Mathematical Geology

, Volume 39, Issue 1, pp 113–134 | Cite as

Optimized Sample Schemes for Geostatistical Surveys



Sample schemes used in geostatistical surveys must be suitable for both variogram estimation and kriging. Previously schemes have been optimized for one of these steps in isolation. Ordinary kriging generally requires the sampling locations to be evenly dispersed over the region. Variogram estimation requires a more irregular pattern of sampling locations since comparisons must be made between measurements separated by all lags up to and beyond the range of spatial correlation. Previous studies have not considered how to combine these optimized schemes into a single survey and how to decide what proportion of sampling effort should be devoted to variogram estimation and what proportion devoted to kriging

An expression for the total error in a geostatistical survey accounting for uncertainty due to both ordinary kriging and variogram uncertainty is derived. In the same manner as the kriging variance, this expression is a function of the variogram but not of the sampled response data. If a particular variogram is assumed the total error in a geostatistical survey may be estimated prior to sampling. We can therefore design an optimal sample scheme for the combined processes of variogram estimation and ordinary kriging by minimizing this expression. The minimization is achieved by spatial simulated annealing. The resulting sample schemes ensure that the region is fairly evenly covered but include some close pairs to analyse the spatial correlation over short distances. The form of these optimal sample schemes is sensitive to the assumed variogram. Therefore a Bayesian approach is adopted where, rather than assuming a single variogram, we minimize the expected total error over a distribution of plausible variograms. This is computationally expensive so a strategy is suggested to reduce the number of computations required

Key Words

variogram ordinary kriging maximum likelihood 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abt, M., 1999, Estimating the prediction mean squared error in Gaussian stochastic processes with exponential correlation structure: Scand. Jour. Statist., v. 26, no. 4, p. 563–578.CrossRefGoogle Scholar
  2. Abt, M., Welch, W. J., and Sacks, J., 1999, Design and analysis for modelling and predicting spatial contamination: Math. Geol., v. 31, no. 1, p. 1–22.CrossRefGoogle Scholar
  3. Bogaert, P., and Russo, D., 1999, Optimal spatial sampling design for the estimation of the variogram based upon a least squares approach: Water Resour. Res., v. 35, no. 4, p. 1275–1289.CrossRefGoogle Scholar
  4. Deutsch, C. V., and Journel, A. G., 1998, GSLIB: Geostatistical software library and users guide, 2nd ed.: Oxford University Press, New York, 369 p.Google Scholar
  5. Dobson, A. J., 1990, An introduction to generalized linear models, 2nd ed.: Chapman and Hall, London, 174 p.Google Scholar
  6. Heuvelink, G. B. M., 1998, Error propagation in environmental modelling with GIS: Taylor, Francis, London, 127 p.Google Scholar
  7. Heuvelink, G. B. M., Brus, D. J., and De Gruijter, J. J., 2004, Optimization of sample configurations for digital soil mapping with universal kriging: Proceedings of Global Workshop on Digital Soil Mapping, INRA, Montpellier.Google Scholar
  8. Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.Google Scholar
  9. Kitanidis, P. K., 1987, Parametric estimation of covariances of regionalised variables: Water Resources Bulletin, v. 23, no. 4, p. 557–567.Google Scholar
  10. Lark, R. M., 2000, Estimating variograms of soil properties by the method-of-moments and maximum likelihood: Eur. Jour. Soil Sci., v. 51, no. 4, p. 717–728.CrossRefGoogle Scholar
  11. Lark, R. M., 2002, Optimized spatial sampling of soil for estimation of the variogram by maximum likelihood: Geoderma, v. 105, no. 1–2, p. 49–80.CrossRefGoogle Scholar
  12. Marchant, B. P., and Lark, R. M., 2004, Estimating variogram uncertainty: Math. Geology, v. 36, no. 8, p. 867–898.CrossRefGoogle Scholar
  13. Müller, W. G., and Zimmerman, D. L., 1999, Optimal designs for variogram estimation: Environmetrics, v. 10, no. 1, p. 23–37.CrossRefGoogle Scholar
  14. Pardo-Igúzquiza, E., 1997, MLREML: A computer program for the inference of spatial covariance parameters by maximum likelihood and restricted maximum likelihood: Comput. & Geosciences, v. 23, no. 2, p. 153–162.CrossRefGoogle Scholar
  15. Pardo-Igúzquiza, E., 1998, Maximum likelihood estimation of spatial covariance parameters: Math. Geol., v. 30, no. 1, p. 95–108.CrossRefGoogle Scholar
  16. Pardo-Igúzquiza, E., and Dowd, P. A., 2001a, Variance-covariance matrix of the experimental variogram: Assessing variogram uncertainty: Math. Geol., v. 33, no. 4, p. 397–419.CrossRefGoogle Scholar
  17. Pardo-Igúzquiza, E., and Dowd, P. A., 2001b, VARIOG2D: A computer program for estimating the semi-variogram and its uncertainty: Comput. & Geosciences, v. 27, no. 5, p. 549–561.CrossRefGoogle Scholar
  18. Pardo-Igúzquiza, E., and Dowd, P. A., 2003, Assessment of the uncertainty of spatial covariance parameters of soil properties and its use in applications: Soil Sci., v. 168, no. 11, p. 769–782.CrossRefGoogle Scholar
  19. Russo, D., 1984, Design of an optimal sampling network for estimating the variogram: Soil Sc. Soc. Am. J., v. 48, no. 4, p. 708–716.CrossRefGoogle Scholar
  20. Todini, E., 2001, Influence of parameter estimation uncertainty in Kriging: Part 1-Theoretical development: Hydrol. Earth Sci. Syst., v. 5, no. 2, p. 215–223.CrossRefGoogle Scholar
  21. Todini, E., and Ferraresi, M., 1996, Influence of parameter estimation uncertainty in Kriging: J. Hydrol., v. 175, no. 4, p. 555–566.CrossRefGoogle Scholar
  22. van Groenigen, J. W., Siderius, W., and Stein, A., 1999, Constrained optimisation of soil sampling for minimisation of the kriging variance: Geoderma v. 87, no. 3–4, p. 239–259.CrossRefGoogle Scholar
  23. van Groenigen, J. W., 2000, The influence of variogram parameters on optimal sampling schemes for mapping by kriging: Geoderma v. 97, no. 3–4, p. 223–236.CrossRefGoogle Scholar
  24. Warrick, A. W., and Myers, D. E., 1987, Optimization of sampling locations for variogram calculations: Water Resour. Res., v. 23, no. 3, p. 496–500.CrossRefGoogle Scholar
  25. Webster, R., and Oliver, M. A., 2001 Geostatistics for Environmental Scientists: John Wiley, Chichester 217p.Google Scholar
  26. Zimmerman, D. L., and Cressie, N., 1992, Mean squared prediction error in the spatial linear model with estimated covariance parameters: Ann. Inst. Statist. Math. v. 44, no. 1, p. 27–43.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Bioinformatics and Biomathematics DivisionRothamsted ResearchHarpendenUK

Personalised recommendations