Mathematical Geology

, Volume 24, Issue 5, pp 539–553 | Cite as

A non-parametric bivariate entropy estimator for spatial processes

  • Mario Rossi
  • Donato Posa


In this paper, entropy is presented as an alternative measure to characterize the bivariate distribution of a stationary spatial process. This non-parametric estimator attempts to quantify the concept of spatial ordering, and it provides a measure of how Gaussian the experimental bivariate distribution is. The concept of entropy is explained and the classical definition presented, along with some important results. In particular, the reader is reminded that, for a known mean and covariance, the bivariate Gaussian distribution maximizes entropy. A “relative entropy” estimator is introduced in order to measure departure of an experimental bivariate distribution from the bivariate Gaussian. Two case studies are presented as examples.

Key words

entropy spatial disorder bivariate Gaussian distribution maximum entropy non-parametric relative entropy estimator 


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  1. Christakos, G., 1990, A Bayesian/Maximum Entropy View to the Spatial Estimation Problem; Math. Geol. v. 22, n. 7, p. 763–778.Google Scholar
  2. Giordano, R., Salter, S., and Mohanty, K., 1985, The Effects of Permeability Variations on Flow in Porous Media: SPE paper 14365, 60th SPE Annual Conference, Las Vegas, NV.Google Scholar
  3. Gull, S. F., and Skilling, J., 1985, The Entropy of an Image,in C. R. Smith and W. T. Gandy, Jr., (Eds.),Maximum Entropy and Bayesian Methods in Inverse Problems: Reidel, Dordrecht, p. 287–301.Google Scholar
  4. Hyvarinen, L. P., 1970, Information Theory for Systems Engineers: Springer-Verlag, 197 p.Google Scholar
  5. Jaynes, E. T., 1983,in R. D. Rosenkrants. (Ed.), Papers on Probability, Statistics and Statistical Physics.Google Scholar
  6. Jones, D. S., 1979, Elementary Information Theory: Clarendon Press, Oxford, 182 p.Google Scholar
  7. Journel, A. G., and Alabert, F., 1988, Focusing on Spatial Connectivity of Extreme-Valued Attributes: Stochastic Indicator Models of Reservoir Heterogeneities: SPE paper No. 18324.Google Scholar
  8. Justice, J. H. 1984,in J. H. Justice (Ed.), Proceedings of the Fourth Maximum Entropy and Bayesian Methods Congress: University of Calgary.Google Scholar
  9. Perillo, G. M. E., and Marone, E., 1986, Determination of Optimal Numbers of Class Intervals Using Maximum Entropy: Math. Geol. v. 18, n. 4.Google Scholar
  10. Rossi, M. E., 1990,Impact of Spatial Clustering on Geostatistical Analysis: Proceedings XXII APCOM Conference, Berlin, v. 2, p. 589–600.Google Scholar
  11. Shannon, C. E., 1948, A Mathematical Theory of Communication: Bell Syst. Tech. J., v. 27, p. 379–423, 623–656.Google Scholar

Copyright information

© International Association for Mathematical Geology 1992

Authors and Affiliations

  • Mario Rossi
    • 1
  • Donato Posa
    • 2
  1. 1.Fluor Daniel, Inc.Redwood City
  2. 2.IRMA-CNRBariItaly

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