Basics of the GPs Technique: Observation Equations Geoffrey blewitt Department of Geomatics, University of Newcastle Newcastle upon Tyne NEl 7RU, United Kingdom Table of Contents L INTRODUCTION 2. GPS DESCRIPTION 2.2 THE GPS SEGMENTS 3. THE PSEUDORANGE OBSERVABLE…… 3. 1 CODE GENERATION 3.2 AUTOCORRELATION TECHNIQUE 3. 3 PSEUDOR UATIONS 4. POINT POSITIONING USING PSEUDORANGE 15 4. 1 LEAST SQUARES ESTIMATION 4.2 ERROR COMPUTATION 18 5. THE CARRIER PHASE OBSERVABLE 5. I CONCEPTS 5.2 CARRIER PHASE OBSERVATION MODEL. 5.3 DIFFERENCING TECHNIQUES 6. RELATIVE POSITIONING USING CARRIER PHASE 6.1 SELECTION OF OBSERⅤ ATIONS…… 6.2 BASELINE SOLUTION USING DOUBLE DIFFERENCES 6.3 STOCHASTIC MODEL 7. INTRODUCING HIGH PRECISION GPS GEODESY 7. 1 HIGH PRECISION SOFTWARE 7.2 SOURCES OF DATA AND INFORMATION 8. CONCLUSIONS Copyright o 1997 by the author. All rights reser Appears in the textbook "Geodetic Applications of published by the Swedish Land Survey
1 Basics of the GPS Technique: Observation Equations § Geoffrey Blewitt Department of Geomatics, University of Newcastle Newcastle upon Tyne, NE1 7RU, United Kingdom geoffrey.blewitt@ncl.ac.uk Table of Contents 1. INTRODUCTION.....................................................................................................................................................2 2. GPS DESCRIPTION................................................................................................................................................2 2.1 THE BASIC IDEA ........................................................................................................................................................2 2.2 THE GPS SEGMENTS..................................................................................................................................................3 2.3 THE GPS SIGNALS .....................................................................................................................................................6 3. THE PSEUDORANGE OBSERVABLE ................................................................................................................8 3.1 CODE GENERATION....................................................................................................................................................9 3.2 AUTOCORRELATION TECHNIQUE .............................................................................................................................12 3.3 PSEUDORANGE OBSERVATION EQUATIONS..............................................................................................................13 4. POINT POSITIONING USING PSEUDORANGE.............................................................................................15 4.1 LEAST SQUARES ESTIMATION ..................................................................................................................................15 4.2 ERROR COMPUTATION.............................................................................................................................................18 5. THE CARRIER PHASE OBSERVABLE............................................................................................................22 5.1 CONCEPTS................................................................................................................................................................22 5.2 CARRIER PHASE OBSERVATION MODEL...................................................................................................................27 5.3 DIFFERENCING TECHNIQUES....................................................................................................................................32 6. RELATIVE POSITIONING USING CARRIER PHASE...................................................................................36 6.1 SELECTION OF OBSERVATIONS.................................................................................................................................36 6.2 BASELINE SOLUTION USING DOUBLE DIFFERENCES .................................................................................................39 6.3 STOCHASTIC MODEL................................................................................................................................................42 7. INTRODUCING HIGH PRECISION GPS GEODESY......................................................................................44 7.1 HIGH PRECISION SOFTWARE ....................................................................................................................................44 7.2 SOURCES OF DATA AND INFORMATION ....................................................................................................................45 8. CONCLUSIONS .....................................................................................................................................................46 § Copyright © 1997 by the author. All rights reserved. Appears in the textbook “Geodetic Applications of GPS,” published by the Swedish Land Survey
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 1. INTRODUCTION The purpose of this paper is to introduce the principles of GPS theory, and to provide a background for more advanced material. with that in mind some of the theoretical treatment has been simplified to provide a starting point for a mathematically literate user of GPS who wishes to understand how GPs works, and to get a basic grasp of GPs theory and terminology. It is therefore not intended to serve as a reference for experienced researchers however, my hope is that it might also prove interesting to the more advanced reader, who might appreciate some "easy reading "of a familiar story in a relatively short text(and no doubt, from a slightly different angle) 2. GPS DESCRIPTION In this section we introduce the basic idea behind GPS, and provide some facts and statistics to describe various aspects of the global Positionining System. 2.1 THE BASIC IDEA GPS positioning is based on trilateration, which is the method of determining position by measuring distances to points at known coordinates. At a minimum, trilateration requires 3 ranges to 3 known points. GPs point positioning, on the other hand, requires 4 pseudoranges"to 4 satellites This raises two questions: (a)"What are pseudoranges? and(b)"How do we know position of the satellites? Without getting into too much detail at this point, we address second question first 2. 1.1 How do we know position of satellites? a signal is transmitted from each satellite in the direction of the earth. This signal is encoded with the"Navigation Message, which can be read by the user's GPS receivers. The Navigation Message includes orbit parameters (often called the" broadcast ephemeris"),from which the receiver can compute satellite coordinates(X, Y, Z). These are Cartesian coordinates in a geocentric system, known as wGS-84, which has its origin at the Earth centre of mass, Z axis pointing towards the north pole, X pointing towards the Prime Meridian(which crosses Greenwich), and Y at right angles to X and Z to form a right-handed orthogonal coordinate system. The algorithm which transforms the orbit parameters into WGS-84 satellite coordinates at any specified time is called the"Ephemeris Algorithm, which is defined in GPS textbooks e.g., Leick, 1991]. We discuss the Navigation Message in more detail later on. For now, we move on to“ pseudoranges.” 2.1.2 What are pseudoranges? Time that the signal is transmitted from the satellite is encoded on the signal, using the time according to an atomic clock onboard the satellite. Time of signal reception is recorded by receiver using an atomic clock. A receiver measures difference in these times
2 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 1. INTRODUCTION The purpose of this paper is to introduce the principles of GPS theory, and to provide a background for more advanced material. With that in mind, some of the theoretical treatment has been simplified to provide a starting point for a mathematically literate user of GPS who wishes to understand how GPS works, and to get a basic grasp of GPS theory and terminology. It is therefore not intended to serve as a reference for experienced researchers; however, my hope is that it might also prove interesting to the more advanced reader, who might appreciate some “easy reading” of a familiar story in a relatively short text (and no doubt, from a slightly different angle). 2. GPS DESCRIPTION In this section we introduce the basic idea behind GPS, and provide some facts and statistics to describe various aspects of the Global Positionining System. 2.1 THE BASIC IDEA GPS positioning is based on trilateration, which is the method of determining position by measuring distances to points at known coordinates. At a minimum, trilateration requires 3 ranges to 3 known points. GPS point positioning, on the other hand, requires 4 “pseudoranges” to 4 satellites. This raises two questions: (a) “What are pseudoranges?”, and (b) “How do we know the position of the satellites?” Without getting into too much detail at this point, we address the second question first. 2.1.1 How do we know position of satellites? A signal is transmitted from each satellite in the direction of the Earth. This signal is encoded with the “Navigation Message,” which can be read by the user’s GPS receivers. The Navigation Message includes orbit parameters (often called the “broadcast ephemeris”), from which the receiver can compute satellite coordinates (X,Y,Z). These are Cartesian coordinates in a geocentric system, known as WGS-84, which has its origin at the Earth centre of mass, Z axis pointing towards the North Pole, X pointing towards the Prime Meridian (which crosses Greenwich), and Y at right angles to X and Z to form a right-handed orthogonal coordinate system. The algorithm which transforms the orbit parameters into WGS-84 satellite coordinates at any specified time is called the “Ephemeris Algorithm,” which is defined in GPS textbooks [e.g., Leick, 1991]. We discuss the Navigation Message in more detail later on. For now, we move on to “pseudoranges.” 2.1.2 What are pseudoranges? Time that the signal is transmitted from the satellite is encoded on the signal, using the time according to an atomic clock onboard the satellite. Time of signal reception is recorded by receiver using an atomic clock. A receiver measures difference in these times:
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE pseudorange =(time difference)x(speed of light) receiver clocks are far from perfect. How do we correct for clock errors ors because the Note that pseudorange is almost like range, except that it includes clock er 2.1.3 How do we correct for clock errors? Satellite clock error is given in Navigation Message, in the form of a polynomial. The unknown receiver clock error can be estimated by the user along with unknown station ordinates. There are 4 unknowns, hence we need a minimum of 4 pseudorang measurements 2.2 THE GPS SEGMENTS There are four GPS segments the Space Segment, which includes the constellation of GPS satellites, which transmit the signals to the user the Control Segment, which is responsible for the monitoring and operation of the the User Segment, which includes user hardware and processing software for positioning, navigation, and timing applications the Ground Segment, which includes civilian tracking networks that provide the User Segment with reference control, precise ephemerides, and real time services (DGPS) which mitigate the effects of"selective availability"(a topic to be discussed later) Before getting into the details of the GPs signal, observation models, and position computations, we first provide more information on the Space Segment and the Control 2. 2. 1 Orbit Design The satellite constellation is designed to have at least 4 satellites in view anywhere, anytime to a user on the ground. For this purpose, there are nominally 24 gPS satellites distributed in 6 orbital planes. So that we may discuss the orbit design and the implications of that design we must digress for a short while to explain the geometry of the GPs constellation According to Kepler's laws of orbital motion, each orbit takes the approximate shape of an ellipse, with the Earths centre of mass at the focus of the ellipse. For a GPs orbit, th eccentricity of the ellipse is so small (0.02) that it is almost circular. The semi-major axis (largest radius)of the ellipse is approximately 26, 600 km, or approximately 4 Earth radii The 6 orbital planes rise over the equator at an inclination angle of 55 to the equator. The point at which they rise from the Southern to Northern Hemisphere across the equator is ed the ""Right Ascension of the ascending node Since the orbital planes are evenly distributed, the angle between the six ascending nodes is 60
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 3 pseudorange = (time difference) × (speed of light) Note that pseudorange is almost like range, except that it includes clock errors because the receiver clocks are far from perfect. How do we correct for clock errors? 2.1.3 How do we correct for clock errors? Satellite clock error is given in Navigation Message, in the form of a polynomial. The unknown receiver clock error can be estimated by the user along with unknown station coordinates. There are 4 unknowns; hence we need a minimum of 4 pseudorange measurements. 2.2 THE GPS SEGMENTS There are four GPS segments: • the Space Segment, which includes the constellation of GPS satellites, which transmit the signals to the user; • the Control Segment, which is responsible for the monitoring and operation of the Space Segment, • the User Segment, which includes user hardware and processing software for positioning, navigation, and timing applications; • the Ground Segment, which includes civilian tracking networks that provide the User Segment with reference control, precise ephemerides, and real time services (DGPS) which mitigate the effects of “selective availability” (a topic to be discussed later). Before getting into the details of the GPS signal, observation models, and position computations, we first provide more information on the Space Segment and the Control Segment. 2.2.1 Orbit Design The satellite constellation is designed to have at least 4 satellites in view anywhere, anytime, to a user on the ground. For this purpose, there are nominally 24 GPS satellites distributed in 6 orbital planes. So that we may discuss the orbit design and the implications of that design, we must digress for a short while to explain the geometry of the GPS constellation. According to Kepler’s laws of orbital motion, each orbit takes the approximate shape of an ellipse, with the Earth’s centre of mass at the focus of the ellipse. For a GPS orbit, the eccentricity of the ellipse is so small (0.02) that it is almost circular. The semi-major axis (largest radius) of the ellipse is approximately 26,600 km, or approximately 4 Earth radii. The 6 orbital planes rise over the equator at an inclination angle of 55 o to the equator. The point at which they rise from the Southern to Northern Hemisphere across the equator is called the “Right Ascension of the ascending node”. Since the orbital planes are evenly distributed, the angle between the six ascending nodes is 60 o
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Each orbital plane nominally contains 4 satellites, which are generally not spaced evenly around the ellipse. Therefore, the angle of the satellite within its own orbital plane, the"true anomaly, is only approximately spaced by 90. The true anomaly is measured from the point of closest approach to the Earth(the perigee).(We note here that there are other types of anomaly" in GPS terminology, which are angles that are useful for calculating the satellite coordinates within its orbital plane). Note that instead of specifying the satellites anomaly at every relevant time, we could equivalently specify the time that the satellite had passed perigee, and then compute the satellites future position based on the known laws of motion of the satellite around an ellipse. Finally, the argument of perigee is the angle between the equator and perigee. Since the orbit is nearly circular, this orbital parameter is not well defined and alternative parameterisation schemes are often used Taken together (the eccentricity, semi-major axis, inclination, Right Ascension of the ascending node, the time of perigee passing, and the argument of perigee), these six parameters define the satellite orbit. These parameters are known as Keplerian elements Given the Keplerian elements and the current time, it is possible to calculate the coordinates of the satellite GPS satellites do not move in perfect ellipses, so additional parameters are necessary Nevertheless, GPS does use Kepler's laws to its advantage, and the orbits are described in parameters which are Keplerian in appearance. Additional parameters must be added to account for non-Keplerian behaviour. Even this set of parameters has to be updated by the Control Segment every hour for them to remain sufficiently valid 2.2.2 Orbit design consequences Several consequences of the orbit design can be deduced from the above orbital parameters and Kepler's laws of motion. First of all, the satellite speed can be easily calculated to be approximately 4 km/s relative to Earth's centre. All the GPS satellites orbits are progrado which means the satellites move in the direction of Earth's rotation. Therefore. the relative motion between the satellite and a user on the ground must be less than 4 km/s. Typical values around I km/s can be expected for the relative speed along the line of sight(range The second consequence is the phenomena of "repeating ground tracks"every day. It is straightforward to calculate the time it takes for the satellite to complete one orbital olution. The orbital period is approximately t= ll hr 58 min. Therefore a GPS satellite completes 2 revolutions in 23 hr 56 min. This is intentional, since it equals the sidereal day which is the time it takes for the Earth to rotate 360.(Note that the solar day of 24 hr is not 360, because during the day, the position of the Sun in the sky has changed by 1/365.25 of a day, or 4 min, due to the Earth's orbit around the Sun) Therefore, every day(minus 4 minutes), the satellite appears over the same geographical location on the Earth's surface. The"ground track? "is the locus of points on the Earth's surface that is traced out by a line connecting the satellite to the centre of the Earth. The
4 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Each orbital plane nominally contains 4 satellites, which are generally not spaced evenly around the ellipse. Therefore, the angle of the satellite within its own orbital plane, the “true anomaly”, is only approximately spaced by 90 o . The true anomaly is measured from the point of closest approach to the Earth (the perigee). (We note here that there are other types of “anomaly” in GPS terminology, which are angles that are useful for calculating the satellite coordinates within its orbital plane). Note that instead of specifying the satellite’s anomaly at every relevant time, we could equivalently specify the time that the satellite had passed perigee, and then compute the satellites future position based on the known laws of motion of the satellite around an ellipse. Finally, the argument of perigee is the angle between the equator and perigee. Since the orbit is nearly circular, this orbital parameter is not well defined, and alternative parameterisation schemes are often used. Taken together (the eccentricity, semi-major axis, inclination, Right Ascension of the ascending node, the time of perigee passing, and the argument of perigee), these six parameters define the satellite orbit. These parameters are known as Keplerian elements. Given the Keplerian elements and the current time, it is possible to calculate the coordinates of the satellite. GPS satellites do not move in perfect ellipses, so additional parameters are necessary. Nevertheless, GPS does use Kepler’s laws to its advantage, and the orbits are described in parameters which are Keplerian in appearance. Additional parameters must be added to account for non-Keplerian behaviour. Even this set of parameters has to be updated by the Control Segment every hour for them to remain sufficiently valid. 2.2.2 Orbit design consequences Several consequences of the orbit design can be deduced from the above orbital parameters, and Kepler’s laws of motion. First of all, the satellite speed can be easily calculated to be approximately 4 km/s relative to Earth’s centre. All the GPS satellites orbits are prograde, which means the satellites move in the direction of Earth’s rotation. Therefore, the relative motion between the satellite and a user on the ground must be less than 4 km/s. Typical values around 1 km/s can be expected for the relative speed along the line of sight (range rate). The second consequence is the phenomena of “repeating ground tracks” every day. It is straightforward to calculate the time it takes for the satellite to complete one orbital revolution. The orbital period is approximately T = 11 hr 58 min. Therefore a GPS satellite completes 2 revolutions in 23 hr 56 min. This is intentional, since it equals the sidereal day, which is the time it takes for the Earth to rotate 360 o . (Note that the solar day of 24 hr is not 360 o , because during the day, the position of the Sun in the sky has changed by 1/365.25 of a day, or 4 min, due to the Earth’s orbit around the Sun). Therefore, every day (minus 4 minutes), the satellite appears over the same geographical location on the Earth’s surface. The “ground track” is the locus of points on the Earth’s surface that is traced out by a line connecting the satellite to the centre of the Earth. The
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE ground track is said to repeat. From the user's point of view the same satellite appears in the same direction in the sky every day minus 4 minutes. Likewise, the"sky tracks"repeat. In general, we can say that the entire satellite geometry repeats every sidereal day(from the point of view of a ground user) As a corollary, any errors correlated with satellite geometry will repeat from one day to the next. An example of an error tied to satellite geometry is"multipath, which is due to the antenna also sensing signals from the satellite which reflect and refract from nearby objects In fact, it can be verified that, because of multipath, observation residuals do have a pattern that repeats every sidereal day. As a consequence, such errors will not significantly affect the precision, or repeatability, of coordinates estimated each day. However, the accuracy can be significantly worse than the apparent precision for this reason Another consequence of this is that the same subset of the 24 satellites will be observed every day by someone at a fixed geographical location. Generally, not all 24 satellites will be seen by a user at a fixed location. This is one reason why there needs to be a global distribution of receivers around the globe to be sure that every satellite is tracked sufficiently well We now turn our attention to the consequences of the inclination angle of 55. Note that a satellite with an inclination angle of 90 would orbit directly over the poles. Any other inclination angle would result in the satellite never passing over the poles. From the user's point of view, the satellite's sky track would never cross over the position of the celestial pole in the sky. In fact, there would be a"hole"in the sky around the celestial pole where the satellite could never pass. For a satellite constellation with an inclination angle of 55, there would therefore be a circle of radius at least 35 around the celestial pole, through which the sky tracks would never cross. Another way of looking at this, is that a satellite can never rise more than 55 elevation above the celestial equator This has a big effect on the satellite geometry as viewed from different latitudes. An observer at the pole would never see a GPS satellite rise above 55 elevation. Most of the satellites would hover close to the horizon. Therefore vertical positioning is slightly degraded near the poles. An observer at the equator would see some of the satellites passing overhead, but would tend to deviate from away from points on the horizon directly to the north and south Due to a combination of Earth rotation, and the fact that the GPS satellites are moving faster than the Earth rotates, the satellites actually appear to move approximately north-south outh-north to an oberver at the equator, with very little east-west motion. The north component of relative positions are therefore better determined than the east component the closer the observer is to the equator. An observer at mid-latitudes in the Northern Hemisphe would see satellites anywhere in the sky to the south, but there would be a large void towards the north. This has consequences for site selection, where a good view is desirable to the outh, and the view to the north is less critical. For example, one might want to select a site in the Northern Hemisphere which is on a south-facing slope(and visa versa for an observer in the Southern Hemisphere) 2.2. 3 Satellite hardware There are nominally 24 GPS satellites, but this number can vary within a few satellites at any given time, due to old satellites being decommissioned, and new satellites being launched to
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 5 ground track is said to repeat. From the user’s point of view, the same satellite appears in the same direction in the sky every day minus 4 minutes. Likewise, the “sky tracks” repeat. In general, we can say that the entire satellite geometry repeats every sidereal day (from the point of view of a ground user). As a corollary, any errors correlated with satellite geometry will repeat from one day to the next. An example of an error tied to satellite geometry is “multipath,” which is due to the antenna also sensing signals from the satellite which reflect and refract from nearby objects. In fact, it can be verified that, because of multipath, observation residuals do have a pattern that repeats every sidereal day. As a consequence, such errors will not significantly affect the precision, or repeatability, of coordinates estimated each day. However, the accuracy can be significantly worse than the apparent precision for this reason. Another consequence of this is that the same subset of the 24 satellites will be observed every day by someone at a fixed geographical location. Generally, not all 24 satellites will be seen by a user at a fixed location. This is one reason why there needs to be a global distribution of receivers around the globe to be sure that every satellite is tracked sufficiently well. We now turn our attention to the consequences of the inclination angle of 55 o . Note that a satellite with an inclination angle of 90 o would orbit directly over the poles. Any other inclination angle would result in the satellite never passing over the poles. From the user’s point of view, the satellite’s sky track would never cross over the position of the celestial pole in the sky. In fact, there would be a “hole” in the sky around the celestial pole where the satellite could never pass. For a satellite constellation with an inclination angle of 55 o , there would therefore be a circle of radius at least 35 o around the celestial pole, through which the sky tracks would never cross. Another way of looking at this, is that a satellite can never rise more than 55 o elevation above the celestial equator. This has a big effect on the satellite geometry as viewed from different latitudes. An observer at the pole would never see a GPS satellite rise above 55 o elevation. Most of the satellites would hover close to the horizon. Therefore vertical positioning is slightly degraded near the poles. An observer at the equator would see some of the satellites passing overhead, but would tend to deviate from away from points on the horizon directly to the north and south. Due to a combination of Earth rotation, and the fact that the GPS satellites are moving faster than the Earth rotates, the satellites actually appear to move approximately north-south or south-north to an oberver at the equator, with very little east-west motion. The north component of relative positions are therefore better determined than the east component the closer the observer is to the equator. An observer at mid-latitudes in the Northern Hemisphere would see satellites anywhere in the sky to the south, but there would be a large void towards the north. This has consequences for site selection, where a good view is desirable to the south, and the view to the north is less critical. For example, one might want to select a site in the Northern Hemisphere which is on a south-facing slope (and visa versa for an observer in the Southern Hemisphere). 2.2.3 Satellite Hardware There are nominally 24 GPS satellites, but this number can vary within a few satellites at any given time, due to old satellites being decommissioned, and new satellites being launched to
6 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE replace them. All the prototype satellites, known as Block I, have been decommissioned Between 1989 and 1994, 24 Block Il(1989-1994)were placed in orbit. From 1995 onwards these have started to be replaced by a new design known as block IIR. The nominal specifications of the GPS satellites are as follows Life goal: 7.5 years Mass: -l tonne(Block IIR: -2 tonnes) Power: solar panels 7.5 m*+ Ni-Cd batteries Atomic clocks: 2 rubidium and 2 cesium The orientation of the satellites is al ways changing, such that the solar panels face the sun, and the antennas face the centre of the Earth. Signals are transmitted and received by the satellite Ising microwaves. Signals are transmitted to the User Segment at frequencies Ll=1575.42 MHz, and L2=1227. 60 MHz. We discuss the signals in further detail later on. Signals are received from the Control Segment at frequency 1783. 74 Mhz. The flow of information is a follows: the satellites transmit Ll and L2 signals to the user, which are encoded with information on their clock times and their positions. The Control Segment then tracks these signals using receivers at special monitoring stations. This information is used to improve the satellite positions and predict where the satellites will be in the near future. This orbit information is then uplinked at 1783. 74 Mhz to the gps satellites, which in turn transmit this new information down to the users. and so on. the orbit information on board the satellite is updated every hour 2.2. 4 The Control Segment The Control Segment, run by the US Air Force, is responsible for operating GPs. The main Control Centre is at Falcon Air Force Base, Colorado Springs, USA. Several ground stations monitor the satellites Ll and L2 signals and assess the"health""of the satellites. As outlined previously, the Control Segment then uses these signals to estimate and predict the satellite orbits and clock errors, and this information is uploaded to the satellites. In addition, the Control Segment can control the satellites; for example, the satellites can be maneuvered into a different orbit when necessary. This might be done to optimise satellite geometry when a new satellite is launched, or when an old satellite fails. It is also done to keep the satellites to within a certain tolerance of their nominal orbital parameters(e.g, the semi-major axis may need adjustment from time to time). As another example, the Control Segment might switch between the several on-board clocks available, should the current clock appear to be 2. 3 THE GPS SIGNALS We now briefly summarise the characteristics of the GPS signals, the types of information that digitally encoded on the signals, and how the U.S. Department of Defense implements denial of accuracy to civilian users. Further details on how the codes are constructed will be presented in Section 3 2.3.1 Signal Description
6 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE replace them. All the prototype satellites, known as Block I, have been decommissioned. Between 1989 and 1994, 24 Block II (1989-1994) were placed in orbit. From 1995 onwards, these have started to be replaced by a new design known as Block IIR. The nominal specifications of the GPS satellites are as follows: • Life goal: 7.5 years • Mass: ~1 tonne (Block IIR: ~2 tonnes) • Size: 5 metres • Power: solar panels 7.5 m 2 + Ni-Cd batteries • Atomic clocks: 2 rubidium and 2 cesium The orientation of the satellites is always changing, such that the solar panels face the sun, and the antennas face the centre of the Earth. Signals are transmitted and received by the satellite using microwaves. Signals are transmitted to the User Segment at frequencies L1 = 1575.42 MHz, and L2 = 1227.60 MHz. We discuss the signals in further detail later on. Signals are received from the Control Segment at frequency 1783.74 Mhz. The flow of information is a follows: the satellites transmit L1 and L2 signals to the user, which are encoded with information on their clock times and their positions. The Control Segment then tracks these signals using receivers at special monitoring stations. This information is used to improve the satellite positions and predict where the satellites will be in the near future. This orbit information is then uplinked at 1783.74 Mhz to the GPS satellites, which in turn transmit this new information down to the users, and so on. The orbit information on board the satellite is updated every hour. 2.2.4 The Control Segment The Control Segment, run by the US Air Force, is responsible for operating GPS. The main Control Centre is at Falcon Air Force Base, Colorado Springs, USA. Several ground stations monitor the satellites L1 and L2 signals, and assess the “health” of the satellites. As outlined previously, the Control Segment then uses these signals to estimate and predict the satellite orbits and clock errors, and this information is uploaded to the satellites. In addition, the Control Segment can control the satellites; for example, the satellites can be maneuvered into a different orbit when necessary. This might be done to optimise satellite geometry when a new satellite is launched, or when an old satellite fails. It is also done to keep the satellites to within a certain tolerance of their nominal orbital parameters (e.g., the semi-major axis may need adjustment from time to time). As another example, the Control Segment might switch between the several on-board clocks available, should the current clock appear to be malfunctioning. 2.3 THE GPS SIGNALS We now briefly summarise the characteristics of the GPS signals, the types of information that is digitally encoded on the signals, and how the U.S. Department of Defense implements denial of accuracy to civilian users. Further details on how the codes are constructed will be presented in Section 3. 2.3.1 Signal Description
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE The signals from a GPS satellite are fundamentally driven by an atomic clocks(usually cesium, which has the best long-term stability). The fundamental frequency is 10.23 Mhz Two carrier signals, which can be thought of as sine waves, are created from this signal by multiplying the frequency by 154 for the LI channel (frequency= 1575.42 Mhz; wavelength 19.0 cm), and 120 for the L2 channel (frequency=1227. 60 Mhz; wavelength= 24.4 cm). The reason for the second signal is for self-calibration of the delay of the signal in the Earth's ionosphere Information is encoded in the form of binary bits on the carrier signals by a process known as phase modulation. (This is to be compared with signals from radio stations, which are typically encoded using either frequency modulation, FM, or amplitude modulation, AM) The binary digits 0 and I are actually represented by multiplying the electrical signals by either +l or -l, which is equivalent to leaving the signal unchanged, or flipping the phase of the signal by 180. We come back later to the meaning of phase and the generation of the binary code There are three types of code on the carrier signals The C/A code The p code The Navigation Message The C/A (course acquisition )code can be found on the LI channel. As will be described later, this is a code sequence which repeats every I ms. It is a pseudo-random code, which appears to be random, but is in fact generated by a known algorithm. The carrier can transmit the C/a code at 1.023 Mbps(million bits per second). The"chip length", or physical distance between binary transitions(between digits +I and-1), is 293 metres. The basic information that the C/A code contains is the time according to the satellite clock when the signal was transmitted (with an ambiguity of l ms, which is easily resolved, since this corresponds to 293 km). Each satellite has a different C/A code, so that they can be uniquely identified The P("precise")code is identical on both the LI and L2 channel. Whereas C/A is a courser code appropriate for initially locking onto the signal, the P code is better for more precise positioning. The P code repeats every 267 days. In practice, this code is divided into 7 day segments; each weekly segment is designated a"PRN number, and is designated to one of the GPS satellites. The carrier can transmit the P code at 10.23 Mbps, with a chip length of 29.3 metres. Again, the basic information is the satellite clock time or transmission, which is identical to the C/A information, except that it has ten times the resolu Unlike the C/A code, the p code can be encrypted by a process known as"anti-spoo or“A/S”(see below) The Navigation Message can be found on the ll channel, being transmitted at a very slow rate of 50 bps. It is a 1500 bit sequence, and therefore takes 30 seconds to transmit. The Navigation Message includes information on the broadcast Ephemeris(satellite orbital ) satellite clock corrections, almanac data(a crude ephemeris for ionosphere information, and satellite health status 2.3. 2 Denial ofaccuracy
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 7 The signals from a GPS satellite are fundamentally driven by an atomic clocks (usually cesium, which has the best long-term stability). The fundamental frequency is 10.23 Mhz. Two carrier signals, which can be thought of as sine waves, are created from this signal by multiplying the frequency by 154 for the L1 channel (frequency = 1575.42 Mhz; wavelength = 19.0 cm), and 120 for the L2 channel (frequency = 1227.60 Mhz; wavelength = 24.4 cm). The reason for the second signal is for self-calibration of the delay of the signal in the Earth’s ionosphere. Information is encoded in the form of binary bits on the carrier signals by a process known as phase modulation. (This is to be compared with signals from radio stations, which are typically encoded using either frequency modulation, FM, or amplitude modulation, AM). The binary digits 0 and 1 are actually represented by multiplying the electrical signals by either +1 or −1, which is equivalent to leaving the signal unchanged, or flipping the phase of the signal by 180 o . We come back later to the meaning of phase and the generation of the binary code. There are three types of code on the carrier signals: • The C/A code • The P code • The Navigation Message The C/A (“course acquisition”) code can be found on the L1 channel. As will be described later, this is a code sequence which repeats every 1 ms. It is a pseudo-random code, which appears to be random, but is in fact generated by a known algorithm. The carrier can transmit the C/A code at 1.023 Mbps (million bits per second). The “chip length”, or physical distance between binary transitions (between digits +1 and −1), is 293 metres. The basic information that the C/A code contains is the time according to the satellite clock when the signal was transmitted (with an ambiguity of 1 ms, which is easily resolved, since this corresponds to 293 km). Each satellite has a different C/A code, so that they can be uniquely identified. The P (“precise”) code is identical on both the L1 and L2 channel. Whereas C/A is a courser code appropriate for initially locking onto the signal, the P code is better for more precise positioning. The P code repeats every 267 days. In practice, this code is divided into 7 day segments; each weekly segment is designated a “PRN” number, and is designated to one of the GPS satellites. The carrier can transmit the P code at 10.23 Mbps, with a chip length of 29.3 metres. Again, the basic information is the satellite clock time or transmission, which is identical to the C/A information, except that it has ten times the resolution. Unlike the C/A code, the P code can be encrypted by a process known as “anti-spoofing” , or “A/S” (see below). The Navigation Message can be found on the L1 channel, being transmitted at a very slow rate of 50 bps. It is a 1500 bit sequence, and therefore takes 30 seconds to transmit. The Navigation Message includes information on the Broadcast Ephemeris (satellite orbital parameters), satellite clock corrections, almanac data (a crude ephemeris for all satellites), ionosphere information, and satellite health status. 2.3.2 Denial of Accuracy
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE The U.S. Department of Defense implements two types of denial of accuracy to civilian users: Selective Availability (S/A), and Anti-Spoofing(A/S). S/A can be thought of as intentional errors imposed on the gPs signal. A/S can be thought of as encryption of the P There are two types of S/A: epsilon, and dither. Under conditions of S/A, the user should be able to count on the position error not being any worse than 100 metres. Most of the time, the induced position errors do not exceed 50 metres Epsilon is implemented by including errors in the satellite orbit encoded in the Navigation Message. Apparently, this is an option not used, according to daily comparisons made between the real-time broadcast orbits, and precise orbits generated after the fact, by the International GPS Service for Geodynamics(IGS). For precise geodetic work, precise orbits are recommended in any case, and therefore epsilon would have minimal impact on precise users. It would, however, directly impact single receiver, low-precision users. Even then, the effects can be mitigated to some extent by using technology known as"differential GPS where errors in the gps signal are computed at a reference station at known coordinates, and are transmitted to the user who has appropriate radio receiving equipment Dither is intentional rapid variation in the satellite clock frequency(10.23 MHz). Dither therefore, looks exactly like a satellite clock error, and therefore maps directly into pseudorange errors. Dither is switched on at the moment(1997), but recent U.S. poli statements indicate that it may be phased out within the next decade. As is the case for epsilon, dither can be mitigated using differential GPS. The high precision user is minimally effected by S/A, since relative positioning techniques effectively eliminate satellite clock erro (as we shall see later) Anti-Spoofing(A/S)is encryption of the P-code. The main purpose of A/s is prevent"the enemy"from imitating a GPS signal, and therefore it is unlikely to be switched off in the foreseeable future. A/S does not pose a signficant problem to the precise user, since precise GPS techniques rely on measuring the phase of the carrier signal itself, rather than the pseudoranges derived from the P code. However, the pseudoranges are very useful for various algorithms, particularly in the rapid position fixes required by moving vehicles and kinematic surveys. Modern geodetic receivers can, in any case, form 2 precise pseudorange observabl on the LI and l2 channels, even if A/S is switched on. ( We briefly touch on how this is done n the next section). As a consequence of not having full access to the P code, the phase noise on measuring the L2 carrier phase can be increased from the level of I mm to the level of I cm for some types of receivers. This has negligible impact on long sessions for static positioning, but can have noticeable effect on short sessions, or on kinematic position Larger degradation in the signal can be expected at low elevations(up to 2 cm) where signal strength is at a minimum 3. THE PSEUDORANGE OBSERVABLE
8 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE The U.S. Department of Defense implements two types of denial of accuracy to civilian users: Selective Availability (S/A), and Anti-Spoofing (A/S). S/A can be thought of as intentional errors imposed on the GPS signal. A/S can be thought of as encryption of the P code. There are two types of S/A: epsilon, and dither. Under conditions of S/A, the user should be able to count on the position error not being any worse than 100 metres. Most of the time, the induced position errors do not exceed 50 metres. Epsilon is implemented by including errors in the satellite orbit encoded in the Navigation Message. Apparently, this is an option not used, according to daily comparisons made between the real-time broadcast orbits, and precise orbits generated after the fact, by the International GPS Service for Geodynamics (IGS). For precise geodetic work, precise orbits are recommended in any case, and therefore epsilon would have minimal impact on precise users. It would, however, directly impact single receiver, low-precision users. Even then, the effects can be mitigated to some extent by using technology known as “differential GPS”, where errors in the GPS signal are computed at a reference station at known coordinates, and are transmitted to the user who has appropriate radio receiving equipment. Dither is intentional rapid variation in the satellite clock frequency (10.23 MHz). Dither, therefore, looks exactly like a satellite clock error, and therefore maps directly into pseudorange errors. Dither is switched on at the moment (1997), but recent U.S. policy statements indicate that it may be phased out within the next decade. As is the case for epsilon, dither can be mitigated using differential GPS. The high precision user is minimally effected by S/A, since relative positioning techniques effectively eliminate satellite clock error (as we shall see later). Anti-Spoofing (A/S) is encryption of the P-code. The main purpose of A/S is prevent “the enemy” from imitating a GPS signal, and therefore it is unlikely to be switched off in the foreseeable future. A/S does not pose a signficant problem to the precise user, since precise GPS techniques rely on measuring the phase of the carrier signal itself, rather than the pseudoranges derived from the P code. However, the pseudoranges are very useful for various algorithms, particularly in the rapid position fixes required by moving vehicles and kinematic surveys. Modern geodetic receivers can, in any case, form 2 precise pseudorange observables on the L1 and L2 channels, even if A/S is switched on. (We briefly touch on how this is done in the next section). As a consequence of not having full access to the P code, the phase noise on measuring the L2 carrier phase can be increased from the level of 1 mm to the level of 1 cm for some types of receivers. This has negligible impact on long sessions for static positioning, but can have noticeable effect on short sessions, or on kinematic positioning. Larger degradation in the signal can be expected at low elevations (up to 2 cm) where signal strength is at a minimum. 3. THE PSEUDORANGE OBSERVABLE
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE In this section, we go deeper into the description of the pseudorange observable, an give some details on how the codes are generated. We develop a model of the pseudorange observation, and then use this model to derive a least-squares estimator for positioning. We discuss formal errors in position, and the notion of"Dilution of Precision,, which can be used to assess the effect of satellite geometry on positioning precision 3.1 CODE GENERATION It helps to understand the pseudorange measurement if we first take a look at the actual generation of the codes. The carrier signal is multiplied by a series of either +l or-1, which are seperated by the chip length(293 m for C/A code, and 29.3 m for P code). This series of +I and-I multipliers can be interpreted as a stream of binary digits(0 and 1) How is this stream of binary digits decided? They are determined by an algorithm, known as a linear feedback register. To understand a linear feedback register, we must first introduce the XOR binary function 3.1.XOR:The“ Exclusive OR” Binary Function a binary function takes two input binary digits, and outputs one binary digit (0 or 1). More familiar binary functions might be the"AND and"OR" functions. For example, the AND function gives a value of I if the two input digits are identical, that is(,0), or(1, 1). If the input digits are different, the AND function gives a value of 0. The OR function gives a value of l if either of the two input digits equals 1, that is(0, 1),(1,0), or(1, 1) The XOR function gives a value of I if the two inputs are different, that is (1, 0)or(0, 1). If the two inputs are the same,(0,0)or(0, 1), then the value is 0 What is XOR(A, B)? Remember this: Is A different to B? If so, the answer is 1. IfA≠B, then Xor(A,B)= If A=B, then XOR(A, B)=0 The XOR function can be represented by the truth table shown in Table I Input Input Output B XOR(A, B) 0 Table 1. Truth table for the xor function 3.1.2 Linear Feedback Registers
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE 9 In this section, we go deeper into the description of the pseudorange observable, an give some details on how the codes are generated. We develop a model of the pseudorange observation, and then use this model to derive a least-squares estimator for positioning. We discuss formal errors in position, and the notion of “Dilution of Precision”, which can be used to assess the effect of satellite geometry on positioning precision. 3.1 CODE GENERATION It helps to understand the pseudorange measurement if we first take a look at the actual generation of the codes. The carrier signal is multiplied by a series of either +1 or -1, which are seperated by the chip length (293 m for C/A code, and 29.3 m for P code). This series of +1 and -1 multipliers can be interpreted as a stream of binary digits (0 and 1). How is this stream of binary digits decided? They are determined by an algorithm, known as a linear feedback register. To understand a linear feedback register, we must first introduce the XOR binary function. 3.1.1 XOR: The “Exclusive OR” Binary Function A binary function takes two input binary digits, and outputs one binary digit (0 or 1). More familiar binary functions might be the “AND” and “OR” functions. For example, the AND function gives a value of 1 if the two input digits are identical, that is (0,0), or (1,1). If the input digits are different, the AND function gives a value of 0. The OR function gives a value of 1 if either of the two input digits equals 1, that is (0,1), (1,0), or (1,1). The XOR function gives a value of 1 if the two inputs are different, that is (1,0) or (0,1). If the two inputs are the same, (0,0) or (0,1), then the value is 0. What is XOR(A,B)? Remember this: Is A different to B? If so, the answer is 1. • If A ≠ B, then XOR(A,B) = 1 • If A = B, then XOR(A,B) = 0 The XOR function can be represented by the “truth table” shown in Table 1. Input A Input B Output XOR(A,B) 0 0 0 0 1 1 1 0 1 1 1 0 Table 1. Truth table for the XOR function. 3.1.2 Linear Feedback Registers
GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Linear feedback registers are used to generate a pseudorandom number sequence. The sequence is pseudorandom, since the sequence repeats after a certain number of digits(which, as we shall see, depends on the size of the register). However, the statistical properties of the sequence are very good, in that the sequence appears to be white noise. We return to these properties later, since they are important for understanding the measurement process. For now, we look at how the register works I Cycle, N AN=XOR(AN-I) BN=AN. nitialise 1 XOR(1,1)=0 XOR(0,1)=1 4 XOR(1,1)=0 XOR(00)=0 0 XOR(0,1)=1 0 XOR(1,0)=1 8(=1) Table 2. A 3 stage linear feedback register. The output is in column C Table 2 illustrates a simple example: the "3 stage linear feedback register. The"state"of the register is defined by three binary numbers(A, B, C). The state changes after a specific time interval. To start the whole process, the intial state of a feedback register is always filled with 1; that is, for the 3 stage register, the initial state is(1, 1, 1). The digits in this state are now shifted to the right, giving(blank, 1, 1). The digit (1) that is pushed off the right side is the output from the register. The blank is replaced by taking the XOR of the other two digits (1, 1). The value, in this case, equals 0. The new state is therefore(0, 1, 1). This process is then repeated, so that the new output is(1), and the next state is(1,0, 1). The next output is (1)and the next state is(1, 1, 0). The next output is(0), and the next state is(0, 1, 1), and so In the above example, the outputs can be written(1, 1, 1, O,.). This stream of digits known as the linear feedback register sequence. This sequence will start to repeat after a while. It turns out that during a complete cycle, the feedback register will produce every possible combination of binary numbers, except for (0, 0, 0). We can therefore easily calculate the length of the sequence before it starts to repeat again. For a 3 stage register there are 8 possible combinations of binary digits. This means that the sequence will repeat after 7 cycles. The sequence length is therefore 7 bits. More generally, the sequence length L(N)=2 where n is the size of the register(number of digits in the state). For example, a 4 state linear feedback register will have a sequence length of 15 bits 3.1.3 C/A Code
10 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Linear feedback registers are used to generate a pseudorandom number sequence. The sequence is pseudorandom, since the sequence repeats after a certain number of digits (which, as we shall see, depends on the size of the register). However, the statistical properties of the sequence are very good, in that the sequence appears to be white noise. We return to these properties later, since they are important for understanding the measurement process. For now, we look at how the register works. Cycle, N AN = XOR(AN-1,C N-1) BN = AN-1 CN = BN-1 1 initialise: 1 1 1 2 XOR(1,1) = 0 1 1 3 XOR(0,1) = 1 0 1 4 XOR(1,1) = 0 1 0 5 XOR(0,0) = 0 0 1 6 XOR(0,1) = 1 0 0 7 XOR(1,0) = 1 1 0 8 (=1) XOR(1,0) = 1 1 1 (pattern repeats) Table 2. A 3 stage linear feedback register. The output is in column C. Table 2 illustrates a simple example: the “3 stage linear feedback register.” The “state” of the register is defined by three binary numbers (A, B, C). The state changes after a specific time interval. To start the whole process, the intial state of a feedback register is always filled with 1; that is, for the 3 stage register, the initial state is (1, 1, 1). The digits in this state are now shifted to the right, giving (blank, 1, 1). The digit (1) that is pushed off the right side is the output from the register. The blank is replaced by taking the XOR of the other two digits (1,1). The value, in this case, equals 0. The new state is therefore (0, 1, 1). This process is then repeated, so that the new output is (1), and the next state is (1, 0, 1). The next output is (1) and the next state is (1, 1, 0). The next output is (0), and the next state is (0, 1, 1), and so on. In the above example, the outputs can be written (1, 1, 1, 0, ....). This stream of digits is known as the “linear feedback register sequence.” This sequence will start to repeat after a while. It turns out that during a complete cycle, the feedback register will produce every possible combination of binary numbers, except for (0, 0, 0). We can therefore easily calculate the length of the sequence before it starts to repeat again. For a 3 stage register, there are 8 possible combinations of binary digits. This means that the sequence will repeat after 7 cycles. The sequence length is therefore 7 bits. More generally, the sequence length is: L(N) = 2 N −1 where N is the size of the register (number of digits in the state). For example, a 4 state linear feedback register will have a sequence length of 15 bits. 3.1.3 C/A Code
