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On Robust Estimation of Variograms in Geostatistics

  • Rudolf Dutter
Part of the Lecture Notes in Statistics book series (LNS, volume 109)

Abstract

The distribution of regionalized variables in the field of spatial statistics is mainly characterized by the variogram (or the covariance-function), which essentially is the variance of the differences of the variables in space. The usual estimate (empirical variance) of this function is highly non-robust.

Several alternative (robust) proposals for the estimation can be found in the literature. The definition often does not follow an obvious intuition (e.g., the estimator of Cressie and Hawkins). The present paper considers different estimators of the variogram (including the application of very new scale estimators). The investigation is mainly done by simulation in the one-dimensional case: data on regular and irregular grids. The results are rather surprising. The behavior of the estimators sometimes is unexpected. The main reasons seem to be the high dependence between the data values and the small sample size if the spatial distribution is irregular, both, however, correspond to practical situations often met.

Key words and phrases

Geostatistics estimation of the variogram simulation robustness 

AMS 1991 subject classifications

86A32 62G35 62H11 65C05 

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References

  1. [1]
    Cressie, N. and Hawkins, D.M. (1980): Robust estimation of the variogram. Math. Geol. 12(2) 115–125.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Croux, C. and Rousseeuw, P.J. (1992): Time-efficient algorithms for two highly robust estimators of scale. In Computational Statistics (Y. Dodge and J. Whittaker, eds.), Vol. 1, 411–428. Physica-Verlag, Heidelberg.Google Scholar
  3. [3]
    Dowd, P.A. (1984): The variogram and kriging: Robust and resistent estimators. In Geostatistics for Natural Resource Characterization (G. Verly et al., eds.), 91–106. D. Reidel, Dordrecht.Google Scholar
  4. [4]
    Dutter, R. (1985): Geostatistik. Eine Einführung mit Anwendungen. B.G. Teubner, Stuttgart.zbMATHGoogle Scholar
  5. [5]
    Harmacek, P. (1992): Geostatistics: Robust Estimation for the Semivariogram — A Simulation Study. Master’s Thesis, Inst. f. Statistik u. Wahrscheinlichkeitstheorie, Technical University, Vienna. In German.Google Scholar
  6. [6]
    Journel, A.G. and Huijbregts, Ch.J. (1978): Mining Geostatistics. Academic Press, New York.Google Scholar
  7. [7]
    Matheron, G. (1971): The Theory of Regionalized Variables and Its Applications. Les Cahiers du Centre de Morphologie Mathématique de Paris, Fontainebleau, Frankreich.Google Scholar
  8. [8]
    Rousseeuw, P.J. (1983): Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications Fourth Pannonian Symposium on Mathematical Statistics and Probability, Bad Tatzmannsdorf, Austria, September 4–10, 1983 (I. Vincze, W. Grossmann, G. Pflug and W. Wertz, eds.), Vol. B, 283–297, Budapest, 1985. Akadémiai Kiadó.Google Scholar
  9. [9]
    Rousseeuw, P.J. and Croux, C. (1991): Alternatives to the median absolute deviation. Technical Report 91-43, Universitaire Instelling Antwerpen, Antwerpen, Belgium.Google Scholar
  10. [10]
    Rousseeuw, P.J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88(424) 1273–1283.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Rousseeuw, P.J. and Leroy, A.M. (1987): Robust Regression and Outlier Detection. John Wiley, New York.CrossRefzbMATHGoogle Scholar
  12. [12]
    Wurzer, F. (1990): Geostatistics: Exploratory, Resistant, and Robust Approaches. Ph.D. Thesis, Technical University, Vienna.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Rudolf Dutter
    • 1
  1. 1.University of TechnologyViennaUSA

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