Comparing sampling patterns for kriging the spatial mean temporal trend

  • C. J. F. ter Braak
  • D. J. Brus
  • E. J. Pebesma
Article

Abstract

In monitoring the environment one often wishes to detect the temporal trend in a variable that varies across a region. A useful executive summary is then the temporal trend in the spatial mean. In this article, the best linear unbiased predictor of the spatial mean temporal trend and its variance are derived under a universal kriging model. Five different, spatially explicit sampling patterns are compared in terms of this variance. For small spatial ranges, the time-separation pattern is optimal, regardless of the temporal range. For larger spatial ranges, the best pattern depends on the temporal range. If the temporal range is small (less than ca. 1/20 of the spatial range), then the always-revisit pattern is best. For larger temporal ranges (between ca. 1/20 and ca. 1/4 of the spatial range) the serially alternating pattern is best, whereas the time-separation pattern is best for larger temporal ranges. The gain in using the serially alternating pattern or time-separation pattern instead of the always-revisit pattern can be substantial. The potential loss with respect to the always-revisit pattern is only minor for the serially alternating pattern, but can be substantial for the time-separation pattern. Unless one has good knowledge of the spatial and temporal range, the serially alternating pattern is thus a good choice. This work extends that by Urquhart and Kincaid and gives further support to their plea for the serially alternating pattern.

Key Words

Dynamic design Geometric anisotropy Rotational design Sampling design Serially alternating design Static design Universal kriging 

References

  1. Abramowitz, M., and Stegun, I. A. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications.MATHGoogle Scholar
  2. Bogaert, P., and Christakos, G. (1997), “Spatiotemporal Analysis and Processing of Thermometric Data Over Belgium,” Journal of Geophysical Research, 102, 25831–25846.CrossRefGoogle Scholar
  3. Breidt, F. J., and Fuller, W. A. (1999), “Designs of Supplemented Panel Surveys with Application to the National Resources Inventory,” Journal of Agricultural, Biological and Environmental Statistics, 4, 391–402.CrossRefMathSciNetGoogle Scholar
  4. Bronswijk, J. J. B., Groot, M. S. M., Fest, P. M. J., and van Leeuwen, T. C. (2003) Landelijk meetnet bodemkwaliteit,” Technical Report 714801031/2003, Rijksinstituut voor Volksgezondheid en Milieu (RIVM), Bilthoven, Netherlands.Google Scholar
  5. Carr, J. R., and Palmer, J. A. (1993), “Revisiting the Accurate Calculation of Block-Sample Covariances using Gauss Quadrature,” Mathematical Geology, 25, 507–524.CrossRefGoogle Scholar
  6. Clifford, D. (2005), “Computation of Spatial Covariance Matrices,” Journal of Computational and Graphical Statistics, 14, 155–167.CrossRefMathSciNetGoogle Scholar
  7. de Gruijter, J. J., Brus, D. J., Bierkens, M. F. P. and Knotters, M. (2006), Sampling for Natural Resource Monitoring, New York, Springer.Google Scholar
  8. Gneiting, T. (2002), “Nonseparable, Stationary Covariance Functions for Space-Time Data,” Journal of the American Statistical Association, 97, 590–600.MATHCrossRefMathSciNetGoogle Scholar
  9. Heuvelink, G. B. M., Musters, P., and Pebesma, E. J. (1997), “Spatio-Temporal Modelling of Soil Water Content,” in Geostatistics Wollongong (vol. 2) eds. E. Baafi and N. Schofield, Dordrecht: Kluwer Academic Publ., pp. 1020–1030.Google Scholar
  10. Journel, A. G., and Huijbregts, C. J. (1978) Mining Geostatistics, New York, Academic Press.Google Scholar
  11. Knotters, M., and Bierkens, M. F. P. (2001), “Predicting Water Table Depths in Space and Time Using a Regionalised Time Series Model,” Geoderma, 103 51–77.CrossRefGoogle Scholar
  12. Kyriakidis, P. C., and Journel, A. G. (1999), “Geostatistical Space-Time Models: A Review,” Mathematical Geology, 31, 651–684.MATHCrossRefMathSciNetGoogle Scholar
  13. Ma, C. (2002), “Spatio-Temporal Covariance Functions Generated by Mixtures,” Mathematical Geology, 34, 965–975.MATHCrossRefMathSciNetGoogle Scholar
  14. — (2003), “Families of Spatio-Temporal Stationary Covariance Models,” Journal of Statistics Planning and Inference, 116, 489–501.MATHCrossRefGoogle Scholar
  15. Marchant, B., and Lark, R. M. (2006), “Adaptive Sampling and Reconnaissance Surveys for Geostatistical Mapping of Soil,” European Journal of Soil Science, 57, 831–845.CrossRefGoogle Scholar
  16. R Development Core Team (2003), R: A Language and Environment for Statistical Computing, Vienna: R Foundation for Statistical Computing.Google Scholar
  17. Solna, K., and Switzer, P. (1996), “Time Trend Estimation for a Geographic Region,” Journal of the American Statistical Association, 91, 577–589.CrossRefMathSciNetGoogle Scholar
  18. Stein, M. L. (1999), Interpolation of Spatial Data. Some Theory for Kriging, New York: Springer.MATHGoogle Scholar
  19. Stein, A., and Corsten, L. C. A. (1991), “Universal Kriging and Cokriging as a Regression Procedure,” Biometrics, 47, 575–587.CrossRefGoogle Scholar
  20. Urquhart, N. S., and Kincaid, T. M. (1999), “Designs for Detecting Trend from Repeated Surveys of Ecological Resources,” Journal of Agricultural, Biological and Environmental Statistics, 4, 404–414.CrossRefMathSciNetGoogle Scholar
  21. Wiens, D. P. (2005), “Robustness in Spatial Studies ii: Min max Design,” Environmetrics, 16, 205–217.CrossRefMathSciNetGoogle Scholar
  22. Wikle, C. K., and Royle, J. A. (1999), “Space-Time Dynamic Design of Environmental Monitoring Networks,” Journal of Agricultural, Biological, and Environmental Statistics, 4, 489–507.CrossRefMathSciNetGoogle Scholar
  23. Zhu, Z., and Stein, M. L. (2006), “Spatial Sampling Design for Prodiction with Estimated Parameters,” Journal of Agricultural, Biological, and Environmental Statistics, 11, 24–44.CrossRefGoogle Scholar
  24. Zhu, Z., and Zhang, H. (2006), “Spatial Sampling Under the Infill Asymptotic Framework,” Environmetrics, 17, 323–337.CrossRefMathSciNetGoogle Scholar
  25. Zimmerman, D. L. (2006), “Optimal Network Design for Spatial Prediction, Covariance Parameter Estimation, and Empirical Prediction,” Environmetrics, 17, 635–652.CrossRefMathSciNetGoogle Scholar

Copyright information

© International Biometric Society 2008

Authors and Affiliations

  • C. J. F. ter Braak
    • 1
    • 2
  • D. J. Brus
    • 2
  • E. J. Pebesma
    • 3
  1. 1.Wageningen UniversityWageningenNetherlands
  2. 2.Wageningen University and Research CentreWageningenNetherlands
  3. 3.Department of Physical Geography, Geosciences FacultyUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations