A formal ontology for categorical gradation in geographic space and for objects with indeterminate boundaries, together with computational representations, is needed to extend the capabilities of GIS and geospatial computation. Standard GIS data models deal with either well-defined geospatial objects with distinct boundaries, or with single-valued fields of continuous numerical variation. But many geographic phenomena do not fit into either of these models, either in fact or in their conceptualization by people. The delineation of spatial entities themselves is fundamental to any subsequent analysis.  This problem has been recognized before Chrisman, 1982; Goodchild and Dubuc, 1987; Burrough and Frank, 1996), Although some solutions were proposed in the Burrough and Frank book and elsewhere, almost a decade later GIS users do not have solutions or ready-to-hand methods.. This may be the most significant gap in the representational power of current GIS software, and thus we propose that it should be highlighted as a UCGIS Research Challenge.

The problem of categorical gradation is not identical to the problem of objects with indeterminate boundaries, but the two problems have a hand-in-glove relationship. Solutions to the gradation problem will automatically provide some solutions to the problem of objects with indeterminate boundaries, since being part of an object is itself a categorical spatial variable.

Current representations of spatial entities can be classified into object- and field-based representations.  Object-based representations include points, lines and polygons, while fields can be represented by raster grids, TINs or polygons coverages. It is our contention that these representations are not adequate for representing many kinds of interesting geographical entities.

The delineation of a spatial entity often involves translation from a conceptual entity that is not initially defined in the spatial domain.  Particularly in the domain of physical geography, many entities are defined either by using process models or by identifying typical, defining characteristics.  Such definitions are subsequently applied to the geographic domain in order to create spatial entities.  For example, a desert may be defined in terms of rainfall or habitat suitability for typical “desert” species, and its spatial extent determined through a process of analyzing the distribution of environmental characteristics throughout a spatial domain. This process of deriving a spatial entity from a conceptual one will involve a loss of information and/or unnecessary generalization if the available spatial entity types do not conform well to the conceptual entity type.  Entities with indistinct boundaries also are found in studies of cognition and society, in the form of neighborhoods, regions, and places. Spatial gradation, or entities with graded boundaries, may be seen as the spatial reflection of the well-established fact from cognitive science that categories typically show cores of prototypical members, and gradients of similarity based on family resemblance to the core. Ontological and representational formalisms for graded spatial entities are thus necessary to conform to the conceptual entities utilized in the environmental and social sciences.

Priority Areas for Research. Research is needed to characterize and categorize different types of gradation. Some of the issues to be address include uncertainty vs. gradation (probability, fuzziness, etc.); individual fuzzy objects vs. continuous-valued classification of a spatial domain; rules of mutual exclusivity between fuzzy objects; studies of how people delimit objects with indeterminate boundaries, or reason about them in various contexts; and development of queries and other GIS functions involving categorical coverages with gradation, and/or objects with indeterminate boundaries.

Gradation Working Group at Buffalo:

Barry Kronenfeld <bjk3@acsu.buffalo.edu>, David Mark <dmark@geog.buffalo.edu>, Barry Smith <phismith@buffalo.edu>, Jeff Brunskill <jeffb@eng.buffalo.edu>, Chen-Chieh Feng <cfeng@geog.buffalo.edu>, Gaurav Sinha <gsinha@acsu.buffalo.edu>, Alexandre Sorokine <sorokine@geog.buffalo.edu>

References:

Burrough, P. A., and Frank, A. U., editors, 1996. Geographic Objects with Indeterminate Boundaries. Bristol, PA: Taylor & Francis Inc.,

Chrisman, N., 1982.  A theory of cartographic error and its measurement in digital data bases.  Proceedings, Fifth International Symposium on Computer-Assisted Cartography (Auto Carto 5), Falls Church, Virginia: ASPRS and ACSM, 159-168.

Goodchild M. F. and Dubuc, O., 1987. A model of error for choropleth maps with applications to geographic information systems.  Proceedings, Auto Carto 8.  Falls Church, VA: ASPRS/ACSM, 165‑174.