Abstract
Flow conditions (flow discharge, flow depth, and flow velocity) in natural streams are mainly determined via the flow resistance formula such as Manning’s equation. Evaluating the accurate Manning’s roughness coefficient (n), especially in rivers with bed form during floods, to obtain more reliable results has always been of interest to scholars. The interaction between the flow and bed form is very complex since the flow conditions control bed forms, and vice versa. The main goal of the present study is to predict n in rivers with bed forms, using soft computing models, including multilayer perceptron artificial neural network (MLPNN), group method of data handling (GMDH), support vector machine (SVM) model, and genetic programming model (GP). To this end, the energy grade line (\({S}_{f}\)), flow Froude number (Fr), the relative submergence (\(y/{d}_{50}\); y = flow depth and d50 = bed sediment size), and the bed form dimensionless parameters (\(\Delta /{d}_{50}\), \(\Delta /\lambda\), and \(\Delta /y\); ∆ = bed form height and λ = bed form length) were used as the input variables, and n was used as the output variable. The results showed that all the test models have acceptable accuracy, while the SVM model showed the highest level of accuracy with the coefficient of determination \({R}^{2}=0.99\) in the verification stage. The sensitivity analysis of SVM and MLPNN models and the structural analysis of GMDH and GP models indicated that the most important parameters affecting n are Fr, \({S}_{f}\), and \(\Delta /\lambda\).
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Abbreviations
- A :
-
The corresponding coefficients (\(A=\left({a}_{1},{a}_{2}, \cdots {a}_{m}\right)\))
- \(\overline{{a }_{i}}\) :
-
The average of the Lagrange coefficients
- B :
-
The channel width
- b :
-
The constant in the regression function
- C :
-
A positive integer that determines the penalty when a model training error occurs
- \({d}_{50}\) :
-
The average diameter of sediment particles
- \(f\) :
-
The target function
- FFNN:
-
The feed-forward neural network model
- Fr :
-
The flow Froude number
- G :
-
Gravitational acceleration
- GMDH:
-
Group method of data handling
- GP:
-
Genetic programming model
- G s :
-
The relative density of sediment particles
- MAPE:
-
Mean absolute percentage error
- MLP-FFA:
-
Multilayer perceptron-firefly algorithm model
- MLPNN:
-
Multilayer perceptron artificial neural network
- N :
-
The number of samples (data used for training)
- \(n\) :
-
The total Manning roughness coefficient
- \(n''\) :
-
The bed form related to the Manning roughness coefficient
- \(O\) :
-
The observed value
- \(\overline{O }\) :
-
The mean observed value
- \(P\) :
-
The calculated value
- \(\overline{P }\) :
-
The mean calculated value
- R :
-
Correlation coefficient
- Re :
-
The flow Reynolds number
- RMSE:
-
Root mean square error
- R2 :
-
The coefficient of determination
- \({S}_{f}\) :
-
The energy grade line
- SI:
-
The scatter index
- SVM:
-
Support vector machine model
- V :
-
The average flow velocity
- \({W}^{T}\) :
-
The transpose of the coefficient matrix
- x :
-
The input variables (\(x=\left({x}_{1},{x}_{2}, \cdots {x}_{m}\right)\))
- \({X}_{j}\) :
-
The calculated value for the chromosome by fitting function \(j\)
- y :
-
The average flow depth
- \({Y}_{j}\) :
-
The measured value or the expected value of the chromosome by fitting \(j\)
- α :
-
The angle of the upstream side of the bed form relative to the horizon
- θ :
-
The angle of the downstream side of the bed form relative to the horizon
- λ :
-
The length of the bed form
- Δ :
-
The height of the bed form
- \(\varphi\) :
-
The kernel function
- \({\rho }_{s}\) :
-
The specific mass of sediment particles
- \({\rho }_{w}\) :
-
The specific mass of water
- \(\mu\) :
-
The dynamic viscosity of water
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Acknowledgements
We are grateful to the Research Council of Shahid Chamran University of Ahvaz for financial support (SCU.WH1400.31373).
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The funding of this research was provided by the Research Council of Shahid Chamran University of Ahvaz (Grant number: SCU.WH1400.31373).
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MBY, AP, MSB: Data Analysis, Supervision, Validation, Writing-Reviewing, and Editing. MH, MB: Conducting experiments and data collection.
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Yarahmadi, M.B., Parsaie, A., Shafai-Bejestan, M. et al. Estimation of Manning Roughness Coefficient in Alluvial Rivers with Bed Forms Using Soft Computing Models. Water Resour Manage 37, 3563–3584 (2023). https://doi.org/10.1007/s11269-023-03514-z
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DOI: https://doi.org/10.1007/s11269-023-03514-z