Sources of Errors in GPS
The most relevant factor for the inaccuracy of the GPS system is no longer an issue. On May 2, 2000 5:05 am (MEZ) the so-called selective availability (SA) was turned off. Selective availability is an artificial falsification of the time in the L1 signal transmitted by the satellite. For civil GPS receivers that leads to a less accurate position determination (fluctuation of about 50 m during a few minutes). Additionally the ephemeris data are transmitted with lower accuracy, meaning that the transmitted satellite positions do not comply with the actual positions. In this way an inaccuracy of the position of 50 – 150 m can be achieved for several hours. While in times of selective availability the position determination with civil receivers had an accuracy of approximately 10 m, nowadays 20 m or even less is usual. Especially the determination of heights has improved considerably from the deactivation of SA (having been more or less useless before).
The reasons for SA were safety concerns. For example terrorists should not be provided with the possibility of locating important buildings with homemade remote control weapons. Paradoxically, during the first gulf war in 1990, SA had to be deactivated partially, as not enough military receivers were available for the American troops. 10000 civil receivers were acquired (Magellan and Trimble instruments), making a very precise orientation possible in a desert with no landmarks.
Meanwhile SA is permanently deactivated due to the broad distribution and world wide use of the GPS system.
The following two graphs show the improvement of position determination after deactivation of SA. The edge length of the diagrams is 200 m, the data were collected on May 1, 2000 and May 3, 2000 over a period of 24 h each. While with SA 95 % of all points are located within a radius of 45 m, without SA 95 % of all points are within a radius of 6.3 m.
Another factor influencing the accuracy of the position determination is the "satellite geometry". Simplified, satellite geometry describes the position of the satellites to each other from the view of the receiver.
If a receiver sees 4 satellites and all are arranged for example in the north-west, this leads to a “bad” geometry. In the worst case, no position determination is possible at all, when all distance determinations point to the same direction. Even if a position is determined, the error of the positions may be up to 100 – 150 m. If, on the other hand, the 4 satellites are well distributed over the whole firmament the determined position will be much more accurate. Let’s assume the satellites are positioned in the north, east, south and west in 90° steps. Distances can then be measured in four different directions, reflecting a „good“ satellite geometry.
The following graph shows this for the two-dimensional case.
If the two satellites are in an advantageous position, from the view of the receiver they can be seen in an angle of approximately 90° to each other. The signal runtime can not be determined absolutely precise as explained earlier. The possible positions are therefore marked by the grey circles. The point of intersection A of the two circles is a rather small, more or less quadratic field (blue), the determined position will be rather accurate.
If the satellites are more or less positioned in one line from the view of the receiver, the plane of intersection of possible positions is considerably larger and elongated- The determination of the position is less accurate.
The satellite geometry is also relevant when the receiver is used in vehicles or close to high buildings. If some of the signals are blocked off, the remaining satellites determine the quality of the position determination and if a position fix is possible at all. This can be observed in buildings close to the windows. If a position determination is possible, mostly it is not very accurate. The larger the obscured part of the sky, the more difficult the position determination gets.
Most GPS receivers do not only indicate the number of received satellites, but also their position on the firmament. This enables the user to judge, if a relevant satellite is obscured by an obstacle and if changing the position for a couple of meters might improve the accuracy. Many instruments provide a statement of the accuracy of the measured values, mostly based on a combination of different factors (which manufacturer do not willingly reveal).
To indicate the quality of the satellite geometry, the DOP values (dilution of precision) are commonly used. Based on which factors are used for the calculation of the DOP values, different variants are distinguished:
- GDOP (Geometric Dilution Of Precision); Overall-accuracy; 3D-coordinates and time
- PDOP (Positional Dilution Of Precision) ; Position accuracy; 3D-coordinates
- HDOP (Horizontal Dilution Of Precision); horizontal accuracy; 2D-coordinates
- VDOP (Vertical Dilution Of Precision); vertical accuracy; height
- TDOP (Time Dilution Of Precision); time accuracy; time
HDOP-values below 4 are good, above 8 bad. HDOP values become worse if the received satellites are high on the firmament. VDOP values on the other hand become worse the closer the satellites are to the horizon and PDOP values are best if one satellite is positions vertically above and three are evenly distributed close to the horizon. For an accurate position determination, the GDOP value should not be smaller than 5. The PDOP, HDOP and VDOP values are part of the NMEA data sentence $GPGSA.
The satellite geometry does not cause inaccuracies in the position determination that can be measured in meters. In fact the DOP values amplify other inaccuracies. High DOP values just amplify other errors more than low DOP values.
The error in the position determination caused by the satellite geometry also depends on the latitude of the receiver. This is shown below in the two diagrams. The diagram on the left side shows the inaccuracy of the height (at the beginning of the curve with SA), recorded in Wuhan (China). Wuhan is situated on 30.5° northern latitude were ideal satellite constellation can be found at all time. The graph on the right side shows the same interval recorded by the Casey-Station in the Antarctica (66.3° southern latitude). Due to the satellite constellation from time to time the error is much larger. Additionally the falsification by the atmospheric effect gets more significant the closer the position is to the poles (for an explanation see “atmospheric effects”).
Although the satellites are positioned in very precise orbits, slight shifts of the orbits are possible due to gravitation forces. Sun and moon have a weak influence on the orbits. The orbit data are controlled and corrected regularly and are sent to the receivers in the package of ephemeris data. Therefore the influence on the correctness of the position determination is rather low, the resulting error being not more than 2 m.
The multipath effect is caused by reflection of satellite signals (radio waves) on objects. It was the same effect that caused ghost images on television when antennae on the roof were still more common instead of todays satellite dishes.
For GPS signals this effect mainly appears in the neighbourhood of large buildings or other elevations. The reflected signal takes more time to reach the receiver than the direct signal. The resulting error typically lies in the range of a few meters.
Another source of inaccuracy is the reduced speed of propagation in the troposphere and ionosphere. While radio signals travel with the velocity of light in the outer space, their propagation in the ionosphere and troposphere is slower.
In the ionosphere in a height of 80 – 400 km a large number of electrons and positive charged ions are formed by the ionizing force of the sun. The electrons and ions are concentrated in four conductive layers in the ionosphere (D-, E-, F1-, and F2-layer). These layers refract the electromagnetic waves from the satellites, resulting in an elongated runtime of the signals.
These errors are mostly corrected by the receiver by calculations. The typical variations of the velocity while passing the ionosphere for low and high frequencies are well known for standard conditions. Theses variations are taken into account for all calculations of positions. However civil receivers are not capable of correcting unforeseen runtime changes, for example by strong solar winds.
It is known that electromagnetic waves are slowed down inversely proportional to the square of their frequency (1/f2) while passing the ionosphere. This means that electromagnetic waves with lower frequencies are slowed down more than electromagnetic waves with higher frequencies. If the signals of higher and lower frequencies which reach a receiver are analysed with regard to their differing time of arrival, the ionospheric runtime elongation can be calculated. Military GPS receivers use the signals of both frequencies (L1 and L2) which are influenced in different ways by the ionosphere and are able to eliminate another inaccuracy by calculation.
The tropospheric effect is a further factor elongating the runtime of electromagnetic waves by refraction. The reasons for the refraction are different concentrations of water vapour in the troposphere, caused by different weather conditions. The error caused that way is smaller than the ionospheric error, but can not be eliminated by calculation. It can only be approximated by a general calculation model.
The following two graphs visualize the ionospheric error. The left data were collected with a one-frequency receiver without ionospheric correction, the right data were collected with a two-frequency receiver with ionospheric correction. Both diagrams have approximately the same scale (Left: latitude -15 m to +10 m, longitude -10 m to +20 m, Right: latitude -12 m to +8 m, longitude -10 m to +20 m). The right graph clearly shows less outliers, while the mean accuracy of the position for 95 % of the data is not considerably enhanced by the correction of the ionospheric error.
With the implementation of WAAS and EGNOS it s possible to set up „maps“ of the atmospheric conditions over different regions. The correction data are sent to the receivers, enhancing the accuracy considerably.
Clock inaccuracies and rounding errors
Despite the synchronization of the receiver clock with the satellite time during the position determination, the remaining inaccuracy of the time still leads to an error of about 2 m in the position determination. Rounding and calculation errors of the receiver sum up approximately to 1 m.
The following section shall not provide a comprehensive explanation of the theory of relativity. In the normal life we are quite unaware of the omnipresence of the theory of relativity. However it has an influence on many processes, among them is the proper functioning of the GPS system. This influence will be explained shortly in the following.
As we already learned, the time is a relevant factor in GPS navigation and must be accurate to 20 - 30 nanoseconds to ensure the necessary accuracy. Therefore the fast movement of the satellites themselves (nearly 12000 km/h) must be considered.
Whoever already dealt with the theory of relativity knows that time runs slower during very fast movements. For satellites moving with a speed of 3874 m/s, clocks run slower when viewed from earth. This relativistic time dilation leads to an inaccuracy of time of approximately 7,2 microseconds per day (1 microsecond = 10-6 seconds).
The theory of relativity also says that time moves the slower the stronger the field of gravitation is. For an observer on the earth surface the clock on board of a satellite is running faster (as the satellite in 20000 km height is exposed to a much weaker field of gravitation than the observer). And this second effect is six times stronger than the time dilation explained above.
Altogether, the clocks of the satellites seem to run a little faster. The shift of time to the observer on earth would be about 38 milliseconds per day and would make up for an total error of approximately 10 km per day. In order that those error do not have to be corrected constantly, the clocks of the satellites were set to 10.229999995453 Mhz instead of 10.23 Mhz but they are operated as if they had 10.23 MHz. By this trick the relativistic effects are compensated once and for all.
There is another relativistic effect, which is not considered for normal position determinations by GPS. It is called Sagnac-Effect and is caused by the movement of the observer on the earth surface, who also moves with a velocity of up to 500 m/s (at the equator) due to the rotation of the globe. The influence of this effect is very small and complicate to calculate as it depends on the directions of the movement. Therefore it is only considered in special cases.
The errors of the GPS system are summarized in the following table. The individual values are no constant values, but are subject to variances. All numbers are approximative values.
|Ionospheric effects||± 5 meters|
|Shifts in the satellite orbits||± 2.5 meter|
|Clock errors of the satellites' clocks||± 2 meter|
|Multipath effect||± 1 meter|
|Tropospheric effects||± 0.5 meter|
|Calculation- und rounding errors||± 1 meter|
Altogether this sums up to an error of ± 15 meters. With the SA still activated, the error was in the range of ± 100 Meter. Corrections by systems like WAAS and EGNOS, which mainly reduce ionospheric effects, but also improve orbits and clock errors, the overall error is reduced to approximately ± 3 - 5 meters.