Transcript Chapter 2 - Geospatial Analysis
Chapter 4
Part B: Distance and directional operations www.spatialanalysisonline.com
Distance computations
Projected coordinates – Euclidean
d ij
x i
x j
y i
y j
2 Spherical coordinates – spherical or ellipsoidal computations
d ij
2
R
sin 1 sin 2 sin 2 cos
i
cos
j
Problem areas:
where
:
A
i
j
2 ,
B
Planar measures over large distances Surface distances (3D/terrain distance) Network distances Variable cost/friction effects Transects (single or multi-part)
i
j
2 3 rd edition www.spatialanalysisonline.com
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Distance computations
Terrain distances – cross section view 3 rd edition www.spatialanalysisonline.com
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Distance computations
Distance, measure and metric Distance: set of distinct objects plus some real valued measure, d ij , of separation between object pairs, i and j Metric: formal (mathematical) definition: d ij >0 if i j d ij =0 if i=j d ij +d jk ≥d ik d ij =d ji (distinction/separation) (co-location/equivalence) (triangle inequality) (symmetry) 3 rd edition www.spatialanalysisonline.com
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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) L p metrics Network distance
p
x
1 Grid distance
x
2
p
y
1
y
2
p
1/
p
3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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.
3 rd edition www.spatialanalysisonline.com
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Distance computations
Distance transform (DT) 3 rd edition www.spatialanalysisonline.com
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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) 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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) 3 rd edition www.spatialanalysisonline.com
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Distance computations
Distance decay models Simple inverse power models IDW interpolation, demand modelling spatial weights matrices… Trip distribution models With or without constraints Statistical modelling Kernel density modelling GWR Geostatistical modelling Transport modelling 3 rd edition www.spatialanalysisonline.com
z j
f
({
d ij z i
}) , 0
T ij
A i B j O i D j f
(
d ij
)
e
d
2 /2
h
2 ,
or
e
,
or
1
d
2
h
2 2 ,
d
r
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Distance computations
Distance decay models ( =10, d=0.1,0.2,..) A. Inverse distance decay, /d B. Exponential distance decay, e d 3 rd edition www.spatialanalysisonline.com
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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 3 rd edition www.spatialanalysisonline.com
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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: V n =
v
i cos i and V e =
v
i sin
i
Compute resultant (
r
) direction: tan -1 (V e /V n ) Magnitude of resultant
r
V n
2
V e
2 Circular variance and standard deviation 3 rd edition www.spatialanalysisonline.com
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Directional operations
Analysis of linear forms – rose diagrams Example – Streams in Crowe Butt region End point direction rose All segments direction rose 3 rd edition www.spatialanalysisonline.com
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Directional operations
Two variable rose diagram Wind speed and direction histograms Resultant vector 3 rd edition www.spatialanalysisonline.com
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Directional operations
Surfaces – aspect vector plot 3 rd edition www.spatialanalysisonline.com
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Directional operations
Surfaces – windflow model vector plot 3 rd edition www.spatialanalysisonline.com
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Directional operations
Point sets Standard deviational ellipse axes Least squares fit 3 rd edition www.spatialanalysisonline.com
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Directional operations
Point sets Correlated walks (CRW) 3 rd edition A. 500 step CRW, variable (random uniform) step length, directional model N(0,1) degrees www.spatialanalysisonline.com
B. 500 step CRW, variable (random uniform) step length, directional model N(30,15) degrees 25