119.255 GPS, mapping and coordinate systems.
Adapted from:
http://www.lordpercy.com/gps_explained.htm
http://www.kowoma.de/en/gps/index.htm
http://www.whatisgps.net/
http://www.trimble.com
http://www.esri.com/news/arcuser/0103/differential1of2.html
The Global Positioning System uses a fairly basic mathematical principle which goes
by the name of Trilateration and is sometimes called triangulation by those who think
they know more than they do. It works by using snippets of known information to
pinpoint a location, if you got lost in let’s say near Shannon and asked a passing
policeman where you were and he said 120 kms from Wellington you may get a bit
annoyed but this is a useful piece of information. You now know you are within a 120
km radius of Wellington and could draw this circle on a map.
If you asked the same question of another passing good spirited individual who told
you were 60 kms from Palmerston North you could draw a circle around Palmerston
North, these two circles will intersect at two points but you don't know at which one
you are standing. So not good if you plan to navigate using this position and much
worse if you are planning to use it to guide a cruise missile.
So with a third location "20 Km from Levin" you can pinpoint exactly where you are,
simple! Well that's in 2D the GPS system does all this Trilateration in 3D using a
Network of 27 US military satellites, 24 of which are used to make calculations and 3
which are spares.
Orbits of the GPS-Satellites
(distances are to scale)
The satellites orbit the earth with a speed of 3.9 km per second and have a
circulation time of 12 h sidereal time, corresponding to 11 h 58 min earth time. This
means that the same satellite reaches a certain position about 4 minutes earlier each
day. The mean distance from the middle of the earth is 26560 km. With a mean earth
radius of 6360 km, the height of the orbits is then about 20200 km. Orbits in this
height are referred to as MEO – medium earth orbit. In comparison, geostationary
satellites like ASTRA or Meteosat – satellites orbit the earth at 42300 km, which is
about twice the distance of GPS satellites.
The satellites are arranged on 6 planes, each of them containing at least 4 slots
where satellites can be arranged equidistantly. Today, typically more than 24
satellites orbit the earth, improving the availability of the system.
The last component we need for an accurate position is to know how far we are from
each satellite, the GPS system does this by sending a pseudo-random code which
starts every night at exactly midnight, the receiver compares this to its own internal
quartz clock and can determine how delayed the radio signal was when it received it
and hence the distance between it and the satellite.
Of course a quartz clock is no match for the atomic clock inside the satellite but by
using a minimum of 4 satellites the error can be nulled out as each reading should be
incorrect by the same amount. We now have the three pieces of information required
to navigate Longitude, Latitude and Altitude, when combined with an electronic map
running on a navigation device you can pinpoint your position down to 15 meters.
There are many applications for GPS the most common is for in vehicle navigation
systems like Tom Tom and Navman use the GPS signal to feed into PDA or PC
based systems where it is used to overlay your current position onto a map and plot
journeys. For walkers many products exist that will give the current location to aid
map navigation or on some of the high end models you can run maps of the local
terrain although these tend to be more limited and of less detail. The Trimble site has
a number of agricultural examples.
How accurate is GPS? (see also WAAS)
Today's GPS receivers are extremely accurate, thanks to their parallel multi-channel
design. most GPS's 12 parallel channel receivers are quick to lock onto satellites
when first turned on and they maintain strong locks, even in dense foliage or urban
settings with tall buildings. Certain atmospheric factors and other sources of error can
affect the accuracy of GPS receivers. Most GPS receivers are accurate to within 15
meters on average.
Newer GPS receivers with WAAS (Wide Area Augmentation System) capability can
improve accuracy to less than three meters on average. No additional equipment or
fees are required to take advantage of WAAS. Users can also get better accuracy
with Differential GPS (DGPS), which corrects GPS signals to within an average of
three to five meters. A number of agencies around the world operate DGPS
correction services, many are free bur in New Zealand user pays. These system
consists of a network of towers that receive GPS signals and transmit a corrected
signal by beacon transmitters. In order to get the corrected signal, users must have a
differential beacon receiver and beacon antenna in addition to their GPS.
The picture below shows the ground track of satellite BIIR-07 (PRN 18) for a 24 hr
period. The yellow arrow marks the 00:00 ’o clock time point. It can be seen that the
orbit time is slightly shifted (about 4 minutes) in 24 h.
The yellow dot marks the satellite position at 09:30 pm. The satellite is positioned
over Ethiopia. The correlating “zone of sight”, within which the satellite signal can be
received, is marked as the footprint in light blue. The graph was compiled with the
FreeWare software WinOrbit (and slightly modified).,
What's the signal?
GPS satellites transmit two low power radio signals, designated L1 and L2. Civilian
GPS uses the L1 frequency of 1575.42 MHz in the UHF band. The signals travel by
line of sight, meaning they will pass through clouds, glass and plastic but will not go
through most solid objects such as buildings and mountains.
A GPS signal contains three different bits of information — a pseudorandom code,
ephemeris data and almanac data. The pseudorandom code is simply an I.D. code
that identifies which satellite is transmitting information.
Ephemeris data, which is constantly transmitted by each satellite, contains important
information about the status of the satellite (healthy or unhealthy), current date and
time. This part of the signal is essential for determining a position.
The almanac data tells the GPS receiver where each GPS satellite should be at any
time throughout the day. Each satellite transmits almanac data showing the orbital
information for that satellite and for every other satellite in the system.
Sources of GPS signal errors
Factors that can degrade the GPS signal and thus affect accuracy include the
following:
Ionosphere and troposphere delays — The satellite signal slows as it passes
through the atmosphere. The GPS system uses a built-in model that calculates an
average amount of delay to partially correct for this type of error.
Signal multipath — This occurs when the GPS signal is reflected off objects such
as tall buildings or large rock surfaces before it reaches the receiver. This
increases the travel time of the signal, thereby causing errors.
Receiver clock errors — A receiver's built-in clock is not as accurate as the atomic
clocks onboard the GPS satellites. Therefore, it may have very slight timing errors.
Orbital errors — Also known as ephemeris errors, these are inaccuracies of the
satellite's reported location.
Number of satellites visible — The more satellites a GPS receiver can "see," the
better the accuracy. Buildings, terrain, electronic interference, or sometimes even
dense foliage can block signal reception, causing position errors or possibly no
position reading at all. GPS units typically will not work indoors, underwater or
underground.
Satellite geometry/shading — This refers to the relative position of the satellites at
any given time. Ideal satellite geometry exists when the satellites are located at
wide angles relative to each other. Poor geometry results when the satellites are
located in a line or in a tight grouping.
Intentional degradation of the satellite signal — Selective Availability (SA) is an
intentional degradation of the signal once imposed by the U.S. Department of
Defense. SA was intended to prevent military adversaries from using the highly
accurate GPS signals. The government turned off SA in May 2000, which
significantly improved the accuracy of civilian GPS receivers.
COORDINATE SYSTEMS
Today, the most widely used coordinate system is longitude and latitude. The prime
meridian and the equator are the reference planes for the definition of longitude and
latitude.
Latitude
For the following thoughts, lets assume that
Earth is a perfect sphere and on first glance it
looks like being one. Earth revolves around
its axis once every 24 hours. To be precise,
this rotation does define the axis. Thinking
more figurative and assuming a physical
axis, this axis would pierce the earth in two
points. These points are the North Pole and
the South Pole. These to Poles must not be
confused with the magnetic poles as these
are significantly displaced from the turning
axis. This fact troubled and still troubles
Earth with longitude and latitude circles
mariners. The equator is located precisely
between the two poles,. It is aligned
perpendicular to the earth’s axis. This first circle of latitude is the origin for counting
the circles of latitude. Starting at the equator, the angle is measured in degrees to the
north and to the south from 0 to 90. Doing this, more circles of latitude are created
and a Position on Earth can be determined as being on a circle around the earth. It is
still impossible to say where exactly you are knowing only the altitude but you are
somewhere on this circle. The determination of the latitude is achieved quite easy
and mankind knows about this at least since the 15 th century using the position of
the Sun in the sky. Also, on the northern hemisphere, the Polar Star directly indicates
the latitude.
Earth’s radius is 6370 kilometers and the circumference is about 40000 km. From
this, the distance between two whole-numbered latitude circles calculates to about
111 km. One degree divides into 60 arc minutes and one arc minutes consists of 60
arc seconds. The distance of one arc minute on Earth’s surface is 1.85 km and like a
coincidence this is exactly the distance of one nautical mile. Of course this did not
happen by chance, the nautical mile was defined being one arc minute and this
makes it more comfortable to measure distances in maps with a scale in longitudes
and latitudes.
View of the Earth from space at summer solstice (left) and winter solstice (right)
Except for the equator and the poles there are a couple more “important” latitude
circles. There is the Polar Circles. These latitudes are at 66.55° (66° 33’) N and S.
They mark the transition between the moderate climatic regions and the polar
regions. They also mark the latitude where the sun does not set in summer and does
not rise in winter. This is caused by the earth being tilted by 23.45° relative to the axis
on its path through space.
The other two important latitudes are the tropics. The northern tropic, also called
tropic of cancer is at 23.45° north (23° 27’). At this latitude, at noon on June 21 st
(more or less) the sun shines from exactly the zenith. This date is midsummer and
the tropic is called tropic of cancer because long time ago, sun was found at the
constellation of cancer at this time of the year. On the northern hemisphere, at this
time of the year, days are longest and nights are shortest. Being more northern than
the polar circle, the sun does not set at least on this day. It may not set for a longer
period of time, depending how far north you are. The tropic in the south, at 23.45°
south is the tropic of Capricorn. Midwinter or the winter solstice is at December 21 st.
Longitude
In addition to the latitude circles, longitude circles or meridians are being added.
They are circles being orthogonal to the equator and they cross both poles. Different
to the latitude, where the equator is a natural origin, there is no start or end for the
longitude circles. As a consequence, there was more than one prime meridian in the
past. As shortly ago as in 1883, on an international geodetic conference in Rome ,
the prime meridian was set to go through Greenwich in England . One reason why
Greenwich was chosen is, that as a consequence, the international date line runs
through the Pacific. This is convenient since there do not happen to live many
people. Since the rotation of the Earth does not have a start and an end, it took a
very long time for mankind to be able to determine the exact longitude of their
position. This brilliant feat became only possible after invention of clocks that were
very precisely for longer periods of time and even on sea. By determination of time
the sun reached its maximum height in the sky at an unknown position compared to
the time of noon at the prime meridian, a precise determination of the longitude
position became possible.
Longitudes are counted easterly and westerly from 0 to 180° starting at the prime
meridian in Greenwich . Instead of adding letters of direction to the longitude and
latitude positions like N and S, E and W, often negative values are used for the south
and west positions.
Degree values used for directions
The unit degree is not only used for
temperatures, longitudes and latitudes but also
for directions. A direction of 0 (degrees) is
North, navigators count clockwise to 360°.
Doing this, 90° is to the East, 180° is to the
South and 270° is to the West. There are four
main intermediate values with names like NE
(45°) and eight more intermediate values
between them and the main directions like
WNW (292.5°). These intermediate directions
are best known from wind directions.
Characteristics of the New Zealand Map Grid (See last page of this
section)
• The true origin of the projection is placed at latitude 41 degrees south (the whole
degree nearest to the middle latitude of the country).
The true origin is placed at longitude 173 degrees east (the whole degree nearest
to the meridian to which equal perpindiculars can be drawn from the eastemmost and
westernmost points of the country).
The meridian of longitude 173 degrees east, which is not represented by a straight
line, is oriented so that its tangent at the origin is the north-south axis of co-ordinates.
The true origin is assigned arbitrary co-ordinates sufficiently large to render all coordinates positive, or east and north of a so-called "false origin".
In a metre co-ordinate system sufficiently extensive to cover the whole country,
the northing must at some stage reach seven integral figures. In the NZMG the coordinates have seven integral figures in all cases.
The easting is always less than 5 000 000 m, the northing always greater than 5
000 000 m, so that no confusion between easting and northing can arise whichever
one is stated first.
• The co-ordinates assigned to the true origin are also sufficiently large for the grid
to be extended a considerable distance out to sea without departing from the
characteristics stated in 5 and 6.
The true origin is placed at latitude 41 degrees south, longitude 173 degrees east,
and the co-ordinates of this point are 2 510 000 m east, 6 023 150 m north.
References
Hannah, J. 1984. Map Projections: their development and use in New Zealand. NZ
Institute of Surveyors: Auckland.
Mackie, 3. B. 1980. Introduction to Geodesy, 4th Ed. Department of Surveying,
University of Otago: Dunedin.
Information on New Zealand Geodetic Datum is available on:
http://www.linz.govt.nz/core/surveysystem/geodeticinfo/geodeticdatums/index.html