SpatialSTEM Has Deep Mathematical Roots

Issue Date: January - 2012, Posted On: 2/6/2012 Joseph Berry

Beyond Mapping By Joseph Berry Joseph Berry is a principal in Berry & Associates, consultants in GIS technology. He can be reached via e-mail at jkberry@du.edu. Recently my interest has been captured by a new arena and expression for the contention that "maps are data," spatialSTEM (sSTEM for short), as a means for redirecting education in general and GIS education in particular. I suspect GeoWorld readers have heard of STEM (Science, Technology, Engineering and Mathematics) and the educational crisis that puts U.S. students well behind many other nations in these quantitatively based disciplines. Although Googling around the globe makes for great homework in cultural geography, it doesn’t advance quantitative proficiency, nor does it stimulate the spatial-reasoning skills needed for problem solving. A lot of folks, ranging from Fareed Zakaria of Time and CNN to Bill Gates to U.S. President Barack Obama, are looking for ways that the United States can recapture its leadership in the quantitative fields. That’s the premise of spatialSTEM: "maps are numbers first, pictures later," and we do mathematical things to mapped data for insight and better understanding of spatial patterns and relationships within decision-making contexts. Structural Differences Figure 1 outlines the important components of map analysis and modeling within a mathematical structure that has been in play since the 1980s (see "Author’s note," page 11). Of the three disciplines forming geotechnology (remote sensing, GIS and GPS), GIS is at the heart of converting mapped data into spatial information. There are two primary approaches for generating this information: mapping/geo-query and map analysis/modeling. Figure 1. A conceptual overview describes the SpatialSTEM framework. The major differences between the two approaches lie in the structuring of mapped data and their intended use. Mapping and geo-query use a data structure akin to manual mapping in which discrete spatial objects (points, lines and polygons) form a collection of independent, irregular features to characterize geographic space. For example, a water map might contain categories of spring (points), stream (lines) and lake (polygons), with the features scattered throughout a landscape. Map analysis and modeling procedures, however, operate on continuous map variables (i.e., map surfaces) composed of thousands of map values stored in georegistered matrices. Within this context, a water map no longer contains separate and distinct features, but is a collection of adjoining grid cells with a map value indicating the characteristic at each location (e.g., spring = 1, stream = 2 and lake = 3). Vectors and Rasters Figure 2 illustrates two broad types of digital maps, formally termed vector for storing discrete spatial objects and raster for storing continuous map surfaces. In vector format, spatial data are stored as two linked data tables. A "spatial table" contains all the X,Y coordinates defining a set of spatial objects that are grouped by object-identification numbers. For example, the location of the forest polygon identified on the figure’s left side is stored as ID#32, followed by an ordered series of X,Y coordinate pairs delineating its border (connect the dots). Figure 2. A basic data structure is described for vector and raster map types. In a similar manner, the ID#s and X,Y coordinates defining the other cover-type polygons are sequentially listed in the table. The ID#s link the spatial table (where) to a corresponding "attribute table" (what) containing information about each spatial object as a separate record. For example, polygon ID#31 is characterized as a mature 60-year-old Ponderosa Pine (PP) forest stand. The right side of Figure 2 depicts raster storage of the same cover-type information. Each grid space is assigned a number corresponding to the dominant cover type present—the "cell position" in the matrix determines the location (where), and the "cell value" determines the characteristic/condition (what). Fundamental Concepts Figure 3 depicts the fundamental concepts supporting raster data. As a comparison between vector and raster data structures, consider how the two approaches represent an elevation surface. In vector, contour lines are used to identify lines of constant elevation, and contour interval polygons are used to identify specified ranges of elevation. Although contour lines are exacting, they fail to describe the intervening surface configuration. Figure 3. Organizational considerations and terminology are necessary for grid-based mapped data. Contour intervals describe the interiors, but they overly generalize the actual "ups and downs" of the terrain into broad ranges that form an unrealistic stair-step configuration (center-left portion of Figure 3). As depicted in the figure, rock climbers would need to summit each of the contour-interval "200-foot cliffs" rising from presumed flat mesas. Similarly, surface-water flow presumably would cascade like waterfalls from each contour interval "lake" like a Spanish multi-tiered fountain. The remainder of Figure 3 depicts the basic raster/grid organizational structure. Each grid map is termed a map layer, and a set of georegistered layers constitutes a map stack. All the map layers in a project conform to a common analysis frame with a fixed number of rows and columns at a specified cell size that can be positioned anywhere in geographic space. As in the case of the elevation surface in the lower-left portion of Figure 3, a continuous gradient is formed with subtle elevation differences that allow hikers to step from cell to cell while considering relative steepness. In addition, surface water can be mapped to sequentially stream from a location to its steepest downhill neighbor, thereby identifying a flow path. The underlying concept of this data structure is that grid cells for all map layers precisely coincide, and by simply accessing map values at a row/column location, a computer can "drill down" through the map layers, noting their characteristics. Similarly, noting the map values of surrounding cells identifies the characteristics within a location’s vicinity on a given map layer or set of map layers. In fact, the preponderance of spatial data is easily and best represented as grid-based continuous map surfaces that are preconditioned for use in map analysis and modeling. The computer does the heavy lifting of computation—what’s needed is a new generation of creative minds that goes beyond mapping to "thinking with maps" within this less-familiar, quantitative framework—a SpatialSTEM environment. Author’s Note: My involvement in map analysis/modeling began in the 1970s with doctoral work in computer-assisted analysis of remotely sensed data a couple of years before civilian satellites. The extension from digital-imagery classification using multivariate statistics and pattern-recognition algorithms in the 1970s to a comprehensive grid-based mathematical structure for all forms of mapped data in the 1980s was a natural evolution. See www.innovativegis.com, selecting "Online Papers" for a link to a 1986 paper on "A Mathematical Structure for Analyzing Maps" that serves as an early introduction to a comprehensive framework for map analysis/modeling. source: www.geoplace.com