Map Algebra

Map algebra (also called cartographic modeling) is a set of operators, operations, and rules governing the manipulation and analysis of spatial data using the raster data model. The raster data [Page 1825]model represents geographic objects and fields by tessellating space as cells, each of which is associated with a value indicating the identity, rank, or quantity of the phenomenon that occupies it. Map algebra treats raster layers as variables in algebra and calculates new raster layers by applying one or a sequence of operations using expressions.

Map algebra decomposes spatial analytic tasks into elementary operations that can be flexibly recombined. Elementary operations in map algebra are grouped into local, focal, and zonal categories based on the spatial scope of the operations. Local operations calculate on the cell-by-cell bases values in the output raster layer using cell values of the input raster layers at the same location. Focal operations calculate output cell values using the input cell values within their respective neighborhoods. A neighborhood is a set of cells that bear certain relationships to the center of the neighborhood. Zonal operations calculate output cell values using the input cell values within the same zones. A zone is a set of cells sharing the same measurement value. A zone may comprise one contiguous grouping of cells or of multiple groupings of cells.

Map algebra operations have been expanded on since their inception in the late 1970s and early 1980s. While the original spatial scope was confined to two-dimensional space, it has been later extended to three-dimensional spaces, in which two-dimensional cells are replaced by three-dimensional voxels. A similar extension was also made for spatiotemporal data sets, where the third dimension is time. Map algebra operations for vector fields, where cell values are vectors (magnitude plus direction) rather than scalar measurements, were also developed. Spatial analysis in the vector data model does not have an algebra counterpart, mainly because the analysis units in the vector data model are not spatially uniform. Recent research, however, has shown that the map algebra approach is also viable for the vector data model.

Although map algebra is a powerful spatial analysis language widely used in spatial analysis with geographic information systems (GIS), its operations do not comprise a set of atomic operations on which all complex raster analyses can be based. Also, the classification of the operations into local, focal, and zonal groups is rather arbitrary. The search for a generic set of operations for both raster and vector data remains a challenge for researchers in GIS.