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Robustness of variograms and conditioning of kriging matrices

  • Phil Diamond
  • Margaret Armstrong
Article

Abstract

Current ideas of robustness in geostatistics concentrate upon estimation of the experimental variogram. However, predictive algorithms can be very sensitive to small perturbations in data or in the variogram model as well. To quantify this notion of robustness, nearness of variogram models is defined. Closeness of two variogram models is reflected in the sensitivity of their corresponding kriging estimators. The condition number of kriging matrices is shown to play a central role. Various examples are given. The ideas are used to analyze more complex universal kriging systems.

Key words

variogram robustness kriging conditioning number 

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References

  1. Armstrong, M., 1982, The problems with universal kriging: Centre de Géostatisque, Fountainebleau, N-784.Google Scholar
  2. Armstrong, M. and Delfiner, P., 1980, Towards a more robust variogram: A case study on coal: Centre de Géostatisque, Fontainebleau, N-671.Google Scholar
  3. Atkinson, K. E., 1978, An introduction to numerical analysis, John Wiley & Sons, New York.Google Scholar
  4. Broyden, C. G., 1975, Basic matrices: Macmillan, London.Google Scholar
  5. Cressie, N. and Hawkins, D. H., 1980, Robust estimators of the variogram: I: Jour. Math. Geol., v. 12, p. 115–126.Google Scholar
  6. Delfiner, P., 1982a, Excerpt from Basic Introduction to Geostatistics: Centre de Géostatisque: Fontainebleau, C-95.Google Scholar
  7. Delfiner, P., 1982b, The intrinsic model of order k: Centre de Géostatisque, Fontainbleau, C-97.Google Scholar
  8. Hampel, F. R., 1971, A general qualitative definition of robustness, Ann. Math. Stat., v. 42, p. 1887–1896.Google Scholar
  9. Huber, P. J., 1977, Robust statistical procedures, Regional Conference Series in Applied Mathematics, SIAM, No. 27, Pennsylvania.Google Scholar
  10. Huber, P. J., 1981, Robust statistics: John Wiley & Sons, New York.Google Scholar
  11. Lawson, C. L. and Hanson, R. J., 1974, Solving least squares problems: Prentice-Hall, New Jersey.Google Scholar
  12. Matheron, G., 1978, Estimer et choisir: Centre de Géostatisque, Fontainebleau, Fascicule 7.Google Scholar
  13. Wilkinson, J. H., 1967, The solution of ill-conditioned linear equations, Ralston, A. and Wilf, H. S. (Eds.), Mathematical methods for digital computers, vol. 2: John Wiley & Sons, New York.Google Scholar
  14. Wilkinson, J. H., 1971, Modern error analysis, SIAM, v. 13, p. 548–568.Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • Phil Diamond
    • 1
  • Margaret Armstrong
    • 2
  1. 1.Department of MathematicsUniversity of QueenslandSt. LuciaAustralia
  2. 2.Centre de Géostatistique et de Morphologie MathématiqueÉcole des Mines de ParisFontainebleauFrance

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