# 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 kowoma GPS tracker 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?