www.spatialanalysisonline.com chapter 4 part b: distance and directional operations

www.spatialanalysisonline.com

Chapter 4

Part B: Distance and directional operations

3rd edition www.spatialanalysisonline.com 2

Distance computations Projected coordinates – Euclidean

Spherical coordinates – spherical or ellipsoidal computations

Problem areas: Planar measures over large distances Surface distances (3D/terrain distance) Network distances Variable cost/friction effects Transects (single or multi-part)

22jijiij yyxxd

2,

2:

coscossinsinsin2 221

jiji

jiij

BAwhere

BARd


Distance computations

Terrain distances – cross section view



Distance, measure and metric Distance: set of distinct objects plus some real-

valued measure, dij, of separation between object pairs, i and j

Metric: formal (mathematical) definition: dij>0 if ij (distinction/separation)

dij=0 if i=j (co-location/equivalence)

dij+djk≥dik (triangle inequality)

dij=dji (symmetry)



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 Network distance Grid distance

1 2 1 2

1/

( , )pp p

pd a b x x y y



Cost distance Cost – time, effort/friction, generalised costs Cost surfaces and grids

Procedures Accumulated Cost Surface (ACS) – spread

algorithms Distance Transform (DT) – scanning algorithms



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



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



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



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.



Distance transform (DT)



Distance transform (DT) - Example applications – (a) Notting Hill carnival access; (b) selection of geothermal pipeline routing in Iceland (A, B1, B2, C)



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



Buffering – generating buffer areas Vector buffering (Euclidean, Isotropic)

Point, line and polygon buffering Inner, outer and symmetric buffering Distinct or merged buffers



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)



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

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ij

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d

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)( ijjijiij dfDOBAT

2 2/ 2

/

22

2

( ) ,

( ) ,

( ) 1 ,

( ) 0 otherwise

d h

d h

f d e or

f d e or

df d d r

h

f d



Distance decay models (=10, d=0.1,0.2,..)

A. Inverse distance decay, /d B. Exponential distance decay, e‑d


Directional operations

Cyclic data typeAnalysis 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



Analysis of linear forms Issues, cont.:

Nature of cyclic measure Solution:

Compute vector-like measures - northing and easting components: Vn=vi cosi and Ve=vi sini

Compute resultant (r) direction: tan-1(Ve/Vn) Magnitude of resultant Circular variance and standard deviation

22en VV r



Analysis of linear forms – rose diagrams Example – Streams in Crowe Butt region

End point direction rose All segments direction rose



Two variable rose diagram Wind speed and direction histograms Resultant vector



Surfaces – aspect vector plot



Surfaces – windflow model vector plot



Point sets Standard deviational ellipse axes Least squares fit



Point sets Correlated walks (CRW)

A. 500 step CRW, variable (random uniform) step length, directional model N(0,1) degrees

B. 500 step CRW, variable (random uniform) step length, directional model N(30,15) degrees

www.spatialanalysisonline.com chapter 4 part b: distance and directional operations

Documents

metric distance

rd editionwww

com15 distance computations

cost x distance

inverse distance decay

exponential distance

cost grid

e d slide