Pseudorange
Encyclopedia
The pseudorange is the pseudo distance between a satellite
and a navigation satellite receiver (see GNSS positioning calculation#Note) —for instance Global Positioning System
(GPS) receivers.
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.
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
can be computed accurately.
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.
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
.
To describe this imprecision, so-called GDOP
quantities have been defined: geometric dilution of precision (x,y,z,t).
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
(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.
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.
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
and a navigation satellite receiver (see GNSS positioning calculation#Note) —for instance Global Positioning System
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
(GPS) receivers.
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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
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.
Pseudorange and time error estimation
Typically a quartzQuartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
oscillator is used in the receiver to do the timing. The accuracy of quartz clock
Quartz clock
A quartz clock is a clock that uses an electronic oscillator that is regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks are at least an order of magnitude more accurate than good mechanical clocks...
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
can be computed accurately.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
Trilateration
In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...
, but would be more accurately referred to as pseudo-trilateration.
Following the laws of error propagation, neither the receiver position nor the clock offset are computed exactly, but rather estimated through a least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
adjustment procedure known from geodesy
Geodesy
Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...
.
To describe this imprecision, so-called GDOP
Dilution of precision (GPS)
Dilution of precision , or geometric dilution of precision , is a term used in GPS and geomatics engineering to specify the additional multiplicative effect of GPS satellite geometry on GPS precision.-Introduction:...
quantities have been defined: geometric dilution of precision (x,y,z,t).
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
(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.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.