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http://www.geog.ucsb.edu/~good/176b/a05.html#7.%20
UNIVERSAL%20TRANSVERSE%20MERCATOR
GEOGRAPHY 176B: TECHNICAL ISSUES IN GIS
LECTURE 5: GEOREFERENCING
1. THE PROBLEM
2. PLACENAMES
3. ADDRESSES, POSTCODES, AREA CODES
4. THE CADASTER AND PUBLIC LAND SURVEYSYSTEM
5. LATITUDE AND LONGITUDE
6. PROJECTIONS AND COORDINATES
7. UNIVERSAL TRANSVERSE MERCATOR
8. STATE PLANE COORDINATES
Credit: many of the illustrations in this unit were prepared by Peter Dana, University of Texas, Austin, for
the Geographer's Craft project
1. THE PROBLEM
The atom of geographic information
<location, time, attribute>
to communicate, we need standard ways of
dealing with all three
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time
Gregorian calendar
attribute
depends on application
temperature scales
vegetation classifications
highway classifications
location
many ways of referencing
how to specify the
locations of a polygon's points?
120.12456 W, 34.89176 N
909 West Campus Lane,Goleta, CA 93117, USA
5789654N, 314654E
NE 1/4, Section 12,
Township 23 Range 5 of
the Second Principal
Meridian
How to reference a location on the Earth's surface?
in a way that others can understand
with sufficient accuracy for the application
many methods
The act of assigning locations to things
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georeferencing
geocoding
geolocating
within the geographic frame of reference
other frames of reference?
Requirements of a georeferencing system
uniqueness
one code per location
every location gets its own code
over what domain?
the entire globe
universal systems
latitude/longitude
over a limited domain
e.g. unique in the U.S.
e.g. unique to a zone
same geocode repeats in
different places
7-digit phone
number repeats
in every area
code
10-digit phone
number unique in
the U.S. andCanada
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11-digit phone
number (add 1)
unique in the
world
placenames repeat
at least 18 stateshave a city of
Springfield
shared meaning
sender and receiver must both
understand
latitude/longitude is
universally understood
Tobler's postcardmail a letter from China to
909 West Campus Lane,
Goleta, CA 93117, USA
persistent through time
latitude/longitude since the 1890s
area codes often change
spatial resolution
how big is the area covered by a
single georeference?
Rhode Island pins location
down to 3000 sq km
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California only to 410,000
sq km
area or linear measure?
a rectangle of
3000 sq km is
about 55 kmacross
a rectangle of
410,000 sq km is
about 650 kmacross
we tend to work
with both
metric georeferences
measure position with respect tofixed points
can be used to measure distances
between points
coordinates
e.g. Salt Lake City street
addresses
ordered allocation
e.g. New York avenues
2. PLACENAMES
Locations identified by name
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likely the first type of georeference
littering the surface of the Earth
Placename authorities
state, national boards
official authorization
Board on Geographic Names
Geographic Names Information
System
Alexandria Gazetteer
Relative names
e.g. 5 miles west of Greenfield, CA
e.g. between Salinas and Greenfield
the museum collection problem
collected 1 mile north of Cachuma
Saddle
metes and bounds property description
from the big maple tree, 100 ft in a
southerly direction to the crossing of
the creek
Problems with placenames
variable spatial resolution
e.g. Asia
e.g. Eiffel Tower
lost through time
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Where was Camelot?
Only locally defined
The Riviera
downtown
Not officially recognized
The Midwest
Fuzzy or crisp
The Atlantic Ocean
Context-specific
LA
If you live in New York
if you live in RiversideAlphabets
Chinese place names
3. ADDRESSES, POSTCODES, AREA CODES
Devised for specific purposes
Used for many other purposes
Addresses
Delivering mail
registering property
place of residence
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spatial resolution
one mailbox
one property
domain
local area
assumptions
dwelling is a destination for mail
dwellings arrayed along streets
streets have names that are unique
within local area
local areas have names that are
unique within state/country
violations?
give your street address to911?
where do they not work?
Postal codes
ZIP - 93117-4338
unique in US
spatial resolution <1 block
changes made frequently
UK postcode - SE3R 1KW
Canadian postcode - N6G 1R1
widely known and accessible
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mailing lists
widely used for marketing, analysis
Area codes
4. THE CADASTER AND PUBLIC LAND SURVEY
SYSTEM
Cadaster
the map of land ownership
field-like
at any point there is exactly one
owner
not an effective georeference
Public land survey system
devised circa 1830 for surveying the western
lands acquired by the U.S.
extended to Canada
a regular basis for allocating land to new
owners
used for georeferencing in land management
oil and gas
BLM
Elements of the PLSS
the principal meridian
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a north-south line laid out very
accurately
ranges
at 6 mile intervals
townships
6 mile blocks east and west of the
principal meridian
sections
36 blocks of 1 square mile each (640
ac)
regularly numbered
numbering east of the principalmeridian
1 2 3 4 5 6
12 11 10 9 8 7
13 14 15 16 17 18
24 23 22 21 20 19
25 26 27 28 29 30
36 35 34 33 32 31
Reversed west of the principal
meridian
4 quarter sections of 1/4 sq mile (160 ac)
NW NE
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SW SE
quarter-quarter sections of 1/16 sq mi (40 ac)
fixed spatial resolution
can go as fine as needed
hierarchical system
Applications
locations of oil and gas leases
locations of public land locations of agricultural land
Problems
the Earth isn't flat
surveying errors
steep terrainother irregularities
Santa Barbara county
5. LATITUDE AND LONGITUDE
Earth's axis
5 m wobble
Center of mass
Equator
90 degrees to axis, through the center of mass
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Prime meridian
Royal Observatory at Greenwich
plane through the axis and the prime meridian
Longitude
plane through the axis and a point
angle between the two planes
180 W to 180 E
360 degrees in 24 hours
15 degrees per time zone
-180 to +180
history of longitude
Longitude by Dava SobelFigure of the Earth
geoid
isosurface
sea level if continents were porous
accurately measured with satellites
ellipsoid
mathematical surface
rotate an ellipse around the axis
some ellipsoids
WGS84
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NAD83
a=6,378,137 m
b=6,356,752 m
f =1/298.257
Clarke's ellipsoid of 1866
NAD27
List of ellipsoids
Ellipsoid Semi-major axis 1/flattening
Airy 1830 6377563.396 299.3249646
Modified
Airy6377340.189 299.3249646
Australian
National6378160 298.25
Bessel 1841(Namibia)
6377483.865 299.1528128
Bessel 1841 6377397.155 299.1528128
Clarke 1866 6378206.4 294.9786982
Clarke 1880 6378249.145 293.465
Everest
(India 1830)6377276.345 300.8017
Everest
(Sabah
Sarawak)
6377298.556 300.8017
Everest
(India 1956)6377301.243 300.8017
Everest(Malaysia 6377295.664 300.8017
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1969)
Everest
(Malay. &
Sing)
6377304.063 300.8017
Everest
(Pakistan)6377309.613 300.8017
Modified
Fischer 19606378155 298.3
Helmert
1906
6378200 298.3
Hough 1960 6378270 297
Indonesian
19746378160 298.247
International
19246378388 297
Krassovsky
1940 6378245 298.3
GRS 80 6378137 298.257222101
South
American
1969
6378160 298.25
WGS 72 6378135 298.26
WGS 84 6378137 298.257223563
Latitude
angle between the Equator and a line drawn
perpendicular to the ellipsoid
not necessarily through the center of
mass
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90 S to 90 N
-90 to +90
parallel
line of constant latitude
Precision of latitude and longitude
degrees, minutes, seconds
decimal degrees
1 second latitude about 30 m
1 minute latitude = 1 nautical mile
1 degree latitude about 70 miles or 110 km
longitude degrees shorter on the Earth
except at the Equator
depending on latitude
at 30 degrees 0.866
at 45 degrees 0.707
at 60 degrees 0.500
at 90 degrees 0.000
great circle
slice through the center of mass
divides the Earth into two equal
hemispheres
small circle
slice not through the center of mass
Approximate figures of the Earth
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larger than 1:10,000, assume a flat Earth
from 1:10,000 to 1:10,000,000 assume a
spheroid
smaller than 1:10,000,000 assume a sphere
Distances on the sphere
latitude φ
longitude λ
R arccos (sin φ1 sin φ2 + cos φ1 cos φ2 cos(λ1 - λ2))
circumference 2π R
distance from Equator (0,0) to N Pole (90,0)
R arccos (0x1 + 1x0 cos (0 - 0))
R arccos (0)
Rπ/2
WGS84
R = (a+b)/2 = 6,367,444 m
Rπ/2 = 10,039,173 m
6. PROJECTIONS AND COORDINATES
Why flatten the Earth?
paper is flat
input and output
easily printed, copied
rasters are flat
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can't cover a curved surface with
squares
use of rasters in analysis, modeling
photographs are flat
to see all of the Earth at once
Map projection
a transformation from (φ,λ) to ( x, y)
Mercator projection
x = λ
y = ln tan (φ/2 + π/4)
inverse transformations
λ = x
φ = 2(arctan e y - π/4)
How does a GIS make the calculations?
often by expanding functions as series
e.g., e y = 1 + y + y2/2! + y3/3! + ...
make forward transformation followed byinverse
are the results exactly the same?
errors may accumulate
Distortion properties
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angles, areas, directions, shapes and distances
become distorted when transformed from a
curved surface to a plane
examples
all these properties cannot be kept undistorted
in a single projection
usually the distortion in one property
will be kept to a minimum while
other properties become very
distorted
Tissot's Indicatrix
is a convenient way of showing distortion
imagine a tiny circle drawn on the surface of the globe
on the distorted map the circle will
become an ellipse, squashed or
stretched by the projection
height changed by the
vertical scale (k )
width changed by the
horizontal scale (h)
the size and shape of the Indicatrix
will vary from one part of the map to
another
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we use the Indicatrix to display the
distorting effects of projections
Conformal (Orthomorphic)
a projection is conformal if the angles in the
original features are preserved
over small areas the shapes of objects
will be preserved
preservation of shape does not holdwith large regions (e.g. Greenland in
Mercator projection)
a line drawn with constant
orientation (e.g. with respect to
north) will be straight on a conformal
projection, is termed a rhumb line or
loxodrome
conformal projections are good for navigation
parallels and meridians cross each other at
right angles (note: not all projections with thisappearance are conformal)
the Tissot Indicatrix is a circle everywhere, but
its size varies
conformal projections cannot have equal area
properties, so some areas are enlarged
generally, areas near margins have a
larger scale than areas near the center
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Equal area (Equivalent)
the representation of areas is preserved so that
all regions on the projection will be
represented in correct relative size
equal area projections are good for
GIS analysis
equal area maps cannot be conformal, so mostEarth angles are deformed and shapes are
strongly distorted
the Indicatrix has the same area everywhere,
but is always elliptical, never a circle (except at
the standard parallel)
Equidistant
cannot make a single projection over which alldistances are maintained
thus, equidistant projections maintain relative
distances from one or two points only
e.g., in a conic projection all
distances from the center arerepresented at the same scale
Geometric analogy: Developable surfaces
the most common methods of projection can beconceptually described by imagining the
developable surface, which is a surface that can
be made flat by cutting it along certain linesand unfolding or unrolling it
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the points or lines where a developable surface
touches the globe in projecting from the globe
are called standard points and lines, or points
and lines of zero distortion.
at these points and lines, the scale isconstant and equal to that of the
globe, no linear distortion is present
if the developable surface touches
the globe, the projection is called
tangent
e.g. cylindrical
if the surface cuts into the globe, it is
called secant
where the surface and the
globe intersect, there is nodistortion
where the surface is
outside the globe, objects
appear bigger than in
reality - scales are greater
than 1
where the surface is insidethe globe, objects appear
smaller than in reality and
scales are less than 1
note: symbols used in the following: λ - longitude φ -
latitude χ - colatitude (90 - lat)
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h - distortion introduced along lines of
longitude
k - distortion introduced along lines of latitude
(h and k are the lengths of the minor
and major axes of the Indicatrix)
commonly used developable surfaces are:
1. Planar or azimuthal
a flat sheet is placed in contact with a globe,and points are projected from the globe to the
sheet
mathematically, the projection is easily
expressed as mappings from latitude and
longitude to polar coordinates with the origin
located at the point of contact with the paper
formulas for stereographic projection
(conformal) are:
r = 2 tan(χ / 2)
q = λ
h = k = sec2(χ / 2)
Examples:
stereographic projection
gnomic projection
Lambert's azimuthal equal-area projection
orthographic projection
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2. Conic
the transformation is made to the surface of a
cone tangent at a small circle (tangent case) or
intersecting at two small circles (secant case)
on a globe
mathematically, this projection is alsoexpressed as mappings from latitude and
longitude to polar coordinates, but with the
origin located at the apex of the cone
formulas for equidistant conical projection with one
standard parallel (φ0 , colatitude χ0) are:
r = tan(χ0) + tan(χ - χ0)
q = n λ
n = cos(χ0)
h = 1.0
k = n r / sin(χ)
Examples
Alber's conical equal area projection with twostandard parallels
Lambert conformal conic projection with two
standard parallels
equidistant conic projection with one standard parallel
3. Cylindrical
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developed by transforming the spherical
surface to a tangent or secant cylinder
mathematically, a cylinder wrapped around the
equator is expressed with x equal to longitude,
and the y coordinates some function of latitude
formulas for cylindrical equal area projectionare:
x = λ
y = sin(φ)
k = sec(φ)
h = cos(φ)
Examples
Mercator Projection
meridians and parallels intersect at
right angles
straight lines are lines of constant
bearing - projection is useful for
navigation
great circles appear as curves
Plate Carree or unprojected or cylindrical
equidistant
4. Non-Geometric (Mathematical) projections
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some projections cannot be expressed
geometrically
have only mathematical descriptions
Examples
Molleweide
Eckert
7. UNIVERSAL TRANSVERSE MERCATOR (UTM)
UTM is the first of two projection based coordinate
systems to be examined in this unit
UTM provides georeferencing at high levels of
precision for the entire globE
established in 1936 by the International Unionof Geodesy and Geophysics
adopted by the US Army in 1947
adopted by many national and
international mapping agencies,
including NATO
is commonly used in topographic and
thematic mapping, for referencingsatellite imagery and as a basis for
widely distributed spatial databases
Transverse Mercator Projection
results from wrapping the cylinder around the poles rather than around the equator
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the central meridian is the meridian where the
cylinder touches the sphere
theoretically, the central meridian is the line of
zero distortion
by rotating the cylinder around the poles the
central meridian (and area of least distortion)can be moved around the earth
Zone System
in order to reduce distortion the globe is
divided into 60 zones, 6 degrees of longitude
wide
zones are numbered eastward, 1 to 60,
beginning at 180 degrees (W long)
the system is only used from 84 degrees N to80 degrees south as distortion at the poles is
too great with this projection
at the poles, a Universal Polar
Stereographic projection (UPS) is
used
each zone is divided further into strips of 8degrees latitude
beginning at 80 degrees S, are
assigned letters C through X, O and I
are omitted
picture
Distortion
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to reduce the distortion across the area covered
by each zone, scale along the central meridian
is reduced to 0.9996
this produces two parallel lines of zero
distortion approximately 180 km away fromthe central meridian
scale at the zone boundary is approximately
1.0003 at US latitudes
Coordinates
coordinates are expressed in meters
eastings ( x) are displacements
eastward
northings ( y) express displacement
northwardthe central meridian is given an easting of
500,000 m
the northing for the equator varies depending
on hemisphere
when calculating coordinates for
locations in the northern hemisphere,
the equator has a northing of 0 m
in the southern hemisphere, the
equator has a northing of 10,000,000
m
UTM zone 14
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Advantages
UTM is frequently used
consistent for the globe
is a universal approach to accurate
georeferencing
Disadvantages
full georeference requires the hemisphere, zone
number, easting and northing (unless the areaof the data base falls completely within a zone)
rectangular grid superimposed on zones
defined by meridians causes axes on adjacentzones to be skewed with respect to each other
problems arise in working across zone
boundaries
no simple mathematical relationship exists
between coordinates of one zone and an
adjacent zone
8. STATE PLANE COORDINATES (SPC)
SPCs are individual coordinate systems adopted by U.S.
state agencies
originated in the 1930s based on NAD27
revised in 1983 based on NAD83
each state's shape determines which projectionis chosen to represent that state
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e.g. a state extended N/S may use a
Transverse Mercator projection
while a state extended E/W may use
a Lambert Conformal Conic projection (both of these are
conformal)
projections are chosen to minimize distortion
over the state
a state may have 2 or more
overlapping zones, each with its own projection system and grid
Texas zones
units are generally in feet
Advantages
SPC may give a better representation than the
UTM system for a state's area
SPC coordinates may be simpler than those of
UTM
Disadvantages
SPC are not universal from state to state
problems may arise at the boundaries of
projections
Use in GIS
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many GIS have catalogs of SPC projections
listed by state which can be used to choose the
appropriate projection for a given state
REFERENCES
Maling, D.H., 1973. Coordinate Systems and Map
Projections, George Phillip and Son Limited, London.
Robinson, A.H., R.D. Sale, J.L. Morrison and P.C.
Muehrcke, 1984, Elements of Cartography, 5th edition,
John Wiley and Sons, New York. See pages 56-105.
Snyder, J.P., 1987. Map Projections - A Working
Manual , US Geological Survey Professional Paper 1395,
US Government Printing Office, Washington.
Strahler, A.N. and A.H. Strahler, 1987. Modern Physical
Geography, 3rd edition, Wiley, New York. See pages 3-8for a description of latitude and longitude and
various appendices for information on coordinate
systems.
REVIEW QUESTIONS
1. Define the three standard properties of map projections: equal-area, equidistant and conformal.
Discuss the relative importance of each for different
applications. What types of applications require which
properties?
2. What type of projection would you expect to be used
in the following circumstances, and why?
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a. an airline pilot flying the North Atlantic between New
York and London.
b. a submarine navigating under the ice of the North
Pole.
c. an agricultural scientist assembling crop yield data for
Africa.
d. an engineer planning the locations of radiotransmitters across the continental US.
3. What map projections would you choose in designing
a workstation to be used by scientists studying various
aspects of global environmental change?
4. By examining the list of SPC systems adopted by the
states, what can you deduce about the criteria used to
determine the projection adopted and the number of
zones used? You will need a map of the US showing the boundaries of states. Are there any surprising choices?
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