Transcript Map Stack
A Mathematical Structure for Map Analysis/Modeling Geotechnology Technological Tool Mapping/Geo-Query (Discrete, Spatial Objects) RS – GIS – GPS (Continuous, Map Surfaces) Analytical Tool Map Analysis/Modeling Geo-registered Analysis Frame Matrix Map Stack “Map-ematics” of Numbers Maps as Data, not Pictures Vector & Raster — Aggregated & Disaggregated Qualitative & Quantitative …organized set of numbers Spatial Analysis Operations Grid-based Map Analysis Spatial Statistics Operations Toolbox GISer’s Perspective: GISer’s Perspective: Reclassify and Overlay Distance and Neighbors Surface Modeling Spatial Data Mining Mathematician’s Perspective: Statistician’s Perspective: Basic GridMath & Map Algebra Advanced GridMath Map Calculus Map Geometry Plane Geometry Connectivity Solid Geometry Connectivity Unique Map Analytics The SpatialSTEM Framework Traditional math/stat procedures can be extended into geographic space to stimulate those with diverse backgrounds and interests for… “thinking analytically with maps” Basic Descriptive Statistics Basic Classification Map Comparison Unique Map Statistics Surface Modeling Advanced Classification Predictive Statistics Spatial Analysis Operations (Math Examples) Advanced Grid Math — Math, Trig, Logical Functions Map Calculus — Spatial Derivative, Spatial Integral Spatial Derivative MapSurface 2500’ …is equivalent to the slope of the tangent plane at a location Slope draped over MapSurface 500’ Surface Fitted Plane 65% SLOPE MapSurface Fitted FOR MapSurface_slope 0% Curve The derivative is the instantaneous “rate of change” of a function and is equivalent to the slope of the tangent line at a point Dzxy Elevation ʃ Districts_Average Elevation Spatial Integral Advanced Grid Math …summarizes the values on a surface for specified map areas (Total= volume under the surface) Surface Area S_Area= Fn(Slope) …increases with increasing inclination as a Trig function of the cosine of the slope angle COMPOSITE Districts WITH MapSurface Average FOR MapSurface_Davg MapSurface_Davg S_area= cellsize / cos(Dzxy Elevation) The integral calculates the area under the curve for any section of a function. Surface Curve (Berry) Spatial Statistics (Linking Data Space with Geographic Space) Roving Window (weighted average) Geo-registered Sample Data Spatial Distribution Spatial Statistics Discrete Sample Map Non-Spatial Statistics Continuous Map Surface Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data (maps the variation). Standard Normal Curve Average = 22.6 In Geographic Space, the typical value forms a horizontal plane implying the average is everywhere to form a horizontal plane StDev = 26.2 Histogram (48.8) 10 20 30 40 50 Numeric Distribution (Berry) X + 1StDev = 22.6 + 26.2 = In Data Space, a standard normal curve can be fitted to the data to identify the “typical value” (average) 0 …lots of NE locations exceed Mean + 1Stdev 60 70 80 Unusually high values X= 22.6 +StDev Average 48.8 Grid Data Structure A Grid Map consists of a matrix of numbers with a value indicating the characteristic/condition at each grid cell location— …forming a set of geo-registered Map Layers or “Map Stack” Vector Contour Lines Raster (discrete) (continuous) Elevation Contour Interval 2300-2500’ Intervals Grid 3D Surface (stair steps) Analysis Frame The Analysis Frame provides consistent “parceling” needed for map analysis and extends discrete Points, Lines and Polygons… Col 3, Row 22 …to continuous Map Surfaces Map MapStack Stack 3D Surface (gradient) Data listing for a Map Stack Col 1, Row 1 Origin Drill-down