Map Calculus in GIS a proposal and demonstration

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

To appear in the International Journal of Geographical Information Science

Map Calculus in GIS: a proposal and demonstration

Mordechai (Muki) Haklay

Department of Geomatic Engineering University College London (UCL)

Gower Street, London, WC1E 6BT Phone: (+44) 20 7679 2745, Fax: (+44) 7380 0453, Email: m.haklay@ucl.ac.uk

Abstract

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 for this new representation. In Map Calculus, GIS layers are stored as functions,

and new layers can be created by combinations of other functions. This paper explains

the principles of Map Calculus and demonstrates the creation of function-based layers

and their supporting management mechanism. The proposal is based on Church’s

(1941) Lambda Calculus and elements of functional computer languages (such as Lisp or

Scheme).

Introduction

The use of computers for the analysis and representation of geographical information is

now, arguably, approaching its fifth decade and the representation and storage of

geographical information in computers seems to have matured and stabilised. Within the

range of different geographical abstractions, Goodchild’s overview (1993) identifies two

major groups – field models, which represent the geographical space as a continuous

“field” or “surface” (in other words, the analysis of phenomena as being continuous

across space), and object models that represent discreet entities in space. A recent GIS

textbook (Longley et al., 2001, p. 145) lists six common field representations in a GIS:

regularly-spaced sample points, irregularly-spaced sample points, rectangular grid cells

(raster), irregularly-shaped polygons, triangular irregular network (TINs) and polylines

which represent contours. Some other variations of the field model have been proposed

over the years, like Tobler’s (1976) vector fields, or combinations of models, such as

Tomlin’s (1990) storage of lineal data within a raster. As Unwin (1981) notes, the field

model is based on the mathematical concept of a scalar field, which can be represented

using the following formula:

z=f(x,y)