Faculty of Environmental Information, Keio University, 5322 Endo, Fujisawa 252, Japan

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Abstract
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An efficient integration of mathematical modelling and GIS would benefit significantly both modellers and GIS users. Thus far, however, the lack of a consistent framework capable of tying together spatial data modelling, spatial data manipulation, and spatial process modelling, has forced modellers using GIS to switch back and forth between different, often incompatible structures. This process is time-consuming and prone to inconsistencies and errors. A number of approaches have been proposed, for the most part attempting to solve the problems for particular kinds of modelling applications through the development of special GIS tools or high-level programming languages. Realising a generic modelling capability within GIS in a consistent manner is a much more difficult task. This is largely because of the very different modes of representing information in GIS on the one hand, and expressing a mathematical model on the other. While the basic form of high-level information representation in GIS is usually the map layer (or spatial object), mathematical models take the form of equations of numerical, categorical, or symbolic variables. The map layer is a natural metaphor for representing static spatial patterns and structures, but it cannot express rules and regularities seen in spatial relations and processes which are more appropriately represented by equations. From a conceptual point of view, this difference in the modes of representation also reflects two different views of space: absolute space, of which the fundamental building block is the geo-referenced item of information, and relative space, which is made up of various kinds of relationships among things and phenomena.

This research develops Geo-Algebra, a mathematical framework for supporting both dynamic and static geo-computational modelling in conjunction with GIS-based spatial data manipulation capabilities. Geo-Algebra is the mathematical generalisation of the concepts and formal structure of Map Algebra developed by Tomlin (1990), and is organised in a powerful mathematical formalism. In Geo-Algebra both static and dynamic spatial models including spatial interaction, diffusion, and land use models are formulated consistently through a limited set of generic operations on map layers which are used simultaneously for GIS data manipulation and analysis. Geo-Algebra thus overcomes the discrepancy in the modes of representation as well as their underlying concepts of space by a common representational framework for both mathematical models expressing spatial relationships and for data models of geo-referenced information, along with their modes of manipulation and transformation.

A significant advantage of the mathematical approach over other approaches to integration, such as through high-level computational languages, is the development of theoretical concepts of the most general kind, allowing it to derive general properties of these in a deductive manner. Geo-Algebra formalises and extends the notion of map mathematically into the novel concepts of relational and meta-relational maps. In a meta-relational map, the information is geo-referenced as in absolute space, but to each location is attached a relational map representing the location's relationships to other locations. Theoretically, these extensions lead to the novel concepts of space bridging the absolute and relative. As absolute and relative views of space were integrated within Geo-Algebra, absolute and relative views of space-time were also integrated into the concept of spatio-temporal relational and meta-relational maps. With these maps, it became possible to deal with arbitrary relationships among events happening in space-time. Discussions of concepts of space are often based on notions which are rich in theoretical implications but cannot be implemented directly on a computational platform as they lack formal definitions. However, the mathematical approach of Geo-Algebra stimulates and integrates methodologically the conceptual development as well as its computational realisation. In addition to dealing with formal and theoretical problems, this research also implements Geo-Algebra both in a general spatial simulation tool and as a high level computational language. Such implementations would allow modellers and GIS users to apply the theoretical results of Geo-Algebra to the solution of real world problems.