The History Of Navigation Computer Science Essay

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Ever since time began, mankind has been trying to find a reliable way of telling where they were and a way to guide them to where they wanted to go and return again. Sailors followed the coastline to keep them on the right track, however out on the open seas this method was ineffective so had to use the position of the stars to chart their courses instead.

The development of the compass and the sextant signified a great advance in early navigation. The needle of the compass always points north. Although this could not tell them where they were it could at least tell them what direction in which they were travelling. The sextant measures the exact angles of stars, the moon and the sun above the horizon by way of adjustable mirrors. However early sextants could only measure the latitude meaning sailors could still not work out their longitude.

Figure : Compass [1] & Sextant (right) [2]

This was quite a big problem when navigating the oceans so in the seventeenth century, a group of well known scientists were gathered in Britain and were given the name the Board of Longitude. They offered a considerable cash reward to anyone who could come up with a way of working out the longitude of a ship to within an accuracy of 30 nautical miles. A man by the name of John Harrison was successful and in 1761 developed a timepiece called a chronometer. This was basically an extremely precise clock and only lost or gained a second each day. The use of this chronometer in conjunction with a sextant allowed travellers to plot their latitude and longitude.

Figure : Chronometer [3]

The twentieth century brought with it the development of Radio-based navigation and this type of system was used in World War II. With advancements in this technology, both ships and airplanes used ground-based radio-navigation systems. The disadvantage of using such a system is that a choice has to be made between a high-frequency system that is accurate, but does not cover a wide area, and a low-frequency system that covers a wide area, but is not very accurate.[4]

The next developments in navigation involved a global navigation satellite system or GNSS led by Russia and the USA and will be discussed in the next section.

History of GNSS

In 1957, Russia took the upper hand in the space race between themselves and the United States by launching its first satellite, Sputnik, into orbit. The day following the launch, researchers at the Massachusetts Institute of Technology were able to determine the orbit of the satellite. They noticed that as Sputnik orbited the earth, the strength of its radio signal varied i.e. increased as the satellite approached them and decreased as it moved away. This fact led to the realisation that if a satellite's position could be tracked from the ground then an objects whereabouts on the ground could be determined using radio signals from the satellite. This formed the basis of an intriguing new technology: Global Positioning System (GPS).

However, two years preceding Sputnik's launch, Roger L. Easton, Sr., an American scientist, had co-written the Naval Research Laboratory's Project Vanguard. This proposal was chosen to be the official U.S. Satellite program under the Eisenhower Administration. The great competition between the Navy, Army and Air Force in the early stages of the U.S. space effort led to the failure to reach an agreement as to which direction to take. Due to all of this mishandling at the beginning, Project Vanguard experienced a series of launch failures, allowing the Soviets to take advantage and put the world's first manmade satellite into orbit. After the successful launch of Sputnik, Easton applied his invention, the Minitrack tracking system to actively follow unknown orbiting satellites. This led to The Naval Space Surveillance System becoming the first system to detect and track all forms of Earth orbiting objects. In the latter part of his career at the Naval Research Laboratory he invented and oversaw the development of crucial technologies for the United States Global Positioning System, or GPS.

In the early 1960s the US navy began to experiment with satellite navigation and by 1964 the Transit system had been developed for use on submarines carrying nuclear missiles. The Transit system used 6 satellites that circled the earth in polar orbits that would project a navigational fix somewhere in the region of once every hour. The only problem with the Transit system however was that it was considered somewhat inaccurate and was often considered inefficient. In 1967 a second GPS system called Timation was developed and was much more accurate due to the inclusion of an atomic clock. This was able to tell the user the length of time it took for a signal to leave the satellite and reach the receiver, therefore one could tell the distance to the satellite. This also meant that the user did not have to stop to get a reading.

In 1973 the Pentagon was in search of a satellite system that was error proof and could be used for military operations, this led to the development of the Navstar Global Positioning System (GPS). In 1978, the first operational GPS satellite was launched, and by the end of 1995 the system was deemed operational with 24 satellites [5]. Since GPS was designed by the U.S. Department of Defence, this technology was initially used to track potential targets, provide position information on aircraft and ground vehicles and determine the location of casualties during military operations. Just like many other technologies developed in the military it gradually made its way into public use.

In 1983, there was huge public outcry after Korean Airlines Flight 007 was shot down by Soviet jet interceptors after straying into their prohibited airspace. The uproar following this encouraged President Ronald Reagan to issue a directive making GPS available "as a common good" to the civilian population.

Then following on from this in May of 2000, President Clinton ordered the U.S. military to terminate Selective Availability (SA), a technique designed to limit the accuracy of the civilian GPS signal. This allowed the general public using GPS receivers to avail of more precise positioning data that had previously only been available to the military. Further restrictions were removed in 2002 making GPS even more accurate.

GPS Segments

Space segment

There are 3 core segments to NAVSTAR GPS. The first is the satellite segment. This segment consists of a constellation of satellites orbiting the earth at an altitude of approximately 20,000 kilometres. The system was designed to operate with 24 satellites. These satellites operate from six offset orbital planes. Each of these satellites orbits the earth twice a day and can be visible above the flat horizon for eight hours. However anywhere between four and eight of these satellites are usually visible to a receiver on earth at any one time.

Control segment

The second component of the GPS is the control segment. This component is responsible for the tracking, communications, data gathering, integration, analysis and control facilities which are used to observe, maintain and manage the GPS satellites and system. There are five tracking stations spread across the earth, with a master control station in Colorado Springs, USA. Data is gathered from a number of sources by the master control station. This data includes satellite health and status information from each GPS satellite, tracking station, timing data from the US Naval Observatory and earth data from the US Defence Mapping Agency. The master control station synthesizes this information and broadcasts navigation, timing and other data to each satellite. The master control station also signals each satellite as appropriate for course connections, changes in operation or other maintenance.

User segment

The final part of the GPS is the user segment. This component is made up of the individuals with the GPS receivers. The receiver is able to record the data transmitted by each satellite and process it into a three-dimensional coordinate [6].

In fact there is more to the system than this. North America also has a secondary system known as WAAS which stands for the Wide Area Augmentation System. This system is made up of roughly 35 ground stations across North America that monitor GPS satellite data. There is a master station on each coast with other stations spread between them that are all kept synchronized with the same atomic clock as the satellite system. Since the precise position of the ground stations is known, when each one receives GPS satellite information that says it is a certain distance out from where it actually is, the ground station communicates that information to the GPS receiver unit as a correction. These are not errors as such but are due to atmospheric effects of the signal coming through the ionosphere and the troposphere and to a lesser extent satellite positional errors called ephemeris errors. Generically these types of system are known as a satellite based augmentation system (SBAS) and Europe have their own version called the European Geostationary Navigation Overlay Service (EGNOS).

How GPS Works

The following paragraphs explain the basic procedure involved in GPS. The principles behind these steps will be explained in more detail in Chapter 2.


Navstar GPS consists of 24 satellites orbiting the Earth at a distance of approximately 20,000 kilometres, transmitting radio signals [7]. They orbit the Earth in such way that a GPS receiver is visible to and as a result can receive signals from at least 4 satellites. Using these readings GPS can calculate your location using a technique called trilateration.

By very accurately measuring our distance from three satellites, your position anywhere on Earth can be calculated using the technique. If we just ignore how to calculate the exact distance from a satellite for a moment and assume that a receiver is a certain nominal distance away from a satellite. This means that the position of the receiver is anywhere on a sphere of this radius and centred at that satellite. If we know the distance to a second satellite, then the receiver is somewhere along the circle where the two spheres intersect. If we know the distance to a third satellite it narrows down the receiver's position to two points where this third sphere intersects the circle where the first two spheres intersected (although in reality the outline of these circles are larger and don't meet at exact points due to various errors that will be mentioned later). Distances from four satellites will intersect at one point but this isn't always necessary because one of the two points could be considered illogical as it would either be too far from Earth or moving at an unachievable velocity and therefore can be discarded without measurement. However, the measurement from the fourth satellite can be useful for synchronising the receiver with universal time or obtaining an accurate altitude which is useful in the case of aircraft.


Figure : Trilateration Schematic[8]

Measuring Distance

In essence a satellite sends out a radio frequency signal using a pseudo-random code. The GPS receiver also runs this very same pattern. There will be a time lag between when the code is sent from the satellite and when the receiver receives it. The amount of time that it is delayed by will be minute because the speed of the signals in the first place, if the satellite is directly overhead may be in the region of 0.06 seconds. This time delay is equal to the time it takes the satellite radio signal to reach the receiver i.e. the time it left the receiver minus the time the receiver gets it. The GPS receiver then does a simple calculation and multiplies the time by the speed of radio waves (same as the speed of light). This allows the receiver to tell exactly how far it is from the satellite. This process is repeated for each visible satellite.

A drawing showing the comparative arrival of the code pattern from the satellite compared to the replica code generated by the receiver. The drawing shows how the code from the satellite is slightly delayed.

Figure : Time Difference in Pseudo-Random codes [9]

This method is commonly used in systems such as the satellite navigation system in motor vehicles and is capable of accuracies of a few metres however for applications where more accurate positioning is needed, the signal that carries the PRN code is used to gain even greater accuracy which can be all the way down to centimetre level.


To allow the distance to be calculated accurately, extremely accurate timing must be used in both the satellite and the receiver. This timing has to be synchronised down to the nanosecond. One way of achieving this is to use atomic clocks but as they cost in the tens of thousands of Euro, they cannot be put to everyday use. Their use would be fine for one-off devices, such as the satellites themselves but for the GPS receivers an ordinary (cheap) quartz clock is used, which constantly resets. In basic terms, the receiver takes in signals from four or more satellites and uses them to gauge its own accuracy.

The atomic clocks are so accurate that they can always be assumed to have the same time. The receiver on the other hand cannot be assumed to have that time and must therefore be able to synchronise with the correct satellite time, the "current time". While the receiver time is incorrect, the distance calculation in the trilateration will be out by an amount proportional to the time error.

The receiver can easily calculate the necessary adjustment that will cause the four spheres to intersect at one point. This allows it to reset its clock to be in sync with the atomic clock in the satellite. This is constantly occurring as long as the receiver is on which means that it is nearly as accurate as the more expensive atomic clocks that are in the satellites

Satellites' Positions

It is also important to know the positions of the satellites. All GPS receivers on the ground have an almanac programmed into their computers that can tell where every satellite should be at any given moment. This isn't too difficult to achieve as the satellites travel in very high predictable orbits that are quite exact. However, these orbits are constantly monitored by the Department of Defence to check each satellites exact altitude, speed and position with any corrections being transmitted to all GPS receivers as part of the satellite's signal.

The errors that may cause this to happen are known as ephemeris errors and are caused by the gravitational pulls from the moon and sun and by the pressure of solar radiation.

Error Correction

There are several things that can happen to the GPS signal that can cause errors. These errors have to be taken into account if an accurate position is to be calculated.

An earlier assumption was made that the time it takes the satellite radio signal to travel to the receiver would be multiplied by the speed of light to work out distance. However this assumption isn't entirely correct as the figure being used for the speed of light is constant only in a vacuum. As seen in Figure : GPS Error Sources the GPS signal in fact has to travel through the charged particles of the ionosphere and then through the water vapour in the troposphere which slows down the signal. These are often called atmospheric effects.

Multipath error occurs when the GPS signal bounces off obstructions such as tall buildings and large rock surfaces before reaching the receiver causing the travel time of the signal to increase and therefore errors. Similar to this is signal masking such as the signal having to travel through branches of trees and so on. Careful positioning of reference stations and receivers must be taken into consideration to try and limit these effects. There are also orbital errors as mentioned earlier called ephemeris errors which are inaccuracies in the reported position of the satellite that can develop between monitoring times.

The slight inaccuracies that may be present at times in the receiver's clock when compared to that of the atomic clocks in the satellite may produce slight errors. If uncorrected, errors 24 hours after an upload of navigation data can be in the order of 1 to 4 m. This error is progressive as it gets steadily worse over time, like ephemeris, until corrected in the next control segment navigation data upload.

These are some of the main errors that are present in the GPS system and that limit its accuracy. Several techniques are employed to try and reduce these errors as will be seen in the following sections. The figure below shows some of the main errors that may occur.

Figure : GPS Error Sources[10]

The geometry of the satellites/constellation is another major factor in determining the accuracy of a signal. The geometry of the constellation is evaluated for several factors, all of which fall into the category of Geometric Dilution of Precision, or GDOP. This determines the quality of the geometry of the satellite constellation. Positional dilution of precision, PDOP (Horizontal DOP/Vertical DOP) is a three-dimensional dilution of precision that is combined with the Time Dilution of Precision (TDOP) to estimate the overall DOP.

Even though each of these GDOP terms can be calculated individually, they are not independent of each other. This means that a high TDOP for example will cause receiver clock errors which in turn will lead to increased positional errors.

If the satellites are close together the angle between them is small which leads to a poor DOP i.e. a high value of DOP. If the angle between the satellites is large the DOP is value is lower and this provides a more accurate measurement.

Figure : Examples of Good and Bad GDOP [11]

A PDOP that is too high may appear as a message and corresponds to PDOP values >10. Values between 5 and 10 are only just about tolerable. PDOP =1 is ideal.

The PDOP value is multiplied by the total system error (= PDOP * Range Accuracy), where the total system error is made up of satellite clock error, ephemeris error, receiver errors and atmospheric errors.

This basically means that the satellite geometry does not cause inaccuracies in itself that can be measured metres. However the DOP values do amplify other inaccuracies. High DOP values simply amplify other errors more than low DOP values.

GPS Error Reduction Techniques

Due to the errors mentioned above there have been efforts to try and reduce or even eliminate the sources of some of these types of errors by various means. Certain applications make it essential to have supremely accurate GPS and with this increasing accuracy more applications will indeed arise. The following paragraphs give an introduction to some such techniques of increasing accuracy.


Differential GPS (DGPS) expands on the principle of GPS by involving an additional receiver. The fundamental basis of DGPS is that any two receivers that are relatively close together (within 50km) will be subjected to similar errors due to the atmosphere. This is because the satellites are so far out in space that the distances between receivers on earth are comparatively insignificant. One receiver moves around taking position measurements (rover), while the other receiver is stationary (base). This stationary receiver is used to measure the timing errors and then relay the correction information to the rover [12].

Normally the timing is used to calculate position but errors in the timing occur due to many things and consequently lead to errors in the position. The location of the stationary receiver is known, and it can quite easily calculate the receiver's inaccuracy. Therefore as opposed to using timing to calculate position it uses its position to calculate the timing, thus negating the timing error. The correction information is broadcast to all rovers that are in the area. This correction information allows a rover to calculate its position with much greater accuracy. DGPS techniques can produce positional accuracies of between 1-10m [13]. The basic principle is shown in the diagram below, however it does not represent the scale of the difference in distance between the base station and the rover/user.

Figure : Schematic Representation of Differential GPS[14]


Real Time Kinematic GPS is the most accurate system. It differentiates itself from DGPS in the manner in which it makes the timing measurements. A basic GPS system calculates the time taken for the unique PRN codes to travel from the satellite to the receiver by locking into that unique code. The receiver knows what each satellite's unique code is and the satellite transmits it with (modulated in) the signal. At the receiver the code sent by the satellite is delayed until it lines up with the receiver's copy of the code. This delay is the time taken for the signal to reach the receiver.

In general the receivers are able to align the signals to approximately 1% of one bit-width. The Coarse Acquisition code sent on the GPS system transmits a bit every 0.98µs, so 1% of this is approximately 0.01µs. Therefore applying the formula for distance:

Distance Error = speed x time

Distance Error = 3x108 x 0.01x10-6

Distance Error = 3m

Whereas GPS uses the unique PRN codes to determine position correct to several metres, the RTK GPS uses the signals that carry the PRN codes to determine position correct to several centimetres. RTK GPS improves on GPS by locking into the carrier signal itself, as opposed to the code contained within it. The Coarse Acquisition code broadcast in the L1 signal changes phase at 1.023 MHz, but the L1 carrier itself is 1575.42 MHz, over a thousand times faster. If the individual periods of the carrier could be correctly identified, then the error would be:

The carrier period is 1/1575.42 MHz = 634.75psec

1% error = 6.35psec

Synchronise (lock-in) with an error of 1% = 628.40psec

Distance error is now 3x108 m/sec x 628.40psec = 0.188m ≈ 19cm

The use of the carrier phase gives advantages but can also cause a problem. One such advantage is the minimising of receiver errors. With DGPS, satellite errors and some atmospheric errors are minimised by measuring the PRN code at a reference point and sending the results to the rover. The rover could then subtract the errors from its own measurements resulting in a more accurate position. The main problem with this however is the possibility of receiver errors. RTK uses the carrier phase along with complex mathematical techniques to eliminate the receiver errors.

Another advantage is the minimising of atmospheric errors. Ionospheric errors can vary significantly in different areas and thus are very difficult to model for single-frequency receivers. However the ionospheric delay induced in the signal is a very predictable function of frequency. If the delays in both the L1 and L2 carrier frequencies are measured, the ionospheric delay can be accurately modelled and therefore reduced. Due to this, centimetre level equipment usually measures both the L1 and L2 carriers and is able to work at greater distances from the base-station [15].

However the problem arises when the carrier phase ambiguity needs to solved by trial and error. Unlike the PRN codes that are uniquely identifiable, the carrier phase is not time stamped and appears as just one large sine wave, so the challenge is to somehow synchronise to and count the carrier periods (for L1 these are 19cm long and L2 are 24cm). The carrier periods can be measured very accurately but it is very difficult to tell how many of these carrier cycles are between us and the satellite. However in this case we are not too concerned with the distance between us and the satellite and are more concerned with the distance and bearing from the rover to the reference receiver. This distance is much shorter and makes it easier to solve. The method used to work it out involves trying one solution then another until one is found that satisfies all the measurements. This process is known as RTK initialisation or finding integers.

Working alone a single RTK receiver could process this to an error of one or more multiples of the wavelength (minimum error = 19cm). However, if used with another RTK receiver, (i.e. in DGPS RTK form). This means the rover calculates its position with millimetre accuracy (carrier phase error) relative to the base station, whose own positional accuracy (of the order of 1-2 cm) is now the limiting factor.

The table below shows some typical errors along with the effects of these errors and reduction or removal techniques.


Differential GPS






System Errors




Ephemeris /satellite position Data

0.4-0.5 metres



Clock errors of the satellites' clocks

1-1.2 metres






Atmospheric Errors




Ionospheric effects

5 metres

Mostly Removed

Almost all removed [1] 

Tropospheric effects

0.5 metre







1.7-7 metres

0.2-2 metres

0.005-0.01 metres




Ground Errors




Receiver (calculations, synchronisation, etc.)

0.1-3 metres

0.1-3 metres

Almost all removed

Multipath effect

0-10 metres

0-10 metres

Greatly reduced

Table : Typical errors along with the effects of these errors and reduction or removal techniques.

All the above information deals only with RTK correction information from a single reference receiver. The limitation of this is distance. The further away the rover is from the reference station the more the signal paths from the satellites diverge through different parts of the atmosphere creating greater atmospheric variances. These variances are predominantly ionospheric and increase the position error. This distance related error poses more of a problem for carrier phase processing than for PRN code processing. This is due to two reasons:

The basic accuracy of the carrier phase is much better meaning that atmospheric errors will first be noticeable in carrier-phase results.

As the distance between the rover and the reference station increases, eventually the errors will make the fitting of the correct number of carrier waves into the solution unclear causing a sharp increase in positional error [15].

Network RTK GPS

The issues highlighted above can be improved by using several reference receivers in the form of a network. Network RTK (NRTK) involves the use of three or more reference stations to collect GPS data and extract information about the atmospheric and ephemeris errors affecting signals within the network. A central processing facility uses the reference station data to generate corrections that are then relayed to RTK users operating within the coverage region of the network. Single-reference station RTK (similar to DGPS) is widely used for many centimetre-level applications. The distance of the rover to the base station in conventional single-base RTK is limited to distances of around 20 km to a maximum of 50km for long range RTK. Beyond this limit, atmospheric biases degrade results - network RTK helps to overcome this limitation. Network RTK can be implemented using techniques such as the Virtual Reference Station (VRS) or the Master-Auxiliary Concept (MAC) which are the most widely used. This makes highly accurate positioning over distances of several tens of kilometres possible (reference station positioning should generally not exceed 70-100km). This approach requires RTK rovers to send their location to the central processing facility in order to receive a corrected data-stream from the network [16]. These corrections are typically transmitted in the standard Radio Technical Commission for Maritime Services (RTCM) format via radio, mobile phone or wireless internet [17]. RTK networks provide several advantages to users such as enabling fast, centimeter-level positioning anywhere over a large area, eliminating the need to set up a private base station for each job and it provides a common coordinate reference frame. The NRTK system is represented in the following diagram.

Figure : Network RTK System [18]

Alternatives to GPS


GLONASS (Global Orbiting Navigation Satellite System) is the Russian satellite system which became fully operational in December 1995. Like GPS, GLONASS also uses 24 satellites and though slightly more accurate than GPS the number of satellites has decreased to seven in 2001 and has decreased further since. In spite of this Russia are working hard to have the system fully operational again. The satellites are placed in two orbits (although there are three orbits) and system performance is not optimal[19].


GPS III is a newer version of GPS. The main development with this system compared to the current GPS system is that it sends out a more powerful signal. Another improvement in the system is the fact that that it will follow a different orbit which allows countries on higher latitude to get better coverage. Along with this, GPS III will be able to work in tandem with the European satellite system called GALILEO which is a great advantage.


GALILEO is the new satellite system of the European Union. The system was developed so that Europe would not have to be dependent on the US GPS system.

GALILEO will offer better accuracy and coverage than GPS and is aimed at civilian users. The first satellite was launched in 2005 and the completion date has been continually pushed out since. When it is completed it will comprise of 30 satellites orbiting above the earth.

Augmentation Systems

WAAS (Wide Area Augmentation System) for the American continent and EGNOS (European Geostationary Navigation Overlay System) for the European continent are two systems that were launched to make both the GPS and GLONASS systems even more accurate.

Each system consists of the three satellites that send out signals to receivers. Ground reference stations then calculate if the satellite signal has any error and send the correction to two of the three geostationary satellites. These in turn send the correction signal back to earth where WAAS/EGNOS enabled GPS receivers apply that correction to their computed GPS position.


The first satellite for China's BeiDou satellite navigation system was launched in 2000, but that first generation system offered only limited coverage to customers in China and neighbouring regions. Now, to end any reliance on the US GPS system, the second generation of BeiDou system has begun operations. The system is set to gradually increase the number of satellites and aims to be able to cover the whole globe by 2020. The Chinese claim that the system has an accuracy of somewhere between 20m and 100m[20].


Initially this technology was used in the trucking industry. Thousands of trucks carrying valuable inventory travel the highways each day. By installing GPS tracking devices within these vehicles, companies can examine and optimize delivery routes, track the speed and location of each truck, thus saving labour hours and reducing insurance rates.

From here, GPS eventually made its way into smaller vehicles, resulting in the increase of automobile navigation systems. Businesses utilised GPS as a practical and efficient way of managing a large fleet of company vehicles, while individual consumers used GPS tracking systems as a deterrent against theft. They also became useful for parents and indeed insurance companies wanting to track the driving habits of teens.

However, applications for GPS tracking technology are not limited to the transportation industry. People use GPS when hiking, jogging or even kayaking. Families use GPS devices in their cars to direct them to their destination, keep a visual journal of favorite points of interest, locate lost pets and track inanimate objects from luggage to laptops. Some 20 million consumers now use GPS tracking technology on a regular basis. As further uses are developed for GPS, it will become more and more intertwined with our daily lives.

However all the above uses are for normal GPS with an accuracy of only a few metres. The applications become much more interesting when Network RTK is used and can benefit the user in many ways. Some applications that are particularly interesting are the use of Network RTK in structural health monitoring, robotics and precision agriculture.

Above section is very generic, add applications specific to Network RTK and maybe make into a chapter on its own?

Structural monitoring with GPS. By: Duff, Keith, Hyzak, Michael, Public Roads, 00333735, Spring97, Vol. 60, Issue 4

Principles of GPS Positioning

Model Creation

In global positioning systems, as mentioned earlier, there are several error sources that affect the observations. These errors may not be as much of problem when talking about GPS systems in cars (sat nav) for example where centimetre level accuracy isn't essential and an accuracy of a few metres may suffice but for high precision applications this level of accuracy may not be adequate. This section will introduce concepts and models used in order to allow precise Network RTK positioning.

GPS Carrier Phases

A GPS signal consists of a high frequency carrier wave and a lower frequency code that is modulated onto it. There are 3 GPS observables which are carrier phases, pseudo ranges and Doppler shift measurements, all of which will be explained in the following paragraphs starting with the carrier phases.

GPS satellites generate a pure sine wave at a frequency (f0) of 10.23MHz and when this is multiplied by integer factors the L1 and L2 carrier waves are generated.

fL1 = f0 x 154 = 1575.42 MHz with an equivalent wavelength of λL1 ≈ 19cm

fL2 = f0 x 120 = 1227.60 MHz with an equivalent wavelength of λL2 ≈ 24cm

Every satellite transmits signals based on these two microwave radio frequencies in the L band, 1575.42MHz, known as L1 and 1227.60MHz known as L2.

These frequencies are chosen for several reasons. First of all if centimetre accuracy is to be achieved, centimetre wavelengths must be used. The wavelengths of the L1 and L2 waves are 19cm and 24cm respectively. Another reason that these frequencies are chosen is to lessen the effect of the ionosphere on the signals (equation 1.7). The higher the frequency used the smaller the ionospheric error will be however if too high frequencies are used the satellite signal becomes weak. Therefore the selection of the frequencies is a good trade off between signal strength and the lessening of the ionospheric effect.

The deterministic model for the carrier phase in metres, also known as the phase range observable is as follows as long as the ambiguity term N is known:

Equation 1.1

Where: Φ1α is the product of the carrier phase measurement from receiver α to satellite 1 observed in cycles and the nominal wavelength in metres.

c is the speed of the signal (speed of light)

The receiver independent error sources are:

Ι1α is the ionospheric error along the signal path

Τ1α is the Tropospheric error along the signal path

δm1α is the multipath effect

dt1 is the satellite clock offset at the time of signal transmission

δ1 is the satellite hardware delay at the time of signal transmission

t - τ1α is the time of signal transmission, which is the difference between the true GPS time at the signal reception t and the signal's travel time τ1α.

The receiver dependent error sources are:

dtα is the receiver clock offset

δα is the receiver hardware delay at the time of signal reception t

ε1α is the carrier phase measurement noise

Making the assumption that there are no errors in the measurement the observed carrier phase would be the sum of the geometric distance ρ1α - including the orbit errors and the receiver and satellite antenna phase centre offsets and the real valued ambiguity Ν1α in cycles that is a mixture of the integer ambiguity and the uncalibrated phase delays originating in the satellite and the receiver.

The GPS Codes

The lower frequency code that is modulated onto the carrier gives the carrier a rough idea of where the receiver will be positioned. This code is called the pseudorandom noise code or a PRN code. The codes are made up of a series of binary values that make them undistinguishable from other forms of noise and each satellite generates its own unique code as to allow the receiver determine which satellite each code is coming from. These codes are also resistant to interference when codes are received simultaneously.

There are also two ranging codes modulated onto the carrier signal, those being the coarse-acquisition C/A code and the P(Y) code. The C/A codes are 1023 digit long binary sequences which are generated at a rate of 1023 million chips per second, translating to a frequency of 1.023 MHz. Each satellite transmits its own unique C/A code and repeats this every millisecond. This code is modulated on the L1 carrier wave.

The P(Y) code frequency is 10.23 MHz, as is the fundamental frequency f0 and repeats itself at approximately every 266.4 days. Each satellite has its own unique one week segment of the code that is reset every week. The P code is modulated on both the L1 and L2 carrier waves [21].

The Y code is an encrypted form of the P-code only available to authorised users. This code is transmitted instead of the P-code when anti-spoofing is in effect. This code is modulated on both the L1 and L2 carrier waves

In order to measure one way ranges, the GPS receiver generates the same pseudorandom code sequence as the satellite and by lining this replicated sequence up with the received one the amount that both sequences are out of sync can be determined and therefore the travel time can be calculated. This travel time is biased by the satellite and receiver clock offsets. The satellite clock offsets will be transmitted with the Navigation Message for each satellite, but the receiver clock offsets have to be estimated as a parameter in the positioning algorithm. For this reason a minimum of four satellites are needed.

The Navigation Message is superimposed on both the L1 and L2 carriers along with PRN codes and contains various pieces of information such as the predicted satellite paths or ephemerides, the predicted satellite clock correction model coefficients, GPS system status information and the GPS ionospheric model.

The deterministic model for the pseudoranges is given:

Equation 1.2

Pα1 is the pseudorange measurement from receiver α to satellite 1

c is the speed of the signal (speed of light)

The receiver independent error sources that affect the pseudorange measurement are:

Ι1α is the ionospheric error along the signal path

Τ1α is the Tropospheric error along the signal path,

dm1α, is the multipath effect

dt1 is the satellite clock offset at the time of signal transmission

d1 is the satellite hardware delay at the time of signal transmission

t - τ1α is the time of signal transmission, which is the difference between the true GPS time at the signal reception t and the signal's travel time τ1α.

The receiver dependent error sources are:

dtα, is the receiver clock offset

dα is the receiver hardware delay at the time of signal reception t

e1α is the pseudorange measurement noise

ρ1α represents the geometric distance between receiver a to satellite 1, including orbit errors and the receiver and satellite antenna phase centre offsets.

If equations 1.1 and 1.2 are compared, the ionospheric refraction effect is negated. The multipath dm1α, hardware delay dα and d1and measurement noise e1α terms take the place of the equivalent carrier phase terms, and because ambiguity is unique to the phase, it is not represented in the pseudorange observation equation.

Doppler Shift

Due to the fact that the satellite is moving, the frequency of the received signal appears to be slightly different or shifted to the primary carrier frequency, L1 or L2, due to the Doppler Effect. The difference between two subsequent in time phase measurements can be represented in distance units as follows:

Equation 1.3

Where: Φ1α is the product of the carrier phase measurement from receiver α to satellite 1 observed in cycles and the nominal wavelength in metres.

c is the speed of the signal (speed of light)

dt1 is the satellite clock offset at the time of signal transmission

dt1 is?????????????????? Receiver clock offset?

d1 is the satellite hardware delay at the time of signal transmission

t - τ1α is the time of signal transmission, which is the difference between the true GPS time at the signal reception t and the signal's travel time τ1α.

∆t is the change in time t

ε1α is the carrier phase measurement noise

ρ1α represents the geometric distance between receiver a to satellite 1, including orbit errors and the receiver and satellite antenna phase centre offsets.

For short time intervals atmospheric refraction, multipath and equipment delays can be ignored i.e. it assumes that these delays are similar at both times and cancel out. Here the change in carrier phase measurement is generally caused by changes in the satellite and receiver position and to changes in the satellite and receiver clock errors as shown in equation 1.3. As a result the first term on the right hand side can be represented as the rate of change of the satellite to receiver distance through a linear velocity term, and the second and third terms as the frequency deviations of the satellite and receiver oscillations through a linear frequency deviation term [22]. Equation 1.3 can be rewritten as:

Equation 1.4

The left hand side of this equation is known as the Doppler frequency shift.

Where: Φ1α is the product of the carrier phase measurement from receiver α to satellite 1 observed in cycles and the nominal wavelength in metres.

t - τ1α is the time of signal transmission, which is the difference between the true GPS time at the signal reception t and the signal's travel time τ1α.

∆t is the change in time t

ε1α is the carrier phase measurement noise

ρ1 is ………………………??

ρα is ……………………….??

λ is wavelength

δfα is ……………………….??

δf1 is ……………………….??

Error Sources

In order for GPS to provide accurate, reliable positioning the range measurements must be accurate and the GDOP must also be adequate. The measurement acquired will be different to the actual positions due to the errors that are present. The errors that are being referred to here can be seen in equations 1.1 and 1.2 and will be discussed further in the following sections.

Orbit Errors

The errors present in the estimates of the satellite coordinates will propagate directly to the estimation of the parameters. This is the reason why orbit errors must be minimised and is achieved mainly by accurately modelling the satellite orbits. The estimation of the accuracy of orbit errors is essential here and one of the organisations that track them is the IGS, the International GNSS Service.

Relative positioning is central to Network RTK in which case the orbit error can be estimated as a general rule [22]. An error dr in the coordinates of a satellite's orbit can end up contributing to errors db in the coordinates of a baseline with length b as shown below:

Equation 1.5

Where r is the mean distance between station and satellite, which is approximately 20000 km). If the size of the baseline is small in comparison to the distance from the receiver to the satellite the effect of the orbital uncertainty dr is insignificant. Common baseline lengths between reference stations in such a network are generally between 20 km to 300 km. Therefore, the maximum error possible in differential GPS for a baseline of 300 km using broadcast ephemeris (100cm accuracy in real time), using equation 1.5 would be about 0.002m as shown below.

= 0.002m

Satellite and Receiver Clock Errors

The satellite and receiver clocks play a very important role in achieving precise GPS positioning. As seen in the equations 1.1 and 1.2, the receiver and satellite clock error components are multiplied by the speed of light c. Therefore because of the size of the factor c, a small clock error can result in a very large code and phase error. If for example there is a clock error of just 1ns, this will translate to a 0.3m range error or a clock error of 1µs translates to a 300m error.

The satellite clock inaccuracy is modelled by the control segment of the GPS system and is sent to all receivers. The clock behaviour ΔtSV for a satellite SV, is solved using the equation shown below.

Equation 1.6

Where: af0 is the satellite clock bias

af1 is the satellite clock drift

af2 is the satellite clock drift rate

t is the GPS system time in seconds

tα is the reference epoch for the coefficients in seconds in the GPS week.

A relativistic correction term Δtr needs also to be applied to complete the model of equation 1.6 [23].

Using a double differencing technique, receiver and satellite clock errors are eliminated.

Atmospheric Errors

A large part of the error component in GPS is made up of the variety of atmospheric effects on the propagated GPS signal. There are two main types of atmospheric errors that we are concerned with: the ionospheric effect and the tropospheric delay. The ionosphere is the region of atmosphere between 50 and 1000 km above the Earth's surface and its effect is frequency dependent. The tropospheric delay is caused by the lower part of the atmosphere, between the surface and 50 km and is frequency independent. Several sources were referred to in writing the following subsections [22, 24-26].


In the higher parts of this region the air has a density similar to that of a gas in a vacuum tube. The Ionosphere consists of ionized plasma. In plasma the negative free electrons and the positive ions are attracted to each other by the electromagnetic force. Due to the weakness of the atmosphere at these heights electrons are capable of remaining free for a time before they are captured by a positively charged ion.

Figure : Layers of the Atmosphere [27]

The gas density increases at lower levels of the ionosphere with the result that the electrons and ions are closer together causing recombination of them to accelerate. This leads to an inconsistency in the electron content throughout the Ionosphere due to the movement of the electrons caused by electric fields, Earth's magnetic lines of force and neutral winds.

These electrons have the effect of slowing down or speeding up the GPS signals travelling through them. The amount of free electrons present along the signal path between the satellite and receiver through an area of 1 m2 is called the TEC value (Total Electron Content).

This causes the GPS carrier phase measurements appear shorter and the code measurements appear longer when compared to the actual distance between the receiver and satellite. The amount of the difference is the same in both instances as represented in Eqn. 1.7.

Equation 1.7

Where, δion ph is the ionospheric contribution of Eqn. 1.1

δion gr is the ionospheric contribution of Eqn. 1.2

f is ????????????

Following from Eqn. 1.7, the relationship of the ionospheric effect on L1 and L2 frequencies is:

Equation 1.8

Where the κ=f1/f2 coefficient relates the ionospheric bias on L1 with the ionospheric bias on L2.

Usually ionospheric models just provide the vertical TEC values (VTEC). If the signal is not at the zenith angle then the combination of TEC values at the vertical and at an angle off the vertical a mapping function similar to the one in Eqn. 1.9 is used.

Equation 1.9

Where z' is the elevation angle at a mean ionospheric height (usually 350 - 450 km)

If dual frequency receivers are used the first order ionospheric effect can be negated using linear combinations of the observables. Another option would be to estimate the ionospheric bias on the GPS signal.


The troposphere is the lower part of the atmosphere and is non-ionised. This layer is largely made up of oxygen and nitrogen. Radio waves at a frequency of up to 15GHz are affected equally by this layer although above this frequency signals will be affected to different degrees.

This layer therefore causes errors similar to that of the ionosphere. The signals are affected by the neutral atoms and molecules in this region. The delay that is caused to the signal is known as Tropospheric delay (T) and has two components being hydrostatic and wet as shown in Eqn. 1.10.

Equation 1.10

The dry portion constitutes 90% of the tropospheric refraction, whereas the wet portion constitutes 10%. However, the models for the dry troposphere are more accurate than the models for the wet troposphere. Therefore, the errors in the wet troposphere have a larger effect on the pseudorange bias than the errors in the dry troposphere.

The hydrostatic component Th in zenith direction (directly above) is called Zenith Hydrostatic Delay (ZHD). The global average positional error caused by this if not taken into account can be up to 2.5 metres. The Zenith Wet Delay (ZWD) Tw is caused by the water vapour present and is very difficult to model accurately, though it does not cause as large a positional error as the ZHD (approx 0.15 metres). Therefore, the ZHD error has a larger effect on the pseudorange bias than the ZWD error. However it must be noted that as the zenith angle increases so too does the overall error due to the fact that the signal must travel a further distance.

There are various models used for the mitigation of tropospheric refraction but for the case of this thesis the Saastamoinen model was chosen:

The Saastamoinen model derives the tropospheric delay in metres as follows:

Equation 1.11

Where T0 is the temperature at the tracking station

P is the pressure at the observed station in millibars

e is the water vapour pressure in millibars

z is the the true zenith distance i.e. the angle.

If the satellite is not in the zenith angle the signal path increases as does the error. The amount of error can be determined using a single layer mapping function model similar to the one used for the ionospheric error.

Equation 1.12

Antenna Phase Centre Offset and Variation

The electrical phase centre for every GPS antenna is not a physical point that can be measured to meaning that the offset of the phase centre to a physical point on the antenna must be known. The electrical phase centre is not constant and is always changing with the changing direction of the signal from the satellite. Antenna instantaneous phase centre errors are largely dependent on the elevation of the satellite [28], but errors caused by the local environment such as multipath must also be taken into account.

The user selects the point in the antenna which is used to establish the offset in relation to the geodetic point above which the antenna is installed. The antenna reference point (ARP) for instance is defined for each antenna type, which is usually the intersection of where the base of the antenna is mounted to the top of the pole.

As each satellite moves relative to the GPS receiver, the antennas electrical phase centre (for each satellite) moves. The average or mean electrical phase centre is automatically calculated and its position relative to the antenna reference point is represented by a vector, the antenna phase centre offset, XOff shown in Figure : Antenna Phase Centre Offset and Variations [29]. Another type of error originates in the variations of the mean phase centre and the instantaneous phase centre of each observation to a satellite and is a function of azimuth α and elevation e. This effect is called phase centre variation (PCV) and is represented by ΔΦPCV(α,e) in the figure below. These errors affect the height component of positional accuracy in particular.

Figure : Antenna Phase Centre Offset and Variations [29]

In reference to Figure : Antenna Phase Centre Offset and Variations the total phase correction Δεϕ(α,e) for a satellite at azimuth α and elevation e, is calculated using the equation below. The correction resulting from this will be in metres and is added to the geometric distance of the observation equation.

Equation 1.13

Where: Δεϕ(α,e) is the total phase correction

ΔΦPCV(α,e) is the phase centre variation

is ………….??

eaf is the unit vector in the direction of the antenna to the satellite.

The mean phase centre offset and the antenna phase centre variation maps are calculated separately for each type of antenna. The two calibration methods used for GPS Antenna offsets and variations are known as; (i) relative and (ii) absolute calibration.

With relative calibration two GPS antennas are used in a short baseline collecting observations to available GPS satellites. Both antenna orientations must be known and the surrounding area chosen in such a way as to limit the multipath effects.

Absolute calibration has an advantage over the relative calibration method in that it doesn't depend on a specific reference antenna. This method tends to be preferred when longer baselines are involved. The reason behind this is that the curvature of the earth's surface causes the same satellites observed by receivers at each end of the baseline to be seen as being at different elevations [30].

Regardless of the technique used in the calibration process randomes i.e. the protective dome shaped covering on the top of an antenna must be avoided as they can cause height differences of up to a few centimetres. However if randomes are used they are typically made of material that minimally affects the signal transmitted or received by the antenna These domes are typically installed for security reasons or to protect against weather conditions.


Multipath is the error caused by the reflection or bouncing of a GPS signal off a surface prior to reaching a receiver causing the signal to arrive at that receiver via more than one path. It is difficult to completely correct multipath error, with the immediate environment being the main concern along with the type of antenna used but to a lesser degree.

It affects both code and phase measurements. Carrier phase multipath is a fraction of the wavelength, whereas code multipath is limited by the chipping rate.

Interference caused by reflection of the signals

Figure : Multipath Error [31]

Even in high precision GPS like Network RTK, multipath error is a serious concern to the user as it is not negated by the differencing techniques due to it being an extremely localised occurrence. It is not always possible for a user to position themselves in an area away from things that may cause multipath particularly if they have to work in that area for a given reason. However reference station sites may be chosen to try and limit or eliminate completely the multipath effect. Pseudorange multipath can reduce the accuracy of any application that relies on precise pseudoranges, such as differential navigation, ionospheric monitoring, ambiguity resolution and RTK surveying. Pseudorange multipath on both frequencies, is evaluated using the and linear combinations using both pseudorange P1 and P2 and carrier phase L1 and L2 data, to eliminate the effects of receiver and satellite clocks, as well as atmospheric influence on the GPS signal.

These combinations are [22]:

Equation 1.14

Equation 1.15

The term in the curly brackets of Eqns. 1.14 and 1.15 is constant, and it is a combination of the phase ambiguity bias N1 andN2 and the ionospheric scale factor as defined in Eqn. 1.8. This offset can be removed through averaging over an appropriate number of epochs since it changes only if there is a cycle slip in either of the carrier phase observations.

It is essential to avoid multipath as it will cause estimated receiver positions to be off. A few approaches can be used to try and limit the affect of the error such as carefully designed antennas, selection of an appropriate position for the antenna and placing radio frequency absorbing material on any potential problem surfaces near the antenna.

Measurement Noise

Measurement noise is made up of all un-modelled errors and second order effects. The degree to which it affects results depends largely on how much noise accompanies the GPS signals in the tracking loops. The noise either comes from the receiver electronics itself or is picked up by the receiver's antenna. It can affect the code or the pseudorange measurements. Measurement noise is represented by the symbol e1a in the pseudorange observation equation and by ε1a in the carrier phase observation equation. The noise for the phase observations is at the millimetre level and as a result has only a minor effect on Network RTK processing.

GPS Measurements

There are two types of measurements carried out by the GPS signals which are the pseudoranges and carrier phases. If a dual frequency GPS receiver is employed and/or more than one receiver is occupied, then new observables can be produced as a linear combination between frequencies and receivers. There are several methods used for this purpose such as, the wide lane, narrow lane, ionospheric free, geometry free, single, double and triple difference linear combinations. For the purpose of this thesis the single, double and triple difference combinations will be examined in detail.

The idea behind relative positioning is to be able to find the coordinates of an unknown point with respect to a known point. This is achieved by forming simultaneous equations between the code or phase range observations. If we have simultaneous observations from receivers a and b to satellites 1 and 2 we can form linear combinations that result in single, double or triple differences. It is possible to carry out differencing between satellites, between receivers and between time. The basis of all the following differencing equations is Eqn. 1.1.The following between receivers and satellites linear combinations have the same formation for both frequencies, hence the frequency subscript is not explicitly identified.

Single Difference Observable

The single difference observable is formed by subtracting observation equations from two separate receivers to a single satellite and is represented by and is called between receiver single differencing, Φ1ab.

The different time arguments ta and tb on the left hand side of the equation represent the fact that the receiver clock errors at each receiver are not the same. However, since the satellites use highly accurate atomic clocks the assumption can be made that the satellite clock errors dt1 and equipment delays δt1 for the time intervals ta - τ1α and tb - τ1b are equal and therefore cancel when the between receivers single difference is formed.

The mathematical model for the single difference pseudorange observation is shown below:

Equation 1.26

The receiver clock errors and clock bias terms are removed by forming the between satellite single difference for receiver a, Φ12a. An equivalent equation is used for receiver b.

Equation 1.27


The following sections will describe the double and triple difference observable. These are formed from the difference between two single differences for a specific receiver for double differencing and forming the between epoch difference of two double differences for triple differencing.

Double Difference Observable

By taking the difference of two between satellites single differences for a specific receiver the double difference observable Φ12ab is formed in metres. Using this information and equation 1.27 above for two receivers a and b with measurements being taken at the exact same time, the following equation is formed:


Where Φ12α - Φ12b are the single difference carrier phase measurements between receivers a and b to satellites 1 and 2 for an epoch t, observed in cycles.

The receiver independent error sources that affect the carrier phase measurements are:

Ι, the ionospheric bias along the signal path

Τ, the tropospheric bias along the signal path

δm, the multipath effect

ε, the measurement noise

ρ, the double difference geometric distance

In this equation receiver and satellite clock offsets and clock biases cancel. The single difference ambiguities difference Na12 - Nb12 is commonly parameterised as a new ambiguity parameter Na12. Another advantage of double differencing is that the Nb12 parameter is an integer because the non-integer terms in the GPS carrier phase observation, due to clock and hardware delays in the transmitter and receiver are eliminated.

Usually more double difference ambiguities than there are original data can be formed, by differencing parts of other double differences. However, there is no more new information created from the additional equations and such observations should not be processed. The set of double difference equations for a given number of stations and satellites that cannot be formed as a linear combination from the involved equations is the linearly independent set of double differences. For a single baseline the linearly independent set will be s - 1, where s is the number of satellites.