Interval and Fuzzy Kriging Techniques Applied to Geological and Geophysical Variables

  • Alberto Consonni
  • Raffaella Iantosca
  • Paolo Ruffo
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 80)


The geostatistical estimation of geological and geophysical variables is often conditioned by inaccuracy and uncertainty in the data, but the way in which this is usually managed may not be fully satisfactory, in some cases. Conventional kriging techniques are capable of dealing with the inaccuracy associated with under-sampled smaller scale variability and with the uncertainty specified by data measurement errors. However, in some instances, the inaccuracy is associated with lack of knowledge, which leads to an interval [min, max] instead of a crisp value in each data location. On the other hand, the uncertainty may describe the case in which there are value-distributions (e.g. [min, mode, max]) instead of crisp values in each data location. A possible solution to deal with these cases is offered by the Interval and Fuzzy approaches, where Interval Kriging and Fuzzy Kriging are applied to the fiirst and the second case, respectively. As far as these methodologies are concerned, some options, such as the moving neighbourhood, the collocated co-kriging and the non-stationary case, have been added so as to simplify their application to real cases. The non-stationary case is solved using a neural network which finds the intrinsic drift in the data and applying the fuzzy approach to the residuals, producing a de-facto extrapolation outside the input data range (not allowed by the stationary fuzzy approach). Two examples of application are also provided: the Interval Kriging is applied to the problem of map estimation of generated hydrocarbon in a basin; the Fuzzy Kriging is applied to the estimation of the best possible seismic velocity map starting from the value-distributions produced by a previous velocity analysis step.


Membership Function Source Rock Fuzzy Number Random Function Seismic Velocity 
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  1. R. Barnes and T. Johnson. Positive Kriging. In: Verly et al. (eds) Geostatistics for Natural Resources Characterization, pp 231–244, Reidel, Dordrecht, 1984.CrossRefGoogle Scholar
  2. R. Barnes and K. You. Adding Bounds to kriging, Mathematical Geology, 24(2): 171–176, 1992.CrossRefGoogle Scholar
  3. J. P. Chiles and P. Delfiner. Geostatistics, Modeling Spatial Uncertainty, Wiley Interscience, 1999.CrossRefGoogle Scholar
  4. V. Demyanov, M. Kanevsky, S. Chernov, E. Savelieva and V. Timonin. Neural Network Residual Kriging Application for Climatic Data, Journal of Geographic Information and Decision Analysis, 2(2): 234–252, 1998.Google Scholar
  5. C. V. Deutsch and A. G. Journel. GSLIB, Geostatistical Software Library and User’ s Guide, Oxford University Press, 1998.Google Scholar
  6. P. Diamond. Interval-valued random functions and the kriging of intervals, Mathematical Geology, 20(3): 145–165, 1988.CrossRefGoogle Scholar
  7. P. Diamond. Fuzzy Kriging, Fuzzy Sets and Systems, 33: 315–332, 1989.CrossRefGoogle Scholar
  8. J. A. Drakopoulos. Probabilities, possibilities and fuzzy sets, Fuzzy Sets and Systems, 75 : 1–15, 1995.CrossRefGoogle Scholar
  9. D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.Google Scholar
  10. H. J. Zimmermann. Fuzzy Set Theory and Its Application, Kluwer Academic Publishers, Dordrecht, 1991.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alberto Consonni
    • 1
  • Raffaella Iantosca
    • 2
  • Paolo Ruffo
    • 1
  1. 1.ENI-AGIP DivisionSan Donato Milanese (Milano)Italy
  2. 2.Intesa Asset Management SGRMilanoItaly

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