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Kriging with imprecise (fuzzy) variograms. II: Application

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The geostatistical analysis of soil liner permeability is based on 20 measurements and imprecise prior information on nugget effect, sill, and range of the unknown variogram. Using this information, membership functions for variogram parameters are assessed and the fuzzy variogram is constructed. Both kriging estimates and estimation variances are calculated as fuzzy numbers from the fuzzy variogram and data points. Contour maps are presented, indicating values of the kriged permeability and the estimation variance corresponding to selected membership values called levels.

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  1. Bardossy, A., Bogardi, I., and Kelly, W. E., 1990, Kriging with Imprecise (Fuzzy) Variograms. I: Theory.Math. Geol., v. 22, p. 63–79.

  2. Bogardi, I., Kelly, W. E., Bardossy, A., Ghohestani-Bojd, H., and Higgins, B., 1987, Component and Systems Reliability for Composite FML Containment Systems, Phase I Report: CR813433-01, EPA HWERL, Cincinnati, Ohio, p. 94.

  3. Chameau, F. L. and Santamarina, F. C., 1987, Membership Functions I and II:Intern. J. Approx. Reas., v. 2, p. 287–316.

  4. Civanlar, M. R. and Trussel, H. F., 1986, Constructing Membership Functions Using Statistical Data:Fuzzy Sets Sys., v. 18, p. 1–13.

  5. Daniel, D. E., 1984, Predicting Hydraulic Conductivity of Clay Liners:J. Geotech. Eng., v. 110, p. 285–300.

  6. Day, S. R. and Daniel, D. E., 1985, Hydraulic Conductivity of Two Prototype Clay Liners:J. Geotech. Eng., v. 111, p. 957–970.

  7. Delhomme, J. P., 1979, Spatial Variability and Uncertainty in Groundwater Flow Parameters: A Geostatistical Approach:Water Resour. Res., v. 15, p. 269–280.

  8. Dickinson, D. and Ferrell, W. R., 1985, Fuzzy Set Knowledge Representation in a System to Recommend Management Decisions,in Methlie, L. (Ed.),Proceedings of the IFIP Working Conference on Knowledge Representation for Decision Support Systems: North-Holland, Amsterdam, p. 326.

  9. Dubois, D. and Prade, H., 1986, Fuzzy Sets and Statistical Data:Eur. J. Operat. Res., v. 25, p. 345–356.

  10. EPA, 1985a, Design, Construction and Evaluation of Clay Liners for Hazardous Waste Facilities:Draft Tech. Res. Doc., 530-SW-85, p. 320.

  11. EPA, 1985b, Construction Quality Assurance for Hazardous Wasteland Disposal Facilities: (68-02-3992), Washington, D.C., p. 121.

  12. EPA, 1985c, Minimum Technology Guidance on Double Liner Systems for Landfills and Surface Impoundments—Design, Construction and Operation: (SW-85-0L4), SAA, p. 67.

  13. GAO, 1985, EPA's Inventory of Potential Hazardous Waste Sites Is Incomplete: GAO/RCED-85-75, March 26, p. 36.

  14. Freeze, A. R. and Cherry, J. A., 1979,Groundwater: Prentice Hall, Englewood Cliffs, New Jersey, p. 604.

  15. Gaines, B. R., 1976, Foundations of Fuzzy Reasoning:Int. J. Man-Machine Stud., v. 8, p. 623–668.

  16. Gelhar, L. W., 1986, Stochastic Subsurface Hydrology from Theory to Applications:Water Resour. Res., v. 22, p. 1355–1455.

  17. Gesheider, G. A., 1976, Psychophysics:Method and Theory, Lawrence Erlbaum (Ed.). Hillsdale, New Jersey, p. 246.

  18. Hoeksema, R. J. and Kitanidis, P. K., 1985, Analysis of the Spatial Structure of Properties of Selected Aquifers:Water Resour. Res., v. 21, p. 563–572.

  19. Journel, A. and Ch.J. Huijbregts, 1978,Mining Geostatistics: Academic Press, New York, p. 600.

  20. Karwowski, W. and Ayoub, M. M., 1984, Fuzzy Modeling of Stresses in Manual Lifting Tasks:Ergonomics, v. 27, p. 641–649.

  21. Krzysztofowicz, R. and Duckstein, L., 1980, Assessment Errors in Multiattribute Utility Functions:Organizational Behavior and Human Performance, p. 326–348.

  22. de Marsily, 1984, Spatial Variability of Properties in Porous Media: A Stochastic Approach,in Bear, J. (Ed.),Fundamentals of Transport Phenomena in Porous Media: The Hague, p. 349.

  23. Rogowski, A. S., Weinrich, B. E., and Simmons, D. E., 1985, Permeability Assessment in a Compacted Clay Liner:Proc. Eighth Annual Madison Waste Conference, September 18–19, Dept. of Engineering and Applied Science, University of Wisconsin-Extension, Madison, p. 315–336.

  24. Rogowski, A. S. and Simmons, D. E., 1988, Geostatistical Analysis of Field Hydraulic Conductivity in Compacted Clay: Math. Geol., v. 20, p. 423–446.

  25. Soulie, M., Favre, M., and Konrad, J., 1980, Analyse Geostatistique d'un Noyau de Barrage tel que Construit:Can. Geotech. J., v. 20, p. 453–467.

  26. Tversky, A. and Kahneman, D., 1981, The Framing of Decisions and the Psychology of Choice:Science, v. 211, p. 453–458.

  27. Wallstein, T. S. and Buddescu, D. V., 1983, Encoding Subjective Probabilities: A Psychological and Psychometric Review:Management Sci., v. 29, p. 151–173.

  28. Warrick, A. W. and Nielsen, D. R., 1980, Spatial Variability of Soil Physical Properties in the Field,in Hillel, D. (Ed.),Applications of Soil Physics: Academic Press, New York, p. 319–344.

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Bardossy, A., Bogardi, I. & Kelly, W.E. Kriging with imprecise (fuzzy) variograms. II: Application. Math Geol 22, 81–94 (1990). https://doi.org/10.1007/BF00890298

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Key words

  • application
  • permeability prediction
  • soil liner
  • kriging
  • variogram
  • fuzzy set theory