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| The '''pseudorange''' (from [[wikt:pseudo-|pseudo-]] and [[:wikt:range|range]]) is the ''pseudo'' <!--an approximation of the--> distance between a [[satellite]] and a navigation satellite receiver (see [[GNSS_positioning_calculation#Note|GNSS positioning calculation]]) —for instance [[Global Positioning System]] (GPS) receivers.
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| To determine its position, a satellite navigation receiver will determine the ranges to (at least) four satellites as well as their positions at time of transmitting. Knowing the satellites' [[orbital parameters]], these positions can be calculated for any point in time. The pseudoranges of each satellite are obtained by multiplying the [[speed of light]] by the time the signal has taken from the satellite to the receiver. As there are accuracy errors in the time measured, the term ''pseudo''-ranges is used rather than ranges for such distances.
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| ==Pseudorange and time error estimation==
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| Typically a [[quartz]] oscillator is used in the receiver to do the timing. The accuracy of [[quartz clock]]s in general is worse (i.e. more) than one part in a million; if the clock hasn't been corrected for a week, the distance will put you not on the Earth but outside the Moon's orbit. Even if the clock is corrected, a second later the clock is not usable anymore for positional calculation, because after a second the error will be hundreds of meters for a typical quartz clock. But in a GPS receiver the clock's time is used to measure the ranges to different satellites at almost the same time, meaning all the measured ranges have the same error. Ranges with the same error are called pseudoranges. By finding the pseudo-range of an additional fourth satellite for precisely position calculation, the time error can also be estimated. Therefore, by having the pseudoranges and the locations of four satellites, the actual receiver's position along the ''x'', ''y'', ''z'' axes and the time error <math>\Delta t</math> can be computed accurately.
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| The reason we speak of ''pseudo''-ranges rather than ranges, is precisely this "contamination" with unknown receiver clock offset. GPS positioning is sometimes referred to as [[trilateration]], but would be more accurately referred to as ''pseudo-trilateration''.
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| Following the laws of [[error propagation]], neither the receiver position nor the clock offset are computed exactly, but rather ''estimated'' through a [[least squares]] adjustment procedure known from [[geodesy]].
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| To describe this imprecision, so-called [[Dilution of precision (GPS)|GDOP]] quantities have been defined: geometric dilution of precision (x,y,z,t).
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| <!-- Actually, the GDOP is computed from the actual satellite positions and is an indication of the quality of that satellite configuration for the calculated position. Satellites close together will give less quality than satellites spread out. It is correct that pseudorange calculations do not give the exact position, but the mathematical error can be made as small as is required. Every iteration using pseudorange calculations will give an improvement of a factor of over a 1000, so the mathematical accuration can go from 1000 km to 1 km, to 1 meter to 1 mm in only three steps. The calculation is an exact process which is repeatable, with four satellites' signals this is an exact process and given the input should always give the same result and is therefore not an estimate. Mathematically it's exact. The actual calculated position though is an estimate; this is not because of the process but because of a set of errors in the input.
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| How actual GPS receivers work if more than four satellites are available, is kept secret by the manufacturers of those receivers, but what's very probable is that the best GDOP is chosen to calculate the position, or a weighted average is made. It is improbable that a least squares method is used, because the satellite constallations with a "bad" GDOP are not weighted equally.-->
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| Pseudorange calculations therefore use the signals of four satellites to compute the receiver's location and the clock error. A clock with an accuracy of one in a million will introduce an error of one millionth of a second each second. This error multiplied by the speed of light gives an error of 300 meters. For a typical satellite constellation this error will increase by about <math>\textstyle{\sqrt 2}</math> (less if satellites are close together, more if satellites are all near the horizon). If positional calculation was done using this clock and only using three satellites, just standing still the GPS would indicate that you are traveling at a speed in excess of 300 meters per second, (over 1000 km/hour or 600 miles an hour). With only signals from three satellites the GPS receiver would not be able to determine whether the 300m/s was due to clock error or actual movement of the GPS receiver.
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| If the satellites being used are scattered throughout the sky, then the value of geometric dilution of precision (GDOP) is low while if satellites are clustered near each other from the receiver's vantage point the GDOP values are higher. The lower the value of GDOP then the better the ratio of position error to range error computing will be, so GDOP plays an important role in calculating the receiver's position on the surface of the earth using pseudoranges. The larger the number of satellites, the better the value of GDOP will be.
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| == References == | |
| * Peter J. G. Teunissen and Alfred Kleusberg, GPS observation equations and positioning concepts, Lecture Notes in Earth Sciences, 1996, volume 60/1996, pages 175-217
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| [[Category:Navigation]]
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| [[Category:Geodesy]]
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