The error analysis for the Global Positioning System is important for understanding how GPS works, and for knowing what magnitude of error should be expected. The GPS makes corrections for receiver clock errors and other effects but there are still residual errors which are not corrected. GPS receiver position is computed based on data received from the satellites. Errors depend on geometric dilution of precision and the sources listed in the table below.
Overview
+ Sources of User Equivalent Range Errors (UERE) Source Effect (m) Signal arrival C/A ±3 Signal arrival P(Y) ±0.3 Ionospheric effects ±5 Ephemeris errors ±2.5 Satellite clock errors ±2 Multipath distortion ±1 Tropospheric effects ±0.5 C/A ±6.7 P(Y) ±6.0
User equivalent range errors (UERE) are shown in the table. There is also a numerical error with an estimated value, , of about . The standard deviations, , for the coarse/acquisition (C/A) and precise codes are also shown in the table. These standard deviations are computed by taking the square root of the sum of the squares of the individual components (i.e., RSS for root sum squares). To get the standard deviation of receiver position estimate, these range errors must be multiplied by the appropriate dilution of precision terms and then RSS'ed with the numerical error. Electronics errors are one of several accuracy-degrading effects outlined in the table above. When taken together, autonomous civilian GPS horizontal position fixes are typically accurate to about 15 meters (50 ft). These effects also reduce the more precise P(Y) code's accuracy. However, the advancement of technology means that in the present, civilian GPS fixes under a clear view of the sky are on average accurate to about 5 meters (16 ft) horizontally.
The term user equivalent range error (UERE) refers to the error of a component in the distance from receiver to a satellite. These UERE errors are given as ± errors thereby implying that they are unbiased or zero mean errors. These UERE errors are therefore used in computing standard deviations. The standard deviation of the error in receiver position, , is computed by multiplying PDOP (Position Dilution Of Precision) by , the standard deviation of the user equivalent range errors. is computed by taking the square root of the sum of the squares of the individual component standard deviations.
PDOP is computed as a function of receiver and satellite positions. A detailed description of how to calculate PDOP is given in the section Geometric dilution of precision computation (GDOP).
for the C/A code is given by:
The standard deviation of the error in estimated receiver position , again for the C/A code is given by:
The error diagram on the left shows the inter relationship of indicated receiver position, true receiver position, and the intersection of the four sphere surfaces.
Signal arrival time measurement The position calculated by a GPS receiver requires the current time, the position of the satellite and the measured delay of the received signal. The position accuracy is primarily dependent on the satellite position and signal delay.
To measure the delay, the receiver compares the bit sequence received from the satellite with an internally generated version. By comparing the rising and trailing edges of the bit transitions, modern electronics can measure signal offset to within about one percent of a bit pulse width, , or approximately 10 nanoseconds for the C/A code. Since GPS signals propagate at the speed of light, this represents an error of about 3 meters.
This component of position accuracy can be improved by a factor of 10 using the higher-chirprate P(Y) signal. Assuming the same one percent of bit pulse width accuracy, the high-frequency P(Y) signal results in an accuracy of or about 30 centimeters. The civilian L5 signal features the same increase in chirp rate.
Carrier phase measurement
More advanced GPS receivers measures of the phase of the signal's carrier wave in addition to the standard code phase measurement. Because the carriers are at a higher frequency (1.18–1.58 GHz), performing phase measurements on this signal would yield a 0.2 cm error using the same assumption of 1% phase error. The receiver needs to have a way to resolve the pseudorange ambiguity from the phase resetting every 19 cm. This is possible by comparing the phases of multiple satellitesBetween-satellite differencing or with the help of a ground station with known location.
Atmospheric effects Inconsistencies of atmospheric conditions affect the speed of the GPS signals as they pass through the Earth's atmosphere, especially the ionosphere. Correcting these errors is a significant challenge to improving GPS position accuracy. These effects are smallest when the satellite is directly overhead and become greater for satellites nearer the horizon since the path through the atmosphere is longer (see airmass). Once the receiver's approximate location is known, a mathematical model can be used to estimate and compensate for these errors.
Ionospheric delay Ionospheric delay of a microwave signal depends on its frequency. It arises from ionized atmosphere (see Total electron content). This phenomenon is known as dispersion and can be calculated from measurements of delays for two or more frequency bands, allowing delays at other frequencies to be estimated.The same principle, and the math behind it, can be found in descriptions of pulsar timing by astronomers. Some military and expensive survey-grade civilian receivers calculate atmospheric dispersion from the different delays in the L1 and L2 frequencies, and apply a more precise correction. This can be done in civilian receivers without decrypting the P(Y) signal carried on L2, by tracking the carrier wave instead of the modulated code. To facilitate this on lower cost receivers, a new civilian code signal on L2, called L2C, was added to the Block IIR-M satellites, which was first launched in 2005. It allows a direct comparison of the L1 and L2 signals using the coded signal instead of the carrier wave.
The effects of the ionosphere generally change slowly, and can be averaged over time. Those for any particular geographical area can be easily calculated by comparing the GPS-measured position to a known surveyed location. This correction is also valid for other receivers in the same general location. Several systems send this information over radio or other links to allow L1-only receivers to make ionospheric corrections. The ionospheric data are transmitted via satellite in Satellite Based Augmentation Systems (SBAS) such as Wide Area Augmentation System (WAAS) (available in North America and Hawaii), EGNOS (Europe and Asia), Multi-functional Satellite Augmentation System (MSAS) (Japan), and GPS Aided Geo Augmented Navigation (GAGAN) (India) which transmits it on the GPS frequency using a special pseudorandom noise sequence (PRN), so only one receiver and antenna are required.
Weather conditions Humidity also causes a variable delay, resulting in errors similar to ionospheric delay, but occurring in the troposphere. This effect is more localized than ionospheric effects, changes more quickly and is not frequency dependent. These traits make precise measurement and compensation of humidity errors more difficult than ionospheric effects.Navipedia: Troposphere Monitoring
The Atmospheric pressure can also change the signals reception delay, due to the dry gases present at the troposphere (78% N2, 21% O2, 0.9% Ar...). Its effect varies with local temperature and atmospheric pressure in quite a predictable manner using the laws of the ideal gases.Navipedia: Tropospheric Delay
Multipath effects GPS signals can also be affected by multipath issues, where the radio signals reflect off surrounding terrain; buildings, canyon walls, hard ground, etc. These delayed signals cause measurement errors that are different for each type of GPS signal due to its dependency on the wavelength.Navipedia: Multipath
A variety of techniques, most notably narrow correlator spacing, have been developed to mitigate multipath errors. For long delay multipath, the receiver itself can recognize the wayward signal and discard it. To address shorter delay multipath from the signal reflecting off the ground, specialized antennas (e.g., a choke ring antenna) may be used to reduce the signal power as received by the antenna. Short delay reflections are harder to filter out because they interfere with the true signal, causing effects almost indistinguishable from routine fluctuations in atmospheric delay.
Multipath errors are more severe in "urban canyons" where buildings block the view of a satellite, leaving the reflected signal as the only observation of the satellite. Choke ring antennas are also less applicable because signals may bounce off of other buildings (therefore appearing as "from above"). The net result is that when a user stands on the eastern side of a street with the eastern half of view of the sky blocked, multipath effects tend to result in an apparent location that is moved westwards. Uber has deployed an ambiguity resolution method using measured SNR values to "shadow-match" against predicted SNR values around the apparent location, an extension of a method originally developed by Paul Groves.
Multipath effects are much less severe in moving vehicles. When the GPS antenna is moving, the false solutions using reflected signals quickly fail to converge and only the direct signals result in stable solutions.
Ephemeris and clock errors While the ephemeris data is transmitted every 30 seconds, the information itself may be up to two hours old. Variability in solar radiation pressure has an indirect effect on GPS accuracy due to its effect on ephemeris errors. If a fast time to first fix (TTFF) is needed, it is possible to upload a valid ephemeris to a receiver, and in addition to setting the time, a position fix can be obtained in under ten seconds. It is feasible to put such ephemeris data on the web so it can be loaded into mobile GPS devices. See also Assisted GPS.
The satellites' atomic clocks experience noise and clock drift errors. The navigation message contains corrections for these errors and estimates of the accuracy of the atomic clock. However, they are based on observations and may not indicate the clock's current state.
These problems tend to be very small, but may add up to a few meters (tens of feet) of inaccuracy.
For very precise positioning (e.g., in geodesy), these effects can be eliminated by differential GPS: the simultaneous use of two or more receivers at several survey points. In the 1990s when receivers were quite expensive, some methods of quasi-differential GPS were developed, using only one receiver but reoccupation of measuring points. At the TU Vienna the method was named qGPS and post processing software was developed.
The International GNSS Service (IGS) uses its global network of monitoring stations to produce higher-accuracy ephemerides and clock drift data for the GPS satellites. The IGS produces and archives the coarse estimates broadcast by the satellites, a finer ultra-rapid observation and prediction published four times per day, an even finer "rapid" data from daily post-processing, and the finest "final" data published weekly. The ultra-rapid prediction can be used in real time to refine results. The other data can be used on recorded GPS data to produce high-accuracy results after the fact.
Dilution of precision
Dilution of precision describes the effect of satellite geometry on how pseudorange errors propagate into positioning and velocity error.
Selective Availability