This paper provides a new representation for fields (continuous surfaces) in Geographical Information Systems (GIS), based on the notion of spatial functions and their combinations. Following Tomlin’s (1990) Map Algebra, the term “Map Calculus” is used

stored as an array of functions, where each function is stored with a direct reference to

the domain over which it is defined. This will require the GIS to store the tessellation as

part of the function, with a template for the function family. As noted, TIN

representations of fields are implementing such an approach: a set of tessellated triangles,

with linear functions that are defined across them. How much a function-based

representation improves the current methods of DEM representation is a matter for

research.

In general, a Map Calculus-enabled GIS will hold the templates for various spatial

functions, which can be local (like distance), neighbourhood and connectivity functions

(Tomlin, 1990; Samet, 1995). Such a template, as in the example above, will enable the

storage and calculation of distance from a given point, a set of points, lines or polygons.

More sophisticated templates will be needed for spatial analysis techniques such as Bi-

cubic interpolation, Kriging (Oliver & Webster, 1990), or Geographically Weighted

Regression (Brunsdon, Fotheringham & Charlton, 1998), although the development of

the latter functions might be challenging. These templates will be used to instantiate a

function which assigns values to some of the variables and prepares the system to rapidly

calculate the surface value for a required location or locations. Thus, they might include

references to the set of points that are being used as the source of spatial interpolation, as

well as other parameters that are specific to each spatial function, such as the minimum

number of points that participate in the interpolation.

A Map Calculus-enabled GIS should also support mathematical operations between

function-based layers and between function-based layers and other representations of

geographical objects, such as rasters. The user should be able to construct sophisticated

models by stringing together layers and by setting a variety of mathematical operations

which can operate on a single functional layer (unary operators) or between layers. Of

course, the development of operations on functional layers follows the work of Tomlin

(1990) and Berry (1993a) on Map Algebra and Cartographic Modelling. Function-based

layer are very efficient in their storage and even with complex and sophisticated

functions, the storage space required is significantly smaller than that of multiple rasters

layers, due to the symbolic nature of functional representations. While the cost of digital

storage has reduced dramatically in recent years (Figure 1), the storage and management

of multiple rasters is still a technical and practical problem. ArcGIS, for example, cannot