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Earth Coordinates
&
Grid Coordinate Systems
You will always find parallels and meridians on large-
scale maps
This is done to make the map a very close
approximation to the size and shape of the piece of
the ellipsoidal earth that it represents
Horizontal Reference Datums
Datums are the collection of very accurate control
points (points of known accuracy) surveyors use to
geo-reference all other map data
Surveyors determine the precise latitude and
longitude of horizontal control points spread across
the landscape
Horizontal control point monuments are fixed
objects established by surveyors when they
determine the exact location of a point
See Figure 1.15, page 17, Muehrcke
Often found on top of a hill or other prominent
feature
From the 1920s to the early 1980s,these control
points were surveyed relative to the surface of the
Clarke 1866 ellipsoid, together forming what is
called the North American Datum of 1927 (NAD 27)
Topographic maps, nautical and aeronautical charts
and many other large-scale maps of this time period
had graticule lines or ticks based on this datum
See Figure 1.16, page 17, Muehrcke
The Corvallis, Oregon map was published in 1969
and the bottom-right corner of the map has the NAD
27 latitude and longitude
By the early 1980s, better knowledge of the earth’s
shape and size and far better surveying methods let
to the creation of a new horizontal reference datum,
the North American Datum of 1983 (NAD 83)
The NAD 27 control points were corrected for
surveying errors where required, then these were
added to thousands of more recently acquired
points
The latitudes and longitudes of all of these points
were determined relative to the Geodetic Reference
System of 1980 (GRS 80) ellipsoid, which is
essentially identical to the WGS 84 ellipsoid
The change of horizontal reference datum meant
that the coordinates for control points across the
continent changed slightly.
This change had to be shown on large-scale maps
published earlier but still in use
See Figure 1.16, page 17, Muehrcke
On topographic maps the new position of the map
corner is shown by a dashed “plus” sign
Many times the shift is in the 100 meter range and
must be taken into account when plotting on older
maps the latitudes and longitudes obtained from
GPS receivers
Even today, many countries continue to use local
datums or ellipsoids resulting in incompatibilities
The European Terrestrial Reference System of 1989
(ETRS 89) is a way to standardize mapping
coordinates but many countries continue to use
locally-derived ellipsoids
The Earth as a Geoid
When we treat the earth as an oblate ellipsoid, we
neglect mountain ranges, ocean trenches and other
surface features
There is justification for doing this
The earth’s surface is truly smooth when we
compare the surface undulations to the 7,918 mile
diameter of the earth
The greatest relief variation is the approximately
12.3 mile difference between the summit of Mt.
Everest (29,035 feet) and the deepest point in the
Marianas Trench (36,192 feet)
This vertical difference is immense on our human
scale, bit only 1/640th of the earth’s diameter
It has been said that is the earth were a bowling ball,
it would be smoother than the bowling ball
The top of Mt. Everest is located at:
o 27o59’N, 86o56’E, 29,035 feet (8,852 meters)
What is this elevation relative to?
Vertical Reference Datums
Elevations and depths are measured relative to what
is called a vertical reference datum
An arbitrary surface with an elevation of zero
The traditional datum used for land elevation is
mean sea level (MSL)
Surveyors define mean sea level as the average for
all low and high tides at a particular starting location
over a metonic cycle (the 19 year cycle of the lunar
phases and days of the year)
Early surveyors chose this datum because of the
measurement technology of the day
Surveyors first used the method of leveling, where
elevations are determined relative to the point
where mean sea level is defined
They used horizontally aligned telescopes and
vertically aligned leveling rods
A small circular monument was placed in the ground
at each surveyed benchmark elevation point
A benchmark is a permanent monument that
establishes the exact elevation of a place
Later, surveyors could determine elevation by
making gravity measurements at different locations
on the landform and relating them to the strength of
gravity at the point used to define MSL
Gravity differences translate into elevation
differences
Mean sea level is easy to determine along
coastlines, but what about inland locations?
What is needed is to extend MSL across the land
Imagine that the MSL is extended under the
continental land masses
This is the same as extending a surface having the
same strength of gravity as mean sea level
See Figure 1.17, page 18, Muehrcke
This imaginary equal gravity surface doesn’t form a
perfect ellipsoid because differences in topography
and earth density affect gravity’s pull at different
locations
The slightly undulating, nearly ellipsoidal surface
that best fits mean sea level for all the earth’s
surface is called a global geoid
The global geoid rises and falls approximately 100
meters above and below the oblate ellipsoid surface
in an irregular pattern
World maps showing land topography and ocean
bathymetry use land heights and water depths
relative to the global geoid surface
The mean sea level datum based on the geoid is so
convenient that it is used to determine elevations
around the world
It is used as the base for the elevation data found on
nearly all topographic maps and nautical charts
Be aware that the local geoid used in your area is
probably slightly above or below (usually within 2
meters) the global geoid elevations used on world
maps
This difference is caused by mean sea level at one or
more locations being used as the vertical reference
datum for your nation or continent, not the average
sea level for all oceans
In the US you may see elevations relative to the
National Geodetic Vertical Datum of 1929 (NGVD
29) on older topographic maps
This datum was defined by the observed heights of
mean sea level at 26 tide gauges, 21 in the US and 5
in Canada
It was also defined by the set of elevations of all
benchmarks resulting from over 60,000 miles of
leveling across the continent totaling over 500,000
vertical control points
North American Vertical Datum of 1988 (NAVD 88):
In the 1980s surveyors adjusted the 1929 datum
with new data
Mean sea level for the continent was defined at one
tidal station on the St. Lawrence River at Rimouski,
Quebec, Canada
NAVD 88 was a necessary update of the 1929 datum
since about 400,000 miles of leveling was added to
the NGVD since 1929
Additionally, numerous benchmarks had been lost
over the decades and the elevations of others had
been affected by vertical changes caused by rising of
land elevations since the retreat of glaciers at the
end of the last ice age (isostatic rebound) or
subsidence from sedimentation and the extraction
of natural resources like oil and water
GPS has created a second option for measuring
elevation
GPS receivers calculate what is called the ellipsoidal
height, the distance above or below the surface of
the WGS 84 ellipsoid along a line from the surface to
the center of the earth
See Figure 1.17, page 18, Muehrcke
An ellipsoidal height is not an elevation, since it is
not measured relative to the mean sea level datum
for your local geoid
You must convert GPS ellipsoidal height values to
mean sea level datum elevations before you can use
them with existing maps
See Figure 1.18, page 19, Muehrcke
Grid Coordinate Systems
There are several ways to pinpoint locations on a
map
The latitude and longitude graticule has been used
for over 2000 years as the worldwide locational
reference system
Geocentric latitude and longitude coordinates on a
sphere or geodetic latitudes and longitudes on an
oblate ellipsoid, still key to modern position finding,
are not as well suited for making measurements of
length, direction and area on the earth’s surface
The basic difficulty is the fact that latitude-longitude
is a coordinate system giving positions on a rounded
surface
It would be much simpler if we could designate
location on a flat surface using horizontal and
vertical lines spaced at regular intervals to form a
square grid
We could then simply read coordinates from the
square grid of intersecting straight lines
Most maps are created by projecting the earth’s
surface onto a flat surface
The advantage of the flat map projection surface is
that we can locate something by using a two-axis
coordinate reference system
This coordinate system is the basis for the square
grid of horizontal and vertical lines on a map
We call a plane-rectangular coordinate system
based upon and mathematically placed on a map
projection a grid coordinate system
To devise such a system for large areas, we have to
deal somehow with the earth’s curvature
We know that transferring something a spherical to
something flat always introduces geometrical
distortion
We also know that map projection distortion caused
by the earth’s ellipsoid shape is minimal for fairly
small regions
If we superimpose a square grid onto flat maps of
small areas, we can achieve positional accuracy good
enough for many map uses
All geographic grid systems are based on Cartesian
coordinates, invented in 1637 by the famous French
philosopher and mathematician Rene Descartes
Cartesian Coordinates
If you superimpose a square grid on a map, with
divisions on a horizontal x-axis and a vertical y-axis
where the axes cross at the system’s origin, you have
established the Cartesian Coordinate System
See Figure 4.1, page 64, Muehrcke
You can now pinpoint any location on the map
precisely and objectively by giving its two
coordinates (x,y)
The Cartesian Coordinate System is divided into four
quadrants (I-IV) based on whether the values along
the x- and y-axes are positive or negative
Mapmakers use only quadrant I for grid coordinate
systems so that all coordinates will be positive
numbers relative to the (0,0) grid origin
Grid Coordinates
The simplest way of defining map positions based on
Cartesian Coordinates has definite advantages over
using the spherical graticule to define positions
Measuring x- and y-coordinates from horizontal and
vertical axes with equally spaced distance
increments greatly simplifies locating environmental
features because you do not have to deal with the
decreasing separation between meridians
converging towards the poles
Grid coordinates systems based on the Cartesian
Coordinate System are especially handy for such
map analysis procedures as finding the distance or
direction between locations or determining the area
of a mapped feature like a lake
Two popular grid coordinate systems are:
o Universal Transverse Mercator (UTM)
o State Plane Coordinate System
Universal Transverse Mercator System
A grid coordinate system can be used worldwide if
enough zones are defined to insure reasonable
geometric accuracy
The best known is UTM
The UTM grid extends around the world from 84oN
to 80oS
60 north-south zones are used, each 6o longitude
See Figure 4.2, page 65, Muehrcke
Each zone has its own central meridian and uses a
secant case transverse Mercator projection
centered on the zone’s central meridian for each of
the 60 zones
This projection makes it possible to achieve a
geometrical accuracy of one part in 2,500 maximum
scale error
Scale factors ranging from 0.9996 to 1.0004 within
each zone
Each zone is individually numbered from west to
east
Each zone has separate origins for the northern and
southern hemispheres
Easting: the east-west x-coordinate in a grid
coordinate system. That is the distance east from the
origin
In both the northern and southern hemispheres, an
easting value of 500,000 meters (written
500,000mE) is assigned to the central meridian of
each UTM zone
This value, called the false easting, is added to all x-
coordinates so that there are no negative eastings in
the zone
A northing is the north-south y-coordinate in a grid
coordinate system
In the northern hemisphere a northing value of 0mN
is assigned to the equator so all northing values are
positive numbers
In the southern hemisphere, the equator is given a
false northing of 10,000,000mN
There are no negative y-values in the southern UTM
zone because this false northing values places the
origin of the zone very close to the south pole
UTM Example
See Figure 4.3, page 66, Meuhrcke
Zone 10 covers much of the western seaboard
The x-axis follows the equator
The central meridian for the zone is 123oW
The longitude range is 120oW to 126oW
The origin lies on the equator 500,000 meters west
of the central meridian at 123oW
UTM Coordinates of the 68th St Entrance to the North
Building
587390.92mE, 4513546.43mN, Zone 18 North
The near global extent of the UTM grid makes is a
valuable worldwide referencing system
The UTM grid is indicated on many foreign maps and
on all recent USGS maps
All GPS vendors program the UTM specifications
into their receivers
Note that UTM coordinates will differ when different
datums are used
You should check the datum information in the GPS
receiver to assure that the coordinates are being
recorded in the correct system
Because meridians and not stat boundaries delimit
UTM zones, it usually takes more than one UTM
zone to cover a state completely
Universal Polar Stereographic System
UTM grid zones extend from 80oS to 84oN
To complete the global coverage, a complementary
rectangle coordinate system called the universal
polar stereographic (UPS) system was created
UPS grid consists of a north zone and a south zone
Each zone is superimposed on a secant case polar
stereographic projection that covers a circular
region over each pole
See Figure 4.4, page 66, Muehckre
The north zone extends from 84oN to the north pole
The UPS coordinate at grid center is:
o 2,000000mE, 2,000,000mN
This assures all coordinates are positive
Virtually all large-scale maps of these high latitudes
are based on the UPS grid
The UTM grid system was not extended to the poles
because the 60 zones converge at the poles meaning
that a new zone would be encountered every few
miles
State Plane Coordinate (SPC) System
The state plane coordinate system was created in
the 1930s by the land surveying profession in the US
as a way to define property boundaries that would
simplify computation of land parcel perimeters and
areas
The idea was to completely cover the US and its
territories with grids laid over map projection
surfaces so that the maximum scale distortion error
would not exceed 1 part in 10,000
A distance measured over a 10,000 foot course
would be accurate to within a foot of the true
distance
This level of accuracy could not be achieved if only
one grid covered the whole country, because the
area is too large
The solution was to divide each state into one or
more zones and make a separate grid for each zone
The US was originally divided into 125 zones
Each has its own projection surface based on the
Clarke 1866 ellipsoid and NAD 27 geodetic latitudes
and longitudes
Most states have several zones
See Figure 4.5, page 67
Secant case Lambert conformal conic projections
are used for states of predominantly east-west
extent
Secant case transverse Mercator projections are
used for states of greater north-south extent
States with more than one zone use the names
North, South, East, West and Central to identify
zones
California uses Roman numerals
In recent years the following states have combined
zones into a single one:
o Nebraska
o South Carolina
o Montana
The logic of the SPC system is simple
Zone boundaries follow state and county
boundaries because surveyors have to register land
surveys in a particular county
Each zone has its own central meridian that defines
the vertical axis for the zone
An origin is established to the west and south of the
zone
Usually 2,000,000 feet west of the central meridian
for the Lambert conformal conic zones and
500,000 feet for the transverse Mercator zones
This means that the central meridians will usually
have an x-coordinate of either 500,000 feet for the
transverse Mercator or 2,000,000 for the Lambert
conformal conic zones
These large numbers for zone centers were selected
so that all x-coordinates will be positive numbers
Although different for each zone, the origin is always
at a parallel to the south of the zone to ensure that
all y-coordinates will be positive numbers
SPC coordinates are read in the same manner as
UTM coordinates – first to the east and then to the
north of the zone’s origin
For example the location of the state capitol dome in
Madison Wisconsin is:
o 2,164,600 ft E, 392,280 ft N, Wisconsin, south
zone
In 1983 the SPC system was modernized by
switching to NAD 83 and the GRS 80 ellipsoid
Zones were redefined in metric units
The SPC system served the needs of states when it
was created and state plane coordinates have been
widely used for public works and land surveys
The SPC system is now largely obsolete as far as
surveyors are concerned
One reason is modern surveying methods can
exceed the accuracy
Also, each SPC zone is a separate entity with its own
grid definition – a fact that frustrate and discourages
uses across zone boundaries
Other Grid Systems
US state grids – combine UTM zones into a single
state grid
European grid coordinate systems
British National Grid coordinate system
Swiss coordinate system