Map Calculus in GIS a proposal and demonstration(2)

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

in which for every location, denoted by the co-ordinates x and y, there is a function f that

yields the value z. The function f is a mathematical function which operates on different

locations; it can also be termed a “spatial function”. The extension of this form to a

higher dimension is possible with the general form of:

z=f(location)

where location describes a set of co-ordinates.

There are two main classes of functional descriptions of geographical phenomena:

piecewise and global. Piecewise functions are defined over sub-domains, which are

constructed through a tessellation of the field. A good example for this is Triangular

Irregular Network (TIN), where a set of linear functions describe the shape of the area.

Piecewise functions are characterised by a set of constraints and limitations on the

function’s continuity at sub-domain boundaries. Global functions are defined in the same

way across the domain. The Euclidean distance function from a point or the Inverse

Distance Weighting (IDW) function are examples for global functions. The same

function is used for every location in space.

In recent years, there has been new interest in radically different approaches to the

representation of fields in GIS. While the foundations of these approaches can be traced

to the 1970s and earlier, the capabilities of current computers and software enable their

implementation. For example, Wood (1998) and Rana and Morley (2002) have made

advances in the consideration of surface networks to represent topography based on

concepts suggested by Pfaltz (1976); Cheng and Molenaar (1999) explored the

representation of fuzzy objects in a GIS, linking their work to Burrough (1989); while

Frank (1999) has suggested the development of algebra to analytically manipulate

geographical objects, an approach that is somewhat similar to the one suggested in the

ROSE algebra (Guting & Schneider, 1995). Such approaches and suggestions for

geographical representations may deal with a specific domain of geographical objects, for

example the relationship between surface networks and topography, or deal with the

more general representational issues, as in the case of the ROSE algebra, and the

representation of fuzzy objects. On a more theoretical level, Miller and Wentz (2003)

made the explicit link between representational frameworks and their use in GIS and

spatial analysis. In other disciplines that deal with solutions to partial differential

equations (such as oceanography or atmospheric science), fields are represented using