Mathematical Geology

, Volume 35, Issue 6, pp 681–698 | Cite as

Gaussian Cosimulation: Modelling of the Cross-Covariance

  • Dean S. Oliver


Whenever two or more random fields are assumed to be correlated in reservoir characterization, it is necessary to generate valid cross-covariance models to describe the relationship. The standard methods for constructing covariance matrices for correlated random fields are not very general. In particular, they do not allow one to specify different auto-covariance models for the two fields. It is not possible, for example, for one field to have a Gaussian auto-covariance and the other an exponential auto-covariance, unless the two fields are uncorrelated. The standard approaches also do not allow for nonsymmetric cross-covariance functions. In this report, I present a straightforward method of cosimulation based on the square root of the auto-covariances. The same approach is used for constructing cross-covariance models for the variables. The approach is quite general and does not require symmetry of the cross-covariance. The modelling of the cross-covariance is illustrated with gamma ray and spontaneous potential logs.

coregionalization cross-covariance filtering covariogram 


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  1. Alabert, F., 1987, The practice of fast conditional simulations through the LU decomposition of the covariance matrix: Math. Geol. v.19, no.5, p. 369-386.Google Scholar
  2. Cressie, N. A. C., 1993, Statistics for spatial data: Wiley, New York, 900p.Google Scholar
  3. Davis, M., 1987, Production of conditional simulations via the LU decomposition of the covariance matrix: Math. Geol: v.19, no.2, p. 91-98.Google Scholar
  4. Goovaerts, P., 1994, On a controversial method for modeling a coregionalization: Math. Geol. v.26, no.2, p. 197-204.Google Scholar
  5. Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 500p.Google Scholar
  6. Goulard, M., and Voltz, M., 1992, Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix: Math. Geol. v.24, no.3, p. 269-286.Google Scholar
  7. Haas, T. C., 2002, New systems for modeling, estimating, and predicting a multivariate spatio-temporal process: Environmetrics, v.13, no.4, p. 311-332.Google Scholar
  8. Journel, A., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, New York, 600p.Google Scholar
  9. Kendall, M. G., 1976, Time-series: 2nd ed: Hafner, New York, 197p.Google Scholar
  10. Meyers, D. E., 1984, Co-kriging new developments, in Verly, G., Michel, D., Journel, A.6. and Marechal, A. eds., Geostatistics for natural resources characterization: D. Reidel, Dordrecht, p. 295-305, 1029 p.Google Scholar
  11. Meyers, D. E., 1989, Vector conditional simulation, in Armstrong, M., ed., Geostatistics, Kluwer Academic, Dordrecht, p. 283-293.Google Scholar
  12. Oliver, D. S., 1995, Moving averages for Gaussian simulation in two and three dimensions: Math. Geol. v.27, no.8, p. 939-960.Google Scholar
  13. Rao, C. R., 1973, Linear statistical inference and its applications; 2nd edr: Wiley, New York, 625p.Google Scholar
  14. Tidwell, V. C., Gutjahr, A. L., and Wilson, J. L., 1999, What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports: Water Resourc. Res. v.35, no.1, p. 43-54.Google Scholar
  15. Webster, R., and Oliver, M., 2001, Geostatistics for environmental scientists: Wiley, New York, 271p.Google Scholar
  16. Yao, T., and Journel, A. G., 1998, Automatic modeling of (cross) covariance tables using fast fourier transform: Math. Geol. v.30, no.6, p. 589-616.Google Scholar

Copyright information

© International Association for Mathematical Geology 2003

Authors and Affiliations

  • Dean S. Oliver
    • 1
  1. 1.Mewbourne School of Petroleum and Geological EngineeringUniversity of OklahomaNorman

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