Cubic Map Algebra Functions for Spatio-Temporal Analysis
Mennis, Jeremy, Viger, Roland, Tomlin, C. Dana, Cartography and Geographic Information Science
The term "map algebra" was first introduced in the late 1970s (Tomlin and Berry 1979) and has since been used in loose reference to a set of conventions, capabilities, and analytical techniques that have been widely adopted for raster-based geographic information systems (GIS) (Tomlin 1990; 1991). Map algebra attempts to accommodate a wide variety of GIS applications in a clear and consistent manner by decomposing data, data processing capabilities, and data processing control techniques into elemental components that can then be recomposed with both ease and flexibility. The resulting algebra-like language is one in which single-factor map layers are treated as variables that can be transformed or combined into new variables by way of primitive operations invoked through expressions conforming to a well defined syntax. Map algebra has been incorporated in many GIS and remote sensing image processing packages, and it has been extended in areas ranging from cellular automata (Takeyama and Couclelis 1997) to environmental modeling (van Deursen 1995; Hofierka and Neteler 2001; Pullar 2001) to topographic analysis (Caldwell 2000). It is widely recognized as one of the most influential analytical frameworks for GIS-based raster data handling (Longley et al. 2001; DeMers 2003).
Like most of the analytical frameworks embodied in current GIS packages, map algebra is primarily oriented toward data that are static. Each layer is associated with a particular moment or period of time, and analytical capabilities are intended to deal with spatial relationships. In its original form, map algebra was never intended to handle spatial data with a temporal component. However, as the availability of spatio-temporal data has increased dramatically in recent years due to the growth of satellite remote sensing and other technologies, and as the sophistication of things such as video games and animation in the motion picture industry has raised popular expectations for spatio-temporal processing capabilities there has also been an increasing demand for the spatio-temporal extension of GIS.
One of the reasons for the widespread adoption of conventional map algebra for raster processing in GIS is its simple syntax and the ability to string together multiple functions to create more complex models. These features provide a simple yet powerful toolbox for raster data manipulation analysis. The inclusion of a library of temporal map algebra functions in GIS packages would be just as useful and facile for analyzing the multitude of spatio-temporal raster data now being generated. We note that despite the growing volume of research on spatio-temporal data models over the past dozen or so years (e.g., Langran 1992; Peuquet 2001), the extension of map algebra to the temporal dimension has been largely ignored by the spatio-temporal GIS research community. The present research is intended as a first step toward the development of such a temporal map algebra library by providing a conceptual foundation for temporal map algebra and the implementation of a select set of map algebra functions that may be applied to spatio-temporal data.
In the following section a framework for the extension of map algebra to the temporal dimension is described. This design is then demonstrated through a prototype implementation of certain temporal map algebra functions which we call "cube functions." Whereas conventional map algebra functions operate on data layers representing two-dimensional space, cube functions operate on data cubes representing two-dimensional space over a third-dimensional period of time. A case study is used to demonstrate how cube functions can be utilized. This case study analyzes the spatio-temporal variability of remotely sensed, southeastern U.S. vegetation character over various land covers and during different El Nino/ Southern Oscillation (ENSO) phases.
A variety of approaches for storing and analyzing spatio-temporal data in GIS have been proposed and implemented (cf. …