Fuzzy-probabilistic calculations of water-balance uncertainty

Original Paper

Abstract

Hydrogeological systems are often characterized by imprecise, vague, inconsistent, incomplete, or subjective information, which may limit the application of conventional stochastic methods in predicting hydrogeologic conditions and associated uncertainty. Instead, predictions and uncertainty analysis can be made using uncertain input parameters expressed as probability boxes, intervals, and fuzzy numbers. The objective of this paper is to present the theory for, and a case study as an application of, the fuzzy-probabilistic approach, combining probability and possibility theory for simulating soil water balance and assessing associated uncertainty in the components of a simple water-balance equation. The application of this approach is demonstrated using calculations with the RAMAS Risk Calc code, to assess the propagation of uncertainty in calculating potential evapotranspiration, actual evapotranspiration, and infiltration—in a case study at the Hanford site, Washington, USA. Propagation of uncertainty into the results of water-balance calculations was evaluated by changing the types of models of uncertainty incorporated into various input parameters. The results of these fuzzy-probabilistic calculations are compared to the conventional Monte Carlo simulation approach and estimates from field observations at the Hanford site.

Keywords

Water balance Uncertainty Fuzzy-probabilistic approach Fuzzy calculations 

References

  1. Arora VK (2002) The use of the aridity index to assess climate change effect on annual runoff. J Hydrol 265:164–177CrossRefGoogle Scholar
  2. Budyko MI (1974) Climate and life. Academic Press, San Diego, CAGoogle Scholar
  3. Chang N-B (2005) Sustainable water resources management under uncertainty. Stochast Environ Res Risk Assess 19:97–98CrossRefGoogle Scholar
  4. Cooper JA, Ferson S, Ginzburg L (2006) Hybrid processing of stochastic and subjective uncertainty data. Risk Anal 16(6):785–791CrossRefGoogle Scholar
  5. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Stat 28:325–339Google Scholar
  6. DOE (1996) Final environmental impact statement for the tank waste remediation system. Hanford Site, Richland, Washington, DOE/EIS-0189. http://www.globalsecurity.org/wmd/library/report/enviro/eis-0189/app_i_3.htm
  7. Dubois D, Prade H (1981) Additions of interactive fuzzy numbers. IEEE Trans Autom Control 26:926–936CrossRefGoogle Scholar
  8. Dubois D, Prade H (1994) Possibility theory and data fusion in poorly informed environments. Control Eng Pract 2(5):811–823CrossRefGoogle Scholar
  9. Ferson S (2002) RAMAS risk calc 4.0 software: risk assessment with uncertain numbers. CRC Press, Boca RatonGoogle Scholar
  10. Ferson S, Ginzburg L (1995) Hybrid arithmetic. In: Proceedings of the 1995 joint ISUMA/NAFIPS conference. IEEE Computer Society Press, Los Alamitos, California, pp 619–623Google Scholar
  11. Ferson S, Kreinovich V, Ginzburg L, Myers DS, Sentz K (2003) Constructing probability boxes and Dempster-Shafer structures, SAND report, SAND2002-4015Google Scholar
  12. Gee GW, Fayer MJ, Rockhold ML, Campbell MD (1992) Variations in recharge at the Hanford Site. Northwest Sci 66:237–250Google Scholar
  13. Gee GW, Oostrom M, Freshley MD, Rockhold ML, Zachara JM (2007) Hanford site vadose zone studies: an overview. Vadose Zone J 6:899–905CrossRefGoogle Scholar
  14. Guyonne D, Dubois D, Bourgine B, Fargier H, Côme B, Chilès J-P (2003) Hybrid method for addressing uncertainty in risk assessments. J Environ Eng 129:68–78CrossRefGoogle Scholar
  15. Kaufmann A, Gupta MM (1985) Introduction to fuzzy arithmetic. Van Nostrand Reinhold, New YorkGoogle Scholar
  16. Maher K, DePaolo DJ, Conrad MS, Serne RJ (2003) Vadose zone infiltration rate at Hanford, Washington, inferred from Sr isotope measurements. Water Resour Res 39(8):1204–1217CrossRefGoogle Scholar
  17. Meyer PD, Rockhold ML, Gee GW (1997) Uncertainty Analysis of infiltration and subsurface flow and transport for SDMP sites, NUREG/CR-6565, US Nuclear Regulatory CommissionGoogle Scholar
  18. Möller B, Beer M (2005) Fuzzy randomness. Uncertainty in Civil Engineering and Computational Mechanics. Comput Mech 36(1):83CrossRefGoogle Scholar
  19. Neitzel DA (1996) Hanford site national environmental policy act (NEPA) characterization. PNL-6415, Rev. 8. Pacific Northwest National Laboratory, Richland, WashingtonGoogle Scholar
  20. Neuman SP (2003) Maximum likelihood Bayesian averaging of alternative conceptual-mathematical models. Stochast Environ Res Risk Assess 17(5):291–305CrossRefGoogle Scholar
  21. Neuman SP, Wierenga PJ (2003) A comprehensive strategy of hydrogeologic modeling and uncertainty analysis for nuclear facilities and sites, NUREG/CR-6805. US Nuclear Regulatory Commission, Washington, DCGoogle Scholar
  22. Orr S, Meystel AM (2005) Approaches to optimal aquifer management and intelligent control in a multiresolutional decision support system. Hydrogeol J 13:223–246CrossRefGoogle Scholar
  23. Penman HL (1963) Vegetation and hydrology, Technical communication no. 53. Commonwealth Bureau of Soils, Harpenden, England, 125 ppGoogle Scholar
  24. Potter NJ, Zhang L, Milly PCD, McMahon TA, Jakeman AJ (2005) Effects of rainfall seasonality and soil moisture capacity on mean annual water balance for Australian catchments. Water Resour Res 41, W06007Google Scholar
  25. Routson RC, Johnson VG (1990) Recharge estimates for the Hanford site 200 areas plateau. Northwest Sci 64(3):150–158Google Scholar
  26. Sankarasubramanian A, Vogel RM (2003) Hydroclimatology of the continental United States. Geophys Res Lett 30(7):1363CrossRefGoogle Scholar
  27. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton, New Jersey, USAGoogle Scholar
  28. Smets P (1990) The combination of evidence in the transferable belief model. IEEE Pattern Anal Mach Intell 12:447–458CrossRefGoogle Scholar
  29. Wagener T, Gupta HV (2005) Model identification for hydrological forecasting under uncertainty. Stoch Environ Res Risk Assess 19:378–387CrossRefGoogle Scholar
  30. Ward AL, Freeman EJ, White MD, Zhang ZF (2005) STOMP: subsurface transport over multiple phases, version 1.0. Addendum: Sparse vegetation evapotranspiration model for the water-air-energy operational mode, PNNL-15465Google Scholar
  31. Winter CL (2004) Stochastic hydrology: practical alternatives exist. Stochast Environ Res Risk Assess 18:271–273CrossRefGoogle Scholar
  32. Yager R, Kelman A (1996) Fusion of fuzzy information with considerations for compatibility, partial aggregation, and reinforcement. Int J Approx Reason 15(2):93–122CrossRefGoogle Scholar
  33. Ye M, Neuman SP, Meyer PD (2004) Maximum likelihood Bayesian averaging of spatial variability models in unsaturated fractured tuff. Water Resour Res 40(5):W05113CrossRefGoogle Scholar
  34. Ye M, Neuman SP, Meyer PD, Pohlmann K (2005) Sensitivity analysis and assessment of prior model probabilities in MLBMA with application to unsaturated fractured tuff. Water Resour Res 41:W12429CrossRefGoogle Scholar
  35. Zadeh L (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28CrossRefGoogle Scholar
  36. Zadeh LA (1986) A Simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination. AI Mag 7:85–90Google Scholar

Copyright information

© U.S. Government 2010

Authors and Affiliations

  1. 1.Lawrence Berkeley National LaboratoryBerkeleyUSA

Personalised recommendations