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Territories, Environment, Remote Sensing & Spatial Information Joint Research Unit Cemagref - CIRAD - ENGREF

GIS: concepts, methods & toolsGIS: concepts, methods & tools

Georeferencing and projectionsGeoreferencing and projections

METIER METIER GraduateGraduate Training Course no. 2 Training Course no. 2 –– Montpellier Montpellier -- FebruaryFebruary 20072007

Information Management in Information Management in EnvironmentalEnvironmental SciencesSciences

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GEOREFERENSING AND PROJECTIONSGEOREFERENSING AND PROJECTIONS

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The problemThe problem

surface ≅ sphere ���� plane surface (map)

M(λ,ϕ) → m(x,y)

m

x

y

– modelling the surface to be projecteddetermining the shape and dimensions of the surface

– projection of the surface on a planeestablishing a projection algorithm

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System of geographic coordinatesSystem of geographic coordinates

• Given that:the Earth presents heterogeneities due to:

• the non-sphericity of the terrestrial volume

• heterogeneity of its density

• the variable altitude of its surface

• Modelling of the terrestrial surface:– geodetic reference base (position/centre of the Earth)

• Cartesian coordinates

–ellipsoid (longitude, latitude):

• mathematically defined envelope

• geographic coordinates (polar)

– geoid (Z):

• gravitational equipotential surface

• mean sea level

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Geoid and Geoid and

ellipsoidsellipsoids

•• Mean sea level:

This is the reference level for altitude measurements. It corresponds to the mean height of

the sea surface calculated for all tidal stages.

• Altitude:

Difference of gravitational potential between a point (or an object) on the Earth’s surface

and a reference surface (normally the mean sea level).

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Definition: Equipotential surface of the Earth’s gravity field that best coincides with the

mean sea level.

This corresponds to the topography that a terrestrial surface would have if it was covered by oceans at rest, only subject to terrestrial gravity, i.e., a surface on which

water does not flow.

The geoid The geoid -- 11

Exaggerated altitudes

×××× 15,000

(ref. Geiger, 1987

in U. Frei et al., 1993)

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The ellipsoid The ellipsoid –– geographic geographic

coordinatescoordinates

X

Y

Z

λ

φO

a

b

X,Y,Z: geocentric coordinates

λ,φ, OM: geographic coordinates

M

m

ellipsoid: a,b

a=equatorial radius

b=semi axis at the poles

The ellipsoid is often defined by:

- flattening: f = (a-b)/a

- square of the eccentricity: e²=(a²-b²)/a²

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Some ellipsoids & their parametersSome ellipsoids & their parameters

Geodesic

system

Associated

ellipsoid

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Changing the ellipsoidChanging the ellipsoid

X

Y

Z

O

X’

Y’

Z’

O’

OO’: ∆X, ∆Y, ∆Z

Rotations: δX, δY, δZ

Scale: ∆e

Normal reference: WGS 84

Use of parameters which geometrically

define the ellipsoid:

the DATUM

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Projections: plane coordinatesProjections: plane coordinates

• The coordinates in projection of M are cartesian coordinates (E,N) of

point m, image of M in the plane projection equipped with an

orthonormal frame of reference (O;e,n)

• The cartographic projection is defined by two functions f and g such that:

E = f (λ,ϕ) et N = g (λ,ϕ)

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Defining (or changing) a system of Defining (or changing) a system of

projectionprojection

• Ellipsoid + datum + projection function

• Going from one system to another:

→ (x,y) projection 1 → (ϕ,λ) ellipsoid 1

→ (ϕ,λ) ellipsoid 1 →(X,Y,Z) datum 1

→ (X,Y,Z) datum 1 → (X,Y,Z) datum 2

→ (X,Y,Z) datum 2 → (ϕ,λ) ellipsoid 2

→ (x,y) projection 2

• In practice… the GIS takes care of it

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Plane projections (azimuthal)Plane projections (azimuthal)

Polar Equatorial Oblique

Gnomonic Stereographic Orthogonal

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Conic projectionsConic projections

Origin

Tangent Standard //

Origin

Secants Standard //

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Cylindric ProjectionsCylindric Projections

direct transverse oblique

Origin = the

equator

Origin =

a meridian

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Projections Projections –– gridgrid

cylindical conic azimuthal

Shape of the projection surface / representation of meridians and parallels

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Projections Projections -- typestypes

• Projection = deformation!– ellipsoid ���� plane

– all projections lead to deformations:• Depending on the properties of the projection, some will conserve the angles,

others the areas.

• Distances are never conserved over the entire map.

• Projection type � projection quality– conformal projections:

conservation of angles, distortion of areas

– equivalent projections (also called equal-area projections):

conservation of areas, modification of angles

– aphylactic projections:

do not conserve areas, nor angles; more ‘aesthetic’

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Projection Projection -- typestypes

Il is impossible to conserve both:– directions (navigation)

– areas (political or statistical maps)

Tissot’s Indicatrice Ellipses

Identical circles are drawn at different latitudes then projected:

otherequivalentconforming

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Conforming projectionConforming projection

Mercator projection: x=λ, y= tan(ϕ/2 + π/4)

(conservation of angles, distortion of areas)

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Equivalent projectionEquivalent projection

Lambert’s equivalent projection

(conservation of areas, modification of angles)

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Aphylactic projectionAphylactic projection

‘Square and flat’ projection (also called equirectangular): x=λ, y=ϕ

(does not conserve areas, nor angles)

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Deformations caused by projectionDeformations caused by projection

DIRECT MERCATOR TRANSVERSE MERCATOR

ALBERS CONIC LAMBERT CONIC

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ExampleExample of of conicconic projection : Lambertprojection : Lambert

Zone 1

Zone 2

Zone 3

Zone 4

4 projection cones

Paris Paris

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Lambert projection Lambert projection parametersparameters

(exemple France)(exemple France)

Ellipsoïde Clarke 1880 (6378249.145 293.465)

Datum NTF (-168 -60 320 0 0 0)

200 000200 000200 000200 000False northing

600 000600 000600 000600 000False easting

42°46’03’’44°59’45’’47°41’46’’50°23’45’’Standard parallel 2

41°33’37’’43°11’57’’45°53’56’’48°35’55’’Standard parallel 1

2°20’14’’2°20’14’’2°20’14’’2°20’14’’Origine longitude

42°09’54’’44°06’46°48’49°30’Origine latitude

4321Zone

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Transverse Mercator projectionTransverse Mercator projectionor ‘Gauss conformal projection’ or ‘Lambert conformal cylindricaor ‘Gauss conformal projection’ or ‘Lambert conformal cylindrical l

projection’projection’

Transverse cylindrical projection

The cylinder is tangent or secant to a meridian that we call the prime meridian.

The image of the prime meridian and of the equatorare perpendicular lines.

The parallels and the meridians are curves but are orthogonal between themselves.

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Universal Transverse Mercator (UTM) projectionUniversal Transverse Mercator (UTM) projection

System of 122 Transverse Mercator projections.

- Defined in about 1950 by the US army to represent

the entire Earth.

- made up by the juxtaposition of 120 conformal transverse Mercator projections:

- 60 meridian zones 6° wide to cover the entire globe (between 80° South and 80° North),

- 2 projections for each zone (North and South).

The poles are represented by

azimuthal projections.

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UTM projectionUTM projection

Origine latitude Equator

Origine longitude 3° Est (zone 31)

(False Easting) 500 000 m

(False Northing) 0 m (10 000 km in south hemisphere)

Scale factor 0,9996

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Reference geid

Lambert coord.

UTM coord.

Geo. coord.

(latitude)

CoordinateCoordinate systemssystems on a on a mapmap