Spatial Variability

  • Mario E. Rossi
  • Clayton V. Deutsch


An essential aspect of geostatistical modeling is to establish quantitative measures of spatial variability or continuity to be used for subsequent estimation and simulation. The modeling of the spatial variability has become a standard tool of mineral resource analysts. In the last 20 years or so, the traditional experimental variogram has given way to more robust measures of variability. Details of how to calculate, interpret and model variograms or their more robust alternatives are contained in this chapter.


Variogram Model Nugget Effect Experimental Variograms Estimation Domain Geometric Anisotropy 
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  1. Armstrong M (1984) Improving the estimation and modeling of the variogram. In: Verly G et al (ed) Geostatistics for natural resources characterization, part I, Reidel, Dordrecht, pp 1–19Google Scholar
  2. Babakhani M, Deutsch CV (2012) Standardized pairwise relative variogram as a robust estimator of spatial structure. Unpublished CCG Research Paper 2012–310Google Scholar
  3. Barnes, RJ (1991) The variogram sill and the sample variance. Math Geol 23(4):673–678CrossRefGoogle Scholar
  4. Christakos G (1984). On the problem of permissible covariance and variogram models. Water Resour Res 20(2):251CrossRefGoogle Scholar
  5. Clark I (1986) The art of cross-validation in geostatistical applications. In: Proceeding of the 19th APCOM, pp 211–220Google Scholar
  6. Clark I, Harper WV (2000) Practical geostatistics. Ecosse North America, Columbus, p 340Google Scholar
  7. Cressie N (1985) Fitting variogram models by weighted least squares. Math Geol 17(5):563–586Google Scholar
  8. Cressie N (1991) Statistics for spatial data. Wiley, New York, p 900. Reprinted 1993Google Scholar
  9. David M (1977) Geostatistical ore reserve estimation. Elsevier, AmsterdamGoogle Scholar
  10. Davis BM (1987) Uses and abuses of cross-validation in geostatistics. Math Geol 17(5):563–586Google Scholar
  11. Deutsch CV, Journel AG (1997) GSLIB: geostatistical software library and user’s guide, 2nd ed. Oxford University Press, New York, p 369Google Scholar
  12. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York, p 483Google Scholar
  13. Gringarten E, Deutsch CV (1999) Methodology for variogram interpretation and modelling for improved reservoir characterization. In: Paper SPE 56654, presented at the SPE annual technical conference and exhibition held in Houston, 3–6 OctoberGoogle Scholar
  14. Isaaks EH (1999) SAGE2001 User’s Manual, Software license and documentation.
  15. Isaaks E, Srivastava RM (1988) Spatial continuity measures for probabilistic and deterministic geostatistics. Math Geol 20(4):313–341CrossRefGoogle Scholar
  16. Isaaks EH, Srivastava RM (1989). An introduction to applied geostatistics. Oxford University Press, New York, p 561.Google Scholar
  17. Journel A (1988) New distance measures: the route towards truly non-Gaussian geostatistics. Math Geol 20(4):459–475CrossRefGoogle Scholar
  18. Journel AG (1987) Geostatistics for the environmental sciences: An introduction. U.S. Environmental Protection Agency, Environmental Monitoring Systems LaboratoryGoogle Scholar
  19. Journel AG, Huijbregts ChJ (1978) Mining geostatistics. Academic, New YorkGoogle Scholar
  20. Myers DE (1991) Pseudo-cross variograms, positive definiteness and cokriging. Math Geol 23:805–816CrossRefGoogle Scholar
  21. Omre H (1984) The variogram and its estimation. In: Verly G, David M, Journel AG, Marechal A (eds) Geostatistics for natural resource characterization: NATO ASI series C, v. 122—Part I. Reidel Publication. Co., Dordrecht, pp 107–125Google Scholar
  22. Reed M, Simon B (1975) Methods of modern mathematical physics, vol II. Academic Press, New YorkGoogle Scholar
  23. Rossi ME, Parker HM (1993) Estimating recoverable reserves: is it hopeless? Presented at the Forum ‘Geostatistics for the Next Century’, Montreal, 3–5 JuneGoogle Scholar
  24. Solow AR (1990) Geostatistical cross-validation: a cautionary note. Math Geol 22:637–639CrossRefGoogle Scholar
  25. Srivastava RM (1987) A non-ergodic framework for variogram and covariance functions. Unpublished Master of Science Thesis, Stanford University, Stanford, CAGoogle Scholar
  26. Srivastava RM, Parker HM (1988) Robust measures of spatial continuity. In: Amstrong M (ed) Geostatistics. Reidel, Dordrecht, pp 295–308Google Scholar
  27. Zhu H, Journel A (1993) Formatting and integrating soft data: stochastic imagining via the Markov-Bayes algorithm. In: Soares A (ed) Geostatistics-Troia. Kluwer, Dordrecht, pp 1–12Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Geosystems International, Inc.Delray BeachUSA
  2. 2.University of AlbertaEdmontonCanada

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