Chapter 6 Part A: Surface analysis – geometrical methods

3 rd editionwww.spatialanalysisonline.com2 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

3 rd editionwww.spatialanalysisonline.com3 Surface analysis – geometrical methods Modelling surfaces - surfaces and fields Mt St Helens – rendered gridMt St Helens – wireframe

3 rd editionwww.spatialanalysisonline.com4 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)

3 rd editionwww.spatialanalysisonline.com5 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

3 rd editionwww.spatialanalysisonline.com6 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

3 rd editionwww.spatialanalysisonline.com7 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)

3 rd editionwww.spatialanalysisonline.com8 Surface analysis – geometrical methods Modelling surfaces – raster models Cell neighbourhoods and derivatives Second order partial derivatives (8-cell finite difference version)

3 rd editionwww.spatialanalysisonline.com9 Surface analysis – geometrical methods Modelling surfaces – raster models Cell neighbourhoods and derivatives Local surface models Fit quadratic polynomial to local neighbourhood (OLS) z=ax 2 +by 2 +cxy+dx+ey+f (6 parameters) Analytically differentiate Aspect: A=tan 1 (e/d) Slope: S t =tan 1 (e 2 +d 2 ) Curvatures: see later slides OR Fit partial quartic polynomial to local neighbourhood (exactly) z=ax 2 y 2 +bx 2 y+cxy 2 +dx 2 +ey 2 +fxy+gx+hy+i (9 parameters) Curvatures: see later slides

3 rd editionwww.spatialanalysisonline.com10 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

3 rd editionwww.spatialanalysisonline.com11 Surface analysis – geometrical methods Modelling surfaces – vector models A. Source rasterB. Contour - derivedC. TIN - derived

3 rd editionwww.spatialanalysisonline.com12 Surface analysis – geometrical methods Modelling surfaces – mathematical models

3 rd editionwww.spatialanalysisonline.com13 Surface analysis – geometrical methods Modelling surfaces – statistical and fractal models A. Random uniformB. Random NormalC. Ridged multi-fractal

3 rd editionwww.spatialanalysisonline.com14 Surface analysis – geometrical methods Modelling surfaces – hybrid (pseudo-random) models

3 rd editionwww.spatialanalysisonline.com15 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)

3 rd editionwww.spatialanalysisonline.com16 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

3 rd editionwww.spatialanalysisonline.com17 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

3 rd editionwww.spatialanalysisonline.com18 Surface analysis – geometrical methods Surface geometry – profiles Single profiles Linear transects Polygonal transects

3 rd editionwww.spatialanalysisonline.com19 Surface analysis – geometrical methods Surface geometry – profiles Multiple profiles Baselines are average across entire grid

3 rd editionwww.spatialanalysisonline.com20 Surface analysis – geometrical methods Surface geometry – morphology

3 rd editionwww.spatialanalysisonline.com21 Surface analysis – geometrical methods Surface geometry – curvature Coordinate systems 1.Original grid coordinates (x,y,z) 2.Rotated grid coordinates (x-rot,y-rot,z) in direction of aspect 3.Tangential coordinates (surface normal, surface tangential plane) Curvature computation and naming wrt alternative coordinate systems

3 rd editionwww.spatialanalysisonline.com22 Surface analysis – geometrical methods Surface geometry – profile curvature Math model: Quadratic model: Quartic model:

3 rd editionwww.spatialanalysisonline.com23 Surface analysis – geometrical methods Surface geometry – plan curvature Math model: Quadratic model: Quartic model:

3 rd editionwww.spatialanalysisonline.com24 Surface analysis – geometrical methods Surface geometry – tangential curvature

3 rd editionwww.spatialanalysisonline.com25 Surface analysis – geometrical methods Surface geometry – additional quadratic curvatures Longitudinal: Cross-sectional: Min, Max and mean:

3 rd editionwww.spatialanalysisonline.com26 Surface analysis – geometrical methods Surface geometry – directional derivatives Computed for direction : First derivative: Second derivative:

3 rd editionwww.spatialanalysisonline.com27 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

3 rd editionwww.spatialanalysisonline.com28 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)

3 rd editionwww.spatialanalysisonline.com29 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)

3 rd editionwww.spatialanalysisonline.com30 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

3 rd editionwww.spatialanalysisonline.com31 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

3 rd editionwww.spatialanalysisonline.com32 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 Line of sight direction lines Lines of sight – yellow= visible from source, red=not visible Viewshed: dark blue=visible area

3 rd editionwww.spatialanalysisonline.com33 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)

3 rd editionwww.spatialanalysisonline.com34 Surface analysis – geometrical methods Visibility – viewsheds and RF propagation A. Source topographyB. Simple optical viewshed (pink=not visible) Mobile phone mast

3 rd editionwww.spatialanalysisonline.com35 Surface analysis – geometrical methods Visibility – Isovist analysis Analysis of visibility in the plane One or more source points Complex optimisation problem Sample point – green areas show visible street areas Near optimal locations for cameras providing full coverage of streets

3 rd editionwww.spatialanalysisonline.com36 Surface analysis – geometrical methods Visibility – Space syntax Analysis of visibility in the built environment

3 rd editionwww.spatialanalysisonline.com37 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

3 rd editionwww.spatialanalysisonline.com38 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

3 rd editionwww.spatialanalysisonline.com39 Surface analysis – geometrical methods Watersheds and drainage – D-infinity Max gradient of 8 facets identified Flows assigned to cells (pixels) in proportions:

3 rd editionwww.spatialanalysisonline.com40 Surface analysis – geometrical methods Watersheds and drainage – case study Pit filled DEMFlow accumulations and watersheds

3 rd editionwww.spatialanalysisonline.com41 Surface analysis – geometrical methods Watersheds and drainage – case study Flats and extremaStream basins