Mathematical Geosciences

, Volume 40, Issue 1, pp 101–128 | Cite as

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

  • Pierre Goovaerts


This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the point-support model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.

This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a naïve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.


Simulation Change of support Choropleth map Disaggregation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Armstrong M (1998) Basic linear geostatistics. Springer, Berlin, 172 p Google Scholar
  2. Avruskin GA, Jacquez GM, Meliker JR, Slotnick MJ, Kaufmann AM, Nriagu JO (2004) Visualization and exploratory analysis of epidemiologic data using a novel space time information system. Int J Health Geogr 3(26). doi: 10.1186/1476-072X-3-26
  3. Barabás N, Goovaerts P, Adriaens P (2001) Geostatistical assessment and validation of uncertainty for three-dimensional dioxin data from sediments in an estuarine river. Environ Sci Technol 35(16):3294–3301 CrossRefGoogle Scholar
  4. Berke O (2004) Exploratory disease mapping: kriging the spatial risk function from regional count data. Int J Health Geogr 3(18). doi: 10.1186/1476-072X-3-18
  5. Chiles JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York, 720 p Google Scholar
  6. Christakos G, Lai J (1997) A study of the breast cancer dynamics in North Carolina. Soc Sci Med 45(10):1503–1517 CrossRefGoogle Scholar
  7. Christakos G, Serre ML (2000) Spatiotemporal analysis of environmental exposure-health effect associations. J Expo Anal Environ Epidemiol 10(2):168–187 CrossRefGoogle Scholar
  8. Collins JB, Woodcock CE (1996) Explicit consideration of multiple landscape scales while selecting spatial resolutions. In: Mowrer HT, Czaplewski RL, Hamre RH (eds) Spatial accuracy assessment in natural resources and environmental sciences: second international symposium. Technical report RM-GTR-277, United States department of Agriculture, Fort Collins, Colorado, pp 121–128 Google Scholar
  9. Collins JB, Woodcock CE (1999) Geostatistical estimation of resolution-dependent variance in remotely sensed images. Photogramm Eng Remote Sensing 65(1):41–50 Google Scholar
  10. Cressie N (1993) Statistics for spatial data. Wiley, New York, 900 p Google Scholar
  11. Croner CM, De Cola L (2001) Visualization of disease surveillance data with geostatistics. Presented at UNECE (United Nations Economic Commission for Europe) work session on methodological issues involving integration of statistics and geography, Sept 2001, Tallinn.
  12. Curran PJ, Atkinson PM (1999) Issues of scale and optimal pixel size. In: Stein A, van der Meer F, Gorte B (eds) Spatial statistics for remote sensing. Kluwer Academic, Dordrecht, pp 115–133 Google Scholar
  13. Deutsch CV, Journel AG (1998) GSLIB: Geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York, 369 p Google Scholar
  14. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York, 483 p Google Scholar
  15. Goovaerts P (2005) Geostatistical analysis of disease data: estimation of cancer mortality risk from empirical frequencies using Poisson kriging. Int J Health Geogr 4(31). doi: 10.1186/1476-072X-4-31
  16. Goovaerts P (2006) Geostatistical analysis of disease data: accounting for spatial support and population density in the isopleth mapping of cancer mortality risk using area-to-point Poisson kriging. Int J Health Geogr 5(52). doi: 10.1186/1476-072X-5-52
  17. Goovaerts P, Jacquez GM, Greiling D (2005) Exploring scale-dependent correlations between cancer mortality rates using factorial kriging and population-weighted semivariograms: a simulation study. Geogr Anal 37(2):152–182 CrossRefGoogle Scholar
  18. Gotway CA, Young LJ (2002) Combining incompatible spatial data. J Am Stat Assoc 97(459):632–648 CrossRefGoogle Scholar
  19. Gotway CA, Young LJ (2004) A geostatistical approach to linking geographically-aggregated data from different sources. Technical report # 2004-012, Department of Statistics, University of Florida Google Scholar
  20. Gotway CA, Young LJ (2005) Change of support: an inter-disciplinary challenge. In: Renard Ph, Demougeot-Renard H, Froidevaux R (eds) geoENV V—Geostatistics for Environmental Applications. Springer, Berlin, pp 1–13 CrossRefGoogle Scholar
  21. Houlding S (1999) Direct volume estimation—a geostatistical technique for mine planning and grade control. Comput Geosci 25(10):1113–1123 CrossRefGoogle Scholar
  22. Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New York, 561 p Google Scholar
  23. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London, 600 p Google Scholar
  24. Kelsall J, Wakefield J (2002) Modeling spatial variation in disease risk: a geostatistical approach. J Am Stat Assoc 97(459):692–701 CrossRefGoogle Scholar
  25. Kupfersberger H, Deutsch CV, Journel AG (1998) Deriving constraints on small-scale variograms due to variograms of large-scale data. Math Geol 30(7):837–852 CrossRefGoogle Scholar
  26. Kyriakidis P (2004) A geostatistical framework for area-to-point spatial interpolation. Geogr Anal 36(2):259–289 CrossRefGoogle Scholar
  27. Mockus A (1998) Estimating dependencies from spatial averages. J Comput Graph Stat 7(4):501–513 CrossRefGoogle Scholar
  28. Monestiez P, Dubroca L, Bonnin E, Durbec JP, Guinet C (2006) Geostatistical modelling of spatial distribution of Balenoptera physalus in the Northwestern Mediterranean Sea from sparse count data and heterogeneous observation efforts. Ecol Model 193(3–4):615–628 CrossRefGoogle Scholar
  29. Oliver MA, Webster R, Lajaunie C, Muir KR, Parkes SE, Cameron AH, Stevens MCG, Mann JR (1998) Binomial cokriging for estimating and mapping the risk of childhood cancer. IMA J Math Appl Med Biol 15(3):279–297 CrossRefGoogle Scholar
  30. Pardo-Iguzquiza E (1999) VARFIT: a Fortran-77 program for fitting variogram models by weighted least squares. Comput Geosci 25(3):251–261 CrossRefGoogle Scholar
  31. Pickle LW, Mungiole M, Jones GK, White AA (1999) Exploring spatial patterns of mortality: the new Atlas of United States mortality. Stat Med 18(23):3211–3220 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geology 2007

Authors and Affiliations

  1. 1.BioMedwareAnn ArborUSA

Personalised recommendations