Position Determination with GPS
In a considerably simplified approach, each satellite is sending out signals with the following content: I am satellite X, my position is Y and this information was sent at time Z. In addition to its own position, each satellite sends data about the position of other satellites. These orbit data (ephemeris und almanac data) are stored by the GPS receiver for later calculations.
For the determination of its position on earth, the GPS receiver compares
the time when the signal was sent by the satellite with the time the signal
was received. From this time difference the distance between receiver
and satellite can be calculated.
If data from other satellites are taken into account, the present position can be calculated by trilateration (meaning the determination of a distance from three points). This means that at least three satellites are required to determine the position of the GPS receiver on the earth surface. The calculation of a position from 3 satellite signals is called 2D-position fix (two-dimensional position determination). It is only two dimensional because the receiver has to assume that it is located on the earth surface (on a plane two-dimensional surface). By means of four or more satellites, an absolute position in a three dimensional space can be determined. A 3D-position fix also gives the height above the earth surface as a result.
Simplified, the position determination by means of a GPS works on the sample principle as the distance of thunderstorms can be judged: the time is measured between lightning and the following thunder. The speed of light is so high that the delay between the time where the flash hits the ground and the time the observer sees the flash can be neglected. The speed of sound in the earth’s atmosphere is approximately 340 m/s. This means that for example a difference of 3 seconds between lightning and thunder corresponds to approximately 1 km distance to the thunderstorm.
However, this procedure is not yet a determination of a position, but only a determination of a distance. If different people on fixed positions would determine the time span between lightning and thunder, this would allow the determination of the position where the flash hit the ground!
In the following an explanation is given, how the position determination by GPS works. For simplification, in the first step we assume that the earth is a two-dimensional disk. This allows us to do some understandable sketches for illustration. The principle can then be transferred to the model of a three-dimensional globe.
In the example on the left, the time needed by a signal to travel from
the first of two satellites to the receiver was determined to be 4 s.
(In reality this value is far too high. As the signals travel with the
speed of light (299 792 458,0 m/s), the actual time span for signals from
the satellite to the receiver lies in the range of 0.07 s.)
Based on this information, we can at state that the receiver is positioned somewhere on a circle with a radius of 4 s around the first satellite (left circle).
If we perform the same procedure with a second satellite (right circle), we get two points of intersection. On one of the two points the receiver must be situated. Now we have used two satellites. But the process is called trilateration, not dilateration so don't we need a third satellite? We may use a third satellite but we could also assume that the receiver is located somewhere close to the earth's surface and not deep in space, so we can neglect point B and know that the receiver must be found on point A. The area in the picture above which shaded grey is the region in which GPS signals are supposed to be “realistic”. Positions outside this area are discarded, so is point B.
This assumption replaces the third satellite which would in theory be required for the process of trilateration. In this example an unequivocal position is obtained from only two satellites.
So we just need a third satellite for a third dimension and that's it?
Well, in principle yes. But…
The problem lies in the determination of the exact runtime of signals. As explained above, satellites impose a sort of time stamp on each transmitted data package. We know that all clocks of satellites are absolutely precise (they are atomic clocks after all) but the problem is the clock in our GPS receiver. Atomic clocks being too expensive, our GPS receivers are based on conventional quartz clocks which are comparatively inaccurate. What does this mean in practice?
Let's stick to our example and suppose the clock in our receiver is 0.5
seconds early compared to the clock in the satellite. The runtime of the
signal seems to be 0.5 s longer than it actually is. This leads to the
assumption that we are on point B instead of point A. The circles
that intersect in point B are called pseudoranges. They are called
“pseudo” as long as no correction of the synchronisation errors
(bias) of the clocks has been performed.
Depending on the accuracy of the clock in the GPS receiver, the determined position will be more or less wrong. For the practice of GPS based navigation this would mean that no determined position can ever be of any use, as the runtimes of the signals are so short, that any clock error has an overwhelming influence on the result.
A clock error of 1/100 second, which is difficult to imagine but quite common from car races or skiing races, would in GPS navigation lead to a mistake in the position of about 3000 km. To achieve an accuracy of 10 m of the position, the runtime of the signal must be precise to 0.00000003 seconds.
As atomic clocks are no option in GPS receivers, the problem is solved
in another and quite elegant way:
If a third satellite is taken into account for the calculation of the
position, another intersection point is obtained: in case that all clocks
are absolutely precise, point A would be obtained, corresponding to the
actual position of the receiver.
In case of the receiver clock being 0.5 s early, the three intersection points B are obtained. In this case the clock error stands out immediately. If now the time of the receiver clock is shifted until the three intersection points B merge to A, the clock error is corrected and the receiver clock is synchronized with the atomic clocks in the satellites.
The GPS receiver can now be regarded as an atomic clock itself. The distances to the satellites, formerly regarded as pseudoranges, now correspond to the actual distances and the determined position is accurate.
In case of the example – a two dimensional disc world – we therefore need three satellites for an unequivocal determination of our position. In the real world which has one additional dimension, we would need a fourth satellite.
Well, then why is it always said that three satellites are enough?
In practice you get a two-dimensional position determination (2D-fix) with three satellites. The position is bound to be located on the earth's surface. The fourth satellite is the geocenter; the distance to the “fourth satellite” corresponds to 6360 km (the radius of the globe). Therewith the fourth satellite necessary for the calculation is given, but the calculation is restricted to locations on the earth surface. However the earth is not a perfect sphere. The surface of the earth in this case means the earth geoid, corresponding to sea level. If the receiver is located on a mountain, the determined position again is afflicted with an inaccuracy, as the runtime of the satellite signals is wrong.
By constantly recalculating its position, the GPS receiver can additionally determine the speed and direction of a movement (referred to as "ground speed" and "ground track").
Another possibility of determining the speed is by using the Doppler's effect which occurs due to the movement of the receiver while receiving the signals. The principle is the same as for a moving siren on a police car: the tune is higher when the car moves towards the listener and it is lower when the car moves away.