GeoWorld February 2012

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cell calculates the surface area of the "inclined plane" at each grid location. The difference between planimetric area represented by traditional maps and surface area based on terrain steepness can be dramatic and greatly affect the characterization of "catchment areas" in envi- ronmental and engineering models of surface runoff. Map calculus expresses such functions as the deriva- tive and integral within a spatial context. The derivative traditionally identifies a measure of how a mathematical function changes as its input changes by assessing the slope along a curve in 2-D abstract space. The spatial equivalent calculates a "slope map" de- picting the rate of change in a continuous map variable in 3-D geographic space. For an elevation surface, slope depicts the rate of change in elevation. For an accumula- tion cost surface, its slope map represents the rate of change in cost (i.e., a marginal cost map). For a travel- time accumulation surface, its slope map indicates the relative change in speed, and its aspect map identifies the direction of optimal movement at each location. Also, the slope map of an existing topographic slope map (i.e., second derivative) will characterize surface roughness (i.e., areas where slope itself is changing). Traditional calculus identifies an integral as the net signed area of a region along a curve expressing a mathematical function. In a somewhat analogous procedure, areas under portions of continuous map surfaces can be characterized. In the spatial integral, the net sum of the numeric values for portions of a continuous map surface (3-D) is calculated in a manner comparable to calculating the area under a curve (2-D). Traditional geometry defines distance as "the shortest straight line between two points" and rou- tinely measures it using the Pythagorean Theorem. Map geometry extends the concept of distance to simple proximity by relaxing the requirement of just "two points" for distances to all locations surround- ing a point or other map feature, such as a road. A further extension involves effective proximity, which relaxes the "straight line" to consider absolute and relative barriers to movement. For example, effec- tive proximity might consider just uphill locations along a road or a complex set of variable hiking conditions that impede movement from a road as a function of slope, cover type and water barriers. The result is that the "shortest but not necessarily straight distance" is assigned to each grid location. Because a straight-line connection can't be assumed, optimal path routines in plane-geometry connectivity (2-D space) are needed to identify the actual shortest routes. Solid-geometry connectivity (3-D space) involves line-of-sight connections that identify visual exposure among locations. A final class of operations involves unique map analytics, such as size, shape, intactness and contiguity of map features. Figure 2. An example demonstrates basic grid math and algebra. Grid-based map analysis takes us well beyond traditional mapping … as well as taking us well beyond traditional procedures and paradigms of mathematics. The next installment of spatialSTEM discussion considers the extension of traditional statistics to spatial statistics. Author's Notes: A table of Web links to further readings on the grid-based map analysis/modeling concepts, terminology, considerations and procedures described in this three-part series on spatialSTEM is posted at basis/MapAnalysis/MA_Intro/sSTEM/sSTEMreading.htm. Figure 3. Several graphics describe spatial-analysis operations. FEBRUAR Y 2O12 / WWW . GEOPLA CE . COM 11

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