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Advanced Distance and Directional Operations in Spatial Analysis

This chapter focuses on distance computations essential for spatial analysis, covering methodologies like Euclidean and spherical coordinates. It addresses problem areas such as large distance measures, terrain distances, and network distances. The text details concepts like Accumulated Cost Surfaces (ACS), Distance Transform (DT), and cost-distance modeling, discussing their applications and procedural techniques. It highlights the significance of metrics in geospatial analysis, alternative measures, and the challenges of distance decay models, providing a comprehensive overview for professionals in the field.

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Advanced Distance and Directional Operations in Spatial Analysis

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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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