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

2. 3 rd editionwww.spatialanalysisonline.com2 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)

3. 3 rd editionwww.spatialanalysisonline.com3 Distance computations Terrain distances – cross section view

4. 3 rd editionwww.spatialanalysisonline.com4 Distance computations Distance, measure and metric Distance: set of distinct objects plus some real- valued measure, d ij, of separation between object pairs, i and j Metric: formal (mathematical) definition: d ij >0if i j (distinction/separation) d ij =0if i=j (co-location/equivalence) d ij +d jk d ik (triangle inequality) d ij =d ji (symmetry)

5. 3 rd editionwww.spatialanalysisonline.com5 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) L p metrics Network distance Grid distance

6. 3 rd editionwww.spatialanalysisonline.com6 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

7. 3 rd editionwww.spatialanalysisonline.com7 Distance computations ACS – simplified version Select start point – current position Take Queens 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

8. 3 rd editionwww.spatialanalysisonline.com8 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

9. 3 rd editionwww.spatialanalysisonline.com9 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

10. 3 rd editionwww.spatialanalysisonline.com10 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.

11. 3 rd editionwww.spatialanalysisonline.com11 Distance computations Distance transform (DT)

12. 3 rd editionwww.spatialanalysisonline.com12 Distance computations Distance transform (DT) - Example applications – (a) Notting Hill carnival access; (b) selection of geothermal pipeline routing in Iceland (A, B1, B2, C)

13. 3 rd editionwww.spatialanalysisonline.com13 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

14. 3 rd editionwww.spatialanalysisonline.com14 Distance computations Buffering – generating buffer areas Vector buffering (Euclidean, Isotropic) Point, line and polygon buffering Inner, outer and symmetric buffering Distinct or merged buffers

15. 3 rd editionwww.spatialanalysisonline.com15 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)

16. 3 rd editionwww.spatialanalysisonline.com16 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

17. 3 rd editionwww.spatialanalysisonline.com17 Distance computations Distance decay models ( =10, d=0.1,0.2,..) A. Inverse distance decay, /d B. Exponential distance decay, e d

18. 3 rd editionwww.spatialanalysisonline.com18 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

19. 3 rd editionwww.spatialanalysisonline.com19 Directional operations Analysis of linear forms Issues, cont.: Nature of cyclic measure Solution: Compute vector-like measures - northing and easting components: V n = v i cos i and V e = v i sin i Compute resultant (r) direction: tan -1 (V e /V n ) Magnitude of resultant Circular variance and standard deviation

20. 3 rd editionwww.spatialanalysisonline.com20 Directional operations Analysis of linear forms – rose diagrams Example – Streams in Crowe Butt region End point direction rose All segments direction rose

21. 3 rd editionwww.spatialanalysisonline.com21 Directional operations Two variable rose diagram Wind speed and direction histograms Resultant vector

22. 3 rd editionwww.spatialanalysisonline.com22 Directional operations Surfaces – aspect vector plot

23. 3 rd editionwww.spatialanalysisonline.com23 Directional operations Surfaces – windflow model vector plot

24. 3 rd editionwww.spatialanalysisonline.com24 Directional operations Point sets Standard deviational ellipse axes Least squares fit

25. 3 rd editionwww.spatialanalysisonline.com25 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