www.spatialanalysisonline.com chapter 6 part a: surface analysis – geometrical methods
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www.spatialanalysisonline.com
Chapter 6
Part A: Surface analysis – geometrical methods
3rd edition 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
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Surface analysis – geometrical methods
Modelling surfaces - surfaces and fields
Mt St Helens – rendered grid Mt St Helens – wireframe
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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)
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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
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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
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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)
y
zz
yz
xzz
xz SNWE
2,
2 y
zz
yz
x
zz
xz
2,
21,01,00,10,1
yxzzzz
yxz
y
zzz
y
z
x
zzz
x
z SWSENWNESNWE
4,
*2,
*2 2
22
2
22
2
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Surface analysis – geometrical methods
Modelling surfaces – raster models Cell neighbourhoods and derivatives
Second order partial derivatives (8-cell finite difference version)
x
zzzzzz
xz
8
22 1,10,11,11,10,11,1
y
zzzzzz
yz
8
22 1,11,01,11,11,01,1
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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
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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
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Surface analysis – geometrical methods
Modelling surfaces – vector models
A. Source raster B. Contour - derived C. TIN - derived
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Surface analysis – geometrical methods
Modelling surfaces – mathematical models
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Surface analysis – geometrical methods
Modelling surfaces – statistical and fractal models
A. Random uniform B. Random Normal C. Ridged multi-fractal
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Surface analysis – geometrical methods
Modelling surfaces – hybrid (pseudo-random) models
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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)
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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
22
yF
xF
S
22
22
yzz
xzz
S SNWE
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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
yz
xz
A ,tan2360
270 1
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Surface analysis – geometrical methods
Surface geometry – profiles Single profiles
Linear transects Polygonal transects
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Surface analysis – geometrical methods
Surface geometry – profiles Multiple profiles
Baselines are average across entire grid
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Surface analysis – geometrical methods
Surface geometry – morphology
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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
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Surface analysis – geometrical methods
Surface geometry – profile curvature Math model:
Quadratic model:
Quartic model:
pqyz
xz
p
pq
yz
y
zyz
xz
yxz
xz
x
z
pr
1 ,
,
2
22
2/3
2
2
222
2
2
2 2
3/ 22 2 2 2
200
1pr
ad be cde
e d d e
2 2
2 2
200pr
dg eh fgh
g h
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Surface analysis – geometrical methods
Surface geometry – plan curvature Math model:
Quadratic model:
Quartic model:
222 2 2
2 2
3/ 2
22
2
pl
z z z z z z zx x y x y yx y
p
z zp
x y
2 2
3/ 22 2
200pl
bd ae cde
e d
2 2
2 2
200pl
dh eg fgh
g h
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Surface analysis – geometrical methods
Surface geometry – tangential curvature
tg
pl
z z z z z z zx x y x y yx y
pq
pq
z zp q p
x y
222 2 2
2 2
1/ 2
1/ 2
22
2
, where
, and 1
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Surface analysis – geometrical methods
Surface geometry – additional quadratic curvatures Longitudinal:
Cross-sectional:
Min, Max and mean:
2 2
2 2
200lon
ad be cde
e d
2 2
2 2
200cro
bd ae cde
e d
22min )( cbaba
22max )( cbaba
max min( )/ 2mean
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Surface analysis – geometrical methods
Surface geometry – directional derivatives Computed for direction :
First derivative:
Second derivative:
cos( ) sin( )dz z zds x y
d z z zx yds x
z
y
2 2 22
2 2
22
2
cos ( ) 2 cos( )sin( )
sin ( )
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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
2 2
2 2
3 22 2 /
x y y xt tt t(t)x yt t
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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)
1 2 1
2 4 2
1 2 1
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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)
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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
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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
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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
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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)
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Surface analysis – geometrical methods
Visibility – viewsheds and RF propagation
A. Source topography B. Simple optical viewshed (pink=not visible)
Mobile phone mast
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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
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Surface analysis – geometrical methods
Visibility – Space syntax Analysis of visibility in the built environment
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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
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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
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Surface analysis – geometrical methods
Watersheds and drainage – D-infinity Max gradient of 8 facets identified Flows assigned to cells (pixels) in proportions:
21
22
21
11 ,
pp
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Surface analysis – geometrical methods
Watersheds and drainage – case studyPit filled DEM Flow accumulations and watersheds
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Surface analysis – geometrical methods
Watersheds and drainage – case studyFlats and extrema Stream basins