Abstract
Suppose data {Z(s i ):i=1, ..., n} are observed at spatial locations {s i :i=1, ..., n}. From these data, an unknownZ(s 0) is to be predicted at a known locations 0c, or, ifZ(s0) has a component of measurement error, then a smooth versionS(s 0) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.
Similar content being viewed by others
References
Adler, R., 1981, The Geometry of Random Fields: New York, John Wiley & Sons, 280 p.
Besag, J., 1974, Spatial Interaction and the Statistical Analysis of Lattice Systems (with discussion), J. Roy. Stat. Soc., Series B, v. 36, p. 192–236.
Cressie, N., 1985, Fitting Variogram Models by Weighted Least Squares, Math. Geol., v. 17, p. 563–586.
Cressie, N., 1986, Kriging Nonstationary Data, J. Amer. Stat. Assoc., v. 81, p. 625–634.
Cressie, N. and Chan, N., 1986, Spatial Modeling of Regional Variables, Preprint 86-54, Statistical Laboratory, Iowa State University, Ames, Iowa, 34 p.
Cressie, N. and Glonek, G. F., 1984, Median Based Covariogram Estimators Reduce Bias, Stat. Prob. Lett., v. 2, p. 299–304.
Gandin, L. S., 1963, Objective Analysis of Meteorological Fields: GIMIZ, Leningrad (reprinted in 1965 by the Israel Program for Scientific Translations, Jerusalem), 238 p.
Journel, A. G., 1983, Nonparametric Estimation of Spatial Distributions, Math. Geol., v. 15, p. 445–468.
Journel, A. G. and Huijbregts, C. J., 1978, Mining Geostatistics: Academic Press, London, 600 p.
Jowett, G. H., 1952, The Accuracy of Systematic Sampling from Conveyor Belts, Appl. Stat., v. 1, p. 50–59.
Karlin, S., 1969, A First Course in Stochastic Processes: New York, Academic Press, 502 p.
Kolmogorov, A. N., 1941a, The Local Structure of Turbulence in an Incompressible Fluid at Very Large Reynolds Numbers: Dokl. Akad. Nauk SSR, v. 30, p. 229–303. (Reprinted in Turbulence: Classic Papers on Statistical Theory, eds. S. K. Friedlander and L. Topping, Interscience Publishers, New York 1961).
Kolmogorov, A. N., 1941b, Interpolation and Extrapolation of Stationary Random Sequences: Izv. Akad. Nauk SSR, Ser. Math., v. 5, no. 3.
Lindgren, B. W., 1976, Statistical Theory: New York, MacMillan, 614 p.
Matern, B., 1960, Spatial Variation: Medd. Statens Skogsforskningsinst., v. 49, no. 5, 144 p.
Matheron, G., 1963, Principles of Geostatistics, Econ. Geol., v. 58, p. 1246–1266.
Matheron, G., 1971, The Theory of Regionalized Variables and its Applications: Cahiers du Centre de Morphologie Mathématique no. 5, Fontainebleau, France, 211 p.
Matheron, G., 1976, A Simple Substitute for Conditional Expectation: The Disjunctive Kriging, in M. Guarascio, M. David, and C. Huijbregts, eds, Advanced Geostatistics in the Mining Industry: D. Reidel, Holland, p. 221–236.
Morrison, D. F., 1978, Multivariate Statistical Methods, 2nd ed: McGraw-Hill, Tokyo, 415 p.
Stein, M. L., 1987, Minimum Norm Quadratic Estimation of Spatial Variograms, Jour. Amer. Stat. Assoc., v. 82, p. 765–772.
Wiener, N., 1949, Extrapolation, Interpolation and Smoothing of Stationary Time Series: MIT Press, Cambridge, 312 p.
Yaglom, A. M., 1962, An Introduction to the Theory of Stationary Random Functions, reprint: Dover Publications, New York, 235 p.
Yakowitz, S. J. and Szidarovsky, F., 1985, A Comparison of Kriging with Nonparametric Regression Methods: Jour. Multivar. Anal., v. 16, p. 21–53.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cressie, N. Spatial prediction and ordinary kriging. Math Geol 20, 405–421 (1988). https://doi.org/10.1007/BF00892986
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00892986