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Chapter 4

Chapter 4. Part B: Distance and directional operations. Distance computations. Projected coordinates – Euclidean Spherical coordinates – spherical or ellipsoidal computations Problem areas: Planar measures over large distances Surface distances (3D/terrain distance) Network distances

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Chapter 4

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  1. Chapter 4 Part B: Distance and directional operations www.spatialanalysisonline.com

  2. Distance computations • Projected coordinates – Euclidean • Spherical coordinates – spherical or ellipsoidal computations • Problem areas: • Planar measures over large distances • Surface distances (3D/terrain distance) • Network distances • Variable cost/friction effects • Transects (single or multi-part) www.spatialanalysisonline.com

  3. Distance computations • Terrain distances – cross section view www.spatialanalysisonline.com

  4. Distance computations • Distance, measure and metric • Distance: set of distinct objects plus some real-valued measure, dij, of separation between object pairs, i and j • Metric: formal (mathematical) definition: • dij>0 if ij (distinction/separation) • dij=0 if i=j (co-location/equivalence) • dij+djk≥dik (triangle inequality) • dij=dji(symmetry) www.spatialanalysisonline.com

  5. Distance computations • Metrics and geospatial analysis • Objects may not be truly point-like/distinct • Triangle inequality may not hold • Symmetry condition may not hold • Alternative measures • Ellipsoidal (Vincenty algorithm) • Lp metrics • Network distance • Grid distance www.spatialanalysisonline.com

  6. Distance computations • Cost distance • Cost – time, effort/friction, generalised costs • Cost surfaces and grids • Procedures • Accumulated Cost Surface (ACS) – spread algorithms • Distance Transform (DT) – scanning algorithms www.spatialanalysisonline.com

  7. Distance computations • ACS – simplified version • Select start point – current position • Take Queen’s move (8-point) grid steps • Accumulate cost x distance (1 or 1.414 units) • Cost often ‘shared’ 50:50 between cells • Select cell with least accumulated cost and move current position to this cell and repeat – record list of visited cells for path information • ACS – generalised • Extend above to a spread process (all directions) • Cell entries are least accumulated cost at each stage www.spatialanalysisonline.com

  8. Distance computations • ACS – example – ArcGIS Spatial Analyst • Create a source grid with 0s in source cells and -1 elsewhere • Create a cost grid with every cell assigned a cost or friction value • Execute the ACS procedure, tracking paths • Define a target grid (as per source grid) • Generate least cost paths from source(s) to target(s) using tracked paths www.spatialanalysisonline.com

  9. Distance computations • ACS • Example accumulated cost surface and paths • Some Issues: • Grid resolution and metric • Barriers • Tracked not steepest paths • Is cost modelling sufficient? • Force modelling • Vector fields • Gradients www.spatialanalysisonline.com

  10. Distance computations • Distance transform (DT) • Derived from high-speed image processing • Provides improved (or exact) Euclidean distances over a grid • Very simple, fast algorithm • Can readily incorporate barriers, gradient and curvature constraints for paths, absolute rise and fall of routes etc. www.spatialanalysisonline.com

  11. Distance computations • Distance transform (DT) www.spatialanalysisonline.com

  12. Distance computations • Distance transform (DT) - Example applications – (a) Notting Hill carnival access; (b) selection of geothermal pipeline routing in Iceland (A, B1, B2, C) www.spatialanalysisonline.com

  13. Distance computations • Network distance • Requires a topologically validated network • Typically uses shortest or least time between vertices • Computed using generic SPA • Static tables (complete from/to) often stored • Takes account of asymmetric links, barriers and turn restrictions • May incorporate traffic models/data www.spatialanalysisonline.com

  14. Distance computations • Buffering – generating buffer areas • Vector buffering (Euclidean, Isotropic) • Point, line and polygon buffering • Inner, outer and symmetric buffering • Distinct or merged buffers www.spatialanalysisonline.com

  15. Distance computations • Buffering • Raster buffering • ‘Euclidean’ distance (Grid versions) • Cost-distance (ACS and DT procedures) • Network buffering • Drive time zones • Very processor intensive • Uniform ‘costs’ • Variable (e.g. road type, multi-modal) www.spatialanalysisonline.com

  16. Distance computations • Distance decay models • Simple inverse power models • IDW interpolation, demand modelling • spatial weights matrices… • Trip distribution models • With or without constraints • Statistical modelling • Kernel density modelling • GWR • Geostatistical modelling • Transport modelling www.spatialanalysisonline.com

  17. Distance computations • Distance decay models (=10, d=0.1,0.2,..) A. Inverse distance decay, /d B. Exponential distance decay, e‑d www.spatialanalysisonline.com

  18. Directional operations • Cyclic data type • Analysis of linear forms • Lines, polylines (may or may not be directed) • Issues: • Data modelling process • Generalisation (e.g. point weeding effects) • Nature of cyclic measure • Methods: • End-node to end-node; linear best fit; disaggregated (component) analysis; weighted analysis www.spatialanalysisonline.com

  19. Directional operations • Analysis of linear forms • Issues, cont.: • Nature of cyclic measure • Solution: • Compute vector-like measures - northing and easting components: Vn=vi cosi and Ve=vi sini • Compute resultant (r) direction: tan-1(Ve/Vn) • Magnitude of resultant • Circular variance and standard deviation www.spatialanalysisonline.com

  20. Directional operations • Analysis of linear forms – rose diagrams • Example – Streams in Crowe Butt region End point direction rose All segments direction rose www.spatialanalysisonline.com

  21. Directional operations • Two variable rose diagram • Wind speed and direction histograms • Resultant vector www.spatialanalysisonline.com

  22. Directional operations • Surfaces – aspect vector plot www.spatialanalysisonline.com

  23. Directional operations • Surfaces – windflow model vector plot www.spatialanalysisonline.com

  24. Directional operations • Point sets • Standard deviational ellipse axes • Least squares fit www.spatialanalysisonline.com

  25. Directional operations • Point sets • Correlated walks (CRW) A. 500 step CRW, variable (random uniform) step length, directional model N(0,1) degrees B. 500 step CRW, variable (random uniform) step length, directional model N(30,15) degrees www.spatialanalysisonline.com

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