (ANN) and multivariate adaptive regression splines (MARS), to predict the discharge capacity of compound open channels, finding that the divided channel method yielded a coefficient of determination (R2) value of 0.76 and a root mean square error (RMSE) value of 0.162, exhibiting superior performance over other analytical methods evaluated. Their research further revealed that the precision of the applied soft computing models surpassed that of analytical methods by delivering an R2 value of 0.97 and an RMSE value of 0.03, indicating that the MARS model has a higher level of accuracy than the MLP model. In an exploration conducted by Parsaie et al. (2017a) to evaluate the precision of various empirical techniques, including the Single Channel Method (SCM), Divided Channel Method (DCM), and Coherence Method (CM), in approximating the flow discharges in a heterogeneous compound channel roughness, they found that if assuming the virtual perpendicular line as the boundary line between sub-sections, DCM could result in a coefficient of determination greater than 0.90, demonstrating its effective performance through empirical approaches. By using the most important characteristics, such as the width ratio, relative flow depth, aspect ratio, Reynolds number, and Froude number, Khuntia et al. (2018) created an artificial neural network (ANN) model to forecast boundary shear stress distribution in straight compound channels, revealing that back-propagation neural network (BPNN) models could function effectively over wide parameter spaces. Das and Khatua (2018b) designed a numerical model that relies on the momentum balance principle to forecast water surface elevations of compound channels with the converging floodplains. The numerical model was solved with the finite difference method in MATLAB, and the simulation results demonstrated a high level of concurrence with the experimental datasets. In addition, the model could yield the lowest level of errors in terms of such metrics as the mean absolute error, mean absolute percentage error, and root mean square error. To calculate flow magnitudes and velocities in the upper and lower main channels, Das et al. (2018) have enhanced the conventional independent subsection method (ISM) framework, which features the establishment of four distinct subsections that are delineated by both vertical and horizontal division lines. These lines correspond to the vertical interface between the primary channel and floodplains, as well as the horizontal interface between upper and lower main channels. The computation results demonstrate that the method possesses the ability to accurately forecast the discharge distributions in both floodplains and main channels. Das and Khatua (2018a) have conducted experiments on symmetric compound channels in diverging and converging floodplains with various relative flow depths, in order to examine the resistance characteristics of overbank flows in non-prismatic sections. A multivariable regression model was constructed, with consideration given to the geometric and hydraulic parameters previously mentioned, for the purpose of estimating the Manning’s roughness coefficient in non-prismatic compound channels. This model demonstrated favourable outcomes in comparison to the alternative methodologies across diverse experimental channels and field data, exhibiting reduced errors. Das et al. (2019) have developed a formula for calculating the flow rate in a system of compound channels with both divergent and convergent branches. Das et al. (2020) employed artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) techniques to forecast the discharge in compound channels featuring converging and diverging sections. The discharge was influenced by several significant input parameters, including the friction factor ratio, hydraulic radius ratio, relative flow depth, and bed slope. The ANFIS model demonstrated a favourable level of performance in comparison to the ANN model, as evidenced by a high coefficient of determination of 0.86 and a low root mean square error of 0.005. In order to estimate the water surface profile in converging compound channels by using non-dimensional variables, Naik et al. (2022) suggested a unique equation based on GEP. Yonesi et al., (2022) have modelled the discharges of compound open channels with non-prismatic floodplains (CCNPF) by utilizing some soft computation models, such as multivariate adaptive regression splines (MARS) and group method of data handling (GMDH). The outcomes were subsequently compared with the data from multilayer perceptron neural networks (MLPNN). The statistical error indices indicate that the developed models can deliver sufficient accuracy in estimating the flow discharge in CCNPF. Additionally, an analysis of the architecture of advanced GMDH and MARS models revealed that the following parameters exerted the most significant influence on the modelling and estimation of discharges: roughness, area, hydraulic radius, flow aspect ratio, depth, and angle of convergence or divergence of floodplain. In a study conducted by Bijanvand et al. (2023), soft computing models such as the Multi-Layer Perceptron Neural Network (MLPNN), Group Method of Data Handling (GMDH), Neuro Fuzzy Group Method of Data Handling (NF-GMDH) and Support Vector Machine (SVM) were utilised to predict water surface elevation of compound channels with converging and diverging floodplains. In this research, the SVM model exhibited superior performance in the testing phase, as evidenced by the statistical indicators; the utilization of the adaptive fuzzy methodology in the development of the Group Method of Data Handling (GMDH) model resulted in a noteworthy enhancement of precision, as evidenced by the statistical metrics during the testing phase; The optimal performance of the activation and kernel functions in the construction of the MLPNN model and the SVM was found to be associated with the sigmoid and radial tangent functions. In their study, Kaushik and Kumar (2023a) utilized machine learning techniques to forecast the water surface profile of a compound channel with a converging floodplains based on a combination of geometric and flow variables. The results indicate that the proposed artificial neural network (ANN) model could deliver a high degree of accuracy in forecasting the water surface profile, as compared to gene expression programming (GEP), support vector machine (SVM), and previously established techniques. Moreover, Kaushik and Kumar (2023b) have also employed gene expression programming to formulate a novel equation for compound channels with converging floodplains. Expressed in terms of nondimensional geometric and flow variables, the equation can be used to estimate the boundary shear force conveyed by floodplains. Finally, it is found that the anticipated proportion of the floodplain-supported shear force as identified through GEP exhibits a satisfactory level of concurrence with the empirical evidence.

This study will employ Support Vector Machine (SVM) modelling technique to predict the water surface profile of non-prismatic compound channels with converging floodplains. This study, along with other researches on similar channel configurations, utilizes experimental datasets to investigate the impacts of diverse non-dimensional geometric and flow parameters on water surface elevations. Other different methods were previously utilized by other researchers to compute the water surface profile in non-prismatic compound channels; and they will be subjected to effectiveness or error analysis to assess the accuracy of the model proposed in this study.

2 Materials and methodologies

2.1 Data sources

Numerous explorations have been conducted on the distribution of shear force, transfer of momentum, and discharge mechanisms of compound channels with non-prismatic floodplains. This study will utilize the data from Rezaei (2006) and Naik and Khatua (2016), as well as the experiments of this paper’s author. Table 1 showcases the data pertaining to the experimental channel dimensions of composite waterway with diverse geometric characteristics.

Table 1 Channel dimensions used (Kaushik & Kumar, 2023a)
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2.2 The experimental setup

The investigations for this study were conducted at a hydraulic laboratory of the Department of Civil Engineering at Delhi Technological University, India, with the experimental procedures operated in a masonry flume 12 m long, 1 m wide, and 0.8 m deep. A cross-sectional compound configuration was established with brick masonry in the flume, and the main channel was 0.5 m in width and 0.25 m in depth. The section of the channel to exhibit convergence was constructed with brick masonry. The maintenance of the geometry was achieved through the utilization of a converging angle of 4° in the channel. The compound channel under investigation exhibited a non-prismatic geometry, with a converging length of 3.6 m. The state of subcritical flow was measured under various circumstances within the two-stage channel, which featured a longitudinal bed slope of 0.001. Manning’s n value was estimated based on the data obtained from both in-bank and over-bank flows in the flooded areas and primary channel. The water supply was recycled through a mechanism that pumps water from an underground sump to a reservoir located in the experimental channel. The flow originating from the conduit was accumulated within a volumetric receptacle equipped with a V-shaped notch; subsequently, it returned to the sump situated below. Figure 1 presents an exposition of the geometric characteristics and a sectional view of the composite waterway.

Fig. 1
figure 1

Cross section of the compound channel

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The experimental setup designed for the proposed research is depicted in Fig. 2 for its top view. A tailgate was implemented at the terminal point of the flume to regulate the profile of water surface and enforce a predetermined flow depth within the flume segment. The height of the water surface was determined with a point gauge, which could achieve an accuracy of 0.1 mm. An Acoustic Doppler Velocimeter (ADV) was employed to identify velocity distributions in three dimensions. The data obtained via the Acoustic Doppler Velocimeter (ADV) was subject to filtration in the Horizon ADV software. The lateral distributions of boundary shear stress were measured with a Preston tube of an outer diameter of 5 mm at the identical sections where the velocity distributions were examined. The pressure difference was measured with a digital manometer. The utilization of Patel (1965) calibration equations was subsequently employed to compute the shear stress values. Measurements of local boundary shear stress and velocity were conducted along the wetted perimeter at intervals of 2.5 cm and 10 cm in the vertical and horizontal directions, respectively, as depicted in Fig. 1.

Fig. 2
figure 2

Plan view of the experimental setup

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2.3 Theoretical background

For river specialists, the ability to make reliable projections of the contour of the top of the water during flood flows is essential for constructing drainage canals, flood defence systems, river training works, and floodplain management. As flood flow conditions in natural rivers are unpredictable, it may be difficult to acquire accurate and complete field data. Understanding the water surface profile in composite channels, especially those including both prismatic and non-prismatic floodplains, requires extensive laboratory experiments. Naik and Khatua (2016) have utilized multivariate analyses based on Eqs. (1–21) for forecasting the water surface profile in a converging compound channel, and Naik et al. (2022) have used the GEP approach based on Eq. (22).

$$\Psi =1.06{\alpha }^{0.22}$$
(1)
$$\Psi =1.16{\alpha }^{0.22}$$
(2)
$$\Psi =1.21{\alpha }^{0.29}$$
(3)
$$\Psi =0.07\alpha +1.78$$
(4)
$$\Psi =0.05\alpha +1.28$$
(5)
$$\Psi =0.13\alpha +1.25$$
(6)
$$\Psi =-0.14{X}_{r}+1.22$$
(7)
$$\Psi =-0.15{X}_{r}+1.32$$
(8)
$$\Psi =-0.22{X}_{r}+1.37$$
(9)
$$\Psi =-0.15{X}_{r}+2.01$$
(10)
$$\Psi =-0.16{X}_{r}+1.45$$
(11)
$$\Psi =-0.21{X}_{r}+1.67$$
(12)
$$\Psi =-1.22+2.27{\alpha }^{0.22}+0.18{X}_{r}$$
(13)
$$\Psi =-1.21+2.28{\alpha }^{0.22}+0.19{X}_{r}$$
(14)
$$\Psi =-0.58+1.63{\alpha }^{0.29}+0.18{X}_{r}$$
(15)
$$\Psi =-0.66+0.29\alpha +0.12{X}_{r}$$
(16)
$$\Psi =0.86+0.29\alpha +0.11{X}_{r}$$
(17)
$$\Psi =0.86+0.29\alpha +0.12{X}_{r}$$
(18)
$${\Psi }^{*}(\theta )=\frac{Actual \Psi }{Eq. (12)}$$
(19)
$$\frac{Actual \Psi }{Eq. (12)}={e}^{0.0017\theta }$$
(20)
$$\Psi ={e}^{0.0017\theta }\left(-1.21+2.25{\alpha }^{0.22}+0.18{X}_{r}\right)$$
(21)
$$\Psi =\left[\frac{3.72+\delta \beta +S\beta }{3.72-S-0.12\alpha }+\frac{9.63-\alpha \delta }{\left(5.73+\theta \right)\times \left({\delta }^{2}\beta +2\delta {X}_{r}\right)}-14.75\delta S-\alpha \delta S\right]$$
(22)

A predictive model was formulated to estimate the configuration of the water’s free surface in a composite waterway in convergent flooded areas. As anticipated, the fluid flow exhibits non-uniform behaviour within the non-prismatic region prior to its entry into the straight zone. The surface of both the primary waterway and the flooded zones showcases a consistently smooth texture. The mean value of Manning’s n for the aforementioned smooth surfaces is 0.013. When assessing the water surface configuration of composite waterways in the narrowing floodplains, several significant factors shall be considered, including but not limited to: the width ratio (α), relative depth ratio (β), aspect ratio (δ), converging angle (θ), relative distance (Xr), and longitudinal bed slope (So).

The requisite equation in a dimensionless form can be formulated as follows:

$$\Psi =F(\alpha , \beta , \delta , \theta , {X}_{r}, {S}_{o})$$
(23)

2.4 SVM model

The Support Vector Regression (SVR) algorithm is a commonly employed machine learning technique for the purpose of regression analysis. The method in question operates on the principles analogous to those of support vector machines (SVMs) for the classification purpose, albeit with the objective of forecasting a continuous output variable on the basis of input variables. The fundamental concept underlying Support Vector Regression (SVR) is to identify a function that can effectively map input variables to output variables, with minimal errors incurred, while also satisfying a predetermined margin. The process of attaining this objective involves the conversion of the input variables into a space of higher dimensions via a kernel function. Subsequently, the regression line that optimizes the margin while maintaining the error under a predetermined tolerance level is determined (Parsaie et al., 2015). SVR exhibits a notable benefit in its capacity to effectively manage non-linear associations between input and output variables. This feature is facilitated by such kernel functions as polynomial, radial basis function (RBF), and sigmoid function, which enable the capture of intricate relationships between variables. Support Vector Regression (SVR) has been effectively utilized in diverse domains, including finance, transportation, and medical imaging. It is of utmost significance to meticulously choose and adjust the kernel functions and other associated parameters, in order to attain optimal performance. One common method for transforming a linear classifier into a non-linear classifier utilizes a non-linear function to map the input space x into a feature space F. An alternative approach involves the utilization of a non-linear function for mapping. The separating function in space F can be expressed as below (Parsaie et al., 2015):

$$f\left(x\right)= {w}^{T}\varphi \left(x\right)+b$$
(24)

Within the subspace, the statistical methodology denoted by the algebraic function f (x, w) can be described as follows:

$$w=\sum_{i=1}^{n}{\alpha }_{i}{x}_{i}$$
(25)
$$f\left(x,w\right)=\sum_{j=1}^{n}{\alpha }_{i}{x}_{i}{\varphi }_{i}\left(x\right)+b$$
(26)
$$f\left(x\right)=\sum_{i=1}^{n}{\alpha }_{i}{x}_{i}^{T}x+b$$
(27)

In the feature space F, these expressions take the following forms:

$$f\left(x\right)=\sum_{i=1}^{n}{\alpha }_{i}\varphi {\left({x}_{i}\right)}^{T}\varphi \left(x\right)+b 0\le {\alpha }_{i}\le C$$
(28)
$$k\left(x, {x}{\prime}\right)=\varphi {\left(x\right)}^{T}\varphi ({x}{\prime})$$
(29)
$$f\left(x\right)=\sum_{i=1}^{n}{\alpha }_{i}k\left(x,{x}_{i}\right)+b$$
(30)

This study finds that among the numerous kernel functions present in SVM, the radial basis kernel function is the most efficacious. Conversely, alternative kernel functions exist that can be utilized to provide assistance in a broad sense.

Linear kernel: \(k\left({x}_{i},{x}_{j}\right)={x}_{i}^{T}{x}_{j}\)

Polynomial kernel: \(k\left({x}_{i},{x}_{j}\right)={(\gamma {x}_{i}^{T}{x}_{j}+r)}^{d}, \gamma >0\)

RBF kernel: \(k\left({x}_{i},{x}_{j}\right)=\mathrm{exp}{(-\gamma \Vert {x}_{i}-{x}_{j}\Vert )}^{2}, \gamma >0\)

Sigmoid kernel: \(k\left({x}_{i},{x}_{j}\right)=\mathrm{tan}h(\gamma {x}_{i}^{T}{x}_{j}+r), \gamma >0\)

The SVM model employs parameters C, γ, r, and d. The efficacy of the Support Vector Machine (SVM) model is contingent upon the calibre of the selected settings for the said parameters, in addition to the kernel parameters. The selection of values for parameters C, γ, and r plays a crucial role in regulating the complexity of the regression model. However, the process of determining the optimal parameters would be further complicated by the intricate nature of their selection. The kernel functions are utilized to reduce the dimensionality of the input space and facilitate classification. The Support Vector Machine (SVM) model is further refined through its exposure to data sets, a process that is analogous to the refinement of other neural network models. This study utilized a reliable and high-quality experimental dataset to evaluate the water surface profile of composite waterways with convergent floodplains. Prior to building the SVM model, the data were partitioned into a training set and a testing set at a 70:30 ratio in MATLAB R (2019).

2.5 Statistical measures

To further test the accuracy of the developed SVM model, various metrics of error analysis, such as the coefficient of determination (R2), mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE), are measured by using the following equations (Kaushik & Kumar, 2023a, 2023b):

$${R}^{2} =\frac{\sum_{i=1}^{N}{\left({a}_{i}-\overline{a }\right)}^{2} {\left({p}_{i}-\overline{p }\right)}^{2}}{\sum_{i=1}^{N}{\left({a}_{i}-\overline{a }\right)}^{2} \sum_{i=1}^{N}{\left({p}_{i}-\overline{p }\right)}^{2}}$$
(31)
$$\mathrm{MAE }=\frac{1}{N} \sum_{i=1}^{N}\left|{p}_{i}-{a}_{i}\right|$$
(32)
$$\mathrm{MAPE}(\mathrm{\%}) =\frac{1}{N} \sum_{i=1}^{N}\left(\frac{\left|{p}_{i}-{a}_{i}\right|}{{a}_{i}} \times 100\right)$$
(33)
$$\mathrm{RMSE }=\sqrt{\frac{1}{N} \sum_{i=1}^{N}{\left({p}_{i}-{a}_{i}\right)}^{2}}$$
(34)

The above equations utilize the actual and predicted values, which are denoted by a and p, respectively. Additionally, the means of the actual and predicted values, represented by ā and \(\overline{p }\), respectively, are also utilized in the computation. The total number of datasets is denoted by N.

3 Results and discussion

In Fig. 3, the surface profile of water in a non-prismatic composite channel is depicted for various relative flow depths in conjunction with the channel length. In the straight segment of the waterway, the water level remains constant. However, in the convergent section, particularly in the latter half of the transition, the velocity of the flow would become faster, and the water level would be decreased. The flow in the lower section of the waterway exhibits a degree of consistency, albeit with some variations. Figure 4 illustrates the variation of the non-dimensional water surface profile (Ψ) in relation to the width ratio, relative depth, aspect ratio, and relative distance at a convergence angle of θ = 4° and relative flow depths ranging from β = 0.10 to 0.60. An increase in flow depth would result in a wider width ratio, which would in turn broaden the water surface profile. The water surface always changes in the same way as the width ratio, even at extreme convergence angles. For a wide range of convergent angles, the water surface would rise in a way not linearly connected to the depths at which the two flows are interacting with each other. The water surface profile across the non-prismatic compound channel exhibits a decreasing aspect ratio as the water traverses various sections of the channel. The influence of distance on two-dimensional representations of the water surface would diminish with increased separations between the spots. This demonstrates that the velocity head would quickly rise up as the potential head is lowered and the transition converges. Given sharper angles of convergence in floodplains, the decline is very dramatic.

Fig. 3
figure 3

Water surface profile of the two-stage waterway

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Fig. 4
figure 4

Variation of non-dimensional water surface profile with various non-dimensional parameters

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Figure 5 shows a scatter plot to compare the predicted and actual values for the SVM model. The close proximity of the values to the line verifies an excellent agreement, remarkably indicating the SVM model's ability to deliver accurate predictions. The fact that this model may offer analytical, non-parametric equations that can be used in real-world situations makes it much more useful. When making forecasts, the value of the SVM approach would yield more promising results, since the projected values are extremely near to the best-fitting line. Table 2 showcases the statistical metrics used to assess the performance of the SVM model, including R2, MSE, RMSE, MAE, and MAPE. As shown in the table, SVM (R2 = 0.99 and RMSE = 0.0199) predicts a better water surface profile in non-prismatic compound channels than other approaches, such as those proposed by Naik et al. (2022) and Naik and Khatua (2016). SVM is found to be the most suitable approach for predicting the surface water profile of composite waterways with the narrowing floodplains, with a mean absolute percentage error (MAPE) of 1.263, which is better that that of other methods.

Fig. 5
figure 5

Comparison between the predicted Ψ and the observed Ψ

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Table 2 Error analysis of SVM-predicted Ψ against previously developed methods
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4 Conclusion

This study demonstrates the applicability of Support Vector Machine (SVM), a machine learning technique, in detecting the elevation of water surface of two-stage channels with the narrowing floodplains. The proposed models are derived from high-quality laboratory datasets, which feature dimensionless geometric and flow characteristics and pertain to composite waterways that exhibit diverse convergent angles and relative flow depths. Several findings and implications are drawn from this study.

Certain parameters, such as the width ratio, relative flow depth, aspect ratio, angle of convergent, relative longitudinal distance, and longitudinal bed slope, appear to influence the proposed model. The phenomenon of decreasing flow depths along the channel length is attributable to the converged geometry of the channel. In fact, this trend has been observed across various floodplain convergence angles and at higher relative depths. It is also revealed that the non-dimensional profile of water surface exhibits an upward trend with hiking width ratio and relative depth, while showing a downward trend in terms of the aspect ratio and relative distance. The alteration of water surface's elevation in two-stage waterways exhibits the similar trend across the spectrum of narrowing angles, as demonstrated in other previous investigations. By exploring the correlation between the elevation of water surface and the non-dimensional geometric and hydraulic features of a narrowing composite waterway, this study identified non-linear correlations among the parameters. The models generated in this study deliver superior performance in terms of R2, MAE, RMSE, and MAPE metrics over previous methodologies, such as those proposed by Naik and Khatua (2016) and Naik et al. (2022). These findings validate that the Support Vector Machine (SVM) approach can yield a satisfactory estimation of the water surface profile in composite waterways, as determined by the assessment metrics. The SVM model proposed in this study exhibits superior performance over other models, as evidenced by its R value of 0.99, RMSE value of 0.0199, and MAPE value of 1.263. These results prove the models' satisfactory performance and practical usefulness across the range of non-dimensional factors examined in this study. The utilization of statistical regression as the sole methodology is considered obsolete and should be improved. The aforementioned process is operated with a technique that could translate knowledge into a feature space of high dimensionality. Subsequently, various methods can be employed to identify interrelated fragments of information within this framework. The comprehensive mapping results in categorized linkages. The physical, process-oriented research in riverine hydraulics has predicted that various parameters have an impact on energy losses, velocity, and discharge. The challenges in obtaining reliable hydraulic data in riverine environments pose significant obstacles to in-depth research in this field.