Skip to main content
SpringerLink
Log in

Discharge coefficient estimation of modified semi-cylindrical weirs using machine learning approaches

  • ORIGINAL PAPER
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Based on the principles design of hydrofoil weirs, Modified Semi-Cylindrical Weirs (MSCWs) incorporate an innovative tangential ramp along the downstream crest contour, thereby significantly enhancing their performance compared to conventional semi-cylindrical weirs. A pivotal parameter in the calculation of flow discharge over the weir is the discharge coefficient (Cd). This study involves a comprehensive comparative analysis of various Cd estimation methodologies for MSCWs, employing a range of machine learning-based models, notably including Artificial Neural Network (ANN), Multivariate Adaptive Regression Splines (MARS), M5 tree, Locally Weighted Polynomial Regression (LWPR), and Support Vector Machine (SVM) models. To begin, a feature selection analysis utilizing the Gamma Test (GT) method was conducted to identify the optimal input configuration for modeling the discharge of MSCWs. The results of the feature selection revealed that the Cd of the MSCWs is primarily influenced by the ratio of upstream flow depth (yup) to crest radius (R), while showing negligible sensitivity to the slope of the downstream ramp (θ). The dataset was partitioned into two segments: 70% were assigned to the training stage, while the remaining 30% were allocated to the testing stage. The precision of Cd predictions is evaluated through four key statistical metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Square Error (RMSE), Correlation Coefficient (R2), and Nash –Sutcliff Efficiency (NSE). The outcomes reveal that, for the training and testing phases, the R2 values for the ANN, MARS, M5 tree, LWPR and SVM models are respectively 0.967, 0.931, 0.974, 0.937, and 0.933, and 0.925, 0.953, 0.953, 0.980, and 0.954. Notably, the LWPR model outperforms the ANN, MARS, M5 tree, and SVM models, boasting MAE, MSE, RMSE, and NSE values of 0.0167, 0.0005, 0.0217, and 0.942 during training, and 0.0107, 0.0002, 0.0136, and 0.949 during testing. As a result, the LWPR model clearly emerges as the superior model, followed by the M5 model tree.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

All the data gathered for this study are available within the manuscript.

References

  • Afaridegan E, Amanian N, Parsaie A, Gharehbaghi A (2023b) Hydraulic investigation of modified semi-cylindrical weirs. Flow Meas Instrum 93:102405. https://doi.org/10.1016/j.flowmeasinst.2023.102405

    Article  Google Scholar 

  • Afaridegan E, Heidarpour M, Fallahi B et al (2023c) Laboratory Investigation of the Effect of Air Injection and Trapezoidal Collar on Reducing Local Scouring around Bridge Pier. J Hydraul Eng 149:4023043. https://doi.org/10.1061/JHEND8.HYENG-13342

    Article  Google Scholar 

  • Afaridegan E, Amanian N, Haghiabi A, et al (2023a) Numerical Investigation of Modified Semi-Cylindrical Weirs. Water Resour Manag 1–14. https://doi.org/10.1007/s11269-023-03523-y

  • Akbari M, Salmasi F, Arvanaghi H et al (2019) Application of Gaussian process regression model to predict discharge coefficient of Gated Piano Key Weir. Water Resour Manag 33:3929–3947

    Article  Google Scholar 

  • Akoz MS, Gumus V, Kirkgoz MS (2014) Numerical simulation of flow over a semicylinder weir. J Irrig Drain Eng 140:4014016

    Article  Google Scholar 

  • Alberg D, Last M, Kandel A (2012) Knowledge discovery in data streams with regression tree methods. Wiley Interdiscip Rev Data Min Knowl Discov 2:69–78

    Article  Google Scholar 

  • Azma A, Tavakol Sadrabadi M, Liu Y et al (2023) Boosting ensembles for estimation of discharge coefficient and through flow discharge in broad-crested gabion weirs. Appl Water Sci 13:45

    Article  Google Scholar 

  • Bagheri S, Kabiri-Samani A (2020) Overflow characteristics of streamlined weirs based on model experimentation. Flow Meas Instrum 73:101720. https://doi.org/10.1016/j.flowmeasinst.2020.101720

    Article  Google Scholar 

  • Baharvand S, Jozaghi A, Fatahi-Alkouhi R et al (2021) Comparative study on the machine learning and regression-based approaches to predict the hydraulic jump sequent depth ratio. Iran J Sci Technol Trans Civ Eng 45:2719–2732

    Article  Google Scholar 

  • Barenblatt GI (1987) Dimensional analysis. CRC Press

    Google Scholar 

  • Bhattacharya B, Solomatine DP (2005) Neural networks and M5 model trees in modelling water level–discharge relationship. Neurocomputing 63:381–396

    Article  Google Scholar 

  • Bhattacharyya D, Deka P, Saha U (2024) Applicability of statistical and deep-learning models for rainfall disaggregation at metropolitan stations in India. J Hydrol Reg Stud 51:101616. https://doi.org/10.1016/j.ejrh.2023.101616

    Article  Google Scholar 

  • Chanson H (2004) Hydraulics of open channel flow. Elsevier

    Google Scholar 

  • Chanson H, Montes JS (1997) Overflow characteristics of cylindrical weirs. Res. Rep. No. CE154, Dept. of Civ. Eng., University of Queensland, Australia

  • Cherkassky V, Ma Y (2004) Practical selection of SVM parameters and noise estimation for SVM regression. Neural Netw 17:113–126. https://doi.org/10.1016/S0893-6080(03)00169-2

    Article  Google Scholar 

  • Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. J Am Stat Assoc 74:829–836

    Article  Google Scholar 

  • Cleveland WS, Devlin SJ (1988) Locally weighted regression: an approach to regression analysis by local fitting. J Am Stat Assoc 83:596–610

    Article  Google Scholar 

  • Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20:273–297. https://doi.org/10.1007/BF00994018

    Article  Google Scholar 

  • Craven P, Wahba G (1978) Smoothing noisy data with spline functions. Numer-Ische Mathematik 31(4):377–403

    Article  Google Scholar 

  • Dariane AB, Behbahani MM (2023) Development of an efficient input selection method for NN based streamflow model. J Appl Water Eng Res 11:127–140. https://doi.org/10.1080/23249676.2022.2088631

    Article  Google Scholar 

  • Deng H, Liu Y, Li P et al (2018) Integrated probabilistic modeling method for transient opening height prediction of check valves in oil-gas multiphase pumps. Adv Eng Softw 118:18–26

    Article  Google Scholar 

  • Emamgolizadeh S, Bateni SM, Shahsavani D et al (2015) Estimation of soil cation exchange capacity using genetic expression programming (GEP) and multivariate adaptive regression splines (MARS). J Hydrol 529:1590–1600

    Article  CAS  Google Scholar 

  • Ettema R, Arndt R, Roberts P, Wahl T (2000) Hydraulic modeling concepts and practice. ASCE manuals and reports on engineering practice, 97, Reston

  • Fatahi-Alkouhi R, Shanehsazzadeh A, Hashemi M (2023) Enhanced Physically Based Models for Pressure Characteristics at Plunge Pool Bottoms. J Hydraul Eng 149:4023044. https://doi.org/10.1061/JHEND8.HYENG-13398

    Article  Google Scholar 

  • Fatahi-Alkouhi R, Shanehsazzadeh A, Hashemi M (2022) Empirical models for hydrodynamic pressure at plunge pool bottoms due to high-velocity jet impact. Iran J Sci Technol Trans Civ Eng 46:1359–1374. https://doi.org/10.1007/s40996-021-00608-9

    Article  Google Scholar 

  • Friedman JH (1991) Multivariate Adaptive Regression Splines. Ann Stat 19:1–67. https://doi.org/10.1214/aos/1176347963

    Article  Google Scholar 

  • Gharehbaghi A, Ghasemlounia R, Afaridegan E, et al (2023) A comparison of artificial intelligence approaches in predicting discharge coefficient of streamlined weirs. J Hydroinformatics jh2023063. https://doi.org/10.2166/hydro.2023.063

  • Gharehbaghi A, Ghasemlounia R (2022) Application of AI Approaches to Estimate Discharge Coefficient of Novel Kind of Sharp-Crested V-Notch Weirs. J Irrig Drain Eng 148:4022001

    Article  Google Scholar 

  • Guo B, Gunn SR, Damper RI, Nelson JDB (2008) Customizing Kernel Functions for SVM-Based Hyperspectral Image Classification. IEEE Trans Image Process 17:622–629. https://doi.org/10.1109/TIP.2008.918955

    Article  CAS  Google Scholar 

  • Haghiabi AH, Parsaie A, Ememgholizadeh S (2018) Prediction of discharge coefficient of triangular labyrinth weirs using adaptive neuro fuzzy inference system. Alexandria Eng J 57:1773–1782

    Article  Google Scholar 

  • Härdle W (1984) Robust regression function estimation. J Multivar Anal 14:169–180

    Article  Google Scholar 

  • Hastie T, Tibshirani R, Friedman JH, Friedman JH (2009) The elements of statistical learning: data mining, inference, and prediction. Springer

    Book  Google Scholar 

  • Hill T, Lewicki P, Lewicki P (2006) Statistics: methods and applications: a comprehensive reference for science, industry, and data mining. General Linear Models 18: 245–276. StatSoft, Tulsa (OK) Chapter

  • Jin H, Lu H, Zhao Y et al (2024) Integration of an improved transformer with physical models for the spatiotemporal simulation of urban flooding depths. J Hydrol Reg Stud 51:101627. https://doi.org/10.1016/j.ejrh.2023.101627

    Article  Google Scholar 

  • Kang F, Li J (2020) Displacement model for concrete dam safety monitoring via Gaussian process regression considering extreme air temperature. J Struct Eng 146:5019001

    Article  Google Scholar 

  • Kang F, Han S, Salgado R, Li J (2015) System probabilistic stability analysis of soil slopes using Gaussian process regression with Latin hypercube sampling. Comput Geotech 63:13–25

    Article  Google Scholar 

  • Karami H, Karimi S, Bonakdari H, Shamshirband S (2018) Predicting discharge coefficient of triangular labyrinth weir using extreme learning machine, artificial neural network and genetic programming. Neural Comput Appl 29:983–989

    Article  Google Scholar 

  • Kendall MG, Stuart A, Ord JK (1987) Kendall’s advanced theory of statistics, 5th edn. Oxford University Press, New York

  • Kumar M, Kumar P, Kumar A et al (2022) Modeling stage–discharge–sediment using support vector machine and artificial neural network coupled with wavelet transform. Appl Water Sci 12:87. https://doi.org/10.1007/s13201-022-01621-7

    Article  Google Scholar 

  • Kwon H, Lall U, Moon Y, et al (2006) Episodic interannual climate oscillations and their influence on seasonal rainfall in the Everglades National Park. Water Resour Res 42(11):W11404. https://doi.org/10.1029/2006WR005017

  • Lall U, Sangoyomi T, Abarbanel HDI (1996) Nonlinear dynamics of the Great Salt Lake: Nonparametric short-term forecasting. Water Resour Res 32:975–985

    Article  Google Scholar 

  • Lall U, Moon Y, Kwon H, Bosworth K (2006) Locally weighted polynomial regression: parameter choice and application to forecasts of the Great Salt Lake. Water Resour Res 42:W05422. https://doi.org/10.1029/2004WR003782

  • Li K-C (1985) From Stein’s unbiased risk estimates to the method of generalized cross validation. Ann Stat 13:1352–1377

  • Mazarei Behbahani MR, Mazarei A (2023) A new criteria for determining the best decomposition level and filter for wavelet-based data-driven forecasting frameworks- validating using three case studies on the CAMELS dataset. Stoch Environ Res Risk Assess 37:4827–4842. https://doi.org/10.1007/s00477-023-02531-z

    Article  Google Scholar 

  • Mohammadzadeh-Habili J, Heidarpour M, Afzalimehr H (2013) Hydraulic characteristics of a new weir entitled of quarter-circular crested weir. Flow Meas Instrum 33:168–178

    Article  Google Scholar 

  • Nelder JA, Mead R, Press WH, et al (1986) Numerical recipes: the art of scientific computing. Cambridge University Press

  • Norouzi R, Arvanaghi H, Salmasi F et al (2020) A new approach for oblique weir discharge coefficient prediction based on hybrid inclusive multiple model. Flow Meas Instrum 76:101810

    Article  Google Scholar 

  • Nouri M, Sihag P, Kisi O et al (2023) Prediction of the Discharge Coefficient in Compound Broad-Crested-Weir Gate by Supervised Data Mining Techniques. Sustainability 15:433

    Article  Google Scholar 

  • Pal M (2006) M5 model tree for land cover classification. Int J Remote Sens 27:825–831

    Article  Google Scholar 

  • Parsaie A, Haghiabi AH (2021) Mathematical expression for discharge coefficient of Weir-Gate using soft computing techniques. J Appl Water Eng Res 9:175–183

    Article  Google Scholar 

  • Parsaie A, Haghiabi AH, Saneie M, Torabi H (2016) Prediction of energy dissipation on the stepped spillway using the multivariate adaptive regression splines. ISH J Hydraul Eng 22:281–292

    Article  Google Scholar 

  • Quinlan JR (1992) Learning with continuous classes. In: 5th Australian joint conference on artificial intelligence. Hobart, pp 343–348

  • Ramamurthy AS, Vo N-D (1993) Characteristics of circular-crested weir. J Hydraul Eng 119:1055–1062

    Article  Google Scholar 

  • Roushangar K, Alami MT, Shiri J, Asl MM (2018) Determining discharge coefficient of labyrinth and arced labyrinth weirs using support vector machine. Hydrol Res 49:924–938

    Article  Google Scholar 

  • Salmasi F, Nahrain F, Abraham J, Taheri Aghdam A (2023) Prediction of discharge coefficients for broad-crested weirs using expert systems. ISH J Hydraul Eng 29:1–11

    Article  Google Scholar 

  • Sarvarinezhad SB, Bina M, Afaridegan E et al (2022) The hydraulic investigation of inflatable weirs. Water Supply 22:4639–4655. https://doi.org/10.2166/ws.2022.123

    Article  CAS  Google Scholar 

  • Schmocker L, Halldórsdóttir BR, Hager WH (2011) Effect of Weir Face Angles on Circular-Crested Weir Flow. J Hydraul Eng 137:637–643. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000346

    Article  Google Scholar 

  • Shafiei S, Najarchi M, Shabanlou S (2020) Prognostication of discharge coefficient of labyrinth weir using outlier robust extreme learning machine. Flow Meas Instrum 71:101681

    Article  Google Scholar 

  • Sheikhi Y, Ashrafi SM, Nikoo MR, Haghighi A (2023) Enhancing daily rainfall prediction in urban areas: a comparative study of hybrid artificial intelligence models with optimization algorithms. Appl Water Sci 13:232. https://doi.org/10.1007/s13201-023-02036-8

    Article  Google Scholar 

  • Smola AJ (1996) Regression estimation with support vector learning machines. In: Proc 13th Int Conf Neural Inf Process Syst, pp 55–61

  • Stefánsson A, Končar N, Jones AJ (1997) A note on the Gamma test. Neural Comput Appl 5:131–133. https://doi.org/10.1007/BF01413858

    Article  Google Scholar 

  • Uncuoglu E, Citakoglu H, Latifoglu L et al (2022) Comparison of neural network, Gaussian regression, support vector machine, long short-term memory, multi-gene genetic programming, and M5 Trees methods for solving civil engineering problems. Appl Soft Comput 129:109623. https://doi.org/10.1016/j.asoc.2022.109623

    Article  Google Scholar 

  • Vapnik V (1999) The nature of statistical learning theory. Springer science & business media. Springer, Berlin. https://doi.org/10.1007/978-1-4757-3264-1

  • Wang F, Zheng S, Ren Y et al (2022) Application of hybrid neural network in discharge coefficient prediction of triangular labyrinth weir. Flow Meas Instrum 83:102108. https://doi.org/10.1016/j.flowmeasinst.2021.102108

    Article  Google Scholar 

  • Wang Q, Yue C, Li X et al (2023) Enhancing robustness of monthly streamflow forecasting model using embedded-feature selection algorithm based on improved gray wolf optimizer. J Hydrol 617:128995. https://doi.org/10.1016/j.jhydrol.2022.128995

    Article  Google Scholar 

  • Zaji AH, Bonakdari H, Shamshirband S (2016) Support vector regression for modified oblique side weirs discharge coefficient prediction. Flow Meas Instrum 51:1–7. https://doi.org/10.1016/j.flowmeasinst.2016.08.006

    Article  Google Scholar 

  • Zamani MG, Nikoo MR, Rastad D, Nematollahi B (2023) A comparative study of data-driven models for runoff, sediment, and nitrate forecasting. J Environ Manage 341:118006. https://doi.org/10.1016/j.jenvman.2023.118006

    Article  CAS  Google Scholar 

  • Zhang WG, Goh ATC (2013) Multivariate adaptive regression splines for analysis of geotechnical engineering systems. Comput Geotech 48:82–95. https://doi.org/10.1016/j.compgeo.2012.09.016

    Article  Google Scholar 

  • Zupan J (1994) Introduction to artificial neural network (ANN) methods: what they are and how to use them. Acta Chim Slov 41:327

    CAS  Google Scholar 

Download references

Acknowledgements

Appreciations are extended to Sahar Salehipour Bavarsad for her skilled scientific and technical assistance.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. Department of Civil Engineering and Transportation, University of Isfahan, Isfahan, Iran

    Reza Fatahi-Alkouhi

  2. Department of Civil Engineering, Faculty of Engineering, Yazd University, Yazd, Iran

    Ehsan Afaridegan & Nosratollah Amanian

Authors
  1. Reza Fatahi-Alkouhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehsan Afaridegan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nosratollah Amanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Study's conception and design involved significant contributions from all authors. Reza Fatahi Alkouhi and Ehsan Afaridegan were in charge of material preparation, data collection, and analysis. The initial manuscript draft was written by Reza Fatahi Alkouhi, Ehsan Afaridegan, and Nosratollah Amanian, and all authors provided feedback on previous versions. Ultimately, the final manuscript was reviewed and approved by all authors after careful consideration.

Corresponding author

Correspondence to Nosratollah Amanian.

Ethics declarations

Ethical approval

This study has been conducted in full compliance with ethical principles, ensuring that no harm or damage has been inflicted upon animals or humans.

Consent to participate

Not applicable.

Consent to publish

The authors affirm that all the relevant data provided for this research were consented for publication.

Competing interests

On behalf of all authors, the corresponding author states that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fatahi-Alkouhi, R., Afaridegan, E. & Amanian, N. Discharge coefficient estimation of modified semi-cylindrical weirs using machine learning approaches. Stoch Environ Res Risk Assess (2024). https://doi.org/10.1007/s00477-024-02739-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00477-024-02739-7

Keywords

Search

Navigation