Mathematical Geology

, Volume 19, Issue 5, pp 387–405 | Cite as

Gaussian approximations to conditional distributions for multi-Gaussian processes

  • Michael Stein


Suppose a multi-Gaussian process is observed at some set of sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center of mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means.

Key words

Change of support conditional prediction error random field 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, T. W., 1984, An Introduction to Multivariate Statistical Analysis: John Wiley & Sons, New York, 675 p.Google Scholar
  2. Carrier, G. F.; Krook, M.; and Pearson, C. E., 1966, Functions of a Complex Variable, Theory and Technique: McGraw-Hill, New York, 438 p.Google Scholar
  3. Cressie, N., 1985, When Are Relative Variograms Useful in Geostatistics?: Math. Geol., v. 17, p. 693–702.Google Scholar
  4. Delfiner, P., 1976, Linear Estimation of Non-Stationary Phenomenon,in M. Guarascio et al. (Eds.), Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht, p. 49–68.Google Scholar
  5. Journel, A. G. and Huijbregts, C. T., 1978, Mining Geostatistics: Academic Press, London, 600 p.Google Scholar
  6. Matheron, G., 1965, Les Variables Régionalisées et Leur Estimation: Masson, Paris, 305 p.Google Scholar
  7. Matheron, G., 1973, The Intrinsic Random Functions and Their Applications: Adv. Appl. Prob., v. 5, p. 439–468.Google Scholar
  8. Matheron, G., 1985, Change of Support for Diffusion-type Random Functions: Math. Geol., v. 17, p. 137–166.Google Scholar
  9. Rao, C. R., 1973, Linear Statistical Inference and Its Applications (2nd ed.): John Wiley & Sons, New York, 625 p.Google Scholar
  10. Verly, G., 1983, The Multigaussian Approach and its Applications to the Estimation of Local Reserves: Math. Geol., v. 15, p. 259–286.Google Scholar
  11. Verly, G., 1984, The Block Distribution Given a Point Multivariate Normal Distribution,in G. Verly et al. (Eds.), Geostatistics for Natural Resources Characterization: Reidel, Dordrecht, Netherlands, p. 495–515.Google Scholar
  12. Yakowitz, S. and Szidarovszky, F., 1985, A Comparison of Kriging with Nonparametric Regression Methods: J. Multivar. Anal., v. 16, p. 21–53.Google Scholar

Copyright information

© International Association for Mathematical Geology 1987

Authors and Affiliations

  • Michael Stein
    • 1
  1. 1.Department of StatisticsThe University of ChicagoChicago

Personalised recommendations