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Computing the Distance Between Your Position and the GPS Satellites |
GPS
determines distance between a GPS satellite and a GPS
receiver by measuring the amount of time it takes a radio
signal (the GPS signal) to travel from the satellite to
the receiver. Radio waves travel at the speed of light,
which is about 186,000 miles per second. So, if the amount
of time it takes for the signal to travel from the satellite
to the receiver is known, the distance from the satellite
to the receiver (distance = speed x time) can be determined.
If the exact time when the signal was transmitted and
the exact time when it was received are known, the signal's
travel time can be determined.
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In
order to do this, the satellites and the receivers use
very accurate clocks which are synchronized so that they
generate the same code at exactly the same time. The code
received from the satellite can be compared with the code
generated by the receiver. By comparing the codes, the
time difference between when the satellite generated the
code and when the receiver generated the code can be determined.
This interval is the travel time of the code. Multiplying
this travel time, in seconds, by 186,000 miles per second
gives the distance from the receiver position to the satellite
in miles.
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Four (4) Satellites to give a 3D position |
Three
measurements can be used to locate a point, assuming the
GPS receiver and satellite clocks are precisely and continually
synchronized, thereby allowing the distance calculations
to be accurately determined. Unfortunately, it is impossible
to synchronize these two clocks, since the clocks in GPS
receivers are not as accurate as the very precise and
expensive atomic clocks in the satellites. The GPS signals
travel from the satellite to the receiver very fast, so
if the two clocks are off by only a small fraction, the
determined position data may be considerably distorted.
The atomic clocks aboard the satellites
maintain their time to a very high degree of accuracy.
However, there will always be a slight variation in clock
rates from satellite to satellite. Close monitoring of
the clock of each satellite from the ground permits the
control station to insert a message in the signal of each
satellite which precisely describes the drift rate of
that satellite's clock. The insertion of the drift rate
effectively synchronizes all of the GPS satellite clocks.
The same procedure cannot be applied
to the clock in a GPS receiver. Therefore, a fourth variable
(in addition to x, y and z), time, must be determined
in order to calculate a precise location. Mathematically,
to solve for four unknowns (x,y,z, and t), there must
be four equations. In determining GPS positions, the four
equations are represented by signals from four different
satellites.
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What are cold/warm/hot fixes? |
When
you first fire up a GPS receiver it has no data with which
to work. It doesn't know about what satellites to use,
where they are or even what the time is. All of that information
has to be received before a fix can be calculated. Fixes
that start with NO information are called "cold"
fixes. After receiving the time signal from one satellite,
the GPS receiver can set its internal clock. It then listens
for any satellite it can "hear" to send out
almanac or ephemeris data. Once the almanac is received
the receiver can then listen specifically for the satellites
that are near the one satellite it can presently hear
at the time. If it hears them, it knows the almanac is
relatively current. Finally, the GPS receiver gets the
ephemeris data about where the satellites are located
in space and with this information and the time signals
from the satellites it can calculate its location and
present the first "fix" to your mapping software.
The process of a cold fix can take as much as 20 minutes,
but may also be done in as little as 3-6 minutes. If you
move the receiver during this first fix, the time may
be extended significantly. The problem with movement is
that if the GPS loses contact with a satellite in the
middle of receiving an ephemeris or almanac string of
data it has to wait until the next full cycle of the signals
before it gets a new chance.
A "warm" fix is one
where the receiver has relatively current almanac data
and just needs the ephemeris update and time signals.
Modern receivers have a small battery and small memory
space internal to them where the data from the last good
fix is held. When power is re-applied, if this data is
still there the receiver uses it as a starting point and
if it is verified by signals it receives, the receiver
can get a new fix is 1-2 minutes, or less. If the data
is not there (for example, if the little battery had died
as most do in a few days or so) or if it is inaccurate
(for example, you take the GPS receiver over 200 miles
from the last fix with it powered off-think airline flight
for a concrete example of this) then the GPS receiver
has to do a cold fix.
A "hot" fix is one where
the receiver has lost the signal from the satellite for
a very brief time (driving through a tunnel, in an urban
canyon, under trees in a forest), but the ephemeris and
almanac data is still valid. In this case the simple acquisition
of time signals is all the receiver needs to relocate
itself. Hot fixes typically only take a few seconds, 5
or less, and can actually happen in less than one second.
Why can't I get a fix?
Many things can interfere with your ability to get a fix:
- Moving the receiver before it gets ephemeris/almanac
data
- Signal blocked by walls, buildings, trees, car roofs,
bridges, tunnels
- Poor satellite geometry - although every effort is made
to have the satellites in "good" orbits, if
you happen to be somewhere where the constellation is
not amenable to a fix, you may have to wait a few minutes
for the satellites to move to a more friendly arrangement
in the sky.
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The
accuracy with which a position can be determined using
GPS in navigation mode depends, on the one hand, on the
accuracy of the individual pseudo-range measurements,
and on the other, on the geometrical configuration of
the satellites used. This is expressed in a scalar quantity,
which in navigation literature is termed DOP (Dilution
of Precision).
There are several DOP designations in current use:
- GDOP: Geometrical DOP (position in 3-D space, incl.
time deviation in the solution)
- PDOP: Positional DOP (position in 3-D space)
- HDOP: Horizontal DOP (position on a plane)
- VDOP: Vertical DOP (height only)
The accuracy of any measurement is proportionately dependent
on the DOP value. This means that if the DOP value doubles,
the error in determining a position increases by a factor
of two.
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PDOP
can be interpreted as a reciprocal value of the volume
of a tetrahedron, formed by the positions of the satellites
and user. The best geometrical situation occurs when
the volume is at a maximum and PDOP at a minimum. Some
GPS receivers can analyse the positions of the satellites
available, based upon the almanac, and choose those
satellites with the best geometry in order to make the
DOP as low as possible. Another important GPS receiver
feature is to be able to ignore or eliminate GPS readings
with DOP values that exceed user-defined limits. Other
GPS receivers may have the ability to use all of the
satellites in view, thus minimizing the DOP as much
as possible.
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The
GPS system has been designed to be as nearly accurate
as possible. However, there are still errors. Added together,
these errors can cause a deviation of +/- 50 -100 meters
from the actual GPS receiver position. There are several
sources for these errors, the most significant of which
are discussed below:
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Atmospheric
Conditions
The
ionosphere and troposphere both refract the GPS signals.
This causes the speed of the GPS signal in the ionosphere
and troposphere to be different from the speed of the
GPS signal in space. Therefore, the distance calculated
from "Signal Speed x Time" will be different
for the portion of the GPS signal path that passes through
the ionosphere and troposphere and for the portion that
passes through space.
Ephemeris Errors/Clock Drift/Measurement Noise
As mentioned earlier,
GPS signals contain information about ephemeris (orbital
position) errors, and about the rate of clock drift for
the broadcasting satellite. The data concerning ephemeris
errors may not exactly model the true satellite motion
or the exact rate of clock drift. Distortion of the signal
by measurement noise can further increase positional error.
The disparity in ephemeris data can introduce 1-5 meters
of positional error, clock drift disparity can introduce
0-1.5 meters of positional error and measurement noise
can introduce 0-10 meters of positional error.
Selective
Availability
Ephemeris errors should not be
confused with Selective Availability (SA), which is the
intentional alteration of the time and epherimis signal
by the Department of Defense. SA can introduce 0-70 meters
of positional error. Fortunately, positional errors caused
by SA can be removed by differential correction.
Multipath
A GPS signal bouncing off a reflective
surface prior to reaching the GPS receiver antenna is
referred to as multipath. Because it is difficult to completely
correct multipath error, even in high precision GPS units,
multipath error is a serious concern to the GPS user.
The chart below lists the most common sources of error
in GPS positions. This chart is commonly known as the
GPS Error Budget:
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| Source
Uncorrected |
Error
Level |
| Ionosphere |
0-30
meters |
| Troposphere |
0-30
meters |
| Measurement
Noise |
0-10
meters |
| Ephemeris
Data |
1-5
meters |
| Clock
Drift |
0-1.5
meters |
| Multipath |
0-1
meter |
| Selective
Availability |
0-70
meters |
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Using Differential GPS to Increase Accuracy |
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As
powerful as GPS is, +/-50 - 100 meters of uncertainty
is not acceptable in many applications. How can we obtain
higher accuracies?
Differential GPS, or DGPS, has
been developed to improve GPS accuracy to within a few
meters. DGPS was originally initiated by the U.S. Coast
Guard to counter the accuracy degradation caused by Selective
Availability. Even with S/A now eliminated, DGPS continues
to be a key tool for highly precise navigation on land
and sea. DGPS technology adds a land-based reference receiver,
located at an accurately surveyed site, to the other GPS
components. This non-moving DGPS reference station knows
where the satellites are located in space at any given
moment, as well as its own exact location. This allows
the station to compute theoretical distance and signal
travel times between itself and each satellite. When those
theoretical measurements are compared to actual satellite
transmissions, any differences represent the error in
the satellite's signal. All the DGPS reference station
has to do is transmit the error factors to your DGPS receiver,
which gives the information to the GPS receiver so it
can use the data to correct its own measurements and calculations.
After differential correction, the GPS Error Budget changes
as follows:
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| Source
Uncorrected |
Error
Level |
With differential |
| Ionosphere |
0-30
meters |
Mostly Removed |
| Troposphere |
0-30
meters |
All Removed |
| Measurement
Noise |
0-10
meters | All Removed |
| Ephemeris
Data |
1-5
meters | All Removed |
| Clock
Drift |
0-1.5
meters | All Removed |
| Multipath |
0-1
meter |
Not Removed |
| Selective
Availability |
0-70
meters | All Removed |
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