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Parameter Estimation and Predictive Uncertainty Quantification in Hydrological Modelling

Living reference work entry

Abstract

The majority of hydrological and environmental models contain parameters that must be specified before the model can be used. Parameter estimation is hence a very common problem in environmental sciences and has received tremendous amount of research and industry attention. This chapter reviews some of the key principles of parameter estimation, with a focus on calibration approaches and uncertainty quantification. The distinct approaches of manual calibration, optimization, multi-objective optimization, and probabilistic approaches are described in terms of key theory and representative applications. Advantages and limitations of these strategies are listed and discussed, with a focus on their ability to represent parametric and predictive uncertainties. The role of posterior diagnostics to check calibration and model assumptions that impact on parameter estimation is emphasized. Auxiliary tricks and techniques are described to simplify the process of parameter estimation in practical applications. The chapter concludes with an outline of directions for ongoing and future research. It is hoped that this chapter will help hydrologists and environmental modellers get to the current state of research and practice in model calibration, parameter estimation, and uncertainty quantification.

Keywords

Hydrological model Parameter estimation Model calibration Optimization Bayesian inference Uncertainty quantification 

References

  1. M.B. Abbott, V.M. Babovic, J.A. Cunge, Reply to comment by Beven et al on “Towards the hydraulics of the hydroinformatics era” by Abbott et al. J. Hydraul. Res. 41(3), 333–336 (2003)Google Scholar
  2. C. Albert, A mechanistic dynamic emulator. Nonlinear Anal. Real World Appl. 13(6), 2747–2754 (2012)CrossRefGoogle Scholar
  3. A.H.-S. Ang, W.H. Tang, Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering (Wiley, Hoboken, 2007)Google Scholar
  4. S.A. Archfield, M. Clark, B. Arheimer, L.E. Hay, H. McMillan, J.E. Kiang, J. Seibert, K. Hakala, A. Bock, T. Wagener, W.H. Farmer, V. Andréassian, S. Attinger, A. Viglione, R. Knight, S. Markstrom, T. Over, Accelerating advances in continental domain hydrologic modeling. Water Resour. Res. 51(12), 10078–10091 (2015)CrossRefGoogle Scholar
  5. J.G. Arnold, N. Fohrer, SWAT2000: Current capabilities and research opportunities in applied watershed modelling. Hydrol. Process. 19(3), 563–572 (2005)CrossRefGoogle Scholar
  6. M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50(2), 174–188 (2002). Special Issue on Monte Carlo Methods for Statistical Signal ProcessingCrossRefGoogle Scholar
  7. M. Asadzadeh, B.A. Tolson, Pareto archived dynamically dimensioned search with hypervolume-based selection for multiobjective optimization. Eng. Optim. 45(12), 1489–1509 (2013)CrossRefGoogle Scholar
  8. K. Beven, TOPMODEL: A critique. Hydrol. Process. 11(9), 1069–1085 (1997)CrossRefGoogle Scholar
  9. K. Beven, On the concept of model structural error. Water Sci. Technol. 52, 167–175 (2005)CrossRefGoogle Scholar
  10. K.J. Beven, A manifesto for the equifinality thesis. J. Hydrol. 320, 18–36 (2006)CrossRefGoogle Scholar
  11. K.J. Beven, A.M. Binley, The future of distributed models: Model calibration and uncertainty prediction. Hydrol. Process. 6, 279–298 (1992)CrossRefGoogle Scholar
  12. K. Beven, I. Westerberg, On red herrings and real herrings: Disinformation and information in hydrological inference. Hydrol. Process. 25, 1676–1680 (2011)CrossRefGoogle Scholar
  13. K. Beven, P. Smith, I. Westerberg, J. Freer, Comment on “Pursuing the method of multiple working hypotheses for hydrological modeling” by P. Clark et al. Water Resour. Res. 48, W11801 (2012)Google Scholar
  14. G.E.P. Box, G.C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1992)CrossRefGoogle Scholar
  15. G.O. Brown, Henry Darcy and the making of a law. Water Resour. Res. 38(7), 1–12 (2002)CrossRefGoogle Scholar
  16. N. Bulygina, H. Gupta, Estimating the uncertain mathematical structure of a water balance model via Bayesian data assimilation. Water Resour. Res. 45, W00B13 (2009)Google Scholar
  17. T.G. Chapman, Optimization of a rainfall-runoff model for an arid zone catchment, in I.A.S.H.-UNESCO Symposium on the Results of Research on Representative and Experimental Basins, (IASH-AISH Publ, Wellington, 1970), pp. 126–144Google Scholar
  18. F.H. Chiew, L. Siriwardena, Estimation of SIMHYD parameter values for application in ungauged catchments, in MODSIM 2005 International Congress on Modelling and Simulation, ed. by A. Zerger, R.M. Argent (Modelling and Simulation Society of Australia and New Zealand, Melbourne, Australia, 2005), pp. 2883–2889Google Scholar
  19. F.H.S. Chiew, M.J. Stewardson, T.A. McMahon, Comparison of six rainfall-runoff modelling approaches. J. Hydrol. 147, 1–36 (1993)CrossRefGoogle Scholar
  20. M.P. Clark, A.G. Slater, D.E. Rupp, R.A. Woods, J.A. Vrugt, H.V. Gupta, T. Wagener, L.E. Hay, Framework for understanding structural errors (FUSE): A modular framework to diagnose differences between hydrological models. Water Resour. Res. 44, W00B02 (2008).  https://doi.org/10.1029/2007WR006735CrossRefGoogle Scholar
  21. M.P. Clark, D. Kavetski, F. Fenicia, Pursuing the method of multiple working hypotheses for hydrological modeling. Water Resour. Res. 47, W09301 (2011)Google Scholar
  22. M.P. Clark, D. Kavetski, F. Fenicia, Reply to comment by K. Beven et al. on “Pursuing the method of multiple working hypotheses for hydrological modeling”. Water Resour. Res. 48, W11802 (2012)CrossRefGoogle Scholar
  23. M.P. Clark, B. Nijssen, J.D. Lundquist, D. Kavetski, D.E. Rupp, R.A. Woods, J.E. Freer, E.D. Gutmann, A.W. Wood, L.D. Brekke, J.R. Arnold, D.J. Gochis, R.M. Rasmussen, A unified approach for process-based hydrologic modeling: 1. Modeling concept. Water Resour. Res. 51(4), 2498–2514 (2015)CrossRefGoogle Scholar
  24. H.L. Cloke, F. Pappenberger, Ensemble flood forecasting: A review. J. Hydrol. 375, 613–626 (2009)CrossRefGoogle Scholar
  25. J. Craig, et al., Raven User’s and Developer’s manual v2.7, http://www.civil.uwaterloo.ca/jrcraig/Raven/. (University of Waterloo, 2017)
  26. B. de Finetti, Foresight: Its logical laws, its subjective sources, in Studies in Subjective Probability, ed. by H.E. Kyburg (Wiley, New York, 1964), pp. 93–158Google Scholar
  27. N. De Vleeschouwer, V.R.N. Pauwels, Assessment of the indirect calibration of a rainfall-runoff model for ungauged catchments in Flanders. Hydrol. Earth Syst. Sci. 17, 2001–2016 (2013)CrossRefGoogle Scholar
  28. J. Demargne, L. Wu, S.K. Regonda, J.D. Brown, H. Lee, M. He, D.J. Seo, R. Hartman, H.D. Herr, M. Fresch, J. Schaake, Y. Zhu, The science of NOAA’s operational hydrologic ensemble forecast service. Bull. Am. Meteorol. Soc. 95(1), 79–98 (2014)CrossRefGoogle Scholar
  29. J. Doherty, Ground water model calibration using pilot points and regularization. Ground Water 41, 170–177 (2003)CrossRefGoogle Scholar
  30. J. Doherty, PEST: Model Independent Parameter Estimation, 5th edn. (Watermark Numerical Computing, Brisbane, 2005)Google Scholar
  31. Q. Duan, S. Sorooshian, V. Gupta, Effective and efficient global optimization for conceptual rainfall-runoff models. Water Resour. Res. 28(4), 1015–1031 (1992)CrossRefGoogle Scholar
  32. Q. Duan, J. Schaake, V. Andreassian, S.W. Franks, G. Goteti, H.V. Gupta, Y.M. Gusev, F. Habets, A. Hall, L. Hay, T. Hogue, M. Huang, G. Leavesley, X. Liang, O.N. Nasonova, J. Noilhan, L. Oudin, S. Sorooshian, T. Wagener, E.F. Wood, Model parameter estimation experiment (MOPEX): An overview of science strategy and major results from the second and third workshops. J. Hydrol. 320(1–2), 3–17 (2006)CrossRefGoogle Scholar
  33. A. Efstratiadis, D. Koutsoyiannis, One decade of multi-objective calibration approaches in hydrological modelling: A review. Hydrol. Sci. J. 55(1), 58–78 (2010)CrossRefGoogle Scholar
  34. G. Evin, D. Kavetski, M. Thyer, G. Kuczera, Pitfalls and improvements in the joint inference of heteroscedasticity and autocorrelation in hydrological model calibration. Water Resour. Res. 49, 4518–4524 (2013)CrossRefGoogle Scholar
  35. G. Evin, M. Thyer, D. Kavetski, D. McInerney, G. Kuczera, Comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity. Water Resour. Res. 50, 2350–2375 (2014)CrossRefGoogle Scholar
  36. F. Fenicia, H.H.G. Savenije, P. Matgen, L. Pfister, Understanding catchment behavior through stepwise model concept improvement. Water Resour. Res. 44, W01402 (2008)Google Scholar
  37. F. Fenicia, S. Wrede, D. Kavetski, L. Pfister, L. Hoffmann, H. Savenije, J.J. McDonnell, Impact of mixing assumptions on mean residence time estimation. Hydrol. Process. 24(12), 1730–1741 (2010). (Special Issue on Residence Times and Preferential Flows)CrossRefGoogle Scholar
  38. F. Fenicia, D. Kavetski, H.H.G. Savenije, Elements of a flexible approach for conceptual hydrological modeling: Part 1. Motivation and theoretical development. Water Resour. Res. 47, W11510 (2011)CrossRefGoogle Scholar
  39. F. Fenicia, D. Kavetski, H.H.G. Savenije, P. L, From spatially variable streamflow to distributed hydrological models: Analysis of key modeling decisions. Water Resour. Res. 52, 954–989 (2016)CrossRefGoogle Scholar
  40. F. Fenicia, D. Kavetski, P. Reichert, C. Albert, Signature-domain calibration of hydrological models using approximate Bayesian computation: Empirical analysis of fundamental properties. Water Resour. Res. in press,  https://doi.org/10.1002/2017WR021616 (2018)
  41. C.W. Fetter, Applied Hydrogeology, 3rd edn. (Prentice-Hall, Upper Saddle River, 1994)Google Scholar
  42. J. Freer, K. Beven, B. Ambroise, Bayesian estimation of uncertainty in runoff prediction and the value of data: An application of the GLUE approach. Water Resour. Res. 32(7), 2161–2173 (1996)CrossRefGoogle Scholar
  43. J.E. Freer, H. McMillan, J.J. McDonnell, K.J. Beven, Constraining dynamic TOPMODEL responses for imprecise water table information using fuzzy rule based performance measures, J. Hydrol. 291(3–4), 254–277 (2004)CrossRefGoogle Scholar
  44. R.A. Freeze, R.L. Harlan, Blueprint for a physically-based, digitally-simulated hydrologic response model. J. Hydrol. 9, 237–258 (1969)CrossRefGoogle Scholar
  45. A. Gelb (ed.), Applied Optimal Estimation (MIT Press, Cambridge, MA, 1974)Google Scholar
  46. A. Gelman, J.B. Carlin, H.S. Stern, D.B. Rubin, Bayesian Data Analysis (Chapman and Hall, London, 1998)Google Scholar
  47. P.E. Gill, W. Murray, M.H. Wright, Practical Optimization (Academic, London, 1981)Google Scholar
  48. L. Giustarini, R. Hostache, D. Kavetski, M. Chini, G. Corato, S. Schlaffer, P. Matgen, Probabilistic flood mapping using synthetic aperture radar data. IEEE Trans. Geosci. Remote Sens. 54(12), 6958–6969 (2016)CrossRefGoogle Scholar
  49. R.S. Govindaraju, Artificial neural networks in hydrology. I: Preliminary concepts. J. Hydrol. Eng. 5(2), 115–123 (2000)CrossRefGoogle Scholar
  50. R.B. Grayson, I.D. Moore, T.A. McMahon, Physically based hydrologic modeling: 2. Is the concept realistic? Water Resour. Res. 28(10), 2659–2666 (1992)CrossRefGoogle Scholar
  51. V.K. Gupta, S. Sorooshian, The automatic calibration of conceptual catchment models using derivative-based optimization algorithms. Water Resour. Res. 21(4), 473–485 (1985)CrossRefGoogle Scholar
  52. H.V. Gupta, S. Sorooshian, P.O. Yapo, Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resour. Res. 34(4), 751–763 (1998)CrossRefGoogle Scholar
  53. H.V. Gupta, T. Wagener, Y. Liu, Reconciling theory with observations: Elements of a diagnostic approach to model evaluation. Hydrol. Process. 22, 3802–3813 (2008)CrossRefGoogle Scholar
  54. T.T. Hailegeorgis, K. Alfredsen, Regional flood frequency analysis and prediction in ungauged basins including estimation of major uncertainties for mid-Norway. J. Hydrol. 9, 104–126 (2017)Google Scholar
  55. A.W. Harbaugh, MODFLOW-2005, the U.S. Geological Survey modular ground-water model – the Ground-Water Flow Process, U.S. Geological Survey Techniques and Methods 6-A16 (2005)Google Scholar
  56. M.C. Hill, D. Kavetski, M.P. Clark, M. Ye, M. Arabi, D. Lu, L. Foglia, S. Mehl, Practical use of computationally frugal model analysis methods. Groundwater 54(2), 159 (2015)CrossRefGoogle Scholar
  57. R. Hostache, X. Lai, J. Monnier, C. Puech, Assimilation of spatially distributed water levels into a shallow-water model. Part II: Use of a remote sensing image of Mosel River. J. Hydrol. 390(3–4), 257–268 (2010)CrossRefGoogle Scholar
  58. M. Hrachowitz, H.H.G. Savenije, G. Blöschl, J.J. McDonnell, M. Sivapalan, J.W. Pomeroy, B. Arheimer, T. Blume, M.P. Clark, U. Ehret, F. Fenicia, J.E. Freer, A. Gelfan, H.V. Gupta, D.A. Hughes, R.W. Hut, A. Montanari, S. Pande, D. Tetzlaff, P.A. Troch, S. Uhlenbrook, T. Wagener, H.C. Winsemius, R.A. Woods, E. Zehe, C. Cudennec, A decade of predictions in ungauged basins (PUB) – A review. Hydrol. Sci. J. 58(6), 198–1255 (2013)CrossRefGoogle Scholar
  59. M. Hrachowitz, O. Fovet, L. Ruiz, T. Euser, S. Gharari, R. Nijzink, J. Freer, H.H.G. Savenije, C. Gascuel-Odoux, Process consistency in models: The importance of system signatures, expert knowledge, and process complexity. Water Resour. Res. 50(9), 7445–7469 (2014)CrossRefGoogle Scholar
  60. D. Huard, A. Mailhot, Calibration of hydrological model GR2M using Bayesian uncertainty analysis. Water Resour. Res. 44, W02424 (2008)CrossRefGoogle Scholar
  61. R.P. Ibbitt, T. O’Donnell, Designing conceptual catchment models for automatic fitting methods, in Mathematical Models in Hydrology Symposium, IAHS-AISH Publication No. 101(2) (1971), pp. 461–475Google Scholar
  62. V.Y. Ivanov, E.R. Vivoni, R.L. Bras, D. Entekhabi, Catchment hydrologic response with a fully distributed triangulated irregular network model. Water Resour. Res. 40(11), W11102 (2004).  https://doi.org/10.1029/2004WR003218CrossRefGoogle Scholar
  63. A.J. Jakeman, G.M. Hornberger, How much complexity is warranted in a rainfall-runoff model? Water Resour. Res. 29(8), 2637–2649 (1993)CrossRefGoogle Scholar
  64. R.E. Kalman, A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)CrossRefGoogle Scholar
  65. D. Kavetski, Analysis of input data uncertainty and numerical robustness in conceptual rainfall-runoff modelling, PhD Thesis, Faculty of Engineering and Built Environment, University of Newcastle (2005)Google Scholar
  66. D. Kavetski, M.P. Clark, Ancient numerical daemons of conceptual hydrological modeling. Part 2: Impact of time stepping schemes on model analysis and prediction. Water Resour. Res. 46, W10511 (2010).  https://doi.org/10.1029/2009WR008896CrossRefGoogle Scholar
  67. D. Kavetski, G. Kuczera, Model smoothing strategies to remove microscale discontinuities and spurious secondary optima in objective functions in hydrological calibration. Water Resour. Res. 43, W03411 (2007).  https://doi.org/10.1029/2006WR005195CrossRefGoogle Scholar
  68. D. Kavetski, S. Franks, G. Kuczera, Confronting input uncertainty in environmental modelling, in Calibration of Watershed Models. Water Science and Application Series 6, ed. by Q.Y. Duan, H.V. Gupta, S. Sorooshian, A. Rousseau, R. Tourcotte. (American Geophysical Union, Washington, DC, 2002), pp. 49–68Google Scholar
  69. D. Kavetski, G. Kuczera, S.W. Franks, Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory. Water Resour. Res. 42(3), W03407 (2006a)Google Scholar
  70. D. Kavetski, G. Kuczera, S.W. Franks, Calibration of conceptual hydrological models revisited: 1. Overcoming numerical artefacts. J. Hydrol. 320(1–2), 173–186 (2006b)CrossRefGoogle Scholar
  71. D. Kavetski, G. Kuczera, S.W. Franks, Calibration of conceptual hydrological models revisited: 2. Improving optimisation and analysis. J. Hydrol. 320(1–2), 187–201 (2006c)CrossRefGoogle Scholar
  72. D. Kavetski, G. Kuczera, M. Thyer, B. Renard, Multistart Newton-type optimisation methods for the calibration of conceptual hydrological models, In Proceedings of Oxley, L. and Kulasiri, D. (eds) MODSIM 2007 International Congress on Modelling and Simulation, Christchurch, New Zealand. (Modelling and Simulation Society of Australia and New Zealand, 2007)Google Scholar
  73. D. Kavetski, F. Fenicia, P. Reichert, C. Albert, Signature-domain calibration of hydrological models using approximate Bayes computation: Theory and comparison to existing applications. Water Resour. Res. in press,  https://doi.org/10.1002/2017WR020528 (2018)
  74. G.B. Kingston, H.R. Maier, M.F. Lambert, Bayesian model selection applied to artificial neural networks used for water resources modeling. Water Resour. Res. 44, W04419 (2008)CrossRefGoogle Scholar
  75. J.W. Kirchner, Getting the right answers for the right reasons: Linking measurements, analyses, and models to advance the science of hydrology. Water Resour. Res. 42(3), W03S04 (2006).  https://doi.org/10.1029/2005WR004362CrossRefGoogle Scholar
  76. L.F. Konikow, J.D. Bredehoeft, Ground-water models cannot be validated. Adv. Water Resour. 15, 75–83 (1992)CrossRefGoogle Scholar
  77. V. Koren, M. Smith, Q. Duan, Use of a priori parameter estimates in the derivation of spatially consistent parameter sets of rainfall-runoff models, in Calibration of Watershed Models, ed. by Q. Duan, H.V. Gupta, S. Sorooshian, A.N. Rousseau, R. Turcotte (AGU Press, Washington, DC, 2003)Google Scholar
  78. R. Krzysztofowicz, Bayesian theory of probabilistic forecasting via a deterministic hydrologic model. Water Resour. Res. 35(9), 2739–2750 (1999)CrossRefGoogle Scholar
  79. G. Kuczera, S. Franks, Testing hydrologic models: Fortification or falsification? in Mathematical Modelling of Large Watershed Hydrology, ed. by V.P. Singh, D.K. Frevert (Water Resources Publications, Littleton, 2002)Google Scholar
  80. G. Kuczera, D. Kavetski, S.W. Franks, M. Thyer, Towards a Bayesian total error analysis of conceptual rainfall-runoff models: Characterising model error using storm-dependent parameters. J. Hydrol. 331(1–2), 161–177 (2006)CrossRefGoogle Scholar
  81. E. Laloy, B. Rogiers, J.A. Vrugt, D. Mallants, D. Jacques, Efficient posterior exploration of a high-dimensional groundwater model from two-stage Markov chain Monte Carlo simulation and polynomial chaos expansion. Water Resour. Res. 49(5), 2664–2682 (2013)CrossRefGoogle Scholar
  82. J. Le Coz, B. Renard, L. Bonnifait, F. Branger, R. Le Boursicaud, Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach. J. Hydrol. 509, 573–587 (2014)CrossRefGoogle Scholar
  83. D.R. Legates, G.J. McCabe Jr., Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour. Res. 35(1), 233–241 (1999)CrossRefGoogle Scholar
  84. J. Lerat, C. Pickett-Heaps, D. Shin, S. Zhou, P. Feikema, U. Khan, R. Laugesen, N. Tuteja, G. Kuczera, M. Thyer, D. Kavetski, Dynamic streamflow forecasts within an uncertainty framework for 100 catchments in Australia, in Hydrology and Water Resources Symposium: The Art and Science of Water, (Engineers Australia, Barton, ACT, Australia, 2015), pp. 1396–1403Google Scholar
  85. G. Lindstrom, B. Johansson, M. Persson, M. Gardelin, S. Bergstrom, Development and test of the distributed HBV-96 hydrological model. J. Hydrol. 201, 272–288 (1997)CrossRefGoogle Scholar
  86. D.P. Loucks, J.R. Stedinger, D.A. Haith, Water Resource Systems Planning and Analysis (Prentice-Hall, Englewood Cliffs, 1981)Google Scholar
  87. D.R. Maidment, Handbook of Hydrology (McGraw-Hill, New York, 1993)Google Scholar
  88. P. Mantovan, E. Todini, Hydrological forecasting uncertainty assessment: Incoherence of the GLUE methodology. J. Hydrol. 330(1–2), 368–381 (2006)CrossRefGoogle Scholar
  89. P. Mantovan, E. Todini, M.L.V. Martina, Reply to comment by Keith Beven, Paul Smith and Jim Freer on “Hydrological forecasting uncertainty assessment: Inconherence of the GLUE methodology”. J. Hydrol. 338, 319–324 (2007)CrossRefGoogle Scholar
  90. A. Marchi, E. Salomons, A. Simpson, A. Zecchin, H. Maier, Z. Wu, C. Stokes, W. Wu, G.C. Dandy, The battle of the water networks II (BWN-II). J. Water Resour. Plann. Manage. 140, 04014009:04014001–04014009:04014014 (2014)Google Scholar
  91. E.S. Martins, J.R. Stedinger, Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res. 36(3), 737–744 (2000)CrossRefGoogle Scholar
  92. D. McInerney, M. Thyer, D. Kavetski, J. Lerat, G. Kuczera, Improving probabilistic prediction of daily streamflow by identifying Pareto optimal approaches for modeling heteroscedastic residual errors. Water Resour. Res. 53, 2199–2239 (2017)CrossRefGoogle Scholar
  93. H. McMillan, B. Jackson, M. Clark, D. Kavetski, R. Woods, Rainfall uncertainty in hydrologic modelling: An evaluation of multiplicative error models. J. Hydrol. 400, 83–94 (2011)CrossRefGoogle Scholar
  94. M. Merriman, On the history of the method of least squares. Analyst 4(2), 33–36 (1877)CrossRefGoogle Scholar
  95. D.A. Miller, R.A. White, A conterminous United States multi-layer soil characteristics data set for regional climate and hydrology modeling. Earth Interact. 2, 2 (1999)CrossRefGoogle Scholar
  96. A. Montanari, E. Toth, Calibration of hydrological models in the spectral domain: An opportunity for scarcely gauged basins? Water Resour. Res. 43, W05434 (2007)CrossRefGoogle Scholar
  97. M. Morawietz, C.-Y. Xu, L. Gottschalk, L.M. Tallaksen, Systematic evaluation of autoregressive error models as post-processors for a probabilistic streamflow forecast system. J. Hydrol. 407(1–4), 58–72 (2011)CrossRefGoogle Scholar
  98. J.E. Nash, J.V. Sutcliffe, River flow forecasting through conceptual models. Part 1 – A discussion of principles. J. Hydrol. 10, 282–290 (1970)CrossRefGoogle Scholar
  99. J.C. Neal, P.M. Atkinson, H.C. W, Flood inundation model updating using an ensemble Kalman filter and spatially distributed measurements. J. Hydrol. 336, 401–415 (2007)CrossRefGoogle Scholar
  100. J. Neal, G. Schumann, P. Bates, A subgrid channel model for simulating river hydraulics and floodplain inundation over large and data sparse areas. Water Resour. Res. 48, W11506 (2012)CrossRefGoogle Scholar
  101. D.J. Nott, L. Marshall, J. Brown, Generalized likelihood uncertainty estimation (GLUE) and approximate Bayesian computation: What's the connection? Water Resour. Res. 48, W12602 (2012)CrossRefGoogle Scholar
  102. W.L. Oberkampf, J.C. Helton, C.A. Joslyn, S.F. Wojtkiewicz, S. Ferson, Challenge problems: Uncertainty in system response given uncertain parameters. Reliab. Eng. Syst. Saf. 85(1–3), 11–19 (2004)CrossRefGoogle Scholar
  103. A. O’Hagan, J. Oakley, Probability is perfect, but we can’t elicit it perfectly. Reliab. Eng. Syst. Saf. 85(1–3), 239–248 (2004)CrossRefGoogle Scholar
  104. F. Pappenberger, K.J. Beven, Ignorance is bliss: Or seven reasons not to use uncertainty analysis. Water Resour. Res. 42, W05302 (2006).  https://doi.org/10.1029/2005WR004820CrossRefGoogle Scholar
  105. C. Perrin, C. Michel, V. Andreassian, Does a large number of parameters enhance model performance? Comparative assessment of common catchment model structures on 429 catchments. J. Hydrol. 242(3–4), 275–301 (2001)CrossRefGoogle Scholar
  106. C. Perrin, C. Michel, V. Andreassian, Improvement of a parsimonious model for streamflow simulation. J. Hydrol. 279(1–4), 275–289 (2003)CrossRefGoogle Scholar
  107. F. Pianosi, L. Raso, Dynamic modeling of predictive uncertainty by regression on absolute errors. Water Resour. Res. 48, W03516 (2012)CrossRefGoogle Scholar
  108. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in Fortran-77: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992)Google Scholar
  109. R. Pushpalatha, C. Perrin, N.L. Moine, V. Andréassian, A review of efficiency criteria suitable for evaluating low-flow simulations. J. Hydrol. 420, 171–182 (2012)CrossRefGoogle Scholar
  110. Y. Qin, D. Kavetski, G. Kuczera, A robust Gauss-Newton algorithm for the optimization of hydrological models: 2. Benchmarking against industry-standard algorithms. Water Resour. Res. in review,  https://doi.org/10.1029/2017WR022489 (2018)
  111. P. Reichert, J. Mieleitner, Analyzing input and structural uncertainty of nonlinear dynamic models with stochastic, time-dependent parameters. Water Resour. Res. 45, W10402 (2009)CrossRefGoogle Scholar
  112. P. Reichert, N. Schuwirth, Linking statistical bias description to multiobjective model calibration. Water Resour. Res. 48, W09543 (2012)CrossRefGoogle Scholar
  113. P. Reichert, S.D. Langhans, J. Lienert, N. Schuwirth, The conceptual foundation of environmental decision support. J. Environ. Manag. 154, 316–332 (2015)CrossRefGoogle Scholar
  114. R.H. Reichle, Data assimilation methods in the Earth sciences. Adv. Water Resour. 31(11), 1411–1418 (2008)CrossRefGoogle Scholar
  115. B. Renard, E. Leblois, G. Kuczera, D. Kavetski, M. Thyer, S. Franks, Characterizing errors in areal rainfall estimates: Application to uncertainty quantification and decomposition in hydrologic modelling. H2009: 32nd Hydrology and Water Resources Symposium, Newcastle (Engineers Australia, Barton ACT, 2009), pp. 505–516Google Scholar
  116. B. Renard, D. Kavetski, M. Thyer, G. Kuczera, S.W. Franks, Understanding predictive uncertainty in hydrologic modeling: Le challenge of identifying input and structural errors. Water Resour. Res. 46, W05521 (2010).  https://doi.org/10.1029/2009WR008328CrossRefGoogle Scholar
  117. B. Renard, D. Kavetski, E.T. Leblois, M. Thyer, G. Kuczera, S.W. Franks, Toward a reliable decomposition of predictive uncertainty in hydrological modeling: Characterizing rainfall errors using conditional simulation. Water Resour. Res. 47(11), W11516 (2011)Google Scholar
  118. B. Revilla-Romero, N. Wanders, P. Burek, P. Salamon, A. de Roo, Integrating remotely sensed surface water extent into continental scale hydrology. J. Hydrol. 543(Pt B), 659–670 (2016)CrossRefGoogle Scholar
  119. J.D. Salas, Analysis and modeling of hydrologic time series, in Handbook of Hydrology, ed. by D.R. Maidment (McGraw-Hill, New York, 1993), pp. 19.11–19.72Google Scholar
  120. L. Samaniego, R. Kumar, S. Attinger, Multiscale parameter regionalization of a grid-based hydrologic model at the mesoscale. Water Resour. Res. 46(5), W05523 (2010)CrossRefGoogle Scholar
  121. H.H.G. Savenije, The art of hydrology. Hydrol. Earth Syst. Sci. 13, 157–161 (2009)CrossRefGoogle Scholar
  122. B. Schaefli, H.V. Gupta, Do Nash values have value? Hydrol. Process. 21(15), 2075–2080 (2007)CrossRefGoogle Scholar
  123. B. Schaefli, D. Kavetski, Bayesian spectral likelihood for hydrological parameter inference. Water Resour. Res. 53, 6857–6884 (2017)CrossRefGoogle Scholar
  124. B. Schaefli, D.B. Talamba, A. Musy, Quantifying hydrological modeling errors through a mixture of normal distributions. J. Hydrol. 332, 303–315 (2007)CrossRefGoogle Scholar
  125. G. Schoups, J.A. Vrugt, A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic and non-Gaussian errors. Water Resour. Res. 46, W10531 (2010)Google Scholar
  126. D.-J. Seo, H.D. Herr, J.C. Schaake, A statistical post-processor for accounting of hydrologic uncertainty in short-range ensemble streamflow prediction. Hydrol. Earth Syst. Sci. 3, 1987–2035 (2006)CrossRefGoogle Scholar
  127. M. Shafii, B.A. Tolson, Optimizing hydrological consistency by incorporating hydrological signatures into model calibration objectives. Water Resour. Res. 51(5), 3796–3814 (2015)CrossRefGoogle Scholar
  128. V.P. Singh, D.A. Woolhiser, Mathematical modeling of watershed hydrology. J. Hydrol. Eng. 7(4), 270–292 (2002)CrossRefGoogle Scholar
  129. M. Sivapalan, G. Bloschl, L. Zhang, R. Vertessy, Downward approach to hydrological prediction. Hydrol. Process. 17(11), 2101–2111 (2003a)CrossRefGoogle Scholar
  130. M. Sivapalan, K. Takeuchi, S.W. Franks, V.K. Gupta, H. Karambiri, V. Lakshmi, X. Liang, J.J. McDonnell, E.M. Mendiondo, P.E. O’Connell, T. Oki, J.W. Pomeroy, D. Schertzer, S. Uhlenbrook, E. Zehe, IAHS decade on predictions in ungauged basins (PUB). Hydrol. Sci. J. 48(6), 857–880 (2003b)CrossRefGoogle Scholar
  131. B.E. Skahill, J. Doherty, Efficient accommodation of local minima in watershed model calibration. J. Hydrol. 329, 122 (2006). in pressCrossRefGoogle Scholar
  132. P. Smith, K.J. Beven, J.A. Tawn, Informal likelihood measures in model assessment: Theoretic development and investigation. Adv. Water Resour. 31(8), 1087–1100 (2008)CrossRefGoogle Scholar
  133. T. Smith, A. Sharma, L. Marshall, R. Mehrotra, S. Sisson, Development of a formal likelihood function for improved Bayesian inference of ephemeral catchments. Water Resour. Res. 46(12), W12551 (2010).  https://doi.org/10.1029/2010WR009514CrossRefGoogle Scholar
  134. S. Sorooshian, J.A. Dracup, Stochastic parameter estimation procedures for hydrologic rainfall-runoff models: Correlated and heteroscedastic error cases. Water Resour. Res. 16(2), 430–442 (1980)CrossRefGoogle Scholar
  135. J.R. Stedinger, R.M. Vogel, S.U. Lee, R. Batchelder, Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resour. Res. 44, W00B06 (2008)CrossRefGoogle Scholar
  136. V.L. Streeter, E.B. Wylie, Fluid Mechanics, First SI Metric Edition. (McGraw-Hill, Singapore, 1983)Google Scholar
  137. L.M. Tallaksen, A review of baseflow recession analysis. J. Hydrol. 165, 349–370 (1995)CrossRefGoogle Scholar
  138. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, Philadelphia, 2005)CrossRefGoogle Scholar
  139. M. Thyer, G. Kuczera, Q.J. Wang, Quantifying parameter uncertainty in stochastic models using the Box-Cox transformation. J. Hydrol. 265(1–4), 246–257 (2002)CrossRefGoogle Scholar
  140. M. Thyer, B. Renard, D. Kavetski, G. Kuczera, S. Franks, S. Srikanthan, Critical evaluation of parameter consistency and predictive uncertainty in hydrological modelling: A case study using Bayesian total error analysis. Water Resour. Res. 45, W00B14 (2009)Google Scholar
  141. B.A. Tolson, C.A. Shoemaker, Dynamically dimensioned search algorithm for computationally efficient watershed model calibration. Water Resour. Res. 43, W01413 (2007)CrossRefGoogle Scholar
  142. A.F.B. Tompson, R. Ababou, L.W. Gelhar, Implementation of the 3-dimensional turning bands random field generator. Water Resour. Res. 25(10), 2227–2243 (1989)CrossRefGoogle Scholar
  143. T. Toni, D. Welch, N. Strelkowa, A. Ipsen, M.P.H. Stumpf, Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009)CrossRefGoogle Scholar
  144. N.K. Tuteja, D. Shin, R. Laugesen, U. Khan, Q. Shao, E. Wang, M. Li, H. Zheng, G. Kuczera, D. Kavetski, G. Evin, M. Thyer, A. MacDonald, T. Chia, B. Le, Experimental Evaluation of the Dynamic Seasonal Streamflow Forecasting Approach (Australian Bureau of Meteorology, Melbourne, 2011)Google Scholar
  145. N.K. Tuteja, S. Zhou, J. Lerat, Q.J. Wang, D. Shin, D.E. Robertson, Overview of communication strategies for uncertainty in hydrological forecasting in Australia, in Handbook of Hydrometeorological Ensemble Forecasting, ed. by Q. Duan, F. Pappenberger, J. Thielen, A. Wood, H.L. Cloke, J.C. Schaake (Springer, Berlin/Heidelberg, 2017), pp. 1–19Google Scholar
  146. R.M. Vogel, Stochastic watershed models for hydrologic risk management. Water Secur. 1, 28–35 (2017)CrossRefGoogle Scholar
  147. J.A. Vrugt, B.A. Robinson, Improved evolutionary optimization from genetically adaptive multimethod search. Proc. Natl. Acad. Sci. U. S. A. 104(3), 708–711 (2007)CrossRefGoogle Scholar
  148. J.A. Vrugt, M. Sadegh, Toward diagnostic model calibration and evaluation: Approximate Bayesian computation. Water Resour. Res. 49(7), 4335–4345 (2013)CrossRefGoogle Scholar
  149. J.A. Vrugt, H.V. Gupta, L.A. Bastidas, W. Bouten, S. Sorooshian, Effective and efficient algorithm for multiobjective optimization of hydrologic models. Water Resour. Res. 39(8), 1214 (2003)Google Scholar
  150. J.A. Vrugt, C.G.H. Diks, H.V. Gupta, W. Bouten, J.M. Verstraten, Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation. Water Resour. Res. 41(1), W01017 (2005)Google Scholar
  151. J.A. Vrugt, C.J.F. ter Braak, M.P. Clark, J.M. Hyman, B.A. Robinson, Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation. Water Resour. Res. 44, W00B09 (2008)Google Scholar
  152. J.A. Vrugt, C.J.F. ter Braak, C.G.H. Diks, B.A. Robinson, J.M. Hyman, D. Higdon, Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int. J. Nonlinear Sci. Numer. Simul. 10(3), 273–290 (2009a)CrossRefGoogle Scholar
  153. J.A. Vrugt, C.J.F. ter Braak, H.V. Gupta, B.A. Robinson, Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stoch. Env. Res. Risk A. 23(7), 1011–1026 (2009b)CrossRefGoogle Scholar
  154. J.A. Vrugt, C.J.F. ter Braak, C.G.H. Diks, G. Schoups, Hydrologic data assimilation using particle Markov chain Monte Carlo simulation: Theory, concepts and applications. Adv. Water Resour. 51, 457–478 (2013)CrossRefGoogle Scholar
  155. Q.J. Wang, D.E. Robertson, Multisite probabilistic forecasting of seasonal flows for streams with zero value occurrences. Water Resour. Res. 47, W02546 (2011)Google Scholar
  156. Q.J. Wang, D.E. Robertson, F.H.S. Chiew, A Bayesian joint probability modeling approach for seasonal forecasting of streamflows at multiple sites. Water Resour. Res. 45(5), W05407 (2009)CrossRefGoogle Scholar
  157. A.H. Weerts, G.Y.H. El Serafy, Particle filtering and ensemble Kalman filtering for state updating with hydrological conceptual rainfall-runoff models. Water Resour. Res. 42, W09403 (2006)CrossRefGoogle Scholar
  158. W.D. Welsh, J. Vaze, D. Dutta, D. Rassam, J.M. Rahman, I.D. Jolly, P. Wallbrink, G.M. Podger, M. Bethune, M.J. Hardy, J. Teng, J. Lerat, An integrated modelling framework for regulated river systems. Environ. Model Softw. 39, 81–102 (2013)CrossRefGoogle Scholar
  159. I. Westerberg, J.-L. Guerrero, J. Seibert, K.J. Beven, S. Halldin, Stage-discharge uncertainty derived with a non-stationary rating curve in the Choluteca River, Honduras. Hydrol. Process. 25(4), 603–613 (2010)CrossRefGoogle Scholar
  160. I.K. Westerberg, H.K. McMillan, Uncertainty in hydrological signatures. Hydrol. Earth Syst. Sci. 19(9), 3951–3968 (2015)CrossRefGoogle Scholar
  161. S. Westra, M. Thyer, M. Leonard, D. Kavetski, M. Lambert, A strategy for diagnosing and interpreting hydrological model nonstationarity, Water Resources Research, 50(6), 5090–5113 (2014)CrossRefGoogle Scholar
  162. D.P. Wright, M. Thyer, S. Westra, Influential point detection diagnostics in the context of hydrological model calibration. J. Hydrol. 527, 1161–1172 (2015)CrossRefGoogle Scholar
  163. K.K. Yilmaz, H.V. Gupta, T. Wagener, A process-based diagnostic approach to model evaluation: Application to the NWS distributed hydrologic model. Water Resour. Res. 44, W09417 (2008)CrossRefGoogle Scholar
  164. P. Young, Data-based mechanistic modelling of environmental, ecological, economic and engineering systems. Environ. Model Softw. 13(2), 105–122 (1998)CrossRefGoogle Scholar
  165. P.C. Young, M. Ratto, A unified approach to environmental systems modeling. Stoch. Env. Res. Risk A. 23(7), 1037–1057 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil, Environmental and Mining EngineeringUniversity of AdelaideAdelaideAustralia
  2. 2.School of EngineeringUniversity of NewcastleCallaghanAustralia
  3. 3.Department of Systems Analysis, Integrated Assessment and Modelling (SIAM), EawagSwiss Federal Institute of Aquatic Science and TechnologyDübendorfSwitzerland

Section editors and affiliations

  • Dmitri Kavetski
    • 1
  • Kuolin Hsu
    • 2
  • Yuqiong Liu
    • 3
  1. 1.School of Civil, Environmental and Mining Engineering, University of AdelaideAdelaideAustralia
  2. 2.Civil & Environmental Engineering, The Henry Samueli School of Engineering, University of CaliforniaIrvineUSA
  3. 3.NASA Goddard Space Flight CenterWashington D.C.USA

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