Combining Predictions and Assessing Uncertainty from Sediment Transport Equations Using Multivariate Bayesian Model Averaging
Publication: Journal of Hydraulic Engineering
Volume 144, Issue 4
Abstract
A key assumption in the use of numerical sediment transport models is the selection and application of an equation to describe sediment transport capacity. This assumption introduces uncertainty in the model predictions because all sediment transport equations are simplifications of the physical processes. Bayesian model averaging (BMA) is a statistical method that characterizes the uncertainty due to the mathematical structure of a model and allows predictions from multiple models to be combined. In this paper, BMA is applied to quantify how the uncertainty originating from the transport equation affects the predictions from a sediment transport model. To apply BMA in this context, the likelihood function is modified to allow consideration of multiple model output variables. In addition, BMA’s assumption that the uncertainty associated with each model remains constant between the calibration and forecast periods is relaxed. The proposed multivariate BMA model is applied to transport equations included in the sedimentation and river hydraulics—one dimension (SRH-1D) model. For two flume experiments, the multivariate BMA model provides improved predictions over the individual transport equations and univariate BMA models, and it provides more realistic descriptions of the predictive uncertainty.
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Acknowledgments
The authors acknowledge the U.S. Bureau of Reclamation Science and Technology Program under Project 1596 for their financial support.
References
Ackers, P., and White, W. R. (1973). “Sediment transport: New approach and analysis.” J. Hydraul. Div., 99(11), 2041–2060.
Ahn, J., and Yang, C. T. (2015). “Determination of recovery factor for simulation of non-equilibrium sedimentation in reservoir.” Int. J. Sediment Res., 30(1), 68–73.
Ahn, J., Yang, C. T., Pridal, D. B., and Remus, J. I. (2013). “Numerical modeling of sediment flushing from Lewis and Clark Lake.” Int. J. Sediment Res., 28(2), 182–193.
Anderson, N. H. (1965). “Averaging versus adding as a stimulus-combination rule in impression formation.” J. Exp. Psychol., 70(4), 394–400.
Bagnold, R. A., and Barndorff-Nielsen, O. (1980). “The pattern of natural size distributions.” Sedimentology, 27(2), 199–207.
Bates, J. M., and Granger, C. W. J. (1969). “The combination of forecasts.” Oper. Res., 20(4), 451–468.
Bertin, X., Fortunato, A. B., and Oliveira, A. (2007). “Sensitivity analysis of a morphodynamic modeling system applied to a Portuguese tidal inlet.” Proc., 5th IAHR Symp. on River, Coastal and Estuarine Morphodynamics, C. M. Dohmen-Janssen and S. J. M. H. Hulscher, eds., Taylor & Francis Group, London, 1–17.
Beven, K. J. (2011). Rainfall-runoff modelling: The primer, Wiley, Chichester, U.K.
Beven, K. J., and Binley, A. (1992). “The future of distributed models: Model calibration and uncertainty prediction.” Hydrol. Processes, 6(3), 279–298.
Brownlie, W. R. (1981). “Prediction of flow depth and sediment discharge in open channels.” Ph.D. dissertation, California Institute of Technology, Pasadena, CA.
Buckland, S. T., Burnham, K. P., and Augustin, N. H. (1997). “Model selection: An integral part of inference.” Biometrics, 53(2), 603–618.
Burnham, K. P., and Anderson, D. R. (2002). Model selection and multimodel inference: A practical information-theoretic approach, 2nd Ed., Springer, New York.
Chahinian, N., and Moussa, R. (2007). “Comparison of different multi-objective calibration criteria of a conceptual rainfall-runoff model of flood events.” Hydrol. Earth Syst. Sci. Discuss., 4(3), 1031–1067.
Clyde, M. A. (1999). “Bayesian model averaging and model search strategies.” Bayesian Statistics, Vol. 6, J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds., Oxford University Press, Oxford, U.K., 157–185.
Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). “Maximum likelihood from incomplete data via the EM algorithm.” J. R. Stat. Soc. Ser. B, 39(1), 1–38.
Downey, C., and Sanner, S. (2010). “Temporal difference Bayesian model averaging: A Bayesian perspective on adapting lambda.” Proc., 27th Int. Conf. on Machine Learning, J. Fürnkranz and T. Cohn, eds., Omnipress, Madison, WI, 311–318.
Einstein, H. A. (1950). “The bed-load function for sediment transportation in open channel flows.”, U.S. Dept. of Agriculture, Urbana, IL.
Engelund, F., and Hansen, E. (1972). A Monograph on sediment transport in alluvial streams, Teknish Forlag, Technical Press, Copenhagen, Denmark.
Gaeuman, D., Andrews, E. D., Krause, A., and Smith, W. (2009). “Predicting fractional bed load transport rates: Application of the Wilcock-Crowe equations to a regulated gravel bed river.” Water Resour. Res., 45(6), W06409.
Givens, G. H., and Hoeting, J. A. (2012). Computational statistic, 2nd Ed., Wiley, Hoboken, NJ.
Granger, C. W., and Ramanathan, R. (1984). “Improved methods of combining forecasts.” J. Forecasting, 3(2), 197–204.
Hansen, B. E. (2007). “Least squares model averaging.” Econometrica, 75(4), 1175–1189.
Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999). “Bayesian model averaging: A tutorial.” Stat. Sci., 14(4), 382–417.
Huang, J. V., and Greimann, B. P. (2010). “Sediment transport and channel morphology model in the San Joaquin River from Friant Dam to Mendota Dam, California.” Proc., 2nd Joint Federal Interagency Conf., Advisory Committee on Water Information (ACWI), Reston, VA, 1–12.
Huang, J. V., and Greimann, B. P. (2013). User’s manual for SRH-1D 3.0, U.S. Dept. of the Interior, Denver.
Kleiber, W., Raftery, A. E., and Gneiting, T. (2011). “Geostatistical model averaging for locally calibrated probabilistic quantitative precipitation forecasting.” J. Am. Stat. Assoc., 106(496), 1291–1303.
Laursen, E. M. (1958). “The total sediment load of streams.” J. Hydraul. Div., 84(1), 1531–1536.
Madden, E. B. (1993). “Modified Laursen method for estimating bed-material sediment load.”, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.
Meyer-Peter, E., and Müller, R. (1948). “Formula for bed-load transport.” Proc., 2nd Int. Association for Hydraulic Research Meeting, IAHR, Delft, Netherlands, 39–64.
Mo, X., and Beven, K. (2004). “Multi-objective parameter conditioning of a three-source wheat canopy model.” Agric. For. Meteorol., 122(1), 39–63.
Nash, J. E., and Sutcliffe, J. V. (1970). “River flow forecasting through conceptual models. I: A discussion of principles.” J. Hydrol., 10(3), 282–290.
Parker, G. (1990). “Surface-based bedload transport relation for gravel rivers.” J. Hydraul. Res., 28(4), 417–436.
Parrish, M. A., Moradkhani, H., and DeChant, C. M. (2012). “Toward reduction of model uncertainty: Integration of Bayesian model averaging and data assimilation.” Water Resour. Res., 48(3), W03519.
Pender, G., Hoey, T. B., Fuller, C., and Mcewan, I. K. (2001). “Selective bedload transport during the degradation of a well sorted graded sediment bed.” J. Hydraul. Res., 39(3), 269–277.
Pinto, L., Fortunato, A. B., and Freire, P. (2006). “Sensitivity analysis of non-cohesive sediment transport formulae.” Cont. Shelf Res., 26(15), 1826–1839.
Raftery, A. E., Gneiting, T., Balabdaoui, F., and Polakowski, M. (2005). “Using Bayesian model averaging to calibrate forecast ensembles.” Mon. Weather Rev., 133(5), 1155–1174.
Ribberink, J. S. (1998). “Bed-load transport for steady flows and unsteady oscillatory flows.” Coastal Eng., 34(1), 59–82.
Rottner, J. (1959). “A formula for bed load transportation.” La Houille Blanche, 14(3), 285–307.
Ruark, M. D., Niemann, J. D., Greimann, B. P., and Arabi, M. (2011). “Method for assessing impacts of parameter uncertainty in sediment transport modeling applications.” J. Hydraul. Eng., 623–636.
Russell, K., et al. (2010). “Klamath reservoir sediment characterization and drawdown impacts for the dam removal investigation.” Proc., 2010 AGU Fall Meeting, American Geophysical Union, Washington, DC, 1053.
Sabatine, S. M., Niemann, J. D., and Greimann, B. P. (2015). “Evaluation of parameter and model uncertainty in simple applications of a 1D sediment transport model.” J. Hydraul. Eng., 04015002.
Schmelter, M. L., Hooten, M. B., and Stevens, D. K. (2011). “Bayesian sediment transport model for unisize bed load.” Water Resour. Res., 47(11), W11514.
Seal, R., Paola, C., Parker, G., Southard, J. B., and Wilcock, P. R. (1997). “Experiments on downstream fining of gravel. I: Narrow-channel runs.” J. Hydraul. Eng., 874–884.
Sloughter, J. M., Gneiting, T., and Raftery, A. E. (2010). “Probabilistic wind speed forecasting using ensembles and Bayesian model averaging.” J. Am. Stat. Assoc., 105(489), 25–35.
van Griensven, A., and Meixner, T. (2007). “A global and efficient multi-objective auto-calibration and uncertainty estimation method for water quality catchment models.” J. Hydroinf., 9(4), 277–291.
van Rijn, L. C. (1989). “Handbook sediment transport by currents and waves.”, Delft Hydraulics Laboratory, Delft, Netherlands.
Veenhuis, B. (2014). “A practical model blending technique based on Bayesian model averaging.” Proc., 22nd Conf. on Probability and Statistics in the Atmospheric Sciences, American Meteor Society, Atlanta, 1–8.
Viallefont, V., Raftery, A. E., and Richardson, S. (2001). “Variable selection and Bayesian model averaging in case-control studies.” Stat. Med., 20(21), 3215–3230.
Vrugt, J. A., Diks, C. G. H., and Clark, M. P. (2008). “Ensemble Bayesian model averaging using Markov chain Monte Carlo sampling.” Environ. Fluid Mech., 8(5–6), 579–595.
Vrugt, J. A., and Robinson, B. A. (2007). “Treatment of uncertainty using ensemble methods: Comparison of sequential data assimilation and Bayesian model averaging.” Water Resour. Res., 43(1), W01411.
Wallingford, H. R. (1990). “Sediment transport, the Ackers and White theory revised.”, HR Wallingford, Wallingford, U.K.
Wilcock, P. R. (2001). “Toward a practical method for estimating sediment-transport rates in gravel-bed rivers.” Earth Surf. Processes Landforms, 26(13), 1395–1408.
Wilcock, P. R., and Crowe, J. C. (2003). “Surface-based transport model for mixed-size sediment.” J. Hydraul. Eng., 120–128.
Wong, M., and Parker, G. (2006). “Reanalysis and correction of bed-load relation of Meyer-Peter and Müller using their own database.” J. Hydraul. Eng., 1159–1168.
Wu, W., Wang, S. S., and Jia, Y. (2000). “Nonuniform sediment transport in alluvial rivers.” J. Hydraul. Res., 38(6), 427–434.
Yang, C. T. (1973). “Incipient motion and sediment transport.” J. Hydraul. Div., 99(10), 1679–1704.
Yang, C. T. (1979). “Unit stream power equations for total load.” J. Hydrol., 40(1), 123–138.
Yang, C. T. (1984). “Unit stream power equation for gravel.” J. Hydraul. Eng., 1783–1797.
Yang, C. T. (1996). Sediment transport: Theory and practice, McGraw-Hill, New York.
Yapo, P. O., Gupta, H. V., and Sorooshian, S. (1998). “Multi-objective global optimization for hydrologic models.” J. Hydrol., 204(1), 83–97.
Ye, M., Neuman, S. P., and Meyer, P. D. (2004). “Maximum likelihood Bayesian averaging of spatial variability models in unsaturated fractured tuff.” Water Resour. Res., 40(5), W05113.
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Published In
Journal of Hydraulic Engineering
Volume 144 • Issue 4 • April 2018
Copyright
©2018 American Society of Civil Engineers.
History
Received: Jan 31, 2017
Accepted: Sep 21, 2017
Published online: Jan 27, 2018
Published in print: Apr 1, 2018
Discussion open until: Jun 27, 2018
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