Mathematical Geology

, Volume 25, Issue 2, pp 219–240 | Cite as

Multivariable spatial prediction

  • Jay M. Ver Hoef
  • Noel Cressie


For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.

Key words

geostatistics kriging cokriging cross-variogram best linear unbiased prediction generalized least squares 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Carr, J. R., and McCallister, P. G., 1985, An Application of Cokriging for Estimation of Tripartite Earthquake Response Spectra: Math. Geol., v. 17, p. 527–545.Google Scholar
  2. Clark, I., Basinger, K. L., and Harper, W. V., 1989, MUCK—A Novel Approach to Cokriging,in B. E. Buxton (Ed.), Proceedings of the Conference on Geostatistical, Sensitivity, and Uncertainty Methods for Ground-Water Flow and Radionuclide Transport Modeling: Battelle Press, Columbus, p. 473–493.Google Scholar
  3. Cressie, N., 1991, Statistics for Spatial Data: John Wiley and Sons, New York, 900 p.Google Scholar
  4. Cressie, N., and Grondona, M. O., 1992, A Comparison of Variogram Estimation with Covariogram Estimation,in K. V. Mardia (Ed.), The Art of Statistical Science: John Wiley and Sons, New York, p. 191–208.Google Scholar
  5. Goldberger, A. S., 1962, Best Linear Unbiased Prediction in the Generalized Linear Regression Model: J. Am. Stat. Assoc., v. 57, p. 369–375.Google Scholar
  6. Hoeksema, R. J., Clapp, R. B., Thomas, A. L., Hunley, A. E., Farrow, N. D., and Dearstone, K. C., 1989, Cokriging Model for Estimation of Water Table Elevation: Water Res. Res., v. 25, p. 429–438.Google Scholar
  7. Huijbregts, C., and Matheron, G., 1971, Universal Kriging (an Optimal Method for Estimating and Contouring in Trend Surface Analysis),in J. I. McGerrigle (Ed.), Proceedings of Ninth International Symposium on Techniques for Decision-Making in the Mineral Industry, Special Vol. 12: The Canadian Institute of Mining and Metallurgy, p. 159–169.Google Scholar
  8. Journel, A. G., and Huijbregts, C. J., 1978, Mining Geostatistics: Academic Press, London, 600 p.Google Scholar
  9. Magnus, J. R., and Neudecker, H., 1988, Matrix Differential Calculus with Applications in Statistics and Econometrics: John Wiley and Sons, New York, 393 p.Google Scholar
  10. Matheron, G., 1969, Le Krigeage Universel: Cahiers du Centre de Morphologie Mathematique, No. 1, Fontainebleau.Google Scholar
  11. Matheron, G., 1971, The Theory of Regionalized Variables and Its Applications: Cahiers du Centre de Morphologie Mathematique, No. 5., Fontainebleau.Google Scholar
  12. Mulla, D. J., 1988, Estimating Spatial Patterns in Water Content, Matric Suction, and Hydraulic Conductivity: Soil Sci. Soc. Am. J., v. 52, p. 1547–1553.Google Scholar
  13. Myers, D. E., 1982, Matrix Formulation of Co-Kriging: Math. Geol., v. 14, p. 249–257.Google Scholar
  14. Myers, D. E., 1984, Cokriging: New Developments,in G. Verly et al. (Eds.), Geostatistics for Natural Resource Characterization: D. Reidel, Dordrecht, p. 295–305.Google Scholar
  15. Myers, D. E., 1988, Some Aspects of Multivariate Analysis,in C. F. Chung et al. (Eds.), Quantitative Analysis of Mineral and Energy Resources: D. Reidel, Dordrecht, p. 669–687.Google Scholar
  16. Myers, D. E., 1989, Vector Conditional Simulation,in M. Armstrong, (Ed.), Geostatistics: D. Reidel, Dordrecht, p. 283–293.Google Scholar
  17. Myers, D. E., 1991, Pseudo-Cross Variograms, Positive-Definiteness, and Cokriging: Math. Geol., v. 23, p. 805–816.Google Scholar
  18. Quimby, W. F., Borgman, L. E., and Easley, D. H., 1986, Selected Topics in Spatial Statistical Analysis—Nonstationary Vector Kriging, Large Scale Conditional Simulation of Three Dimensional Gaussian Random Fields, and Hypothesis Testing in a Correlated Random Field: Report to the Environmental Monitoring Systems Lab, E.P.A., Las Vegas, Nevada, 140 p.Google Scholar
  19. Stein, A., van Dooremolen, W., Bouma, J., and Bregt, A. K., 1988, Cokriging Point Data on Moisture Deficit: Soil Sci. Soc. Am. J., v. 52, p. 1418–1423.Google Scholar
  20. Trangmar, B. B., Yost, R. S., and Uehara, G., 1986, Spatial Dependence and Interpolation of Soil Properties in West Sumatra, Indonesia: II. Co-Regionalization and Co-Kriging: Soil Sci. Soc. Am. J., v. 50, p. 1396–1400.Google Scholar
  21. Yates, S. R., and Warrick, A. W., 1987, Estimating Soil Water Content Using Cokriging: Soil Sci. Soc. Am. J., v. 51, p. 23–30.Google Scholar

Copyright information

© International Association for Mathematical Geology 1993

Authors and Affiliations

  • Jay M. Ver Hoef
    • 1
  • Noel Cressie
    • 1
  1. 1.Iowa State UniversityAmes

Personalised recommendations