Chapter 4
Part B: Distance and directional
operations
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Distance computations
 Projected coordinates – Euclidean
dij 
xi  x j 2  y i  y j 2
 Spherical coordinates – spherical or ellipsoidal
computations
d  2R sin  sin A  sin B  cos  cos  
ij
 Problem areas:
1
where : A 
2
2

i   j
2
,B 
i
j

i   j
2
Planar measures over large distances
Surface distances (3D/terrain distance)
Network distances
Variable cost/friction effects
Transects (single or multi-part)
3rd edition
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Distance computations
Terrain distances – cross section view
3rd edition
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Distance computations
Distance, measure and metric
Distance: set of distinct objects plus some realvalued measure, dij, of separation between
object pairs, i and j
Metric: formal (mathematical) definition:
dij>0 if ij
dij=0 if i=j
dij+djk≥dik
dij=dji
3rd edition
(distinction/separation)
(co-location/equivalence)
(triangle inequality)
(symmetry)
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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)
Lp metrics
d p (a, b) 
Network distance
Grid distance
3rd edition
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x x
1
p
2
 y1  y 2

p 1/ p
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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
3rd edition
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Distance computations
 ACS – simplified version
Select start point – current position
Take Queen’s 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
3rd edition
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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
3rd edition
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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
3rd edition
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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.
3rd edition
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Distance computations
Distance transform (DT)
3rd edition
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Distance computations
 Distance transform (DT) - Example applications –
(a) Notting Hill carnival access; (b) selection of
geothermal pipeline routing in Iceland (A, B1, B2, C)
3rd edition
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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
3rd edition
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Distance computations
Buffering – generating buffer areas
Vector buffering (Euclidean, Isotropic)
Point, line and polygon buffering
Inner, outer and symmetric buffering
Distinct or merged buffers
3rd edition
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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)
3rd edition
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Distance computations
 Distance decay models
Simple inverse power models
 IDW interpolation, demand modelling
 spatial weights matrices…
zj 
f({ zi })
d ij

,  0
Trip distribution models
 With or without constraints
Statistical modelling
 Kernel density modelling
 GWR
 Geostatistical modelling
 Transport modelling
3rd edition
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Tij  Ai B j Oi D j f (d ij )
f (d)  e
d 2 /2 h2
, or
f (d)  e d / h , or
2
 d2 
f (d)   1  2  , d  r
h 

f (d)  0 otherwise 16
Distance computations
Distance decay models (=10, d=0.1,0.2,..)
A. Inverse distance decay, /d
3rd edition
B. Exponential distance decay, e-d
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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
3rd edition
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Directional operations
Analysis of linear forms
Issues, cont.:
Nature of cyclic measure
Solution:
Compute vector-like measures - northing and easting
components: Vn=vi cosi and Ve=vi sini
Compute resultant (r) direction: tan-1(Ve/Vn)
Magnitude of resultant r  Vn2  Ve2
Circular variance and standard deviation
3rd edition
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Directional operations
Analysis of linear forms – rose diagrams
Example – Streams in Crowe Butt region
End point direction rose
3rd edition
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All segments direction rose
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Directional operations
Two variable rose diagram
Wind speed and direction histograms
Resultant vector
3rd edition
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Directional operations
Surfaces – aspect vector plot
3rd edition
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Directional operations
Surfaces – windflow model vector plot
3rd edition
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Directional operations
Point sets
Standard deviational ellipse axes
Least squares fit
3rd edition
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Directional operations
Point sets
Correlated walks (CRW)
A. 500 step CRW,
variable (random uniform)
step length, directional
model N(0,1) degrees
3rd edition
B. 500 step CRW, variable
(random uniform) step length,
directional model N(30,15)
degrees
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