This paper discusses and illustrates the use of kriging techniques for estimating the spatial pattern of contaminants in environmental media, particularly soil. The assumptions underlying kriging are reviewed as are some advantages and disadvantages of the method. Lognormal kriging (kriging applied to logarithmic-transformed data) is illustrated using a set of radionuclide soil concentrations at a nuclear testing area on the Nevada Test Site. This example shows how lognormal kriging can be used to estimate average concentrations at points or for blocks of land, concentration contours over space, confidence bands about these contours, and radionuclide inventory in soil. The validity of kriging estimates depends on the accurate estimation and modeling of the spatial correlation structure of the phenomenon. Accuracy is especially important when lognormal kriging is used and estimates of means and their standard deviations are required in the original, untransformed scale. This paper illustrates the bias that can result when a changing correlation structure over space is ignored.
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Agterberg, F. P.: 1974, Geomathematics, Elsevier Scientific Publishing Company, New York.
Alldredge, J. R. and Alldredge, N. G.: 1978, ‘Geostatistics: A Bibliography’, International Statistical Review 46, 77–88.
Armstrong, M.: 1984, ‘Problems with Universal Kriging’, Mathematical Geology 16, 101–108.
Barnes, M. G.: 1978, Statistical Design and Analysis in the Cleanup of Environmental Radionuclide Contamination, Desert Research Institute, University of Nevada System. NVO-1253-12.
Barnes, M. G.: 1980, ‘The Use of Kriging for Estimating the Spatial Distribution of Radionuclides and other Spatial Phenomena’, TRAN-STAT: Statistics for Environmental Studies, No. 13. Pacific Northwest Laboratory, Richland, WA. PNL-SA-9051.
Bell, G. D. and Reeves, M.: 1979, ‘Kriging and Geostatistics: A Review of the Literature Available in English’, Proc. Austratas. Inst. Min. Metall., No. 269, pp. 17–27.
Brodlie, K. W.: 1980, Mathematical Methods in Computer Graphics and Design, Academic Press, New York.
Chiles, J. P. and Chauvet, P.: 1974, ‘Kriging: A Method for Cartography of the Sea Floor’, International Hydrographic Review 52, 25–41.
Clark, I.: 1979, Practical Geostatistics, Applied Science Publishers, London. Well written, introductory discussion.
Cressie, N. and Hawkins, D. M.: 1980, ‘Robust Estimation of the Variogram: I’, Mathematical Geology 12, 115–125.
Dalenius, T., Hajek, J., and Zubrzycki, S.: 1961, ‘On Plane Sampling and Related Geometrical Problems’, Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 125–150.
David, M.: 1977, Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing Company, New York. Well written, intermediate discussion.
Davis, J. C.: 1973, Statistics and Data Analysis in Geology, John Wiley and Sons, New York.
Delfiner, P. and Delhomme, J. P.: 1975, ‘Optimal Interpolation by Kriging’, in J. C.Davis and M. J.McCullagh (eds.), Display and Analysis of Spatial Data, John Wiley and Sons, New York, pp. 96–114.
Delfiner, P. and Gilbert, R. O.: 1978, ‘Combining Two Types of Survey Data for Estimating Geographical Distribution of Plutonium in Area 13’, in M. G.White and P. B.Dunaway (eds.), Selected Environmental Plutonium Research Reports of the NAEG, U.S. Department of Energy, Nevada Operations Office, Las Vegas, NV. NVO-192. pp. 361–404.
Delfiner, P., Delhomme, J. P., Chiles, J. P., Renard, D., and Irigoin, F.: 1980, BLUEPACK-3D, Ecole Nationale Superieure des mines de Paris, Centre de geostatistique et de morphologie mathematique, Fontainebleau, France.
Delhomme, J. P.: 1978, ‘Kriging in the Hydrosciences’, Advances in Water Resources 1, 251–266.
Delhomme, J. P.: 1979, ‘Spatial Variability and Uncertainty in Groundwater Flow Parameters: A Geostatistical Approach’, Water Resources Research 15, 269–280.
Doctor, P. G.: 1979, An Evaluation of Kriging Techniques for High Level Radioactive Waste Repository Site Characterization, Pacific Northwest Laboratory, Richland, WA. PNL-2903.
Doctor, P. G. and Gilbert, R. O.: 1978, ‘Two Studies in Variability for Soil Concentrations: With Aliquot Size and With Distance’, in M. G.White and P. B.Dunaway (eds.), Selected Environmental Plutonium Research Reports of the NAEG, U.S. Department of Energy, Nevada Operations Office, Las Vegas. NVO-192. pp. 405–449.
Dowd, P. A.: 1982, ‘Lognormal Kriging — The General Case’, Mathematical Geology 14, 475–499. Most complete discussion available on kriging, intermediate to difficult level.
Draper, N. and Smith, H. S.: 1981, Applied Regression Analysis, 2nd edition. John Wiley and Sons, New York.
Eynon, B. and Switzer, P.: 1982, ‘The Variability of Rainfall Acidity’, SIMS Technical Report No. 58. Department of Statistics, Stanford University, Stanford, CA.
Faith, R. and Sheshinski, R.: 1979, ‘Misspecification of Trend in Spatial Random-Function Interpolation with Application to Oxidant Mapping’, SIMS Technical Report No. 28, Department of Statistics, Stanford University, Stanford, CA.
Foley, T. A.Jr.: 1981a, Off-Site Radiological Exposure Review Project: A Computer-Aided Surface Interpolation and Graphical Display, Desert Research Institute, University of Nevada System, Las Vegas. DOE/DP/01253-45021.
Foley, T. A.Jr.: 1981b, NTS Radiological Assessment Project: Comparison of Delta Surface Interpolation with Kriging for the Frenchman Lake Region of Area 5, Desert Research Institute, University of Nevada System, Las Vegas. DOE/DP/01253-45022.
Grivet, C. D.: 1980, ‘Modeling and Analysis of Air Quality Data’, SIMS Technical Report No. 43, Department of Statistics, Stanford University, Stanford, CA.
Haas, A. and Jousselin, C.: 1976, ‘Geostatistics in Petroleum Industry’, Advanced Geostatistics in the Mining Industry, D. Reidel Publ. Co., Holland, Dordrecht, pp. 49–68.
Harbaugh, J. W., Doveton, J. H., and Davis, J. C.: 1977, Probability Methods in Oil Exploration, John Wiley and Sons, New York.
Hawkins, D. M. and Cressie, N.: 1984, ‘Robust Kriging — A Proposal’, Mathematical Geology 16, 3–18.
Henley, S.: 1981, Nonparametric Geostatistics, Halsted Press, John Wiley and Sons, New York.
Journel, A. G.: 1980, ‘The Lognormal Approach to Predicting Local Distributions of Selective Mining Unit Grades’, Mathematical Geology 12, 285–303. Gives correction factors when transforming logarithmic results back to original scale if data are lognormal.
Journel, A. G. and Huijbregts, CH. J.: 1978, Mining Geostatistics, Academic Press, New York. Most complete discussion available on kriging, intermediate to difficult level.
Krige, D. G. and Magri, E. J.: 1982, ‘Studies of the Effects of Outliers and Data Transformation on Variogram Estimates for a Base Metal and a Gold Ore Body’, Mathematical Geology 14, 557–564.
Matérn, B.: 1960, Spatial Variation, Meddelander fran Statens Skogforskningsinstitut, Vol. 49, No. 5, 144 pp.
McBratney, A. B. and Webster, R.: 1981, ‘The Design of Optimal Sampling Schemes for Local Estimation and Mapping of Regionalized Variables. Part II. Program and Examples’, Computers and Geosciences 7, 335–365. Computer program listed.
McBratney, A. B., Webster, R., and Burgess, T. M.: 1981, ‘The Design of Optimal Sampling Schemes for Local Estimation and Mapping of Regionalized Variables. Part I. Theory and Method’, Computers and Geosciences 7, 331–334. Computer program listed.
Mousset-Jones, P. F.: 1980, Geostatistics, McGraw-Hill, New York. Computer program listed.
Paunez, I.: 1978, ‘English Language Publications on the French School of Geostatistics’, Mathematical Geology 10, 253–260.
Piazza, A., Menozzi, P., and Cavalli-Sforza, L.: 1981, ‘The Making and Testing of Geographic Gene-Frequency Maps’, Biometrics 37, 635–659.
Rendu, J. M.: 1978, An Introduction to Geostatistical Methods of Mineral Evaluation, South African Institute of Mining and Metallurgy, Johannesburg. Intermediate to difficult level discussion.
Rendu, J. M.: 1979, ‘Normal and Lognormal Estimation’, Mathematical Geology, Vol. II, pp. 407–422. Gives correction factors when transforming logarithmic results back to original scale if data are lognormal.
Ripley, B. D.: 1981, Spatial Statistics, John Wiley and Sons, New York.
Sophocleous, M., Paschetto, J. E., and Olea, R. A.: 1982, ‘Ground-water Network Design for Northwest Kansas, Using the Theory of Regionalized Variables’, Ground Water 20, 48–58.
White, G. C., Simpson, J. C., and Bostick, K. V.: 1980, Studies of Long-Term Ecological Effects of Exposure to Uranium V, Los Alamos National Laboratory, Los Alamos, NM. LA-8221.
Whitten, E. H. T.: 1975, ‘The Practical Use of Trend-Surface Analyses in the Geological Sciences’, in J. C.Davis and M. J.McCullagh (eds.), Display and Analysis of Spatial Data, John Wiley and Sons, New York, pp. 282–297.
Operated for the U.S. Department of Energy by Battelle Memorial Institute
Work supported by Nevada Applied Ecology Group, U.S. Department of Energy, Nevada Operations Office under Contract DE-AC06-76RLO 1830.
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Gilbert, R.O., Simpson, J.C. Kriging for estimating spatial pattern of contaminants: Potential and problems. Environ Monit Assess 5, 113–135 (1985). https://doi.org/10.1007/BF00395842
- Spatial Pattern
- Test Site
- Correlation Structure