The Nature of Geographic Information

5. Topology


Topology is different from topography. (You’d be surprised how often these terms get mixed up.) In Chapter 2, you read about the various ways that absolute positions of features can be specified in a coordinate system, and how those coordinates can be projected or otherwise transformed. Topology refers to the relative positions of spatial features. Topological relations among features — such as containment, connectivity, and adjacency—don’t change when a dataset is transformed. For example, if an isolated node (representing a household) is located inside a face (representing a congressional district) in the MAF/TIGER database, you can count on it remaining inside that face no matter how you might project, rubber-sheet, or otherwise transform the data. Topology is vitally important to the Census Bureau, whose constitutional mandate is to accurately associate population counts and characteristics with political districts and other geographic areas.

As David Galdi (2005) explains in his white paper “Spatial Data Storage and Topology in the Redesigned MAF/TIGER System,” the “TI” in TIGER stands for “Topologically Integrated.” This means that the various features represented in the MAF/TIGER database—such as streets, waterways, boundaries, and landmarks (but not elevation!)—are not encoded on separate “layers.” Instead, features are made up of a small set of geometric primitives—including 0-dimensional nodes and vertices, 1-dimensional edges, and 2-dimensional faces—without redundancy. That means that where a waterway coincides with a boundary, for instance, MAF/TIGER represents them both with one set of edges, nodes, and vertices. The attributes associated with the geometric primitives allow database operators to retrieve feature sets efficiently with simple spatial queries. The separate feature-specific TIGER/Line Shapefiles published at the county level (such as point landmarks, hydrography, Census block boundaries, and the "All Lines" file you are using in the multi-part "Try This") were extracted from the MAF/TIGER database in that way. Notice, however, that when you examine a hydrography shapefile and a boundary shapefile, you will see redundant line segments where the features coincide. That fact confirms that TIGER/Line Shapefiles, unlike the MAF/TIGER database itself, are not topologically integrated. Desktop computers are now powerful enough to calculate topology “on the fly” from shapefiles or other non-topological data sets.  However, the large batch processes performed by the Census Bureau still benefit from the MAF/TIGER database’s persistent topology.

MAF/TIGER’s topological data structure also benefits the Census Bureau by allowing it to automate error-checking processes. By definition, features in the TIGER/Line files conform to a set of topological rules (Galdi 2005):

  1. Every edge must be bounded by two nodes (start and end nodes).
  2. Every edge has a left and right face.
  3. Every face has a closed boundary consisting of an alternating sequence of nodes and edges.
  4. There is an alternating closed sequence of edges and faces around every node.
  5. Edges do not intersect each other, except at nodes.

Compliance with these topological rules is an aspect of data quality called logical consistency. In addition, the boundaries of geographic areas that are related hierarchically—such as blocks, block groups, tracts, and counties—are represented with common, non-redundant edges. Features that do not conform to the topological rules can be identified automatically, and corrected by the Census geographers who edit the database. Given that the MAF/TIGER database covers the entire U.S. and its territories, and includes many millions of primitives, the ability to identify errors in the database efficiently is crucial.

So how does topology help the Census Bureau assure the accuracy of population data needed for reapportionment and redistricting? To do so, the Bureau must aggregate counts and characteristics to various geographic areas, including blocks, tracts, and voting districts. This involves a process called “address matching” or “address geocoding” in which data collected by household is assigned a topologically-correct geographic location. The following pages explain how that works.

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