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

practice of using raster layers to simulate surfaces. The paper is completed by drawing

conclusions and pointing to future research directions.

Throughout this paper, the term “Map Calculus-enabled GIS” is used to describe a GIS

that has been enhanced with the required procedures and data structures to handle

function-based layers. This paper envisages Map Calculus capabilities as being an

extension to raster and vector-based representations and, while a GIS that is based on

Map Calculus is theoretically possible, the discussion in this paper will focus on a hybrid

implementation where function-based layers extend the vector and raster capabilities of a

common GIS.

Map Calculus in a GIS

A GIS that can handle Map Calculus is somewhat equivalent to mathematical or

statistical software suites (such as Matlab or SAS ). Such a system should have the

capabilities to store and calculate functions in real time, while storing information about

these functions (such as variable values) in a form that facilitates fast and efficient

computation. As previously mentioned, function-based layers are especially suited as an

alternative to the representation of continuous surfaces, which are based on global

functions, as raster layers. To understand how function-based layers works, it is best to

look at three examples – first, a simple distance function; second, a spatial interpolation

based on IDW, and finally a Digital Elevation Model (DEM).

A distance function from a given location (A) with the co-ordinates (xA,yA) is currently

represented in GIS packages as a raster layer with an arbitrary resolution, where each cell

is assigned a value that represents the distance from the centre of the pixel to A. In a

function-based layer, this can be represented by a single formula – naturally, the formula

for the Euclidian distance for any given location of B from A1:

ThisFarA=(xB xA)2+(yB yA)2 1 ThisFar is taken directly from Tomlin (1990) and while Tomlin discusses the layer as the product of

ThisFar operation, here the function itself is stored.