Chapter 6

1. Chapter 6 Part A: Surface analysis – geometrical methods www.spatialanalysisonline.com

2. Surface analysis – geometrical methods • Modelling surfaces - surfaces and fields • Surfaces – typically scalar fields: • Continuous - z-values (magnitude) assumed to exist for every (x,y) coordinate pair • Real valued (may be integer coded, e.g. remote sensing data) and generally positive (may be negative) • Single valued (open or 2D manifold) – multiple values treated as separate surfaces or layers • Surfaces - vector fields: • Magnitude and direction assumed to exist for every (x,y) coordinate pair www.spatialanalysisonline.com

3. Surface analysis – geometrical methods • Modelling surfaces - surfaces and fields Mt St Helens – rendered grid Mt St Helens – wireframe www.spatialanalysisonline.com

4. Surface analysis – geometrical methods • Modelling surfaces - surfaces and fields • Surfaces - Data sources: • Physical surfaces – national mapping agencies, field surveys. DEM, contour, TIN or raster (image) models plus associated attribute data • Sample surveys – point/block samples converted to grids using interpolation procedures • Remote sensing – satellite, aerial • Vector data – e.g. wind strength/direction, magnetic survey data • Programmatically derived surfaces (theoretical models and best fits) www.spatialanalysisonline.com

5. Surface analysis – geometrical methods • Modelling surfaces – raster models • {x,y,z} representation, n x m • Row order – geographic vs mathematical • Treatment of missing and masked data • Coding of cell neighbourhoods www.spatialanalysisonline.com

6. Surface analysis – geometrical methods • Modelling surfaces – raster models • Advantages: • Computationally very convenient • Easy to display visually (2D image and 3D models) • Aligns with some data capture (remote sensing) techniques • Readily available for physical surfaces (DEM) • Disadvantages • Very large storage requirement • Computation can be processor intensive • Fixed grid size, shape, orientation • Representation of certain objects (e.g. lines) may be poor www.spatialanalysisonline.com

7. Surface analysis – geometrical methods • Modelling surfaces – raster models • Cell neighbourhoods and derivatives • First order partial derivatives – finite difference model • Second order partial derivatives (simple version) www.spatialanalysisonline.com

8. Surface analysis – geometrical methods • Modelling surfaces – raster models • Cell neighbourhoods and derivatives • Second order partial derivatives (8-cell finite difference version) www.spatialanalysisonline.com

9. Surface analysis – geometrical methods • Modelling surfaces – raster models • Cell neighbourhoods and derivatives • Local surface models • Fit quadratic polynomial to local neighbourhood (OLS) z=ax2+by2+cxy+dx+ey+f (6 parameters) • Analytically differentiate • Aspect: A=tan‑1(e/d) • Slope: St=tan‑1(e2+d2) • Curvatures: see later slides OR • Fit partial quartic polynomial to local neighbourhood (exactly) z=ax2y2+bx2y+cxy2+dx2+ey2+fxy+gx+hy+i (9 parameters) • Curvatures: see later slides www.spatialanalysisonline.com

10. Surface analysis – geometrical methods • Modelling surfaces – vector models • Principal models: • TIN • Compact, fast to process • Representational detail, complexity of processing • Contour – raster DEM datasets often derived from contour source material • Conversion to-from TIN/DEM www.spatialanalysisonline.com

11. Surface analysis – geometrical methods • Modelling surfaces – vector models www.spatialanalysisonline.com

12. Surface analysis – geometrical methods • Modelling surfaces – mathematical models www.spatialanalysisonline.com

13. Surface analysis – geometrical methods • Modelling surfaces – statistical and fractal models www.spatialanalysisonline.com

14. Surface analysis – geometrical methods • Modelling surfaces – hybrid (pseudo-random) models www.spatialanalysisonline.com

15. Surface analysis – geometrical methods • Surface geometry – gradient, slope, aspect • Gradient: vector measure – 2 components: • Slope – often computed as rise over run (tan) – varies by direction. Usually defined as maximum value at a given point (magnitude component) • Aspect – direction of maximum slope (direction component) www.spatialanalysisonline.com

16. Surface analysis – geometrical methods • Surface geometry – slope models • Rise over run (tan) • Rise over surface distance (sin) • Surface z=F(x,y) analytical differential • Surface – grid differential • Surface – averaging algorithms (D-infinity, 8-point etc.) • TIN model – direct computation or conversion to grid • Slope – resolution, orientation effects www.spatialanalysisonline.com

17. Surface analysis – geometrical methods • Surface geometry – aspect • Direction in degrees from North • Directional bias from grid orientation • Classified aspect – gradation, 8-way, 4-way • Aspect and lighting/thermal modelling www.spatialanalysisonline.com

18. Surface analysis – geometrical methods • Surface geometry – profiles • Single profiles • Linear transects • Polygonal transects www.spatialanalysisonline.com

19. Surface analysis – geometrical methods • Surface geometry – profiles • Multiple profiles Baselines are average across entire grid www.spatialanalysisonline.com

20. Surface analysis – geometrical methods • Surface geometry – morphology www.spatialanalysisonline.com

21. Surface analysis – geometrical methods • Surface geometry – curvature • Coordinate systems • Original grid coordinates (x,y,z) • Rotated grid coordinates (x-rot,y-rot,z) in direction of aspect • Tangential coordinates (surface normal, surface tangential plane) • Curvature computation and naming wrt alternative coordinate systems www.spatialanalysisonline.com

22. Surface analysis – geometrical methods • Surface geometry – profile curvature • Math model: • Quadratic model: • Quartic model: www.spatialanalysisonline.com

23. Surface analysis – geometrical methods • Surface geometry – plan curvature • Math model: • Quadratic model: • Quartic model: www.spatialanalysisonline.com

24. Surface analysis – geometrical methods • Surface geometry – tangential curvature www.spatialanalysisonline.com

25. Surface analysis – geometrical methods • Surface geometry – additional quadratic curvatures • Longitudinal: • Cross-sectional: • Min, Max and mean: www.spatialanalysisonline.com

26. Surface analysis – geometrical methods • Surface geometry – directional derivatives • Computed for direction : • First derivative: • Second derivative: www.spatialanalysisonline.com

27. Surface analysis – geometrical methods • Surface geometry – paths • Paths as plane curves • Paths as space curves • Parametric specification • Path curvature: • Radius of curvature: 1/path curvature=1/ • Smoothing www.spatialanalysisonline.com

28. Surface analysis – geometrical methods • Surface smoothing • Resolution increase/Grid re-calculation • Using a smoothing interpolator (e.g. spline) • Filtering or kernel smoothing (e.g. 3x3 ‘Gaussian’ kernel) www.spatialanalysisonline.com

29. Surface analysis – geometrical methods • Surface geometry – pit filling • Hydrographic modelling • Prior to flow modelling • 8-cell model and other rules • Masked fill • Depression-depth based filling • Error correction • Arising from data collection • Arising from data processing (e.g. interpolation) www.spatialanalysisonline.com

30. Surface analysis – geometrical methods • Surface geometry – volumetric analysis • Profiles – simple cut and fill computations • Surfaces: • Single grid vs reference (base) surface (e.g. z=0) • Grid pairs – grid 1 (upper), grid 2 (lower) • Result – estimate positive or negative volume (relative, and/or wrt base) • Computational procedures • Numerical integration (trapezoidal rule) • Exact computation from TIN • Indirect computation from point or profile data www.spatialanalysisonline.com

31. Surface analysis – geometrical methods • Visibility – Overview • Application areas • Line of sight modelling • Viewshed (visible areas) modelling • Single and multi-point problems • Static vs dynamic problems • Optical vs radio path visibility • Euclidean model • Earth curvature model • Propagation modelling www.spatialanalysisonline.com

32. Surface analysis – geometrical methods • Visibility – line of sight analysis • Simplified form of viewshed • Point source plus direction(s) • Coloured line transect(s) • Tabulated data • Profile plots Point source, offset from surface Viewshed: dark blue=visible area Line of sight direction lines Lines of sight – yellow= visible from source, red=not visible www.spatialanalysisonline.com

33. Surface analysis – geometrical methods • Visibility – viewsheds and RF propagation • Viewshed (visible areas) modelling • Input surface raster • Point set raster – single, multi-point, zones etc • Offsets for observation and target points • Range (distance and angular) constraints • Output – binary or multi-coded raster • RF – selection of propagation model, parameters (e.g. frequency, gain) and clutter modelling (typically surface offsets and obstacles) www.spatialanalysisonline.com

34. Surface analysis – geometrical methods • Visibility – viewsheds and RF propagation Mobile phone mast www.spatialanalysisonline.com

35. Surface analysis – geometrical methods • Visibility – Isovist analysis • Analysis of visibility in the plane • One or more source points • Complex optimisation problem Near optimal locations for cameras providing full coverage of streets Sample point – green areas show visible street areas www.spatialanalysisonline.com

36. Surface analysis – geometrical methods • Visibility – Space syntax • Analysis of visibility in the built environment www.spatialanalysisonline.com

37. Surface analysis – geometrical methods • Watersheds and drainage – assumptions • Uniform precipitation • Flows take place entirely across surfaces which they do not alter; unaffected by absorption or groundwater • Flows grow as a linear function with distance; not altered by slope values, just by direction • No barriers to flow • Study region is complete and meaningful in the context of the analysis www.spatialanalysisonline.com

38. Surface analysis – geometrical methods • Watersheds and drainage – modelling steps • Input (complete/mosaic-ed) DEM • Remove pits • Identify flow directions – D-8, D-infinity or MFM • Output ldd grid • Identify flats and extrema • Accumulate hypothetical flows to generate and merge streams – include pour points • Identify watersheds and stream basins www.spatialanalysisonline.com

39. Surface analysis – geometrical methods • Watersheds and drainage – D-infinity • Max gradient of 8 facets identified • Flows assigned to cells (pixels) in proportions: www.spatialanalysisonline.com

40. Surface analysis – geometrical methods • Watersheds and drainage – case study www.spatialanalysisonline.com

41. Surface analysis – geometrical methods • Watersheds and drainage – case study www.spatialanalysisonline.com