Transcript Chapter 2
Chapter 6
Part A: Surface analysis – geometrical
methods
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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
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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
z z E zW z zN zS
,
x
2x
y
2y
z z1,0 z 1,0 z z 0,1 z 0,1
,
x
2 x
y
2 y
Second order partial derivatives (simple version)
z NE z NW zSE zSW
2 z z E 2 z * zW 2 z z N 2 z * zS 2 z
,
,
xy
4 xy
x 2
x 2
y 2
y 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)
z z1,1 2z1,0 z1,1 z 1,1 2z 1,0 z 1,1
x
8x
z z1,1 2 z 0,1 z 1,1 z1,1 2 z 0,1 z 1,1
y
8y
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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
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B. Contour - derived
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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
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B. Random Normal
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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
2
F
F
S
x
y
2
z zS
z zW
S E
N
2x
2y
2
2
Surface – averaging algorithms (D-infinity, 8-point etc.)
TIN model – direct computation or conversion to grid
Slope – resolution, orientation effects
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Surface analysis – geometrical methods
Surface geometry – aspect
Direction in degrees from North
A 270
z z
360
tan1 ,
2
x y
Directional bias from grid orientation
Classified aspect – gradation, 8-way, 4-way
Aspect and lighting/thermal modelling
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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
2
Math model:
pr
2 z z
2 z z z 2 z
2
2
2
xy x y y
x x
pq 3 / 2
2
z
y ,
2
2
z
z
p , q 1 p
x
y
Quadratic model:
Quartic model:
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pr
pr
200 ad 2 be2 cde
e2 d 2 1 d 2 e2
200 dg 2 eh2 fgh
g
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h2
3/2
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Surface analysis – geometrical methods
Surface geometry – plan curvature
2
Math model:
pl
2 z z
2 z z z 2 z z
2
2
2
x y x y y y
x x
p 3/2
2
z z
p
x y
Quadratic model:
Quartic model:
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2
2
e d
200 dh2 eg 2 fgh
pl
g 2 h2
pl
200 bd 2 ae2 cde
2
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2 3/2
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Surface analysis – geometrical methods
Surface geometry – tangential curvature
2
tg
2 z z
2 z z z 2 z z
2
2
2
x
x
y
x
y
x
y y
pq1/2
2
1/2
p
pl
q
2
, where
2
z z
p
, and q 1 p
x y
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Surface analysis – geometrical methods
Surface geometry – additional quadratic
curvatures
200 ad be cde
Longitudinal:
e d
2
lon
2
2
2
200 bd 2 ae2 cde
Cross-sectional:
cro
Min, Max and mean:
min a b (a b)2 c 2
e
2
d2
max a b (a b)2 c 2
mean (max min )/2
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Surface analysis – geometrical methods
Surface geometry – directional derivatives
Computed for direction :
First derivative:
dz z
z
cos()
sin()
ds x
y
Second derivative:
d 2 z 2 z
2 z
2
2 cos ( ) 2
cos( )sin( )
2
x y
ds
x
2 z 2
2 sin ( )
y
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Surface analysis – geometrical methods
Surface geometry – paths
Paths as plane curves
Paths as space curves
Parametric specification
Path curvature:
x 2 y y 2 x
2
2
t
t
t
t
(t)
3/2
x 2 y 2
t t
Radius of curvature: 1/path curvature=1/
Smoothing
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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)
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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
Viewshed: dark
blue=visible area
Line of sight direction
lines
Lines of sight – yellow=
visible from source,
red=not visible
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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
Near optimal locations for
cameras providing full
coverage of streets
Sample point – green
areas show visible
street areas
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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:
1
2
p1
, p2
1 2
1 2
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Surface analysis – geometrical methods
Watersheds and drainage – case study
Pit filled DEM
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Flow accumulations and watersheds
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Surface analysis – geometrical methods
Watersheds and drainage – case study
Flats and extrema
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Stream basins
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