Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong M (1998) Basic linear geostatistics. Springer, Berlin, 172 p
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
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
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
Chiles JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York, 720 p
Christakos G, Lai J (1997) A study of the breast cancer dynamics in North Carolina. Soc Sci Med 45(10):1503–1517
Christakos G, Serre ML (2000) Spatiotemporal analysis of environmental exposure-health effect associations. J Expo Anal Environ Epidemiol 10(2):168–187
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
Collins JB, Woodcock CE (1999) Geostatistical estimation of resolution-dependent variance in remotely sensed images. Photogramm Eng Remote Sensing 65(1):41–50
Cressie N (1993) Statistics for spatial data. Wiley, New York, 900 p
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. http://www.unece.org/stats/documents/2001/09/gis/25.e.pdf
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
Deutsch CV, Journel AG (1998) GSLIB: Geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York, 369 p
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York, 483 p
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
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
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
Gotway CA, Young LJ (2002) Combining incompatible spatial data. J Am Stat Assoc 97(459):632–648
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
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
Houlding S (1999) Direct volume estimation—a geostatistical technique for mine planning and grade control. Comput Geosci 25(10):1113–1123
Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New York, 561 p
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London, 600 p
Kelsall J, Wakefield J (2002) Modeling spatial variation in disease risk: a geostatistical approach. J Am Stat Assoc 97(459):692–701
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
Kyriakidis P (2004) A geostatistical framework for area-to-point spatial interpolation. Geogr Anal 36(2):259–289
Mockus A (1998) Estimating dependencies from spatial averages. J Comput Graph Stat 7(4):501–513
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
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
Pardo-Iguzquiza E (1999) VARFIT: a Fortran-77 program for fitting variogram models by weighted least squares. Comput Geosci 25(3):251–261
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goovaerts, P. Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units. Math Geosci 40, 101–128 (2008). https://doi.org/10.1007/s11004-007-9129-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-007-9129-1