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Fuzzy Regression Analysis for Sediment Incipient Motion under Turbulent Flow Conditions

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Abstract

Sediment incipient motion in turbulent flow conditions is hard to be determined due to the random turbulence fluctuations as well as the random positioning and orientation of the sediment particles. The most common criteria for the initiation of motion of sediment particles, like the Shields diagram, have a general movement or no-movement functionality. In such criteria, the subjectivity of the researcher on the definition of sediment incipient motion influences the quantification of sediment transport; however, uncertainties in the selection of critical flow conditions may lead to large computational errors. In order to avoid the ambiguity of selecting a unique entrainment threshold, this paper employed fuzzy regression and set a fuzzy band, which offers a transition from sporadic to general sediment entrainment. Two approaches were used, namely the conventional fuzzy regression analysis and a goal programming-based fuzzy regression. In the latter, the modification of the fuzzy linear regression basis, by establishing a distance (error) associated with the underestimation of the left-hand boundary and a distance associated with the overestimation of the right-hand boundary, constitutes a significant improvement in the field since it avoids the significant influence of the outliers. The fuzzy regression analyses employ data from the pertinent literature and consider critical shear stress to depend on a particle Reynolds number and relative roughness. The results show that the terms that contain the particle Reynolds number do not have any uncertainty, while the fuzziness appears in both the coefficient of the relative roughness and the constant terms.

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References

  • Aguirre-Pe J (1975) Incipient erosion in high gradient open channel flow with artificial roughness elements. In: Proceedings of the 16th IAHR Congress, IAHR, vol 2. Sao Paulo, pp 173–180

  • Aksoy S (1973) The influence of relative depth on threshold of grain motion. In: Proceedings of the International Symposium on River Mechanics, IAHR. Bangkok, pp 359–370

  • Andrews E D (1994) Marginal bed load transport in a gravel bed stream, Sagehen Creek, California. Water Resour Res 30(7):2241–2250. doi:10.1029/94WR00553

    Article  Google Scholar 

  • Armanini A, Gregoretti C (2005) Incipient sediment motion at high slopes in uniform flow condition. Water Resour Res 41(12):W12431. doi:10.1029/2005WR004001

    Article  Google Scholar 

  • Ashida K, Bayazit M (1973) Initiation of motion and roughness of flows in steep channels. In: Proceedings of the 15th IAHR Congress, IAHR, vol 1. Istanbul, pp 475–484

  • Bathurst J C, Graf W H, Cao H H (1983) Initiation of sediment transport in steep channels with coarse bed material. In: Sumer BM, Muller A (eds) Mechanics of sediment transport, Balkema, pp 207–213

  • Bathurst J C, Graf W H, Cao H H (1987) Bed load discharge equations for steep mountain rivers. In: Thorne C R, Bathurst J C, Hey R D (eds) Sediment transport in gravel-bed rivers. Wiley, New York, pp 453–477

  • Bettess R (1984) Initiation of sediment transport in gravel streams. Proc Inst Civ Eng Part 2 77(1):79–88. doi:10.1680/iicep.1984.1275

    Google Scholar 

  • Brownlie W R (1981) Prediction of flow depth and sediment discharge in open channels. Tech. Rep. KH-R-43A, W.M. Keck Laboratory of Hydraulics and Water Resources, Pasadena, California

  • Buffington J M (1999) The legend of A. F Shields J Hydraul Eng 125(4):376–387. doi:10.1061/(ASCE)0733-9429(1999)125:4(376)

    Article  Google Scholar 

  • Buffington J M, Montgomery D R (1997) A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers. Water Resour Res 33(8):1993–2029. doi:10.1029/96WR03190

    Article  Google Scholar 

  • Chang N B, Wen C, Wu S (1995) Optimal management of environmental and land resources in a reservoir watershed by multiobjective programming. J Environ Manag 44(2):144–161. doi:10.1006/jema.1995.0036

    Article  Google Scholar 

  • Chen L H, Hsueh C C, Chang C J (2013) A two-stage approach for formulating fuzzy regression models. Knowl-Based Syst 52:302–310. doi:10.1016/j.knosys.2013.08.010

    Article  Google Scholar 

  • Chiew Y M, Parker G (1994) Incipient sediment motion on non-horizontal slopes. J Hydraul Res 32(5):649–660. doi:10.1080/00221689409498706

    Article  Google Scholar 

  • Dancey C L, Diplas P, Papanicolaou A, Bala M (2002) Probability of individual grain movement and threshold condition. J Hydraul Eng 128(12):1069–1075. doi:10.1061/(ASCE)0733-9429(2002)128:12(1069)

    Article  Google Scholar 

  • Everts C H (1973) Particle overpassing on flat granular boundaries. J Waterw Harbors Coastal Eng Div 99(4):425–438

    Google Scholar 

  • Fenton J D, Abbott J E (1977) Initial movement of grains on a stream bed: the effect of relative protrusion. Proc Roy Soc Lond A 352(1671):523–537. doi:10.1098/rspa.1977.0014

    Article  Google Scholar 

  • Ferguson R (2007) Flow resistance equations for gravel- and boulder-bed streams. Water Resour Res 43(5):W05427. doi:10.1029/2006WR005422

    Article  Google Scholar 

  • Ferguson R I (2012) River channel slope, flow resistance, and gravel entrainment thresholds. Water Resour Res 48(5):W05517. doi:10.1029/2011WR010850

    Article  Google Scholar 

  • Fernandez Luque R, van Beek R (1976) Erosion and transport of bed-load sediment. J Hydraul Res 14(2):127–144

    Article  Google Scholar 

  • Ganoulis J (1994) Engineering Risk Analysis of Water Pollution: Probabilities and Fuzzy Sets. VCH

  • Gilbert GK (1914) The transportation of debris by running waters. Prof. Paper 86, U.S. Geological Survey

  • Graf W H, Suszka L (1987) Sediment transport in steep channels. J Hydrosci Hydraul Eng 5(1):11–26

    Google Scholar 

  • Grass A J (1970) Initial instability of fine bed sand. J Hydraul Div 96(3):619–632

    Google Scholar 

  • Hassanpour H, Maleki H R, Yaghoobi M A (2010) Fuzzy linear regression model with crisp coefficients: a goal programming approach. Iran J Fuzzy Syst 7(2):19–39

    Google Scholar 

  • Hojati M, Bector C, Smimou K (2005) A simple method for computation of fuzzy linear regression. Eur J Oper Res 166(1):172–184. doi:10.1016/j.ejor.2004.01.039

    Article  Google Scholar 

  • Hundecha Y, Bardossy A, Theisen H W (2001) Development of a fuzzy logic-based rainfall-runoff model. Hydrol Sci J 46(3):363–376. doi:10.1080/02626660109492832

    Article  Google Scholar 

  • Ikeda S (1982) Incipient motion of sand particles on side slopes. J Hydraul Div 108(1):95–114

    Google Scholar 

  • Katul G, Wiberg P, Albertson J, Hornberger G (2002) A mixing layer theory for flow resistance in shallow streams. Water Resour Res 38(11):32,1–32,8. doi:10.1029/2001WR000817

    Article  Google Scholar 

  • Kennedy J F (1995) The Albert Shields story. J Hydraul Eng 121(11):766–772. doi:10.1061/(ASCE)0733-9429(1995)121:11(766)

    Article  Google Scholar 

  • Kim J S, Whang K S (1998) A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function. Eur J Oper Res 107 (3):614–624. doi:10.1016/S0377-2217(96)00363-3

    Article  Google Scholar 

  • Kirchner J W, Dietrich W E, Iseya F, Ikeda H (1990) The variability of critical shear stress, friction angle, and grain protrusion in water-worked sediments. Sedimentology 37(4):647–672. doi:10.1111/j.1365-3091.1990.tb00627.x

    Article  Google Scholar 

  • Klir G J, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall PTR

  • Kramer H (1935) Sand mixtures and sand movement in fluvial models. Trans ASCE 100(1909):798–838

    Google Scholar 

  • Lamb M P, Dietrich W E, Venditti J G (2008) Is the critical Shields stress for incipient motion dependent on channel-bed slope? J Geophys Res 113:F02008. doi:10.1029/2007JF000831

    Google Scholar 

  • Lavelle J W, Mofjeld H O (1987) Do critical stresses for incipient motion and erosion really exist? J Hydraul Eng 113(3):370–385. doi:10.1061/(ASCE)0733-9429(1987)113:3(370)

    Article  Google Scholar 

  • Liu T Y (1935) Transportation of the bottom load in an open channel. Master’s thesis, Univ. of Iowa, Iowa City

  • Mavis F T, Liu T Y, Soucek E (1937) The transportation of detritus by flowing water, II. Univ. Iowa Studies Eng. 341, Univ. of Iowa, Iowa City

  • Meyer-Peter E, Müller R (1948) Formulas for bed load transport. In: Proceedings of the 2nd Meeting of the International Association for Hydraulic Research. Stockholm, pp 39–64

  • Miller M C, McCave IN, Komar P D (1977) Threshold of sediment motion under unidirectional currents. Sedimentology 24(4):507–527. doi:10.1111/j.1365-3091.1977.tb00136.x

    Article  Google Scholar 

  • Modarres M, Nasrabadi E, Nasrabadi M M (2005) Fuzzy linear regression models with least square errors. Appl Math Comput 163(2):977–989. doi:10.1016/j.amc.2004.05.004

    Google Scholar 

  • Neill C R (1967) Mean velocity criterion for scour of coarse uniform bed-material. In: Proceedings of the 12th IAHR Congress, IAHR, vol 3. Fort Collins, pp 46–54

  • Nezu I, Nakagawa H (1993) Turbulence in open-channel flows. Balkema, Rotterdam

    Google Scholar 

  • Nikora V, Goring D, McEwan I, Griffiths G (2001) Spatially averaged open-channel flow over rough bed. J Hydraul Eng 127(2):123–133. doi:10.1061/(ASCE)0733-9429(2001)127:2(123)

    Article  Google Scholar 

  • Nikora V, Koll K, McEwan I, McLean S, Dittrich A (2004) Velocity distribution in the roughness layer of rough-bed flows. J Hydraul Eng 130(10):1036–1042. doi:10.1061/(ASCE)0733-9429(2004)130:10(1036)

    Article  Google Scholar 

  • Nikuradse J (1933) Strömungsgesetze in rauhen Rohren. Forschungsheft 361, Forschung auf dem Gebiete des Ingenieurwesens

  • Paintal A S (1971) Concept of critical shear stress in loose boundary open channels. J Hydraul Res 9(1):91–113. doi:10.1080/00221687109500339

    Article  Google Scholar 

  • Papadopoulos B K, Sirpi M A (1999) Similarities in fuzzy regression models. J Optimiz Theory App 102(2):373–383. doi:10.1023/A:1021784524897

    Article  Google Scholar 

  • Papanicolaou A N, Diplas P, Evaggelopoulos N, Fotopoulos S (2002) Stochastic incipient motion criterion for spheres under various packing conditions. J Hydraul Eng 128(4):369–380. doi:10.1061/(ASCE)0733-9429(2002)128:4(369)

    Article  Google Scholar 

  • Paphitis D (2001) Sediment movement under unidirectional flows: an assessment of empirical threshold curves. Coast Eng 43(3–4):227–245. doi:10.1016/S0378-3839(01)00015-1

    Article  Google Scholar 

  • Parker G (2008) Transport of gravel and sediment mixtures. In: Garcia MH (ed) ASCE manuals and reports on engineering no. 110, sedimentation engineering processes, measurements, modeling and practice, ASCE. Virginia, pp 165–251

  • Pender G, Shvidchenko A B, Chegini A (2007) Supplementary data confirming the relationship between critical Shields stress, grain size and bed slope. Earth Surf Proc Land 32(11):1605–1610. doi:10.1002/esp.1588

    Article  Google Scholar 

  • Peters G (1994) Fuzzy linear regression with fuzzy intervals. Fuzzy Sets Syst 63 (1):45–55. doi:10.1016/0165-0114(94)90144-9

    Article  Google Scholar 

  • Prager E J, Southard J B, Vivoni-Gallart E R (1996) Experiments on the entrainment threshold of well-sorted and poorly sorted carbonate sands. Sedimentology 43(1):33–40. doi:10.1111/j.1365-3091.1996.tb01457.x

    Article  Google Scholar 

  • Prancevic J P, Lamb M P (2015) Unraveling bed slope from relative roughness in initial sediment motion. J Geophys Res 120(3):474–489. doi:10.1002/2014JF003323

    Article  Google Scholar 

  • Raudkivi A J (1963) Study of sediment ripple formation. J Hydraul Div 89 (6):15–33

    Google Scholar 

  • Recking A (2009) Theoretical development on the effects of changing flow hydraulics on incipient bed load motion. Water Resour Res 45(4):W04401. doi:10.1029/2008WR006826

    Article  Google Scholar 

  • Rouse H (1939) An analysis of sediment transportation in the light of fluid turbulence. Soil Conservation Service Report No. SCS-TP-25, United States Department of Agriculture, Washington, D. C.

  • Shields A F (1936) Application of similarity principles and turbulence research to bedload movement. Mitt. Preuss. Versuchsanst. Wasser., 26, Transl. into English by Ott, W.P. and van Uchelen J.C. California Institute of Technology

  • Shrestha R R, Simonovic S P (2010) Fuzzy nonlinear regression approach to stage-discharge analyses: case study. J Hydrol Eng 15(1):49–56. doi:10.1061/(ASCE)HE.1943-5584.0000128

    Article  Google Scholar 

  • Shvidchenko A B, Pender G (2000) Flume study of the effect of relative depth on the incipient motion of coarse uniform sediments. Water Resour Res 36(2):619–628. doi:10.1029/1999WR900312

    Article  Google Scholar 

  • Spiliotis M, Bellos C (2015) Flooding risk assessment in mountain rivers. In: Proceedings of the EWRA 9th World congress water resources management in a changing World: challenges and opportunities. Istanbul

  • Suszka L (1991) Modification of transport rate formula for steep channels. In: Armanini A, Di Silvio G (eds) Fluvial hydraulics of mountain regions, lecture notes in Earth sciences, vol 37. Springer, Berlin, pp 59–70

  • Tanaka H (1987) Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst 24(3):363–375. doi:10.1016/0165-0114(87)90033-9

    Article  Google Scholar 

  • Tanaka H, Uejima S, Asai K (1982) Linear regression analysis with fuzzy model. IEEE Trans Syst Man Cybern 12(6):903–907

    Article  Google Scholar 

  • Tanaka H, Hayashi I, Watada J (1989) Possibilistic linear regression analysis for fuzzy data. Eur J Oper Res 40(3):389–396. doi:10.1016/0377-2217(89)90431-1

    Article  Google Scholar 

  • Tsakiris G, Tigkas D, Spiliotis M (2006) Assessment of interconnection between two adjacent watersheds using deterministic and fuzzy approaches. Eur Water 15(16):15–22

    Google Scholar 

  • Tsujimoto T (1991) Bed-load transport in steep channels. In: Armanini A, Di Silvio G (eds) Fluvial hydraulics of mountain regions, lecture notes in Earth sciences, vol 37. Springer, Berlin, pp 89–102

  • Turowski J M, Badoux A, Rickenmann D (2011) Start and end of bedload transport in gravel-bed streams. Geophys Res Lett 38(4):L04401. doi:10.1029/2010GL046558

  • USWES (1935) Study of riverbed material and their use with special reference to the Lower Mississippi River. Tech. Rep. 17, U.S. Waterways Experiment Station, Vicksburg, Mississippi

  • Vanoni V A (2006) ASCE manuals and reports on engineering no. 54, Sedimentation engineering, ASCE. Virginia, U.S.A.

  • Vollmer S, Kleinhans M G (2007) Predicting incipient motion, including the effect of turbulent pressure fluctuations in the bed. Water Resour Res 43(5):W05410. doi:10.1029/2006WR004919

    Article  Google Scholar 

  • Wang H F, Tsaur R C (2000) Resolution of fuzzy regression model. Eur J Oper Res 126(3):637–650. doi:10.1016/S0377-2217(99)00317-3

    Article  Google Scholar 

  • Wiberg P L, Smith J D (1987) Calculations of the critical shear stress for motion of uniform and heterogeneous sediments. Water Resour Res 23(8):1471–1480. doi:10.1029/WR023i008p01471

    Article  Google Scholar 

  • Wolman MG, Brush LM (1961) Factors controlling the size and shape of stream channels in coarse noncohesive sands. Prof. Paper 282-G, U.S. Geol. Surv

  • Yaghoobi M A, Tamiz M (2007) A method for solving fuzzy goal programming problems based on MINMAX approach. Eur J Oper Res 177(3):1580–1590. doi:10.1016/j.ejor.2005.10.022

    Article  Google Scholar 

  • Yalin M S, Karahan E (1979) Inception of sediment transport. J Hydraul Div 105(11):1433–1443

    Google Scholar 

  • Zanke U C E (2003) On the influence of turbulence on the initiation of sediment motion. Int J Sediment Res 18(1):17–31

    Google Scholar 

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Acknowledgments

The authors are grateful to the Editor in Chief and two anonymous Reviewers for their constructive and insightful comments, which improved the presentation of the paper. An initial version of this paper has been presented at the 9th World Congress of the European Water Resources Association (EWRA) ”Water Resources Management in a Changing World: Challenges and Opportunities”, Istanbul, Turkey, June 10–13, 2015.

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  1. Division of Hydraulics, Department of Civil Engineering, Istanbul Technical University, Maslak, 34469, Istanbul, Turkey

    Vasileios Kitsikoudis

  2. Department of Civil Engineering, Democritus University of Thrace, Vasil. Sophias 12, Xanthi, 67132, Greece

    Mike Spiliotis & Vlassios Hrissanthou

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Correspondence to Vasileios Kitsikoudis.

Appendix

Appendix

In general terms, one of the most popular forms of goal programming is the following:

$$ \left\{ \begin{array}{l} \max\sum\limits_{j=1}^{J}\left( d_{j}^{-}+d_{j}^{+}\right)\\ \quad\\ \left( AX\right)_{j}+d_{j}^{-}-d_{j}^{+}=b_{j} \ \text{(to express the} "\cong")\\ \\ \text{other constraints}\ \end{array} \right. $$
(15)

in which the equation \(\left (AX\right )_{j}+d_{j}^{-}-d_{j}^{+}=b_{j}\) expresses the j th soft equality constraints:

$$ \left( AX\right)_{j}\cong b_{j} $$
(16)

where b j is the target value for one objective and let the divergence from the equality constraint be measured by the corresponding distances \(d_{j}^{-}\) and \(d_{j}^{+}\geq 0\).

Goal programming can be extended and consider constraints with inequalities, which are enabled to be partially (softly) satisfied. In this case, the goal programming approach can be also utilized with only one distance of divergence for each equality constraint as follows:

$$ \left( AX\right)_{j}-d_{j}^{+}\leq b_{j} \ \left( \text{to express the} "\tilde{\leq}"\right) $$
(17)
$$ \left( AX\right)_{j}+d_{j}^{-}\geq b_{j} \ \left( \text{to express the} "\tilde{\geq}"\right) $$
(18)

Similar to the conventional goal programming, the objective function is the sum of the inequalities diversions. Kim and Whang (1998) proposed a similar modification in order to express the fuzzy inequalities for the case of the fuzzy programming formulation.

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Kitsikoudis, V., Spiliotis, M. & Hrissanthou, V. Fuzzy Regression Analysis for Sediment Incipient Motion under Turbulent Flow Conditions. Environ. Process. 3, 663–679 (2016). https://doi.org/10.1007/s40710-016-0154-2

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