Datums Ellipsoids Projections CoordinateSystems

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Representing

The Shape

of

the

Earth

•Geoids

•Ellipsoids

•Datums

•Coordinate Systems

•Projections

The Shape of the Earth

3 ways

to

model

it

• Topographic surface• the land/air interface

• complex (rivers, valleys, etc) and difficult to model

• Geoid• a theoretical, continuous surface for the earth which is perpendicular at every point

to the direction of gravity (surface to which plumb line is perpendicular)

• approximates mean sea‐level in open ocean without tides, waves or swell

• satellite observation (after 1957) showed it to be quite irregular because of local variations in gravity.

• Spheres and spheroids (3‐dimensional circle and ellipse)

• mathematical

models

which

can

be

used

to

approximate

the

geoid

and

provide

the

basis for accurate location (horizontal) and elevation (vertical) measurement

• sphere (3‐dimensional circle) with radius of 6,370,997m considered ‘close enough’ for small scale maps (1:5,000,000 and below ‐ e.g. 1:7,500,000)

• spheroid (3‐dimensional ellipse) should be used for larger scale maps of 1:1,000,000 or more (e.g. 1:24,000)


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Spheroid, Ellipsoid, and Geoid

• Spheroid is a solid generated by rotating an ellipse about either the major

or minor axis

• Ellipsoid is a solid for which all plane sections through one axis are ellipses

and through the other are ellipses or circles

– If any two of the three axes of that ellipsoid are equal, the figure becomes a

spheroid (ellipsoid of revolution)

– If all three are equal, it becomes a sphere

• Geoid is the equipotential gravity surface of the earth at mean sea level.

At any point it is perpendicular to the direction of gravity

Reference: Smith, James R., Introduction to Geodesy: The history and concepts of modern geodesy. John Wiley & Sons, 1997

What is an Oblate Ellipsoid (Spheroid)?


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Which Spheroid to use?

Hundreds

have

been

defined

depending

upon:

• Available measurement technology

• Area of the globe – e.g North America, Africa

• Map extent – country, continent or global

• Political issues – e.g Warsaw pact versus NATO

• ArcGIS supports 26 different

spheroids!

– conversions via math formulae

Most commonly encountered are:

• Clarke 1866 for North America

• basis for USGS 7.5 Quads

• a=6,378,206.4m b=6,356,583.8m

• GRS80 (Geodetic Ref. System, 1980)

• current North America mapping

• a=6,378,137m b=6,356,752.31414m

• WGS84 (World

Geodetic

Survey,

1984)

• current global choice

• a=6,378,137 b=6,356,752.31

Latitude and Longitude: location on the spheroid

Longitude meridiansPrime meridian is zero: Greenwich, U.K.

International Date Line is 180° E&W

1 degree=69.17 mi at Equator

53.06 mi at 40N/S

0 mi at 90N/S

Latitude parallelsequator is zero

1 degree=68.70 mi at equator

69.41 mi at poles

(1 mile=1.60934km=5280 feet)

Lat / long coordinates for a

location change depending

on spheroid chosen!


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graticule: network of lines on globe or map representing latitude and longitude.Origin is at Equator/Prime Meridian intersection (0,0)grid: set of uniformly spaced straight lines intersecting at right angles.

(XY Cartesian coordinate system)Latitude normally listed first (lat,long), the reverse of the convention for X,Y Cartesian coordinates

Latitude and Longitude Graticule

Latitude

and

Longitude

• The most comprehensive and powerful method

of georeferencing

– Metric, standard, stable, unique

• Uses a well‐defined and fixed reference frame

– Based on the Earth’s rotation and center of mass, and

the Greenwich Meridian


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Definition of longitude. The Earth is seen here from above the North Pole, looking

along the Axis, with the Equator forming the outer circle. The location of Greenwichdefines the Prime Meridian. The longitude of the point at the center of the red cross is

determined by drawing a plane through it and the axis, and measuring the angle

between this plane and the Prime Meridian.

Definition of Latitude

• Requires a model of the Earth’s shape

• The Earth is somewhat elliptical – The N‐S diameter is roughly 1/300 less than the E‐

W diameter

– More accurately modeled as an ellipsoid than a

sphere – An ellipsoid is formed by rotating an ellipse about

its shorter axis (the Earth’s axis in this case)


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The Public Land Survey System (PLSS)


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The (Brief)

History

of

Ellipsoids

• Because the Earth is not shaped precisely as an ellipsoid, initially each country felt free to adopt its own as the most accurate approximation to its own part of the Earth

• Today an international standard has been adopted known as WGS 84 – Its US implementation is the North American Datum of 1983 (NAD 83)

– Many US maps and data sets still use the North American Datum of 1927 (NAD 27)

– Differences can be as much as 200 m


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Latitude and the Ellipsoid

• Latitude (of the red point) is the

angle between a perpendicular

to the surface and the plane of

the Equator

• WGS 84

– Radius of the Earth at the Equator

6378.137 km

– Flattening 1 part in 298.257

Geoid

• A geoid is a representation of the Earth which would

coincide exactly with the mean ocean surface of the

Earth, if the oceans were to be extended through the

continents

• A smooth but highly irregular surface that corresponds

but to

a surface

which

can

only

be

known

through

extensive gravitational measurements and calculations,

not to the actual surface of the Earth's crust


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Geoid

Geoid vs.

an

Ellipsoid

1. Ocean

2. Ellipsoid

3. Local plumb

4. Continent

5. Geoid


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Datums:all location measurement is relative to a specific datum

For the Geodesist

• a set of parameters defining a coordinate system, including:

– the spheroid (earth model)

– a point of origin (ties spheroid to earth)

For the Local Surveyor

• a set of points whose precise location and /or elevation has been determined, which serve as reference points from which other point’s locations can be determined (horizontal datum)

• a surface to which elevations are referenced, usually ‘mean sea level’ (vertical datum)

• points usually marked with brass plates called survey markers or monuments whose identification codes and precise locations (usually in lat/long) are published

North

American

Datums• NAD27

– Clark 1866 spheroid

– Meades Ranch origin

– visual triangulation

– 25,000 stations • (250,000 by 1970)

– NAVD29 (North American Vertical Datum, 1929) provided elevation

– basis for most USGS 7.5 minute quads

NAD83

– satellite (since 1957) and laser distance data showed inaccuracy of NAD27

– 1971 National Academy of Sciences report recommended new datum

– used GRS80 spheroid

(functionally equivalent to WGS84)

– origin: Mass‐center of Earth

– 275,000 stations

– “Helmert blocking” least squares technique fitted 2.5 million other fed, state and local agency points.

– NAVD88 provides vertical datum

– points can differ up to 160m from NAD27, but

seldom more

than

30m,

and

data

from

digit.

map

more inaccurate than datum diff.

– no universal mathematical formulae for conversion from NAD27: See USGS Survey Bulletin # 1875 for conversion tables (in ARC/INFO). Transformations are preformed to local coordinates

– http://www.ngs.noaa.gov/cgi‐bin/nadcon.prl


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Ground‐zero for Geo‐nerds everywhere

Meades Ranch, KS (12 miles north of Lucas,

KS) is the designated geodedetic base point

for the North American Datum 1927 (NAD 27)

Owner of ranch is now Mr. Kyle Brant

•access with permission only

http://www.scottosphere.com/history/meades-ranch.html

The NGS data sheet is here:http://www.ngs.noaa.gov/cgi-bin/ds2.prl?retrieval_type=by_pid&PID=KG0640

State Plane & the NAD 27Calculations for map projections are performed using the parameters of the ellipsoid


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Roadside Marker for the Geodetic Center of North America

Meades Ranch, KS (12 miles north of Lucas, KS) is the designated geodedetic base point

for the North American Datum 1927 (NAD 27)

http://www.worldslargestthings.com/kansas/geodetic.htm


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Map Projections: the concept

A method by which the curved 3‐D surface of the earth is

represented on a flat 2‐D map surface

a two dimensional representation, using a plane coordinate

system, of the earth’s three dimensional sphere/spheroid

location on the 3‐D earth is measured by latitude and longitude

location

on

the

2‐

D

map

is

measured

by

x,y Cartesian

coordinates

unlike choice of spheroid, choice of map projection does not

change a location’s lat/long coords, only its XY coords.

Map projections

• Earth spherical ‐ maps flat!

• Thus all maps have distortions

• A good map has distortions that are predictable

and systematic


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Map projection

Why is it called a

‘projection’?

Because we ‘project’ the

earth’s spherical surface

on a flat surface‐

As if we

were

shining

a light

from center of earth:

Maps can

be:

1. Conformal: Shape correct

2. Equivalent (Equal area): Area correct

3. Azimuthal: Direction correct


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Physical Surface

Use a physical surface to project the sphere

1. Plane

2. Cone

3. Cylinder

Note differences between projections by comparing distortions

of the

lines

of

latitude

and

longitude.

1.

Plane projection

Hold plane against the

surface of the globe

(typically the pole)

Lines of longitude

straight, radiating

Lines of latitudes are

circles


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1. Plane projection

4. Distortion increases

away from the center

("principal point")

5. Good for polar regions

6. But can't show more

than half the world.

2.

Conic projection

• Hold cone over pole


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2. Conic projection

2. Lines of latitude curve

3. Line of longitude are straight, and converge

towards top

4. Distortion increases away from standard

parallel

5. Good for Mid‐latitudes

3.

Cylindrical projection

1. Wrap cylinder around the earth

2. Lines of latitude and longitude are straight, intersect at 90°

3. Distortion

greatest at poles


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3. Cylindrical projection

4. Good for low‐latitude areas

5. Poor representation of poles

Good for navigation

Mercator’s

Projection

Other ‐

mathematical

• Condensed and interrupted projections.


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Projections and Coordinates

• There are many reasons for wanting to project the Earth’s surface onto a plane, rather than deal with the curved surface

– The paper used to output GIS maps is flat

– Flat maps are scanned and digitized to create GIS databases

– Rasters are flat, it’s impossible to create a raster on a curved surface

– The Earth has to be projected to see all of it at once

– It’s much easier to measure distance on a plane

Map Projections:

the inevitability

of

distortion

• because we are trying to represent a 3‐D sphere on a 2‐D

plane, distortion is inevitable

• thus, every two dimensional map is inaccurate with respect to

at least one of the following:

– area

– shape

– distance

– direction

We are trying to represent

this amount of the earth on

this amount of map space.


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Distortions

• Any projection

must

distort

the

Earth

in

some

way

• Two types of projections are important in GIS

– Conformal property: Shapes of small features are preserved: anywhere on the projection the distortion is the same in all directions

– Equal area property: Shapes are distorted, but features have the correct area

– Both types of projections will generally distort distances!

Map Projections: classifications ‐ or

How can

we

think

about

map

projections?

Property Preserved• Equal area projections preserve the

area of features (popular in GIS)

• Conformal projections preserve the shape of small features (good for presentations) , and show localdirections (bearings) correctly (useful for coastal navigation)

• Equidistant projections preserve distances (scale) to places from one point, or along a one or more lines

• True

direction

projections preserve

bearings (azimuths) either locally (in which case they are also conformal) or from center of map.

Geometric Model Used• Planar/Azimuthal/Zenithal : image of

spherical globe is projected onto a map plane which is tangent to (touches) globe at single point

• Conical: image of spherical globe is projected onto a cone which is tangent along a line(s) (usually a parallel of latitude)‐‐ cone is then unfolded to create “flat map”

• Cylindrical: image

of

spherical

globe

is

projected onto a cylinder which also is tangent along a line(s)‐‐again, cylinder is unfolded to create a “flat map”

Classified by Property Preserved or by Geometrical Model


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Geometric Model

Choosing a Map Projection

• Issues to Consider:

• Extent of area to map: city, state, country, world?

• Location: polar, mid‐latitude, equatorial?

• Predominant extent of area to map: E‐W, N‐S, oblique?

• Rules of thumb

• Always record lat/long coords not projected X,Y coords in the GIS database if

possible

• Check project specifications; does it specify a required projection?

• State Plane or UTM often specified for US gov. work.

• Use equal

‐area

projections

for

thematic

or

distribution

maps,

and

as

a general

choice for GIS work

• Use conformal projections in presentations

• For navigational applications, need true distance or direction


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Commonly Encountered Map Projections in GIS

• Albers Conic Equal‐Area – often used for US base maps showing all of the “lower 48” states

– standard parallels set at 29 1/2N and 45 1/2N

• Lambert Conformal Conic – often used for US Base map of all 50 states (including Alaska and Hawaii), with

standard parallels set at 37N and 65N

– also for State Base Map series, with standard parallels at 33N and 45N

– also used in State Plane Coordinate System (SPCS)

• Transverse Mercator – used in SPCS for States with major N/S extent

– Universal Transverse Mercator (UTM) used for world wide military (and other) large

scale mapping

Most commonly, you encounter these 3 projections, along with the SPCS and UTMprojections systems which use them.

Cylindrical Projections

NormalMercator

TransverseTransverse Mercator

ObliqueOblique Mercator

Parallels are line

of tangency

Meridians are line

of tangency


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Conformal Projections

• Preserve

local

shape

• Misrepresent areas

• Graticule lines (lat and long) on globe are perpendicular

• Best used for large‐scale reference maps

• Preserve angles and shapes at points

Do not use for data distribution maps – will distort area, and

therefore, misrepresent

densities

Better for showing routes and locations

Lambert Conformal Conic

Conterminous U.S.

http://howdyyall.com/Texas/Members/Bob/GPS/Projectn.htm


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Equal Area Projections• Preserve the area of displayed features

• Graticule lines (lat and long) on globe may not intersect at right

angles

• In some instances (especially maps of smaller regions) it will not

be obvious that shape has been distorted

• Thus, it will not be obvious that shape has been distorted and

distinguishing from an equal‐area projection from a conformal

projection is difficult (unless documented)

Choose equal area when making thematic maps

Remember – if you are mapping data distributions, choose an equal

area projection!

Albers Equal Area is customized for the continental United States

Albers Equal‐Area Conic

Conterminous U.S.

http://howdyyall.com/Texas/Members/Bob/GPS/Projectn.htm

Has two lines of standard parallel

(two lines that are ‘true’)


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The “Unprojected” Projection

• Assign latitude to the y axis and longitude to the x axis

– A type of cylindrical projection

– Is neither conformal nor equal area

– As latitude increases, lines of longitude are much closer

together on the Earth, but are the same distance apart on the

projection

– Also known as the Plate Carrée or Cylindrical Equidistant

Projection

The Universal Transverse Mercator (UTM) Projection

• A type of cylindrical projection

• Implemented as an internationally standard coordinate system

• Initially devised as a military standard

• Uses a system of 60 zones• Maximum distortion is 0.04%

• Transverse Mercator because the cylinder is wrapped around the Poles, not the Equator


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Zones are each six degrees of longitude, numbered as shown at the top, from W to E

Universal Transverse Mercator (UTM)

• First adopted by US Army in 1947 for large scale maps worldwide

• Used from lat. 84°N to 80°S; Universal Polar Stereographic (UPS) used for polar areas

• Globe divided into 60 N/S zones, each 6° wide; – these are numbered from one to sixty going east from 180th meridian

• Each zone divided into 20 E/W belts, each 8° high lettered from the south pole using C thru X (O and I omitted)

• The meridian halfway between the two boundary meridians for each zone is designated as the central meridian and a cylindrical projection is done for each zone


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• Coordinate origins are at the intersection of the equator and the zone’s central

meridian;

• This origin given a value of 0 meters north, 500,000m east, thus no negative

values

– a false origin at 10,000,000 meters south used for southern hemishere

• Military uses a different system for coordinate location dividing each UTM

primary grid zone into 10km by 10km squares and designates each by a double

letter

UTM in Pennsylvania:2 zones: Zone 17 (84W-78W), Zone 18 (78W-72W)



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Implications of the Zone System

• Each zone defines a different projection

• Two maps of adjacent zones will not fit along their common border

• Jurisdictions that span two zones must make special arrangements

– Use only one of the two projections, and accept the greater‐

than‐normal

distortions

in

the

other

zone

– Use a third projection spanning the jurisdiction

– E.g. Italy is spans UTM zones 32 and 33

UTM Coordinates

• In the N Hemisphere define the Equator as 0 mN

• The central meridian of the zone is given a false

easting of 500,000 mE

• Eastings and northings are both in meters

allowing easy estimation of distance on the

projection

• A UTM georeference consists of a zone number,

a hemisphere, a six‐digit easting and a seven‐

digit northing

– E.g., 14, N, 468324E, 5362789N


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State Plane Coordinates

• Defined in the US by each state

– Some states use multiple zones

– Several different types of projections are used by

the system

• Provides less distortion than UTM

– Preferred for applications needing very high

accuracy, such

as

surveying

State Plane Coordinate System (SPCS)

• Began in 1930s for public works projects; popular with interstate designers

• States divided into 1 or more zones (~130 total for US)

• each zone designed to maintain scale distortion to less than 1 part per 10,000

• Pennsylvania has 2 zones running E/W: north (3701), south (3702)

• Different projections used:

• transverse mercator (conformal) for States with large N/S extent

• Lambert conformal conic for rest (incl. Pennsylvania)

• some states use both projections (NY, FL, AK)

• oblique mercator used for Alaska panhandle

• Each zone also has:

• unique standard parallels (2 for Lambert) or central meridian (1 for mercator)

• false coordinate origins which differ between zones, and use feet for NAD27 and meters for NAD83

• (1m=39.37 in. exact used for conversion)• See Snyder, 1982 USGS Bulletin # 1532, p. 56‐63 for details


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North American Datums

• NAD27 – Clark 1866 spheroid

– Meades Ranch origin

– visual triangulation

– 25,000 stations • (250,000 by 1970)

– NAVD29 (North American Vertical Datum, 1929) provided elevation

– basis for most USGS 7.5 minute quads

NAD83 – satellite (since 1957) and laser distance data

showed inaccuracy of NAD27

– 1971 National Academy of Sciences report recommended new datum

– used GRS80 spheroid

(functionally equivalent to WGS84)

– origin: Mass‐center of Earth

– 275,000 stations

– “Helmert blocking” least squares technique fitted 2.5 million other fed, state and local agency points.

– NAVD88 provides vertical datum

– points can differ up to 160m from NAD27, but seldom more than 30m, and data from digit. map more inaccurate than datum diff.

– no universal mathematical formulae for conversion from

NAD27:

See

USGS

Survey

Bulletin

# 1875

for

conversion tables (in ARC/INFO). Transformations are preformed to local coordinates

– http://www.ngs.noaa.gov/cgi‐bin/nadcon.prl

NAD27 and NAD83 Ellipsoids (Canadian Spacial Reference System, 2006)

Documents

Datums Ellipsoids Projections CoordinateSystems