Achieving an accurate estimation of the flow resistance in open channel flows is crucial for resolving several critical engineering difficulties. In instances when there is excessive flow on both banks of a river, it results in the breach of the primary channel, leading to the discharge of water into the adjacent floodplain. The alteration of floodplain geometry occurs as a consequence of agricultural and developmental practises, leading to the emergence of compound channels that exhibit converging, diverging, or skewed characteristics throughout the course of the flow. The efficacy of conventional equations in accurately forecasting flow resistance is limited due to their heavy reliance on empirical approaches. As a result of this phenomenon, there persists a significant need for methodologies that possess both novelty and precision. The objective of this work is to use the support vector machine (SVM) technique for the estimation of the Manning's roughness coefficient in a compound channel with converging floodplains. Statistical indicators are used to validate the constructed models in the experimental investigation, enabling the assessment of their performance and efficacy. The findings indicate a significant correlation between the Manning's roughness coefficient predicted by SVM and both experimental data and prior research outcomes.

  • Flow resistance is crucial across various engineering contexts, such as planning irrigation systems, managing drainage networks, and constructing flood control structures.

  • This study delved into assessing flow resistance in non-prismatic compound channels by employing support vector machines.

  • The results indicate a strong alignment between the predicted Manning's roughness coefficient and the actual observed values.

The following symbols are used in this paper:

B

total width of compound channel

b

width of the main channel

h

height of the main channel

H

flow depth

α

width ratio (B/b)

β

relative flow depth [(H–h)/H]

δ

aspect ratio (b/h)

θ

converging angle

Xr

relative distance (x/L)

L

converging length

x

distance between two consecutive sections

So

longitudinal bed slope

Se

energy slope

Qr

discharge ratio (Q/Qb)

Q

discharge at any depth

Qb

bankfull discharge

Fr

Froude number

Re

Reynolds number

A

flow area

P

wetted perimeter

R

hydraulic radius (A/P)

n

Manning's roughness coefficient

C

Chezy's roughness coefficient

f

Darcy–Weisbach resistance coefficient

a

actual values

p

predicted values

ā

mean of actual values

mean of predicted values

N

number of datasets

k

number of variables

y/D

ratio of water depth to culvert diameter

SVM

support vector machine

GEP

gene expression programming

ANN

artificial neural network

ANFIS

adaptive neuro-fuzzy inference system

MARS

multivariate adaptive regression splines

MLP

multilayer perceptron

GMDH

group method of data handling

MLPNN

multilayer perceptron neural networks

NF-GMDH

neuro-fuzzy group method of data handling

CCNPF

compound open channels with non-prismatic floodplains

PS

prismatic section

NPS

non-prismatic section

ADV

acoustic Doppler velocimeter

CM

Cox method

EBM

Einstein and Banks method

LM

Lotter method

SVR

support vector regression

RBF

radial basis function

BPNN

back-propagation neural network

GPR

Gaussian process regression

HDPE

high density polyethylene

R2

coefficient of determination

MSE

mean squared error

RMSE

root mean squared error

MAE

mean absolute error

MAPE

mean absolute percentage error

SI

scatter index

E

coefficient of efficiency

AIC

Akaike information criterion

The consideration of flow resistance is of paramount importance in compound open channels, as it has a direct influence on the hydraulic behavior and overall efficiency of the channel system. Flow resistance is the phenomenon that arises as water moves inside a channel, resulting in the counteracting force against its motion. This resistance induces changes in velocity and causes a dissipation of energy. The comprehension and precise estimation of flow resistance are of utmost importance in a range of engineering applications, including the design and administration of irrigation systems, drainage networks, and flood control structures. The impact of flow resistance on water conveyance capacity is a significant aspect to consider in compound open channels. The velocity of water is substantially influenced by flow resistance, which therefore impacts the total flow rate that can be sustained by the channel. Engineers have the ability to enhance the efficiency of water distribution for irrigation or flood channeling by designing channels that optimize water conveyance via precise quantification of flow resistance. The significance of flow resistance is crucial in the determination of water surface profiles. The presence of flow resistance induces energy losses that have a direct impact on both the water surface slope and the distribution of water depths inside the channel. The precise forecasting of water surface profiles has significant importance in the fields of hydraulic design, floodplain management, and flood control operations. The comprehension of flow resistance is crucial for engineers as it enables them to develop models and simulations to analyze channel behavior. This, in turn, facilitates the accurate prediction of floods and the development of efficient strategies for flood mitigation planning. Several researchers have conducted studies on the flow, velocity, and shear stress distribution on compound sections (Kar 1977; Knight & Demetriou 1983; Myers 1987; Bhattacharya 1995; Patra 1999; Patra & Kar 2000; Patra et al. 2004; Yang & Lim 2005; Khatua et al.2011a, 2011b; Mohanty 2013). Traditional channel division techniques are used for the assessment of discharge capacity in compound channels. It is postulated that the compound section is delineated by hypothetical interface planes located at the confluence of the main channel and the floodplain. The absence of shear stress in these regions suggests a lack of momentum transmission. The momentum transfer phenomena were first examined by Sellin (1964) via laboratory research. Extensive investigations were conducted in laboratory flumes to study both prismatic and meandering compound channel designs. Nevertheless, a comparison between the compound section data of prismatic compound channels and non-prismatic compound channels revealed significant discrepancies in the estimation of shear force carried by floodplains. These errors can be attributed to the omission of additional mass and momentum transfer, as elucidated by Bousmar & Zech (2002), Bousmar et al. (2004), Rezaei (2006), and Rezaei & Knight (2009). The additional momentum exchange associated with this phenomenon must be considered when doing flow modeling for non-prismatic compound channels. Yang et al. (2005) carried out the estimation of the Manning and Darcy–Weisbach resistance coefficients in composite waterway channels. The experimental results suggest that the Darcy–Weisbach resistance coefficient for a compound channel is dependent on the Reynolds number. Bjerklie et al. (2005) developed equations to estimate in-bank river discharge, which are constructed based on theoretical considerations and calibrated and confirmed using a comprehensive experimental database, rely on the Manning's and Chezy equations. The study conducted by Proust et al. (2006) investigated the phenomenon of asymmetric geometry, specifically focusing on its enhanced convergence rate. An augmented convergence angle of 22° leads to heightened levels of mass transfer and head loss. Cao et al. (2006) introduces novel methodologies for the computation of flow resistance and momentum flux in compound open channels. Yang et al. (2007) demonstrated that the methodologies used to investigate the resistance characteristics of inbank and overbank flows in compound channel consisting of a rough main channel and rough floodplains resulted in substantial errors in evaluating the composite roughness. The flow behavior of skewed, two-stage converging and diverging channels was investigated by Chlebek et al. (2010). The researchers noted an elevation in head losses resulting from the transfer of mass and momentum, as well as the homogeneity of velocity on shrinking floodplains. Additionally, they discovered an augmented velocity gradient on the expanding floodplains. The observed variations in flow distribution may be attributed to alterations in the flow force between the main channel and floodplains within different subsections. Rezaei & Knight (2011) conducted a study on compound channels with floodplains that are non-prismatic. The authors examined the depth-averaged velocity, local velocity distributions, and boundary shear stress distributions at different convergence angles. The analysis of depth-averaged velocities provides insights into the impact of contractions on velocity distributions. Notably, these contractions lead to an augmentation of velocity in close proximity to the walls of the main channel. This effect is particularly pronounced in the latter portion of the convergence reach. Additionally, for cases where the relative depth is high, the lateral flow entering the main channel also contributes to these velocity variations. The study conducted by Yonesi et al. (2013) examines the influence of roughness in floodplains on the occurrence of overbank flow in compound channels with non-prismatic floodplains. The rate of change in velocity, as shown by the gradient, exhibited a rise in magnitude as the angle of divergence saw an increase. The magnitude of shear stress increases with the augmentation of surface roughness on the floodplain. Naik et al. (2017) developed novel mathematical equations to predict the spatial variation of boundary shear stress in a composite channel that includes a convergent floodplain. The effective calculation of relationships for stage–discharge with decreased width ratios in compound channels with converging floodplains has been facilitated by the use of developed mathematical expressions. Das & Khatua (2018) conducted experiments on a composite stream that had symmetric geometry. This stream had sections that expanded and narrowed, resulting in zones of flood with varying depths of relative flow. The experiments were conducted to examine the resistance properties of the flow in terms of different roughness coefficients such as C, n, and f.

Creating models to estimate Manning's roughness coefficient using various methods complicates the analysis of interdependencies among components. These models become intricate and time-consuming, leading to a significant reduction in the time required for experiments and labor-intensive calculations. SVM, GEP, ANN, ANFIS, and the M5 decision tree models are commonly used for predicting flow in composite channels (Parsaie et al. 2017). Parsaie & Haghiabi (2017) employed analytical methods, specifically ANN and MARS, to forecast the discharge capacity of compound open channels, revealing that the MARS model outperformed the MLP model in accuracy. Mohanta et al. (2018) used the GMDH-NN technique to estimate Manning's n in meandering compound channels. The evaluation of GMDH-NN performance is conducted using two distinct machine learning approaches, namely SVR and MARS, using a range of statistical methodologies. The results demonstrate that the GMDH-NN model accurately predicts Manning's n in comparison to the MARS and SVR models. Khuntia et al. (2018) developed an ANN model using key characteristics to predict boundary shear stress distribution in straight compound channels, showcasing the effectiveness of BPNN models across diverse parameter spaces. Das et al. (2020) utilized ANN and ANFIS techniques to predict discharge in compound channels with converging and diverging sections, with the ANFIS model demonstrating superior performance over the ANN model. Naik et al. (2022) proposed a unique equation using GEP to estimate water surface profile in converging compound channels using nondimensional variables. Yonesi et al. (2022) utilized soft computation models such as MARS and GMDH to model discharge in compound open channels with non-prismatic floodplains (CCNPF), showing sufficient accuracy in flow discharge estimation. The effective parameters of the Manning's n in HDPE culverts were investigated by Matin (2022) by the utilization of intelligence methodologies, including GPR and SVM. The results of the sensitivity analysis indicate that the parameters y/D and Q play a crucial role in predicting the Manning coefficient for HDPE culvert models. Bijanvand et al. (2023) employed soft computing models like MLPNN, GMDH, NF-GMDH, and SVM to predict water surface elevation in compound channels with converging and diverging floodplains, where SVM model demonstrated superior performance in testing. Kaushik & Kumar (2023a) utilized ANN, SVM, and GEP to forecast water surface profile in compound channels with converging floodplains, with the ANN model displaying high accuracy. Kaushik & Kumar (2023b) formulated a novel equation for compound channels with converging floodplains using GEP, estimating the boundary shear force conveyed by floodplains. Kaushik & Kumar (2023c) utilized SVM to approximate water surface profile in compound channels featuring narrowing floodplains, showing a high degree of concordance with empirical data and previous research findings.

Historically, research has been undertaken on the flow characteristics in compound channels with non-prismatic floodplains. However, there has been little focus on the flow resistance in non-prismatic compound channel with converging floodplains using creative methodologies. The current investigation focused on a compound channel including smooth converging floodplains. To estimate the flow resistance in terms of Manning's roughness coefficient using both nondimensional geometric and hydraulic parameters, a supervised learning approach known as SVM was used. The efficacy of the suggested models was evaluated using theoretical methodologies and established approaches using statistical analysis.

Experimental setup

The tests were conducted in the Hydraulics laboratory, which is situated under the Department of Civil Engineering at Delhi Technological University. The trials were carried out inside a masonry flume with dimensions of 12 m in length, 1.0 m in width, and 0.8 m in depth. In this study, a compound cross-section was created using brick masonry, with a main channel width of 0.5 m and a depth of 0.25 m (Figure 1). Figure 1 provides a comprehensive depiction of the geometric attributes associated with a two-stage channel. The construction of the converging portion of the channel included the use of brick masonry, which was employed to achieve a converging angle denoted as θ = 4°. The floodplains were made smooth with the help of trowel finish. The compound channel consists of a prismatic section (PS) of 6 m in length, followed by a non-prismatic section (NPS) spanning 3.6 m, with the remaining portion being the downstream section. The experiments were conducted at six relative flow depths ranging from 0.10 to 0.60. Flow properties, including the stage-discharge relationship, water surface profile, velocity distribution, and shear stress distribution, were measured for each discharge in both the PS and different NPS, as seen in Figure 2. The parts denoted as NPS1 and NPS5 represent the first and final segments of the converging component, respectively. On the other hand, NPS3 corresponds to the central region of the converging section. The NPS2 serves as an intermediary segment situated between NPS1 and NPS3. NPS4 may be regarded as the intermediary segment situated between NPS3 and NPS5.
Figure 1

Cross-section of compound channel.

Figure 1

Cross-section of compound channel.

Close modal
Figure 2

Experimental setup.

Figure 2

Experimental setup.

Close modal

The subcritical flow regime was seen under various circumstances in the two-stage channel, which had a longitudinal bed slope of 0.001. The estimation of Manning's n value was conducted based on data obtained from both in-bank and overbank flows in the floodplains and main channel. The water supply in the experimental channel is sourced from an underground sump and transferred to an above tank by the system. The water originating from the channel is collected inside a volumetric tank that is equipped with a v-notch. The v-notch was appropriately adjusted in order to accurately quantify the discharge emanating from the experimental channel. Subsequently, it facilitates the return of the fluid to the sump situated underneath. Figure 2 depicts the experimental configuration and the assortment of apparatus used in the investigation. A tailgate was constructed at the downstream end of the flume to regulate the water surface profile and enforce a predetermined flow depth inside that section of the flume. A precision point gauge with a precision of 0.1 mm was used to measure the profile of the water surface at distances of 1.0 and 0.3 m in the PS and NPS, respectively. The researchers used an ADV to measure the mean velocity of the cross-section and the three-dimensional velocity distributions around the wetted perimeter. The measurements were taken at vertical intervals of 2.5 cm and horizontal intervals of 10 cm, as seen in the grid format presented in Figure 1. The data obtained via the use of the ADV were processed and refined using the Horizon ADV program. The measurement of lateral distributions of boundary shear stress was conducted using a Preston tube with an outer diameter of 5 mm at the identical portions where the velocity distributions were examined. The pressure differential was measured using a digital manometer. Subsequently, the calibration equations proposed by Patel (1965) were used to compute the shear stress values.

Model development for Manning's n

The lateral variation of channel section roughness is observed along the wetted perimeter in a river that is experiencing flooding. The determination of the conveyance capacity of a compound channel necessitates the consideration of the composite roughness. Despite the varied structures and forms of equations, they can be categorized into distinct groups based on different criteria. These groups include the CM (Cox 1973) (as represented by Equation (1)), the EBM (Einstein & Banks 1950) (as represented by Equation (2)), and the LM (Lotter 1933) represented by Equation (3)). The Equation (4) was formulated by Das & Khatua (2018) through the utilization of multivariate analysis, with consideration of geometric and flow parameters, for a compound channel featuring converging floodplains. The composite roughness equations are
formula
(1)
formula
(2)
formula
(3)
formula
(4)
Composite roughness for compound channel with converging floodplains is calculated by the following expression:
formula
(5)

The determination of Manning's n values is subject to various factors, among which are the width ratio, relative depth ratio, aspect ratio, converging angle, relative distance, longitudinal bed slope, energy slope, discharge ratio, Froude's number, and Reynold's number. Several parameters are utilized to facilitate the application of the model equation to various compound channels. The details of the dataset used in the present investigation are given in Table 1.

Table 1

Statistical characteristics of the dataset used

Range
Minimum 0.84 0.053 0.001 0.00003 0.98 0.05 76912.63 0.0081 
Maximum 2.00 0.600 1.00 0.001 0.00012 2.43 0.18 229804.35 0.0162 
Average 1.48 0.349 0.50 0.001 0.00008 1.58 0.12 135006.63 0.0116 
Median 1.50 0.358 0.50 0.001 0.00009 1.48 0.12 130987.07 0.0117 
Standard deviation 0.34 0.175 0.31 0.00003 0.47 0.04 34764.67 0.0021 
Range
Minimum 0.84 0.053 0.001 0.00003 0.98 0.05 76912.63 0.0081 
Maximum 2.00 0.600 1.00 0.001 0.00012 2.43 0.18 229804.35 0.0162 
Average 1.48 0.349 0.50 0.001 0.00008 1.58 0.12 135006.63 0.0116 
Median 1.50 0.358 0.50 0.001 0.00009 1.48 0.12 130987.07 0.0117 
Standard deviation 0.34 0.175 0.31 0.00003 0.47 0.04 34764.67 0.0021 

The requisite equation in a dimensionless form can be formulated as follows:
formula
(6)

SVM model

SVM is a highly effective and extensively used machine learning technique employed for both classification and regression purposes. This approach has notable efficacy in scenarios when the data exhibits non-linear separability, indicating that the classes cannot be distinctly partitioned by a linear or hyperplane boundary. The SVM algorithm is designed to identify the ideal hyperplane that effectively distinguishes different classes by maximizing the margin. The margin refers to the distance between the hyperplane and the closest data points from each class. An SVR algorithm is a commonly employed machine learning technique utilized for the purpose of regression analysis. The method in question operates on analogous principles to those of SVMs utilized for classification purposes, albeit with the objective of forecasting a continuous output variable on the basis of input variables. The fundamental concept underlying SVR is to identify a function that can effectively map input variables to output variables, with minimal error, 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 through the application of a kernel function. Subsequently, the regression line that optimizes the margin while maintaining the error within a predetermined tolerance level is determined (Drucker et al. 1997). SVR exhibits a notable benefit in its capacity to effectively manage non-linear associations between input and output variables. The attainment of this objective is facilitated through the utilization of kernel functions, including but not limited to polynomial, RBF, and sigmoid functions. These functions enable the capture of intricate relationships between variables (Smola & Schölkopf 2004). SVR has been effectively utilized in diverse domains, including finance (Sharma et al. 2012), transportation (Yeh et al. 2010), and medical imaging (Borges et al. 2016). It is of utmost significance to meticulously choose and adjust the kernel function and other associated parameters to attain optimal performance. One common method for transforming a linear classifier into a non-linear classifier involves the utilization of a non-linear function to map the input space x onto a feature space F. An alternative approach involves the utilization of a non-linear function for the purpose of mapping. The separating function in space f can be expressed as (Kaushik & Kumar 2023a):
formula
(7)
Within the subspace, the statistical methodology denoted by the algebraic function f(x, w) can be articulated as follows:
formula
(8)
formula
(9)
formula
(10)
In the feature space, F, this expression takes the form:
formula
(11)
formula
(12)
formula
(13)
Conversely, there exists a diverse range of supplementary kernel functions that can be employed in various settings to enhance performance.

Linear kernel:

Polynomial kernel:

RBF kernel:

Sigmoid kernel:

The SVM model employs parameters C, γ, r, and d. The C parameter governs the balance between optimizing the margin and minimizing the classification error on the training dataset. A lesser value of C leads to a broader margin, which may let some instances to be misclassified. Conversely, a bigger value of C aims to minimize misclassification by sacrificing margin width. The parameter γ is unique to the RBF kernel. It regulates the impact of specific training samples. A decrease in the gamma value results in a more refined decision boundary, characterized by a higher degree of smoothness. Conversely, an increase in the gamma value introduces more fluctuations and irregularities to the border, potentially leading to overfitting. The parameter denoted as ‘d’ is exclusive to polynomial kernels. The parameter determines the order of the polynomial function used for the transformation. The attainment of advanced academic degrees has the potential to result in decision boundaries of greater complexity. However, it is important to acknowledge that this pursuit may also elevate the likelihood of overfitting. The efficacy of the SVM model is contingent upon the caliber of the selected settings for said parameters, in addition to the kernel parameters. The selection of values for the 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 is further complicated by the intricate nature of their selection. Kernel functions are utilized to reduce the dimensionality of the input space and facilitate classification. The refinement of the SVM model is achieved through its exposure to data sets, a process that is analogous to the refinement of other neural network models. The flow resistance in terms of Manning's roughness coefficient (n) of compound channel with converging floodplains was evaluated utilizing a reliable and high-quality experimental dataset. Prior to building the SVM model, the data are partitioned into training and testing sets using a 70:30 ratio in MATLAB R (2019). Table 2 illustrates the various models that were developed to predict roughness coefficient, utilizing distinct kernel functions and kernel scales while maintaining a consistent relationship and proportion of training and testing datasets.

Table 2

Different models developed using SVR

ModelPresetKernel functionKernel scalePrediction speed (obs/s)Training time (s)
M1 Linear SVM Linear Automatic 330 17.48 
M2 Quadratic SVM Quadratic Automatic 370 16.30 
M3 Cubic SVM Cubic Automatic 360 16.07 
M4 Fine Gaussian SVM Gaussian 0.79 360 15.85 
M5 Medium Gaussian SVM Gaussian 3.2 380 15.58 
ModelPresetKernel functionKernel scalePrediction speed (obs/s)Training time (s)
M1 Linear SVM Linear Automatic 330 17.48 
M2 Quadratic SVM Quadratic Automatic 370 16.30 
M3 Cubic SVM Cubic Automatic 360 16.07 
M4 Fine Gaussian SVM Gaussian 0.79 360 15.85 
M5 Medium Gaussian SVM Gaussian 3.2 380 15.58 

Statistical measures

Various error analysis approaches were used to evaluate the accuracy of the machine learning models. The aforementioned metrics included the coefficient of determination (R2), mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), scatter index (SI), coefficient of efficiency (E), and Akaike Information Criterion (AIC). The relevant equations were used to compute these measures (Kaushik & Kumar 2023a).
formula
(14)
formula
(15)
formula
(16)
formula
(17)
formula
(18)
formula
(19)
formula
(20)
formula
(21)
Figure 3 depicts the longitudinal distance-dependent variation of Manning's roughness coefficient (n) in a compound channel with smooth converging floodplains for different relative flow depths. The decrease in hydraulic radius along the longitudinal distance results in a corresponding reduction in the n values within the PS, extending up to the midpoint of the converging portion. Following the midpoint of the converging section, there is a notable increase in the numerical values of n due to the convergent nature of the channel's geometry. The observed outcomes are attributed to a significant rise in the energy gradient relative to the hydraulic radius. Subsequently, the downstream section experiences a decrease in n values due to the reduction in hydraulic radius and energy slope as the flow progresses further along its length. Similar fluctuation patterns are observed in higher relative flow depths as well as in lower relative flow depths. The study reveals that a significant increase in roughness occurs primarily in the converging regions as the relative depth escalates from 0.20 to 0.60. This phenomenon can be attributed to the transfer of momentum from the main channel to the floodplains. The efficacy of the SVM methodology in generating precise forecasts is evidenced by the scatter diagrams that employ diverse models to juxtapose the projected and actual n values, as depicted in Figure 4. The SVM model exhibits significant potential, as evidenced by the notable indication of the values' proximity to the line representing exceptional concurrence. The SVM predicted values that are dispersed over the best-fitting line exhibit a superior level of concurrence with the empirical data, resulting in elevated R2 values. According to the data presented in Figure 4(e), the SVM model five that was created using a 70:30 ratio of data in training and testing yielded predicted values of Manning's n that were more closely aligned with the best fitted line than other developed models. This suggests that the SVM model five is more accurate in its predictions.
Figure 3

Variation of Manning's n along the longitudinal distance of the non-prismatic compound channel.

Figure 3

Variation of Manning's n along the longitudinal distance of the non-prismatic compound channel.

Close modal
Figure 4

Scatter plot to predict Manning's roughness coefficient (n) for (a) model 1; (b) model 2; (c) model 3; (d) model 4; and (e) model 5.

Figure 4

Scatter plot to predict Manning's roughness coefficient (n) for (a) model 1; (b) model 2; (c) model 3; (d) model 4; and (e) model 5.

Close modal

In order to assess the effectiveness of the SVM models, statistical metrics including R2, MSE, RMSE, MAE, MAPE, SI, E, and AIC were employed, and their corresponding numerical values are displayed in Table 3. The SVM model five has demonstrated superior predictive performance in estimating the Manning's roughness coefficient of composite waterways with converging floodplains, when compared to alternative approaches such as Das & Khatua (2018), as well as established theoretical methods including the Lotter method for composite roughness computation. This is evident from the highest R2 value, lowest MSE value, lowest RMSE value, lowest MAE value, and lowest MAPE value. The SI serves as a standardized measure for the quantification of inaccuracy. A lower SI value is indicative of a higher level of model performance. The SVM model five has a significantly reduced SI value of 0.033 in both the training and testing stages, indicating a minimum level of error. The efficiency measure, represented as E, is assigned a value of 1 when there is a complete correspondence between the model and the observed data. The interval of E extends from negative infinity to 1. The SVM model five has an E value of 0.95 for both the training and testing datasets. The AIC is a widely used criterion in statistical model selection. It is used to choose the best suitable model by evaluating the statistical likelihood function. The model that has the lowest AIC value is regarded as the most optimum model. The SVM model five has the lowest AIC score of −510.754 when compared to the other models proposed. The statistical study of several indices indicates that SVM model five exhibits superior performance compared to other models and established methodologies in forecasting the Manning's roughness coefficient in compound channels with narrowing floodplains.

Table 3

Statistical analysis of predicted n by various approaches

Statistical parametersSVM model 1SVM model 2SVM model 3SVM model 4SVM model 5Das & Khatua Method (2018) Lotter method (1933) 
R2 0.62 0.90 0.82 0.82 0.95 0.80 0.78 
MSE 1.82 × 10−6 4.84 × 10−7 8.82 × 10−7 8.67 × 10−7 2.33 × 10−7 0.0019 0.0025 
RMSE 0.00135 0.000696 0.000939 0.000931 0.000483 0.043 0.05 
MAE 0.00104 0.000516 0.000676 0.000727 0.000386 0.002 0.045 
MAPE 5.168 2.917 4.309 3.728 2.405 10.921 15.813 
SI 0.059 0.042 0.061 0.044 0.033 0.088 0.117 
E 0.62 0.90 0.82 0.82 0.95 0.70 0.78 
AIC −469.873 −505.382 −472.319 −493.169 −510.754 −250.614 −105.291 
Statistical parametersSVM model 1SVM model 2SVM model 3SVM model 4SVM model 5Das & Khatua Method (2018) Lotter method (1933) 
R2 0.62 0.90 0.82 0.82 0.95 0.80 0.78 
MSE 1.82 × 10−6 4.84 × 10−7 8.82 × 10−7 8.67 × 10−7 2.33 × 10−7 0.0019 0.0025 
RMSE 0.00135 0.000696 0.000939 0.000931 0.000483 0.043 0.05 
MAE 0.00104 0.000516 0.000676 0.000727 0.000386 0.002 0.045 
MAPE 5.168 2.917 4.309 3.728 2.405 10.921 15.813 
SI 0.059 0.042 0.061 0.044 0.033 0.088 0.117 
E 0.62 0.90 0.82 0.82 0.95 0.70 0.78 
AIC −469.873 −505.382 −472.319 −493.169 −510.754 −250.614 −105.291 

The study employs SVM to estimate the Manning's roughness coefficient (n) in non-prismatic compound channel with smooth narrowing floodplains. The SVM models are developed using high-quality experimental datasets with varying geometrical and flow variables. The study indicates that the proposed model is influenced by a range of features, such as the width ratio, relative depth of flow, aspect ratio, angle of convergence, relative longitudinal distance, longitudinal bed slope, energy slope, discharge ratio, Froude number, and Reynolds number. In the prismatic portion, the Manning roughness coefficient decreases, and in the NPS, it decreases up to the middle. After then, a precipitous increase in the roughness value is seen. The present study aims to examine the correlation between the Manning's roughness coefficient and the nondimensional geometric and flow variables of a composite waterway with narrowing floodplains. It has been discovered that there is a connection that is not linear between each of the parameters. In comparison to previously developed techniques, the newly created model demonstrates superior performance in terms of statistical parameters for a variety of datasets. In line with the evaluation criteria, the SVM model five could fairly forecast the Manning's coefficient in compound channels with converging floodplains. This was shown by the fact that the technique was successful. Since it had the best R2 and the lowest MSE, RMSE, MAE, MAPE, SI, and AIC, the SVM model developed using Gaussian kernel function demonstrated superior performance in the prediction of Manning's n in non-prismatic compound channels. The models proposed in this paper possess practical implications for non-prismatic rivers, such as the River Main in Northern Ireland, the Brahmaputra River in India, and other analogous river systems. The results of this research will be valuable in shaping the design of flood control and diversion structures, leading to a reduction in both economic and human losses.

The authors express their gratitude for the help received from the Department of Civil Engineering at Delhi Technological University in Delhi, India.

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

The authors declare there is no conflict.

Bhattacharya
A. K.
1995
Mathematical Model of Flow in A Meandering Channel
.
PhD Thesis
,
Indian Institute of Technology Kharagpur
,
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.
Bijanvand
S.
,
Mohammadi
M.
&
Parsaie
A.
2023
Estimation of water's surface elevation in compound channels with converging and diverging floodplains using soft computing techniques
.
Water Supply
23
(
4
),
1684
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