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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(http://www.css.cornell.edu/faculty/dgr2/).

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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1 2 3 4 5 6 7

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Petal width (mm)

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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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DRIVE

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CA

NA

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RO

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TO

WN

RO

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LINE

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RO

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IDG

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GE

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GE

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RD

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ROAD

B RO

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D

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TEETER

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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

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ROAD

POINT

EA

STL AKE R D

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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

RE

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.

D G Rossiter

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

D G Rossiter

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)

D G Rossiter

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