PLSCS/TRES 6200
Spatial Modelling and Analysis
Concepts of Space
D G Rossiter
Cornell UniversitySection of Soil & Crop Sciences
November 27, 2021
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Spatial concepts 1
Topic: Types of spaces
• A vector space in mathematics is any set of variables that form metric axeswith a known orientation to a reference system.
– This allows us to compute a distance and direction between points in thatspace.
– If these variables represent attributes, we have a feature space.– If they represent geographic coördinates, we have a metric geographic
space.
• A non-metric mathematical space can represent topology
– If these are geospatial relations, we have a topological geographic space.
D G Rossiter
Spatial concepts 2
Feature space
This is the vector space formed by any set of variables:
• Axes represent each variable;
• Coördinates are values of variables, possibly transformed or combined;
• Classified variables should be ordinal with some distance measure, butsometimes nominal variables are considered to be in feature space.
• The observations are related in this ‘space’, e.g. the “distance” between themcan be calculated.
• This is the basis of multivariate analysis.
Feature space is sometimes referred to as attribute space.
D G Rossiter
Spatial concepts 3
Scatterplot of a 2D feature space
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Petal width (mm)
Pet
al le
ngth
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This is a visualisation of a 2D feature space using a scatterplot; two attributes ofindividual iris flowers as coördinates.
Source: Anderson, Edgar (1935), The irises of the Gaspé Peninsula, Bulletin of the American. Iris Society, 59, 2-5.
D G Rossiter
Spatial concepts 4
Scatterplot of a 3D feature space
Petal.Length
Petal.Width
Sepal.Length
Anderson Iris data, three attributes as coördinates.
D G Rossiter
Spatial concepts 5
Geographic space
• “Geo” + “graphy” = “Earth” + “mapping”
• Related somehow to the Earth’s surface
• metric vs. topological
D G Rossiter
Spatial concepts 6
Metric geographic space
• a mathematical space where the axes are map coördinates that relate pointsto some reference location on or in the Earth (or another physical body)
• These coördinates are often in some geographic coördinate system that wasdesigned to give each location on (part of) the Earth a unique identification(see lecture on Coördinate Reference Systems)
• However, a local coördinate system can be used, as long as there is a clearrelation between locations and coördinates;
– ungeoreferenced aerial or satellite imagery– photograph of microscope slide (not the Earth’s surface but has metric
geometry)
• Key point: can compute distances, angles of separation, and areas.
D G Rossiter
Spatial concepts 7
Metric space
• Coördinates represent “true” distances along their axes
• Axes are 1D lines or arcs with a unit of measure (e.g. metres, kilometres,degrees . . . )
• One-dimensional: coördinates are on a line with respect to some origin (0):(x1) = x
• Two-dimensional: coördinates are on a grid with respect to some origin (0,0):(x1, x2) = (x,y) = (E,N)
– Latitude-longitude (sometimes called “geographic”) coördinates do not haveequal distances in the two dimensions; they should be transformed to metric(grid) coördinates for geo-statistical analysis.
• Three-dimensional: coördinates are grid and elevation (or depth! a negativeelevation) from a reference elevation: (x1, x2, x3) = (x,y, z) = (E,N,H)
D G Rossiter
Spatial concepts 8
Maps of metric space
• Shows features as abstract objects (points, lines, polygons, grids);
• The features (attributes) are labelled; these are collected in a legend;
– continuous vs. classified attributes/features
• A map shows both metric geographic and feature (attribute) spaces;
• One map can show many attributes
– Special case: contour maps “2.5D”, show the third geographic dimension(height, depth) on a 2D display
D G Rossiter
Spatial concepts 9
Some maps in metric space
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ER
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CK
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ROADRO AD ETNA
DRYDEN
STEVENSON ROAD
STREET
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RO
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EN
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WATROS
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-
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CA
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LOCKE ROADROAD
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LINE
GOODMAN RD
RO
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RD
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ROAD
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ROAD
B RO
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TR
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D
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TEETER
BILL
RIDGE
SH
OR
ED
RIV
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T
BU
SH
LA
NUT
MIL
LIKEN
STATION
ROAD
LANSING -
GENOA TOWN LINE ROAD COUNTY LINE RD
FRENCH HILL
ROAD
BROOK SHILL
RO AD
ALGERINE
L ANSIN
G
STATION ROAD
LUD
LOW
RO
AD
SW
EAZEYROAD
GEO
RGE
ROAD
WH
ITE
TAIL
CR
OS
SIN
G
ROAD
MY
ER
S
RO
AD
RO
AD
LUDL OWVILLE
PO
RT
LAN
D
ROAD
POINT
EA
STL AKE R D
RE
ACH RUN
BE
NS
ON
RO
AD
AUBURN
AU
BU
RN
RO
AD
AU
BU
RN
SH
AR
PS
TE
EN
ROAD
KA
RN
RO
AD
BACON ROAD
OS
TR
AN
DR
OA
D
ROAD
BUCK
ROAD
RO
AD
LUCE
CO
LLINS
RO
AD
WA
RR
EN
FO
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ST
AC
RE
S D
R
BE
AN
HI LL LA
WAR
RE
N D
R
SNYDER
R OAD
BOMAX
DRIVE
RDWEST GROTON
#UNA-2
#
UNA-23#
UNA-22#
UNA-21
#
UNA-3
#
UNA-4#
UNA-5# UNA-6
#UNA-19
#
UNA-17
#
UNA-14
#
UNA-15
# UNA-13
#
UNA-9
#
UNA-12
#UNA-36
#UNA-38
#UNA-11
#UNA-10
#UNA-37
UNA-8
UNA-16
UNA-7
UNA-18
UNA-31
UNA-20
UNA-1
UNA-40UNA-41
#UNA-34
UNA-45UNA-48
#
UNA-42
#
UNA-35
#
UNA-39
UNA-73
#
UNA-80
#
UNA-79
# UNA-43
#
UNA-78
#
UNA-77
#
UNA-44
# UNA-76
#
UNA-46#
UNA-75
#
UNA-74
#
UNA-47
# UNA-33
UNA-32
UNA-49UNA-50
UNA-69#
UNA-68 #
UNA-70
UNA-51
#UNA-29
#
UNA-30
#
UNA-52
UNA-27
#
UNA-28
UNA-54
#
UNA-53
#
UNA-26
#
UNA-25
#
UNA-24
#
UNA-24
#
UNA-55
UNA-56#
UNA-58
#
UNA-57
#
UNA-60
#
UNA-59
#
UNA-95
UNA-61
#UNA-62
UNA-91
#
UNA-93#
UNA-94
UNA-63
#UNA-90
#UNA-64
#
UNA-89
#
UNA-92
#UNA-103
# UNA-97
#UNA-102
#
UNA-65
UNA-67
UNA-88UNA-85
UNA-86
UNA-84
#
UNA-66
#
UNA-87
#
UNA-83
#
UNA-82
UNA-71# UNA-72
#
UNA-81
#
UNA-111
#
UNA-112
UNA-114
#
UNA-113
#
UNA-116
#
UNA-115
#
UNA-122
# UNA-121
UNA-120
UNA-123
UNA-125
#
UNA-109
#
UNA-110
UNA-108
#UNA-107
#
UNA-124
#UNA-106
#UNA-105
#
UNA-130
#UNA-102
# UNA-129
#
UNA-98
#
UNA-99
#
UNA-100
#
UNA-128
# UNA-135
# UNA-131
#
UNA-104
#
UNA-132
#
UNA-134
#
UNA-136
#UNA-137
#
UNA-96
UNA-139
UNA-140
UNA-146
#
UNA-141
# UNA-145
#
UNA-153
#UNA-152
UNA-150
UNA-144
#UNA-151
#
UNA-147
# UNA-149
#UNA-148
#
UNA-163
#
UNA-143
#
UNA-142
UNA-164
UNA-165
#
UNA-167
UNA-166
#
UNA-168
#
UNA-169
#
UNA-170
#
UNA-171
UNA-172
#
UNA-189UNA-188
UNA-186
UNA-191
#
UNA-192
#
UNA-190
#UNA-187
UNA-173
UNA-174
#
UNA-178
UNA-175
UNA-181
UNA-182UNA-180
#
UNA-177
# UNA-179
#
UNA-162
#
UNA-161
#
UNA-160
#
UNA-157
UNA-126
#
UNA-119
#
UNA-133
UNA-117#
UNA-118
UNA-158
#
UNA-159
#
UNA-176
UNA-183
#
UNA-184
UNA-185
#
UNA-176
UNA-155
UNA-156
UNA-154
#
#
VILLAGEOF
LANSING
VILLAGEOF
CAYUGA HEIGHTS
CITY OF ITHACA
#
VILLAGEOF
DRYDEN
VILLAGE OF FREEVILLE
#
VILLAGEOF
TRUMANSBURG
VILLAGE OF GROTON
#
TOWN OF ITHACA
TOWN OF GROTON
TOWN OF DRYDEN
TOWN OF CAROLINE
TOWN OF DANBY
TOWN OF ENFIELD
TOWN OF NEWFIELD
TOWNOF
ULYSSES
TOWN OF LANSING
#UNA-101
UNA-127
34B
34
34B
34 34B
222
38
222
38
38366
392
13
38
1313
366
34
96
89
227
79
32713
13A
96B
#
79
34
96
96B
13
34B
#
UNA-138
Prepared by the Tompkins County Planning Department for the Tompkins County Environmental Management Council
Unique Natural Areas in Tompkins County2 0 2 Miles
1:48000
S
N
EWLast Updated: September 1999
Schuyler County
Tioga County
Tioga County
Seneca County
ChemungCounty
Schu
yler
Cou
nty
Cortl
and
Coun
ty
Tioga County
Cayuga County
UNA boundaries were delineated by field biologists based on a review of air photographs, digital GIS base map data (roads,building footprints, 20 foot contours and streams) and field visits. UNA boundaries are approximate and should be used forgeneral planning purposes only. As a practical matter the county does not warrant the accuracy or completeness of theinformation portrayed. The end user of this map agrees to accept the data "as is" with full knowledge that errors andommissions may exist, and to hold harmless the county for any damages that may result from an inappropriate use of thismap. For further information about the Unique Natural Areas GIS data set see the Unique Natural Areas metadata record.
Unique Natural Area
Municipal BoundaryNew York State RoadRoad
Cayuga Lake
contours; themes: roads (lines); parks& administrative divisions (polygons)
labelled polygons and lines
These are plotted with a metric Coördinate Reference System (CRS), not shownhere.
D G Rossiter
Spatial concepts 10
Soil class maps
Dutch soil survey 1:50k sheet 34E;themes: soil class; ground-water levelclass; subsoil condition (“polkadot”overprint)
SSURGO 1:25k, Edgecombe CountyNC; theme: soil mapping unitsn.b. extremely poor, non-connotative,colour scheme
D G Rossiter
Spatial concepts 11
Navigation maps – 1D and 2D
note: different X and Z scales; X scale aligned with runway 18R “Polderbaan”
D G Rossiter
Spatial concepts 12
Topologic geographic space
• Topologic relations are preserved;
– adjacency, connectivity, containment, intersection . . .
• Distances are not true
D G Rossiter
Spatial concepts 13
True 1D, 2D topology, distorted distances and angles
MAYOR OF LONDON
Online maps are strictly for personal use only. To license the Tube map for commercial use please visit tfl.gov.uk/maplicensing
Tube map
Transp
ort for L
ondon Dec
ember
2017
Transport for London December 2017
Key to lines
Metropolitan
Victoria
Circle
Central
Bakerloo
DLR
London Overground
TfL Rail
London Trams
Piccadilly
Waterloo & City
Jubilee
Hammersmith & City
Northern
District
District open weekends and on some public holidays
Emirates Air Linecable car (special fares apply)
Check before you travel§ Brixton No step-free access from early February until September 2018.---------------------------------------------------------------------------§ Hounslow West Step-free access for manual wheelchairs only.---------------------------------------------------------------------------§ Victoria Step-free access is via the Cardinal Place entrance.---------------------------------------------------------------------------§ Services or access at these stations are subject to variation. Please search ‘TfL stations’ for full details.
Transport for London December 2017
Key to symbols Explanation of zones
1
3
45
6
2
7
8
9
Station in both zones
Station in both zones
Station in both zones
Station in Zone 9
Station in Zone 6
Station in Zone 5
Station in Zone 3Station in Zone 2
Station in Zone 1
Station in Zone 4
Station in Zone 8
Station in Zone 7
Interchange stations
Step-free access from street to train
Step-free access from street to platform
National Rail
Riverboat services
Airport
Victoria Coach Station
Emirates Air Line cable car
A
B
C
D
E
F
1 2 3 4 5 6 7 8 9
1 2 3 4 5 76 8 9
A
B
C
D
E
F
2 2/3 43
22
2
8 8 6 8
2
4
4
65
9
1
3
2
3
3
3
1
1
33
5
3
5 79 7 7Special fares apply
London Tramsfare zone
5
4
4
4
46
Special faresapply
River Thames
Regent’s ParkGoodgeStreet
Bayswater
Warren Street
Aldgate
Farringdon
BarbicanRussellSquare
High Street Kensington
Old Street
Green Park
BakerStreet
NottingHill Gate
Mansion House
Temple
OxfordCircus
TowerHill
PiccadillyCircus
CharingCross
Holborn
Tower Gateway
Monument
Moorgate
Leicester SquareSt Paul’s
Hyde Park Corner
Knightsbridge
Angel
Queensway
Marble Arch
Covent Garden
LiverpoolStreet
Great PortlandStreet
Bank
Chancery Lane
LancasterGateHolland
Park
Cannon Street
Fenchurch Street
GloucesterRoad
EustonSquareEdgware
Road
Edgware Road
Embankment
Blackfriars
TottenhamCourt Road
King’s CrossSt Pancras
MarylebonePaddington
Watford High Street
Watford Junction
Bushey
Carpenders Park
Hatch End
North Wembley
South Kenton
Kenton
Wembley Central
Kensal Green
Queen’s Park
Stonebridge Park
Bethnal Green
Cambridge Heath
London Fields
Harlesden
Willesden Junction
Headstone Lane
Harrow &Wealdstone
Kilburn Park
WarwickAvenue
Maida ValeEuston
NewCross Gate
Imperial Wharf
ClaphamJunction
Crystal Palace Norwood Junction
Sydenham
Forest Hill
Anerley
Penge West
Honor Oak Park
Brockley
Wapping
New Cross
Queens RoadPeckham
Peckham Rye
Denmark Hill
Surrey Quays
Whitechapel
WandsworthRoad
Rotherhithe
ShoreditchHigh Street
Haggerston
Hoxton
Shepherd’sBush
Shadwell
CanadaWater
Fulham Broadway
West Brompton
Parsons Green
Putney Bridge
East Putney
Southfields
Wimbledon Park
Wimbledon
Kensington(Olympia)
AldgateEast
Bethnal Green Mile End
Dalston Kingsland
HackneyWick
Homerton
HackneyCentral
RectoryRoad
HackneyDowns
Theydon Bois
Epping
Debden
Loughton
Buckhurst Hill
Leytonstone
Wood StreetBruce Grove
White Hart Lane
Silver Street
Edmonton Green
Southbury
Turkey Street
Theobalds Grove
Cheshunt
Enfield Town
StamfordHill
Bush HillPark
Highams Park
Chingford
Leyton
Woodford
South Woodford
Snaresbrook
Hainault
Fairlop
Barkingside
Newbury Park
Stratford
RodingValley
GrangeHill
Chigwell
Redbridge
GantsHill
Wanstead
Dalston Junction
Canonbury
Stepney Green
SevenSisters
Highbury &Islington
TottenhamHale
Clapton
St James Street
StokeNewington
Dagenham East
Dagenham Heathway
Becontree
Upney
Upminster
Upminster Bridge
Hornchurch
Elm Park
Ilford
Goodmayes
Chadwell Heath
Romford
Gidea Park
Harold Wood
Shenfield
Brentwood
Seven Kings
HarringayGreenLanes
LeytonstoneHigh Road
LeytonMidland Road
Emerson ParkSouth Tottenham
Barking
East Ham
Plaistow
Upton Park
Upper Holloway
CrouchHill
Gospel Oak
BowChurch
WestHam
BowRoad
Bromley-by-Bow
Island Gardens
Greenwich
Deptford Bridge
South Quay
Crossharbour
Mudchute
Heron Quays
West IndiaQuay
Elverson Road
Devons Road
Langdon Park
All Saints
Canary Wharf
Cutty Sark for Maritime Greenwich
Lewisham
West Silvertown
EmiratesRoyal Docks
EmiratesGreenwichPeninsula
PontoonDock
LondonCity Airport
WoolwichArsenal
King George V
Custom House for ExCeL
Prince Regent
Royal Albert
Beckton Park
Cyprus
Beckton
Gallions Reach
Westferry Blackwall
RoyalVictoria
CanningTown
PoplarLimehouse
EastIndia
StratfordInternational
Star Lane
NorthGreenwich
Maryland
Manor Park
Oakwood
Cockfosters
Southgate
Arnos Grove
Bounds Green
Turnpike Lane
Wood Green
Manor House
FinsburyPark
Arsenal
KentishTown West
Holloway Road
Caledonian Road
Mill Hill East
Edgware
Burnt Oak
Colindale
Hendon Central
Brent Cross
Golders Green
Hampstead
Belsize Park
Chalk Farm
Camden Town
High Barnet
Totteridge & Whetstone
Woodside Park
West Finchley
Finchley Central
East Finchley
Highgate
Archway
Tufnell Park
Kentish Town
MorningtonCrescent
CamdenRoad
CaledonianRoad &
Barnsbury
Amersham
Chorleywood
Rickmansworth
Chalfont &Latimer
Chesham
Moor Park
Croxley
Watford
Northwood
Northwood Hills
Pinner
North Harrow
Harrow-on-the-Hill
NorthwickPark
PrestonRoad
Wembley Park
Rayners Lane
Stanmore
Canons Park
Queensbury
Kingsbury
Neasden
Dollis Hill
Willesden Green
Swiss Cottage
KilburnWestHampstead
Finchley Road
WestHarrow
IckenhamUxbridge
Hillingdon RuislipRuislip Manor
Eastcote
St John’s Wood
HeathrowTerminal 5
HeathrowTerminal 4
Northfields
Boston Manor
SouthEaling
Osterley
Hounslow Central
Hounslow East
Hounslow West
Hatton CrossHeathrowTerminals 2 & 3
Perivale
Hanger Lane
RuislipGardens
South Ruislip
Greenford
Northolt
South Harrow
Sudbury Hill
Sudbury Town
Alperton
Park Royal
North Ealing
EalingBroadway
West Ruislip
Ealing Common
Gunnersbury
Kew Gardens
Richmond
Acton Town
ChiswickPark
TurnhamGreen
StamfordBrook
RavenscourtPark
WestKensington
BaronsCourt
Earl’sCourt
Shepherd’sBush Market
Goldhawk Road
Hammersmith
Wood Lane
WhiteCity
Finchley Road& Frognal
KensalRise
BrondesburyPark
Brondesbury
KilburnHigh Road
SouthHampstead
WestActon
NorthActon
EastActon
Southwark
Waterloo
London Bridge Bermondsey
Vauxhall
Lambeth NorthPimlico
Stockwell
Brixton
Elephant & Castle
OvalKennington
Borough
Clapham North
Clapham High Street
Clapham Common
Clapham South
Balham
Tooting Bec
Tooting Broadway
Colliers Wood
South Wimbledon
Morden
Latimer Road
Ladbroke Grove
Royal Oak
Westbourne Park
PuddingMill Lane
Acton Central
South Acton
HampsteadHeath
Stratford High Street
Abbey Road
WoodgrangePark
WalthamstowQueen’s Road
West Croydon
BeckenhamJunction
Elmers End
Harrington Road
Arena
DundonaldRoad
Merton Park Woodside
Blackhorse Lane
Addiscombe
AvenueRoad
Sandilands
WellesleyRoad
Reeves Corner
Mitcham BeddingtonLane
AmpereWay
WandlePark
Centrale
ChurchStreet
BelgraveWalk
PhippsBridge
MordenRoad
TherapiaLane
WaddonMarsh
GeorgeStreet
LebanonRoad
EastCroydon
BeckenhamRoad
MitchamJunction
Birkbeck
WalthamstowCentral
WansteadPark
Blackhorse Road
Forest Gate
Coombe Lane
Lloyd Park King Henry’sDriveFieldway
NewAddington
GravelHill
AddingtonVillage
WestminsterSouthKensington
SloaneSquare
St James’sParkVictoria
BondStreet
Bakerloo line
National Rail fares apply betweenHarrow & Wealdstone and Kenton.
LT Cards are not valid.
Zone 3Zone 5 Zone 4 Zone 2 Zone 1
KentonSouth
KentonHarrow &
Wealdstone ËNorth
WembleyWembley Central Ë
StonebridgePark
Willesden Junction ËHarlesden
EdgwareRoad
Regent’sPark
LambethNorth
Charing Cross Ë Marylebone Ë
KensalGreen
Queen’s Park Ë
KilburnPark
MaidaVale
WarwickAvenue
Stock code: 463067
Elephant& Castle
Northern
Waterloo Ëfor Waterloo International
JubileeNorthern
Waterloo & CityFestival PierWaterloo Pier
PiccadillyCircus
Piccadilly Northern
OxfordCircus
CentralVictoria
BakerStreet
CircleHammersmith
& CityJubilee
Metropolitan
Paddington Ë
CircleDistrict
Embankment
CircleDistrict
NorthernEmbankment
Pier
Stations with step-free exit from the Bakerloo lineplatforms to the street are shown with this symbol
Stations with step-free exit from the Bakerloo lineplatforms to the street are shown with this symbol
Trains toHeathrow
Ë 100m walk
Note: 3D not correct topology
D G Rossiter
Spatial concepts 14
Typology of spatial data – views of the world
Objects real-world entities: can be discretely identified “in the field” and locatedin geographic space
Fields continuously-varying properties in space
• Represented in a GIS by some discretization, but conceptually continuous• Measured with some spatial support (sample size, instrument field of view,
. . . ), but conceptually continuous
Networks interconnected line and point objects
D G Rossiter
Spatial concepts 15
It’s not so simple . . .
• the conceptual definition of the object may be vague (fuzzy definition)
• It may be difficult to identify an object in the field (fuzzy identification)
• Objects may have fuzzy boundaries or locations
• Object concepts may depend on the map scale
– Roads and buildings are conceived of as polygons at large map scale, linesat small map scale
• Continuous variables are measured with some spatial support (sample size,instrument field of view, . . . ); this is a lower limit of the resolution with whichwe can describe the conceptually-continuous field
D G Rossiter
Spatial concepts 16
Typology of spatial data – conceptual models of spatial objects
Point a single point location, defined by coordinates
Line a set of ordered points, connected by straight line segments
Curves a set of control points (not necessarily on the curve); and a mathematicalfunction of coordinates (e.g., splines)
Polygon an area delineated by one or more lines, possibly containing holes (andholes within holes . . . )
Network a group of curves connected at points (vertices); mathematical “graph”
Grid a collection of points or cells, organised in a regular lattice covering an area
D G Rossiter
Spatial concepts 17
Polygons
Baarle-Nassau (NL), Baarle-Hertog (Baerle-Duc) (B); source Wikimedia commons
“H” polygons are B inside NL (i.e., holes in the surround unlabelled “N”); “N”polygons are N inside B (i.e., holes in a “H” polygon).
D G Rossiter
Spatial concepts 18
Grids
KED−ffreq*dist prediction, log−ppm Zn
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
40 x 40 m square cells conceptually smooth
Cell values could be centre point predictions, block averages, block maxima,block ranges or standard deviations . . .in any case, one number represents the entire grid cell.
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Spatial concepts 19
Typology of spatial data – data models
These are how spatial data are represented inside a GIS – not their conceptualrepresentation
Vector exact mathematical form: 0-dimension = points; 1-dimension = linesegments (which can be joined); 2-dimension = areas; 3-dimension = volumes
• Triangulated Irregular Network (TIN) is a vector data model of a 2-Dcontinuous surface conceptual model
Raster a regular tessellation (e.g., square or hexagonal grid); fixed resolution
• data values may be grid cell averages, maxima, minima . . . or single valuesat the centre point
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Spatial concepts 20
Concepts of scale
1. cartographic (map) scale: relation of map distances to ground distances
• “large” = large area of paper needed to represent a given ground area
2. geographic scale: size of area being studied
• “large”= over a wide (“large”) area
3. process scale: spatial extent / variability of process operating on landscape
• e.g., soil erosion: rill, plot, small catchment, river system . . .
4. measurement (observation) scale: size (“support”) of observations
• e.g., soil ped, core, profile, pit, trench, . . .
5. modelling scale: size of fundamental area at which processes or objects arerepresented in models (“support”)
Blöschel, G., & M. Sivapalan. 1995. Scale issues in hydrological modelling: a review. Hydrol. Proc. 9: 251-290. DOI:
https://dx.doi.org/10.1002/hyp.3360090305D G Rossiter
Spatial concepts 21
Temporal scale
The above-mentioned spatial scales can also be used to describe temporalscales.
(Except for “cartographic”).
D G Rossiter
Spatial concepts 22
Matching scales
Key points:
• modelling scale should match process scale
• information at different measurement scales must be harmonized
– e.g., satellite imagery at different resolutions– e.g., demographic information at census ward vs. postal code vs.
administrative unit (these also at different levels)– up-, down-scaling (see below)
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Spatial concepts 23
Up- and down-scaling
Upscaling from detailed scale (e.g., lab. experiments) to coarse scale (e.g., ag.field, region, continent . . . )
Downscaling from a coarse scale (e.g., general circulation model of theatmosphere) to a detailed scale (e.g., local weather forecast)
These can be either spatial or temporal scales.
Often the inputs to a model do not have the same scale, so some must beadjusted; and/or the input scale does not match the desired output scale.
D G Rossiter
Spatial concepts 24
Issues in spatial scaling
• Upscaling: must compress/summarize information
– e.g., area weighted averaging of properties – but is this meaningful? (e.g.,white car in black parking lot → grey pixel)
– maybe re-run models with coarser scale inputs– maybe interpolate including information (somewhat) outside the upscaled
resolution
• Downscaling: must create new spatially-explicit information at a finer scale
– Just increasing pixel resolution is not creating information!– Example: disaggregating a coarse-resolution soil polygon (soil association
with known landscape relation) to fine-resolution polygons (soilconsociation = more-or-less homogeneous unit); using expert knowledge +covariates (e.g., terrain classification)
• Bui, E. N., & Moran, C. J. (2001). Disaggregation of polygons of surficial geology and soil maps using spatialmodelling and legacy data. Geoderma, 103(1-2), 79-94.https://doi.org/10.1016/S0016-7061(01)00070-2
• Khan, M. R., de Bie, C. A. J. M., van Keulen, H., Smaling, E. M. A., & Real, R. (2010). Disaggregating and mappingcrop statistics using hypertemporal remote sensing. International journal of applied earth observation andgeoinformation, 12(1), 36-46.https://doi.org/10.1016/j.jag.2009.09.010
D G Rossiter
Spatial concepts 25
The modifiable areal unit problem
Issue: the same analysis may give different results, i.e., lead to differentinferences / conclusions, if the data is aggregated at different scales
Example: voting patterns by large → small geographic area (polygon size):
• Vote for B.H. Obama vs. W.M. Romney, US President, 2012:
1. USA (Obama 51.4%)2. NY state (Obama 62.6%)3. NY 23rd congressional district (Romney wins, Obama 48.4%)4. Tompkins County (Obama 68.2%)5. City of Ithaca (Obama 83.3%)6. 5th ward (Obama 85.2%)7. 5th ward 2nd district (Obama 91.6%),
• Summarize predictor variables at the same scales (party registration, censusethnicity, IRS-reported income . . . )
• Do you expect the same predictive model?
D G Rossiter
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