- A Geostatistical Framework for Area-to-Point Spatial Interpolation
The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought-after point predictions. In particular, it is proved that appropriate modeling of all area-to-area and area-to-point covariances required by the geostatistical framework yields coherent (mass-preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal-average (or areal-total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area-to-point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler's smooth pycnophylactic interpolation, on a quasi-infinite domain without non-negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free-space Green's functions (linear in 1-D or logarithmic in 2-D) of Poisson's partial differential equation. In lieu of a formal case study, several 1-D examples are given to illustrate pertinent concepts.
Going from one spatial support (domain informed by each measurement or unknown value) to another is of critical importance to numerous scientific disciplines. Coarse spatial resolution predictions of general circulation models, for example, need to be downscaled to the watershed level (or even finer in the case of spatially distributed [End Page 259] models) for hydrologic impact assessment studies. Similarly, socioeconomic variables reported on census tracts need to be downscaled to smaller regions for detailed modeling. Scaling issues continue to be a critical and vibrant research topic in Geography; a recent review of such issues and some of their geostatistical solutions can be found in Atkinson and Tate (2000).
Area-to-point interpolation is a particular case of change of support, whereby areal data are used to predict point values; these points need not lie on a regular grid or comprise a surface. For a recent comprehensive review of existing approaches for change of support, see Gotway and Young (2002). Alternatively, area-to-point interpolation can be viewed as a special case of areal interpolation (Haining 2003), whereby both source data and target values pertain to the same spatial attribute and are defined respectively over areal units and points. Routine applications of area-to-point interpolation in geography (Lam 1983), however, tend to ignore several critical issues: (i) the explicit account of the different supports of the areal data and sought-after point predictions, for example, the areal data are often incorrectly collapsed into their respective polygon centroids; (ii) the coherence of predictions, for example, the areal average of point predictions within any area comprising an areal average datum, should be equal to that datum (if the latter is assumed error free); and (iii) the uncertainty in the resulting point predictions.
The geostatistical framework for area-to-point prediction presented in this paper is a special case of the original development of Kriging, which was formulated as the spatial prediction of an areal value using available areal data of the same or different variable (Matheron 1971; Journel and Huijbregts 1978; Gotway and Young 2002). Most geosta-tistical textbooks and applications regarding change of support, however, address only the problem of point-to-area interpolation via the use of block Kriging (Isaaks and Srivastava 1989; Cressie 1993; Wackernagel 1995; Goovaerts 1997). This is partially a consequence of the mining practice, where most of the original geostatistical developments...