global positioning systems (GPS)

Navigation is simple - in theory. Determining your position on the surface of the earth requires you to know either your compass bearing to a landmark or the distance separating you from the landmark. Plotting this information on a chart localises your position along a line or arc in relation to the landmark. If you now plot the bearing or distance information of a second landmark, the lines or arcs intersect, and you know that you are located at the intersection point. Simple. Navigators have been using this system for centuries.

The problem with this simple system is that it is not much use if fighting the bar in heavy turbulence with the ATC insisting that you tell you require quick and precise position him where you are, right now. All indication. Getting a compass bearing this, and you don't know where you to a landmark is quick, but not precise. Getting distance information is difficult or impossible - or used to be until a few decades ago. You want to have this information, and be able to plot it, while sitting in a microlight, are on the dinky little map on your knee - Oh boy, do you need GPS!

What are the requirements for a successful position-determining system for you, me and the millions of hikers, 4x4'ers, fishermen and paramedics? Certainly such a system has to be affordable; it also needs to be accurate, easy to use, portable and rugged. Very tough requirements to meet.

Distances between landmarks can be accurately measured by different means. As kids we all used to count the seconds separating the lightning flash and the arrival of the thunderclap. Since we know the speed of sound in air (say 330m/s), we can easily work out the distance between the origin of the flash and the observer. This reasoning of course depends on us seeing the flash at the time that it was generated, which means that we accept that the speed of light is infinite and that there is no delay between the flash being generated and it being observed. In reality, the velocity of light (say 300 000 000 m/s), is high, but it is not infinitely so. If we have an accurate enough timing device, it is possible to determine the time lapse between the generation of a light signal and its reception.

The critical part in this is having a timing device that is accurate enough for the purpose. Using a clock that is accurate to 0,001 second (one millisecond) to determine the time-of flight of a light signal, leads to an error of 300km. Nanosecond accuracy (0,000 000 001 sec) leads to an error of 0,0003km. This 300mm error is much more acceptable, but is totally dependent on the availability of ultra accurate atomic clocks.

As an aside - in 1714 the British Parliament offered a prize of 20 000 pounds to the first person to devise a method of finding longitude at sea with an accuracy of 30 nautical miles. This sum was a great deal of money at the time, but it was claimed only in 1762 by a certain John Harrison, who constructed a clock, the chronometer, that was so accurate that navigators could use it to attain the magic 30 mile precision.

Today we use radio signals instead of pulses of light as the means of determining the distance between a receiver (your GPS equipment and a signal source. Radio signals propagate at the speed of light, and precise distance measurement is possible if very accurate clocks are available in both the transmitter and the receiver. The signal source needs to radiate a signal containing real time, an identification code and a position indicator. If the signal source were an immovable ground station the positional information would not be necessary. The receiver needs to keep its own accurate time so

that the time-of-flight of the signal can be determined, and it also needs to know from which transmitter (where) the signal originated. Since the US military started with the idea of GPS, they are understandably very concerned about the safety and integrity of ground installations. Another problem with ground stations is that there would need to be a very large number to cover the whole earth. The decision was therefore to go with a satellite system, and the Navstar system was born.

The satellites are big and expensive enough to each contain a few atomic clocks (a few, since there have to be spares and they also need to be checking each other), but your hand - held GPS receiver cannot make use of this bulky and non-affordable timing technology. A timing device that is cheap and small, however, is the ordinary quartz crystal oscillator - the same one that is used in just about all wristwatches. The short term accuracy of such a timing device can be of the order of a microsecond (1 000 nanoseconds), but this is not nearly good enough to allow it to be used directly for computing the distance between the receiver and the satellite. The success of the whole system depends on the implementation of successful strategies to compensate for the receiver clock errors.

Radar Radio Direction And Ranging also makes use of the time that a signal takes to travel between target and antenna, but in this case the process is straightforward. With radar the antenna radiates a strong microwave signal which is bounced off the target and which is then reflected back to the antenna. The process involves the determination of the time-of-flight of the signal, dividing this value by two and then computing the distance. Here the interval between sending and receiving a signal is determined by the sender - accurate real-time clocks are not necessary at all. Trivial, compared to the requirements for GPS.

Having more than one satellite available in the sky allows your GPS receiver to compute the distance between itself and various 'visible' satellites. Determining the distances to more than two satellites allows you to plot your position accurately. Since each distance plot places you somewhere on the surface of a sphere centered on a satellite, using data from three satellites allows the position of the receiver to be calculated to being within a three-sided volume of space. The size of this volume depends on the accuracy of the data, and therefore primarily on the accuracy of the clocks. The microsecond accuracy of the receiver clock does not lead to acceptable volume sizes.

Solving the clock error problem is crucial. Determining a position on earth requires values for three unknowns; x, y and z-positions. We know from high school algebra that solving for three unknowns requires at least three independent equations. Three satellites will give us x, y and z, but imprecisely, since we do not know the value of a fourth variable, which is the time error of the GPS receiver. This means that we have an equation with four unknowns: x, y, z and the time error. Getting information from a fourth satellite makes it possible to solve for this fourth unknown. The GPS receiver makes use of an iterative, or repetitive, mathematical algorithm to determine the time error.

This value is then used to re-compute distances to all four (or more) satellites. As a bonus, in addition to the positional information, the GPS system therefore provides time signals accurate to within about 200 nanoseconds. This then also is

the reason why the time readout of a GPS-receiver cannot be re-set by the user - it is continually being computed from the information received from at least three independent atomic clocks, and this is why it always extremely accurate. If your GPS time readout differs from that given by the SABC, the SABC is wrong and you are right!

Signals from four satellites provide 3D (x, y and z) and time information. Three satellites lead to 2D information - x, y and time, while the z or altitude information cannot be computed.

The above description makes all this sound rather easy. If the satellites were fixed in space it would in actual fact be so. But, the satellites are not fixed in space. They circle the earth at 14000 km/h at a distance of 20 190 km, in orbits that take 11 hours 58 minutes to complete, so that they seem to drift, like the stars, four minutes per day. In contrast to this, geostationary satellites (like many communications satellites), are at 36 000 km from the earth and circle the earth once in exactly 24 hours so that they always stay over the same spot on the surface of the earth. Geostationary satellites have one attribute that makes them unsuitable for GPS-use, and that is the fact that they can only be launched into equatorial orbits, which would give very poor visibility from high and low latitudes. It would also mean that they are all in the same plane, making for very poor positional accuracy.

To give good coverage to GPS receivers anywhere on earth, there are always at least 24 satellites (or Space Vehicles, SV's, as they are often referred to) in orbit. Of these, three are spares. Four SV's travel in each of six orbits, each inclined at 55 to the equator so that four or more of the SV's are at all times visible from any point on the surface of the earth. In addition to the atomic clocks and the communications equipment each satellite also contains fuel for its small manoeuvring engines, giving it a limited capability of orbit adjustment.

In addition to the Navstar satellites in space (the 'Space Component'), there is also a ground support system (the 'Control Segment') with a master control station and a number of monitoring stations around the world. The last component of the system is the 'User Segment', which includes us with our GPS receivers.

Once a SV is in space, it does not stay in exactly the same orbit from day to day. There are factors that influence the orbit unpredictably, such as pressure from the solar wind, and predictably, like the gravitational effects of the moon and planets. The ground stations of the Control Segment need to determine the speed and position of each satellite with great accuracy so that it can predict and describe the satellite's orbit unambiguously. Such a description of the satellite clock parameters and its orbital characteristics is called an ephemeris, and this information is uploaded to each individual satellite. The ephemeris information can be updated twice per day as the satellite passes over a ground station. When or if the ground station finds that a satellite has wandered from its predicted position in orbit, the orbit is re-computed and the data uploaded to the satellite. Since there is limited fuel on board, it is only as a last resort that the satellite is physically moved with the aid of its own engines. This happens when the orbit deviates so much from the desired path that accurate prediction is no longer possible. When your GPS receiver acquires information from a satellite, the ephemeris data is received, and this is what allows

it to measure the time difference between when the signal was broadcast and when it was acquired, so that it is possible to compute the distance between satellite and receiver.

All the satellites broadcast information continuously on the same two frequencies, the L1 frequency at 1575.42MHz, and L2 at 1227.6MHz. Cost and bulk considerations cause most or all small receivers to utilise only the L1 frequency. The radiated power of a satellite transmission is not much more than 500W. Compare this with the radio in your microlight, putting out 5W. The difference being that the satellite is at least 20 000 km distant, and if it appears to be low on the horizon, it is much further away than this. The signal arriving at your receiver is therefore barely distinguishable above the background electronic noise.

The small non-directional antenna on the GPS receiver recovers this extremely low level, noisy signal) and passes it to the receiver where spread spectrum technology deciphers the signal. The advantage of spread-spectrum is that it can extract a very low-level signal from background noise, but it can only do so at the expense of speed. Data rates of 50 bits per second are the norm. If a receiver needs to acquire all the information that a satellite broadcasts, which includes ephemeris data and 'almanac' data, the process will take more than 1 0 minutes to complete. Normally the receiver only needs the ephemeris information from the satellite.

So, what influences the signal during the transit between satellite and GPS receiver?

The distance between the satellite and the receiver.  

The relativistic effect of the satellite moving relative to the earth. Einstein demonstrated that time is retarded when velocity increases. Even though the satellites move at a low fraction of light speed, nanosecond accuracy implies that correction is needed for the relativistic effects of satellites either approaching or receding from the receiver.

The rotation of the earth under the satellite displaces the receiver during the time-of-flight of the signal.  

Doppler-shift - this being a function of the rate at which the satellite approaches or recedes from the receiver. This changes the frequency of the received signal in the same way that the frequency of the noise of an approaching vehicle seems to be high, and suddenly becomes lower when the vehicle passes the observer. This doppler-shift is used by the GPS system to determine your speed and the direction that you are moving in. If the satellites were in geo-stationary orbits, this information would not have been available. The TRANSIT satellite system, used by the US Navy, is (was) an earlier-generation positioning system using primarily doppler-shift for determining position. This is not really useful for rapidly moving receivers, since the receiver movement influences the observed doppler-shift and reduces accuracy.

Precession of the earth's rotation axis. The wobble of the axis takes more than 25 000 years to go through a complete rotation, but it is also corrected for in the computation.

The refraction of the radio signals by the ionosphere and the troposphere. Since these layers are inhabited by charged particles, they have an influence on the propagation of the signals The path of the signal through the layers are lengthened or shortened, depending on the apparent angle of the satellite above the horizon. The effect on the signal can be determined by measuring the difference in scattering between the L1 and L2 frequencies. This can then be accurately corrected for. This is what is done in dual-frequency receivers, but in our small single- frequency consumer systems a mathematical algorithm adequately, if not perfectly, corrects for the effect.

This process can take many minutes, even tens of minutes. The same thing may happen if you do not use the unit for some months, and the receiver considers the almanac data to be out of date If the receiver has valid almanac data, it 'knows' which satellites to expect, and where they are located at that specific time. Determining the doppler frequency of the signal is the next step. This is done by listening at each of 20 or to predefined frequencies close to L1, and measuring signal strength at each. This is speeded up by the receiver being able to compute the expected frequencies by using the satellite locations kept in memory, by correcting for its own oscillator error based on previous error values also kept in memory, and predicting its own oscillator frequency based on the current temperature.

Once it has determined the frequency for each satellite it starts to receive data on the L1 frequency. The satellites nearest to the overhead position are preferred for initial acquisition. Each satellite broadcasts its own identification code in the form of a 1023-bit pseudo-random number sequence, repeated every millisecond.

The receiver needs to set its own clock to the correct time slot. It does this by trying all the possible values. This process may take one or two seconds. In many GPS receivers this stage of the proceedings is shown on the display by the satellite acquisition histogram bar becoming visible as a hollow bar.

As soon as it has locked onto and identified a satellite, the downloading of ephemeris data can start. There is approximately 1 500 bits of data in a message and this is sent at 50bps. This download takes about 30 seconds. This needs to be done for at least three satellites to be able to compute x, y and time. If more satellites are in range, data are acquired from them as well. The downloading can be done via more than one channel, and can also be multiplexed amongst the satellites, so that the process need not take more than a minute or so. The successful downloading of ephemeris data for a specific satellite is often indicated on a GPS receiver by the hollow histogram bar turning solid black.

If the receiver is switched on within a few minutes of a previous session, it will get a fix on its position in usually less than 20 seconds. This can happen since it saves the ephemeris data that it held at its previous shutdown. Under these circumstances it will verify that the ephemeris data is still valid and that the relevant satellites are still 'visible'.

Ephemeris data from a satellite is considered to be valid for four hours (the transit time from horizon to horizon), but is in any case updated after two hours. Once this downloading is done, the CPU in the receiver can get to work and determine the position of the receiver. To do this it performs a variety of computational steps.

The dedicated central processing unit in the receiver translates the ephemeris data into a format suitable for its calculations.  
It calculates the satellite positions so that it has accurate elevation and bearing parameters to base its troposphere modelling on.
 
It calculates initial distances (pseudo-range) on which ionosphere modelling is based.
 
These steps are repeated for each satellite in range.
 
It corrects for the earth's rotation, based on the pseudo-range data.
 
Recalculates the receiver position.
 
Corrects the altitude data for geoid height.

Displays the position on the receiver readout.

Corrects the time signal for UTC offset and other factors and displays it.

Continually recalculates the position based on the data from additional (more than four) satellites and displays the best possible solution.
 
Calculates horizontal speed by using the Doppler information. The result of all this is a position in latitude and longitude, altitude, time and horizontal speed. The accuracy of these values depends on two factors.

The inherent limitations of dock accuracy and parameter modelling.

  The factor called Selective Availability (SA). There has always been concern that the Navstar system can be used for military purposes against the US The system is under control of the US Department of Defence, but the non-military use was deemed to be so important that the decision was taken to make the system available for civilian use, but with reduced accuracy.

To this end they introduced a random degradation of the accuracy of the data. Since this is a random degradation, the accuracy is not always the same. Sometimes there is no degradation, at other times there may be a maximum degradation - there is no way to know what the effect is. The result is that the accuracy can only be expressed statistically, and the official description is: Horizontal (long-lat) - within 100m for 95% of the time, and within 300m 99.99% of the time. Vertical (altitude) - within 1 56m for 95% of the time, and within 100m 99.99% of the time.

In practice the values are appreciably better than this. Various measurements, under different conditions,' have been made over the years. Results have shown that average horizontal accuracy have tended to be around 50m.

Human nature being what it is, there has been a great deal of indignation over this degradation of a perfectly good signal. Various methods have been used to improve accuracy. One of the most widely used depends on the presence of ground stations transmitting very accurate time signals from very accurately known positions. The user buys an additional receiver for this signal and feeds in this "Differential" signal into the GPS receiver.

This very accurately corrects for the clock error and the introduced SA and allows average horizontal accuracy of 1 to 4m and altitude averages of less than 10m. These DGPS values have been widely used for accurate navigation and surveying.