Estimation of Nonlinear Muskingum Parameters in Flood Routing Using Graphical Method

Document Type : Research Paper

Authors

1 Water Engineering Department, Faculty of Agriculture, Shahrekord University, Shahrekord, Iran.

2 Department of Water Engineering, Water & Soil Engineering Faculty, Gorgan University of Agricultural Sciences and Natural Resources, Gorgan, Iran.

10.22059/jwim.2025.395793.1233

Abstract

Flood Routing is one of the fundamental topics in water resources system management and flood control engineering. The Muskingum model is among the most well-known and widely used hydrological routing methods. In the nonlinear Muskingum method, three parameters must be estimated: the storage coefficient (K), the weighting factor for inflow and outflow (χ), and the exponent of the storage term (m). In contrast, the linear Muskingum method only involves the first two parameters (K and χ), making it simpler with one less variable. This research focuses on presenting a simple graphical method for estimating the parameters of the nonlinear Muskingum model. To assess the accuracy and reliability of the proposed graphical method, results were compared with those obtained from Excel's SOLVER tool, which is considered a more precise technique. The graphical method was applied to three flood events. The results showed that the parameters derived using the proposed method were very close to those estimated using SOLVER. Notably, across all flood events, both the performance criteria and hydrographs indicated that the graphical method for the nonlinear Muskingum model outperformed its linear counterpart in terms of accuracy. Furthermore, comparisons between observed peak discharge and the estimated peak discharge revealed that, in all cases, the nonlinear Muskingum model provided values closer to the actual recorded data than the linear model. For instance, the Nash-Sutcliffe Efficiency (NSE) values obtained for the Wilson, Wye, and Karun flood events were as follows: Wilson Flood: 0.19 (linear graphical), 0.35 (nonlinear graphical), 0.96 (SOLVER); Wye Flood: 0.86 (linear graphical), 0.95 (nonlinear Graphical), 0.97 (Solver); Karun Flood: 0.62 (linear graphical), 0.98 (nonlinear graphical), 0.99 (SOLVER).

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References

  1. Akan, O. (2006). Open Channel Hydraulics. In Open Channel Hydraulics. https://doi.org/10.1016/B978-0-7506-6857-6.X5000-0
  2. Bazargan, J., & Norouzi, H. (2018). Investigation the Effect of Using Variable Values for the Parameters of the Linear Muskingum Method Using the Particle Swarm Algorithm (PSO). Water Resources Management, 32(14), 4763-4777. https://doi.org/10.1007/s11269-018-2082-6
  3. Farzin, S., Singh, V., Karami, H., Farahani, N., Ehteram, M., Kisi, O., Allawi, M., Mohd, N., & El-Shafie, A. (2018). Flood Routing in River Reaches Using a Three-Parameter Muskingum Model Coupled with an Improved Bat Algorithm. Water, 10(9), 1130. https://doi.org/10.3390/w10091130
  4. Gill, M. A. (1978). Flood routing by the Muskingum method. Journal of Hydrology, 36(3-4), 353-363. https://doi.org/10.1016/0022-1694(78)90153-1
  5. Hsu, M.-H., Fu, J.-C., & Liu, W.-C. (2003). Flood routing with real-time stage correction method for flash flood forecasting in the Tanshui River, Taiwan. Journal of Hydrology, 283(1-4), 267-280.
  6. Jongman, B. (2018). Effective adaptation to rising flood risk. Nature Communications, 9(1), 1-3.
  7. Lu, C., Ji, K., Wang, W., Zhang, Y., Ealotswe, T. K., Qin, W., Lu, J., Liu, B., & Shu, L. (2021). Estimation of the Interaction Between Groundwater and Surface Water Based on Flow Routing Using an Improved Nonlinear Muskingum-Cunge Method. Water Resources Management, 35(8), 2649-2666. https://doi.org/10.1007/s11269-021-02857-9
  8. Mahmodinia, S. H., Javan, M., & Eghbalzadeh, A. (2014). Comparison of different objective function on estimation of linear and non-linear Muskingum model optimum parameters.
  9. Mahmoudinia, S., Javan, M., & Eghbalzade, A. (2014). Comparison of different objective function on estimation of linear and non-linear Muskingum model optimum parameters. WEJ, 7(20), 29-42. http://wej.miau.ac.ir/article%7B%5C_%7D498.html
  10. Mai, X., Liu, H.-B., & Liu, L.-B. (2023). A new hybrid cuckoo quantum-behavior particle swarm optimization algorithm and its application in Muskingum model. Neural Processing Letters, 55(6), 8309-8337.
  11. McCarthy, G. T. (1938). The unit hydrograph and flood routing. Proceedings of Conference of North Atlantic Division, US Army Corps of Engineers, 1938, 608-609.
  12. Mosavi, A., Ozturk, P., & Chau, K. (2018). Flood prediction using machine learning models: Literature review. Water, 10(11), 1536.
  13. Norouzi, H., & Bazargan, J. (2022). Calculation of Water Depth during Flood in Rivers using Linear Muskingum Method and Particle Swarm Optimization (PSO) Algorithm. Water Resources Management, 36(11), 4343-4361.
  14. Nourani, V., Mogaddam, A. A., & Nadiri, A. O. (2008). An ANN‐based model for spatiotemporal groundwater level forecasting. Hydrological Processes: An International Journal, 22(26), 5054-5066.
  15. Orouji, H., Bozorg Haddad, O., Fallah-Mehdipour, E., & Mariño, M. A. (2014). Flood routing in branched river by genetic programming. Proceedings of the Institution of Civil Engineers-Water Management, 167(2), 115-123.
  16. Ponce, V. M., & Lugo, A. (2001). Modeling looped ratings in Muskingum-Cunge routing. Journal of Hydrologic Engineering, 6(2), 119-124.
  17. Todini, E. (1991). Hydraulic and hydrologic flood routing schemes. In Recent advances in the modeling of hydrologic systems (pp. 389-405). Springer.
  18. Yoon, J., & Padmanabhan, G. (1993). Parameter estimation of linear and nonlinear Muskingum models. Journal of Water Resources Planning and Management, 119(5), 600-610.
Water and Irrigation Management
Volume 15, Issue 4
March 2026
Pages 729-743

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