Abstract
The longitudinal dispersion coefficient (Kx) is fundamental to modeling of pollutant and sediment transport in natural rivers, but a general expression for Kx, with applicability in low or high flow conditions, remains a challenge. The objective of this paper is to develop a Pareto-Optimal-Multigene Genetic Programming (POMGGP) equation for Kx by analyzing 503 data sets of channel geometry and flow conditions in natural streams worldwide. In order to acquire reliable data subsets for training and testing, Subset Selection of Maximum Dissimilarity Method (SSMD), rather than the classical trial and error method, was used by a random manipulation of these data sets. A new hybrid framework was developed that integrates SSMD with Multigene Genetic Programming (MGP) and Pareto-front optimization to produce a set of selected dimensionless equations of Kx and find the best equation with wide applicability. The POMGGP-based final equation was evaluated and compared with 8 published equations, using statistical indices, graphical visualization of 95% confidence ellipse, Taylor diagram, discrepancy ratio (DR) distribution, and scatter plots. Besides being simple and applicable to a broad range of conditions, the proposed equation predicted Kx more accurately than did the other equations and can therefore be used for the prediction of longitudinal dispersion coefficient in natural river flows.
Similar content being viewed by others
References
Alizadeh MJ, Shabani A, Kavianpour MR (2017) Predicting longitudinal dispersion coefficient using ANN with metaheuristic training algorithms. Int J Environ Sci Technol 14:2399
Ahmad Z (2013) Prediction of longitudinal dispersion coefficient using laborary and field data: relationship comparisons. Hydrol Res 44(2)
Carr ML, Rehmann CR (2007) Measuring the dispersion coefficent with acoustic doppler current profilers. J HydraulEng-Asce 133(8):977–982
DanandehMehr AD, Kahya E (2017) A Pareto-optimal moving average multigene genetic programming model for daily streamflow prediction. J Hydrol 549:603–615
DanandehMehr AD, Nourani V (2017) A Pareto-optimal moving average-multigene genetic programming model for rainfall-runoff modelling. Environ Model Softw 92:239–251
Deng Z-Q, Singh VP, Bengtsson L (2001) Longitudinal dispersion coefficient in straight rivers. J Hydraul Eng 127:919–927
Elder JW (1959) The dispersion of a marked fluid in turbulent shear flow. J Fluid Mech 5(04):544–560
Fan FM, Fleischmann AS, Collischonn W, Ames DP, Rigo D (2015) Large-scale analytical water quality model coupled with GIS for simulation of point sourced pollutant discharges. Environ Model Softw 64:58–71
Fischer BH, (1975) Discussion of ‘‘simple method for predicting dispersion in streams,’’ by R.S. McQuivey and T.N. Keefer. J Environ Eng Div 101:453
Fischer HB, List EJ, Koh RCY, Imberger J, Brooks NH (1979) Mixing in Inland and Coastal Waters. Academic, New York
Gandomi AH, Alavi AH (2012a) A new multi-gene genetic programming approach to nonlinear system modeling. Part I: materials and structural engineering problems. Neural Comput & Applic 21(1):171–187
Gandomi AH, Alavi AH (2012b) A new multi-gene genetic programming approach to non-linear system modeling. Part II: geotechnical and earthquake engineering problems. Neural Comput & Applic 21(1):189–201
Hadgu LT, Nyadawa MO, Mwangi1 JK, Kibetu PM, Mehari BB (2014) Application of Water Quality Model QUAL2K to Model the Dispersion of Pollutants in River Ndarugu, Kenya. Computational Water, Energy, and Environmental Engineering 3:162–169
Johnson RA, Wichern DW (2007) Multivariate analysis. Encyclopedia of Statistical Sciences, 8. [Chapter 4 (result 4.7 on page 163)
Kashefipour MS, Falconer RA (2002) Longitudinal dispersion coefficients in natural channels. Water Res 36(6):1596–1608
Kennard RW, Stone LA (1969) Computer aided design of experiments. Technometrics 11(1):137–148
Koza JR (1992) Genetic programming: on the programming of computers by means of natural selection (Vol. 1). MIT press
Li X, Liu H, Yin M (2013) Differential evolution for prediction of longitudinal dispersion coefficients in natural streams. Water ResourManag 27:5245–5260
Liu H (1977) Predicting dispersion coefficient of streams. J Environ Eng Div 103:59–69
May RJ, Maier HR, Dandy GC, Fernando TG (2008) Non-linear variable selection for artificial neural networks using partial mutual information. Environ Model Softw 23(10):1312–1326
Moses SA, Janaki L, Joseph S, Joseph J (2016) Water quality prediction capabilities of WASP model for a tropical lake system. Lake and Reservoirs 20(4):285–299
Najafzadeh M, Tafarojnoruz A (2016) Evaluation of neuro-fuzzy GMDH-based particle swarm optimization to predict longitudinal dispersion coefficient in rivers. Environ Earth Sci 75(2):1–12
Noori R, Deng Z, Kiaghadi A, Kachoosangi FT (2016) How reliable are ANN, ANFIS, and SVM techniques forpredicting longitudinal dispersion coefficient in natural rivers? J Hydraul Eng 142:04015039
Noori R, Karbassi A, Farokhnia A, Dehghani M (2009) Predicting the longitudinal dispersion coefficient using support vector machine and adaptive neuro-fuzzy inference system techniques. Environ Eng Sci 26(10):1503–1510
Parveen N, Singh SK (2016) Application of Qual2e Model for River Water Quality Modelling. International Journal of Advance Research and Innovation 4(2):429–432
Rajeev RS, Dutta S (2009) Prediction of longitudinal dispersion coefficients in natural rivers using genetic algorithm. Hydrol Res 40(6):544–552
Riahi-Madvar H, Ayyoubzadeh SA, Khadangi E, Ebadzadeh MM (2009) An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS. Expert Syst Appl 36(4):8589–8596
Sattar AM, Gharabaghi B (2015) Gene expression models for prediction of longitudinal dispersion coefficient in streams. J Hydrol 524:587–596
Searson DP (2015) GPTIPS 2: an open-source software platform for symbolic data mining. In: Handbook of genetic programming applications (pp. 551–573). Springer International Publishing
Seo IW, Cheong TS (1998) Predicting longitudinal dispersion coefficient in natural streams. J Hydraul Eng 124:25
Tayfour G, Singh VP (2005) Predicting longitudinal dispersion coefficient in natural streams by artificial neural network. J Hydraul Eng 131(11):991–1000
Taylor KE (2001) Summarizing multiple aspects of model performance in a single diagram. J Geophys Res-Atmos 106(D7):7183–7192
Wang YF, Huai WX, Wang WJ (2017) Physically sound formula for longitudinal dispersion coefficients of natural rivers. J Hydrol 544:511–523
Wang Y, Huai W (2016) Estimating the longitudinal dispersion coefficient in straight natural rivers. J Hydraul Eng 142(11):04016048
Yapo PO, Gupta HV, Sorooshian S (1998) Multi-objective global optimization for hydrologic models. J Hydrol 204(1–4):83–97
Zhang T, Georgiopoulos M, Anagnostopoulos GC (2017) Pareto-optimal model selection via SPRINT-race. IEEE Transactions on Cybernetics
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Riahi-Madvar, H., Dehghani, M., Seifi, A. et al. Pareto Optimal Multigene Genetic Programming for Prediction of Longitudinal Dispersion Coefficient. Water Resour Manage 33, 905–921 (2019). https://doi.org/10.1007/s11269-018-2139-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11269-018-2139-6