by Robert A. Nelson

On a recent trip to visit the Jet Propulsion Laboratory, I flew from Washington, DC, to Los Angeles on a new Boeing 747-400 airplane. The geographical position of the plane and its relation to nearby cities was displayed throughout the flight on a video screen in the passenger cabin. When I arrived in Los Angeles, I rented a car that was equipped with a navigator. The navigator guided me to my hotel in Pasadena, displaying my position on a map and verbally giving me directions with messages like "freeway exit ahead on the right followed by a left turn." When I reached the hotel, it announced that I had arrived at my destination. Later, when I was to join a colleague for dinner, I found the restaurant listed in a menu and the navigator took me there.

This remarkable navigation capability is made possible by the Global Positioning System (GPS). It was originally designed jointly by the U.S. Navy and the U.S. Air Force to permit the determination of position and time for military troops and guided missiles. However, GPS has also become the basis for position and time measurement by scientific laboratories and a wide spectrum of applications in a multibillion dollar commercial industry. Roughly one million receivers are manufactured each year and the total GPS market is expected to approach $10 billion by the end of next year. The story of GPS and its principles of measurement are the subjects of this article.

Early Methods Of Navigation

The shape and size of the earth has been known from the time of antiquity. The fact that the earth is a sphere was well known to educated people as long ago as the fourth century, B.C. In his book, On the Heavens, Aristotle gave two scientifically correct arguments. First, the shadow of the earth projected on the moon during a lunar eclipse appears to be curved. Second, the elevations of stars change as one travels north or south, while certain stars visible in Egypt cannot be seen at all from Greece.

The actual radius of the earth was determined within 1 percent by Eratosthenes in about 230 B.C. He knew that the sun was directly overhead at noon on the summer solstice in Syene (Aswan, Egypt), since on that day it illuminated the water of a deep well. At the same time, he measured the length of the shadow cast by a column on the grounds of the library at Alexandria, which was nearly due north. The distance between Alexandria and Syene had been well established by professional runners and camel caravans. Thus Eratosthenes was able to compute the earth’s radius from the difference in latitude that he inferred from his measurement. In terms of modern units of length, he arrived at the figure of about 6,400 km. By comparison, the actual mean radius is 6,371 km (the earth is not precisely spherical, as the polar radius is 21 km less than the equatorial radius of 6,378 km).

The ability to determine one’s position on the earth was the next major problem to be addressed. In the second century, A.D., the Greek astronomer Claudius Ptolemy prepared a geographical atlas, in which he estimated the latitude and longitude of principal cities of the Mediterranean world. Ptolemy is most famous, however, for his geocentric theory of planetary motion, which was the basis for astronomical catalogs until Nicholas Copernicus published his heliocentric theory in 1543.

Historically, methods of navigation over the earth’s surface have involved the angular measurement of star positions to determine latitude. The latitude of one’s position is equal to the elevation of the pole star. The position of the pole star on the celestial sphere is only temporary, however, due to precession of the earth’s axis of rotation through a circle of radius 23.5 degrees over a period of 26,000 years. At the time of Julius Caesar, there was no star sufficiently close to the north celestial pole to be called a pole star. In 13,000 years, the star Vega will be near the pole. It is perhaps not a coincidence that mariners did not venture far from visible land until the era of Christopher Columbus, when true north could be determined using the star we now call Polaris. Even then, the star’s diurnal rotation caused an apparent variation of the compass needle. Polaris in 1492 described a radius of about 3.5 degrees about the celestial pole, compared to 1 degrees today. At sea, however, Columbus and his contemporaries depended primarily on the mariner’s compass and dead reckoning.

The determination of longitude was much more difficult. Longitude is obtained astronomically from the difference between the observed time of a celestial event, such as an eclipse, and the corresponding time tabulated for a reference location. For each hour of difference in time, the difference in longitude is 15 degrees.

Columbus himself attempted to estimate his longitude on his fourth voyage to the New World by observing the time of a lunar eclipse as seen from the harbor of Santa Gloria in Jamaica on February 29, 1504. In his distinguished biography, Admiral of the Ocean Sea, Samuel Eliot Morrison states that Columbus measured the duration of the eclipse with an hourglass and determined his position as seven hours and 15 minutes west of Cadiz, Spain, according to the predicted eclipse time in an almanac he carried aboard his ship. Over the preceding year, while his ship was marooned in the harbor, Columbus had determined the latitude of Santa Gloria by numerous observations of the pole star. He made out his latitude to be 18 degrees, which was in error by less than half a degree and was one of the best recorded observations of latitude in the early 16th century, but his estimated longitude was off by some 38 degrees.

Columbus also made legendary use of this eclipse by threatening the natives with the disfavor of God, as indicated by a portent from Heaven, if they did not bring desperately needed provisions to his men. When the eclipse arrived as predicted, the natives pleaded for the admiral’s intervention, promising to furnish all the food that was needed.

New knowledge of the universe was revealed by Galileo Galilei in his book, The Starry Messenger. This book, published in Venice in 1610, reported the telescopic discoveries of hundreds of new stars, the craters on the moon, the phases of Venus, the rings of Saturn, sunspots and the four inner satellites of Jupiter. Galileo suggested using the eclipses of Jupiter’s satellites as a celestial clock for the practical determination of longitude, but the calculation of an accurate ephemeris and the difficulty of observing the satellites from the deck of a rolling ship prevented use of this method at sea. Nevertheless, James Bradley, the third Astronomer Royal of England, successfully applied the technique in 1726 to determine the longitudes of Lisbon and New York with considerable accuracy.

Inability to measure longitude at sea had the potential of catastrophic consequences for sailing vessels exploring the New World, carrying cargo and conquering new territories. Shipwrecks were common. On October 22, 1707, a fleet of 21 ships under the command of Admiral Sir Clowdisley Shovell was returning to England after an unsuccessful military attack on Toulon in the Mediterranean. As the fleet approached the English Channel in dense fog, the flagship and three others foundered on the coastal rocks and nearly 2,000 men perished.

Stunned by this unprecedented loss, the British government in 1714 offered a prize of Pounds 20,000 for a method to determine longitude at sea within a half a degree. The scientific establishment believed that the solution would be obtained from observations of the moon. The German cartographer Tobias Mayer, aided by new mathematical methods developed by Leonard Euler, offered improved tables of the moon in 1757. The recorded position of the moon at a given time as seen from a reference meridian could be compared with its position at the local time to determine the angular position west or east.

Just as the astronomical method appeared to achieve realization, the British craftsman John Harrison provided a different solution through his invention of the marine chronometer. The story of Harrison’s clock has been recounted in Dava Sobel’s popular book, Longitude.

Both methods were tested by sea trials. The lunar tables permitted the determination of longitude within four minutes of arc, but with Harrison’s chronometer, the precision was only one minute of arc. Ultimately, portions of the prize money were awarded to Mayer’s widow, Euler and Harrison.

In the 20th century, with the development of radio transmitters, another class of navigation aids was created using terrestrial radio beacons, including Loran and Omega. Finally, the technology of artificial satellites made possible navigation and position determination using line of sight signals involving the measurement of Doppler shift or phase difference.

Transit

Transit, the Navy Navigation Satellite System, was conceived in the late 1950s and deployed in the mid-1960s. It was finally retired in 1996 after nearly 33 years of service. The Transit system was developed because of the need to provide accurate navigation data for Polaris missile submarines. As related in an historical perspective by Bradford Parkinson, et al. in the journal, Navigation (Spring 1995), the concept was suggested by the predictable but dramatic Doppler frequency shifts from the first Sputnik satellite, launched by the Soviet Union in October 1957. The Doppler-shifted signals enabled a determination of the orbit using data recorded at one site during a single pass of the satellite. Conversely, if a satellite’s orbit were already known, a radio receiver’s position could be determined from the same Doppler measurements.

The Transit system was composed of six satellites in nearly circular, polar orbits at an altitude of 1,075 km. The period of revolution was 107 minutes. The system employed essentially the same Doppler data used to track the Sputnik satellite. However, the orbits of the Transit satellites were precisely determined by tracking them at widely spaced fixed sites. Under favorable conditions, the rms accuracy was 35 to 100 meters. The main problem with Transit was the large gaps in coverage. Users had to interpolate their positions between passes.

Global Positioning System

The success of Transit stimulated both the U.S. Navy and the U.S. Air Force to investigate more advanced versions of a space-based navigation system with enhanced capabilities. Recognizing the need for a combined effort, the Deputy Secretary of Defense established a Joint Program Office in 1973. The Navstar Global Positioning System (GPS) was thus created.

In contrast to Transit, GPS provides continuous coverage. Also, rather than Doppler shift, satellite range is determined from phase difference.

There are two types of observables. One is pseudorange, which is the offset between a pseudorandom noise (PRN) coded signal from the satellite and a replica code generated in the user’s receiver, multiplied by the speed of light. The other is accumulated delta range (ADR), which is a measure of carrier phase.

The determination of position may be described as the process of triangulation using the measured range between the user and four or more satellites. The ranges are inferred from the time of propagation of the satellite signals. Four satellites are required to determine the three coordinates of position and time. The time is involved in the correction to the receiver clock and is ultimately eliminated from the measurement of position.

High precision is made possible through the use of atomic clocks carried onboard the satellites. Each satellite has two cesium clocks and two rubidium clocks, which maintain time with a precision of a few parts in 1013 or 1014 over a few hours, or better than 10 nanoseconds. In terms of the distance traversed by an electromagnetic signal at the speed of light, each nanosecond corresponds to about 30 centimeters. Thus the precision of GPS clocks permits a real-time measurement of distance to within a few meters. With post-processed carrier phase measurements, a precision of a few centimeters can be achieved.

The design of the GPS constellation had the fundamental requirement that at least four satellites must be visible at all times from any point on earth. The tradeoffs included visibility, the need to pass over the ground control stations in the United States, cost and sparing efficiency.

The orbital configuration approved in 1973 was a total of 24 satellites, consisting of eight satellites, plus one spare in each of three equally spaced orbital planes. The orbital radius was 26,562 km, corresponding to a period of revolution of 12 sidereal hours, with repeating ground traces. Each satellite arrived over a given point four minutes earlier each day. A common orbital inclination of 63 degrees was selected to maximize the in-orbit payload mass with launches from the Western Test Range. This configuration ensured between six and 11 satellites in view at any time.

As envisioned 10 years later, the inclination was reduced to 55 degrees, and the number of planes was increased to six. The constellation would consist of 18 primary satellites, which represents the absolute minimum number of satellites required to provide continuous global coverage with at least four satellites in view at any point on the earth. In addition, there would be three in-orbit spares.

The operational system, as presently deployed, consists of 21 primary satellites and three in-orbit spares, comprising four satellites in each of six orbital planes. Each orbital plane is inclined at 55 degrees. This constellation improves on the "18 plus 3" satellite constellation by more fully integrating the three active spares.

Space Segment

There have been several generations of GPS satellites. The Block 1 satellites, built by Rockwell International, were launched between 1978 and 1985. They consisted of 11 prototype satellites, including one launch failure, which validated the system concept. The 10 successful satellites had an average lifetime of 8.76 years.

The Block 2 and Block 2A satellites were also built by Rockwell International. Block 2 consists of nine satellites launched between 1989 and 1990. Block 2A, deployed between 1990 and 1997, consists of 19 satellites with several navigation enhancements. In April 1995, GPS was declared fully operational with a constellation of 24 operational spacecraft and a completed ground segment. The 28 Block 2/2A satellites have exceeded their specified mission duration of six years and are expected to have an average lifetime of more than 10 years.

Block 2R comprises 20 replacement satellites that incorporate autononomous navigation based on crosslink ranging. The satellites are being manufactured by Lockheed Martin. The first launch in 1997 resulted in a launch failure. Currently there is one 2R satellite in orbit, also launched in 1997. The next 2R satellite launch was scheduled for October 5, as we went to press. One to four more launches are anticipated over the next year.

The fourth generation of satellites is the Block 2 follow-on (Block 2F). This program includes the procurement of 33 satellites and the operation and support of a new GPS operational control segment. The Block 2F program was awarded to Rockwell (now a part of Boeing). Further details may be found in a special issue of the Proceedings of the IEEE for January 1999.

Control Segment

The master control station (MCS) for GPS is located at Schriever Air Force Base in Colorado Springs, CO. The MCS maintains the satellite constellation and performs the stationkeeping and attitude control maneuvers. It also determines the orbit and clock parameters with a Kalman filter using measurements from five monitor stations distributed around the world. The orbit error is about 1.5 meters.

GPS orbits are derived independently by various scientific organizations using carrier phase and post-processing. The state of the art is exemplified by the work of the International GPS Service (IGS), which produces orbits with an accuracy of approximately 3 centimeters within two weeks.

The system time reference is managed by the U.S. Naval Observatory in Washington, DC. GPS time is measured from Saturday/Sunday midnight at the beginning of the week. The GPS time scale is a composite "paper clock" that is synchronized to keep step with Coordinated Universal Time (UTC) and International Atomic Time (TAI). However, UTC differs from TAI by an integral number of leap seconds to maintain correspondence with the rotation of the earth, whereas GPS time does not include leap seconds. The origin of GPS time is midnight on January 5/6, 1980 (UTC). At present, TAI is ahead of UTC by 32 seconds, TAI is ahead of GPS by 19 seconds, and GPS is ahead of UTC by 13 seconds. Only 1,024 weeks were allotted from the origin before the system time is reset to zero because 10 bits are allocated for the calendar function (1,024 is the 10th power of 2). Thus, the first GPS rollover occurred at midnight on August 21, 1999. The next GPS rollover will take place May 25, 2019.

Signal Structure

The satellite position at any time is computed in the user’s receiver from the navigation message that is contained in a 50 bps data stream. The orbit is represented for each one hour period by a set of 15 Keplerian orbital elements, with harmonic coefficients arising from perturbations, and is updated every four hours.

This data stream is modulated by each of two code division multiple access, or spread spectrum, pseudorandom noise (PRN) codes: the coarse/acquisition C/A code (sometimes called the clear/access code) and the precision P code. The P code can be encrypted to produce a secure signal called the Y code. This feature is known as the Anti-Spoof (AS) mode, which is intended to defeat deception jamming by adversaries. The C/A code is used for satellite acquisition and for position determination by civil receivers. The P(Y) code is used by military and other authorized receivers.

The C/A is a Gold code of register size 10, which has a sequence length of 1,023 chips and a chipping rate of 1.023 MHz and thus repeats itself every one millisecond. (The term "chip" is used instead of "bit" to indicate that the PRN code contains no information.) The P code is a long code of length 2.3547 x 1014 chips, with a chipping rate of 10 times the C/A code, or 10.23 MHz. At this rate, the P code has a period of 38.058 weeks, but it is truncated on a weekly basis so 38 segments are available for the constellation. Each satellite uses a different member of the C/A Gold code family and a different one-week segment of the P code sequence.

The GPS satellites transmit signals at two carrier frequencies: the L1 component with a center frequency of 1575.42 MHz and the L2 component with a center frequency of 1227.60 MHz. These frequencies are derived from the master clock frequency of 10.23 MHz, with L1 = 154 x 10.23 MHz and L2 = 120 x 10.23 MHz. The L1 frequency transmits both the P code and the C/A code, while the L2 frequency transmits only the P code. The second P code frequency permits a dual-frequency measurement of the ionospheric group delay. The P-code receiver has a two-sigma rms horizontal position error of about 5 meters.

The single frequency C/A code user must model the ionospheric delay with less accuracy. In addition, the C/A code is intentionally degraded by a technique called Selective Availability (SA), which introduces errors of 50 to 100 meters by dithering the satellite clock data. Through differential GPS measurements, however, position accuracy can be improved by reducing SA and environmental errors.

The transmitted signal from a GPS satellite has right-hand circular polarization. According to the GPS Interface Control Document, the specified minimum signal strength at an elevation angle of 5 degrees into a linearly polarized receiver antenna with a gain of 3 dB (approximately equivalent to a circularly polarized antenna with a gain of 0 dB) is -160 dBW for the L1 C/A code; -163 dBW for the L1 P code; and -166 dBW for the L2 P code. The L2 signal is transmitted at a lower power level since it is used primarily for the ionospheric delay correction.

Pseudorange

The fundamental measurement in the Global Positioning System is pseudorange. The user equipment receives the PRN code from a satellite and, having identified the satellite, generates a replica code. The phase by which the replica code must be shifted in the receiver to maintain maximum correlation with the satellite code, multiplied by the speed of light, is approximately equal to the satellite range. It is called the pseudorange because the measurement must be corrected by a variety of factors to obtain the true range.

The corrections that must be applied include signal propagation delays caused by the ionosphere and the troposphere, the space vehicle clock error, and the user’s receiver clock error. The ionosphere correction is obtained either by measurement of dispersion using the two frequencies L1 and L2 or by calculation from a mathematical model, but the tropospheric delay must be calculated since the troposphere is nondispersive. The true geometric distance to each satellite is obtained by applying these corrections to the measured pseudorange.

Other error sources and modeling errors continue to be investigated. For example, a recent modification of the Kalman filter has led to improved performance. Studies have also shown that solar radiation pressure models may need revision and there is some new evidence that the earth’s magnetic field may contribute to a small orbit period variation in the satellite clock frequencies.

Carrier Phase

Carrier phase is used to perform measurements with a precision that greatly exceeds those based on pseudorange. However, a carrier phase measurement must resolve an integral cycle ambiguity, whereas the pseudorange is unambiguous.

The wavelength of the L1 carrier is about 19 centimeters. Thus with a cycle resolution of 1 percent, a differential measurement at the level of a few millimeters is theoretically possible. This technique has important applications to geodesy and analogous scientific programs.

Relativity

The precision of GPS measurements is so great that it requires the application of Albert Einstein’s special and general theories of relativity for the reduction of its measurements. Professor Carroll Alley of the University of Maryland once articulated the significance of this fact at a scientific conference devoted to time measurement in 1979. He said, "I think it is appropriate…to realize that the first practical application of Einstein’s ideas in actual engineering situations are with us in the fact that clocks are now so stable that one must take these small effects into account in a variety of systems that are now undergoing development or are actually in use in comparing time worldwide. It is no longer a matter of scientific interest and scientific application, but it has moved into the realm of engineering necessity."

According to relativity theory, a moving clock appears to run slow with respect to a similar clock that is at rest. This effect is called "time dilation." In addition, a clock in a weaker gravitational potential appears to run fast in comparison to one that is in a stronger gravitational potential. This gravitational effect is known in general as the "red shift" (only in this case it is actually a "blue shift").

GPS satellites revolve around the earth with a velocity of 3.874 km/s at an altitude of 20,184 km. Thus on account of its velocity, a satellite clock appears to run slow by 7 microseconds per day when compared to a clock on the earth’s surface. But because of the difference in gravitational potential, the satellite clock appears to run fast by 45 microseconds per day. The net effect is that the clock appears to run fast by 38 microseconds per day. This is an enormous rate difference for an atomic clock with a precision of a few nanoseconds. To compensate for this large secular rate, the clocks are given a rate offset prior to satellite launch of -4.465 parts in 1010 from their nominal frequency of 10.23 MHz so that on average they appear to run at the same rate as a clock on the ground. The actual frequency of the satellite clocks before launch is thus 10.22999999543 MHz.

Although the GPS satellite orbits are nominally circular, there is always some residual eccentricity. The eccentricity causes the orbit to be slightly elliptical, and the velocity and altitude vary over one revolution. Thus, although the principal velocity and gravitational effects have been compensated by a rate offset, there remains a slight residual variation that is proportional to the eccentricity. For example, with an orbital eccentricity of 0.02, there is a relativistic sinusoidal variation in the apparent clock time having an amplitude of 46 nanoseconds. This correction must be calculated and taken into account in the GPS receiver.

The displacement of a receiver on the surface of the earth due to the earth’s rotation in inertial space during the time of flight of the signal must also be taken into account. This is a third relativistic effect that is due to the universality of the speed of light. The maximum correction occurs when the receiver is on the equator and the satellite is on the horizon. The time of flight of a GPS signal from the satellite to a receiver on the earth is then 86 milliseconds and the correction to the range measurement resulting from the receiver displacement is 133 nanoseconds. An analogous correction must be applied by a receiver on a moving platform, such as an aircraft or another satellite. This effect, as interpreted by an observer in the rotating frame of reference of the earth, is called the Sagnac effect. It is also the basis for a laser ring gyro in an inertial navigation system.

GPS Modernization

In 1996, a Presidential Decision Directive stated the president would review the issue of Selective Availability in 2000 with the objective of discontinuing SA no later than 2006. In addition, both the L1 and L2 GPS signals would be made available to civil users and a new civil 10.23 MHz signal would be authorized. To satisfy the needs of aviation, the third civil frequency, known as L5, would be centered at 1176.45 MHz, in the Aeronautical Radio Navigation Services (ARNS) band, subject to approval at the World Radio Conference in 2000. According to Keith McDonald in an article on GPS modernization published in the September 1999 GPS World, with SA removed the civil GPS accuracy would be improved to about 10 to 30 meters. With the addition of a second frequency for ionospheric group delay corrections, the civil accuracy would become about 5 to 10 meters. A third frequency would permit the creation of two beat frequencies that would yield 1-meter accuracy in real time.

A variety of other enhancements are under consideration, including increased power, the addition of a new military code at the L1 and L2 frequencies, additional ground stations, more frequent uploads, and an increase in the number of satellites. These policy initiatives are driven by the dual needs of maintaining national security while supporting the growing dependence on GPS by commercial industry. When these upgrades would begin to be implemented in the Block 2R and 2F satellites depends on GPS funding.

Besides providing position, GPS is a reference for time with an accuracy of 10 nanoseconds or better. Its broadcast time signals are used for national defense, commercial and scientific purposes. The precision and universal availability of GPS time has produced a paradigm shift in time measurement and dissemination, with GPS evolving from a secondary source to a fundamental reference in itself.

The international community wants assurance that it can rely on the availability of GPS and continued U.S. support for the system. The Russian Global Navigation Satellite System (Glonass) has been an alternative, but economic conditions in Russia have threatened its continued viability. Consequently, the European Union is considering the creation of a navigation system of its own, called Galileo, to avoid relying on the U.S. GPS and Russian Glonass programs.

The Global Positioning System is a vital national resource. Over the past 30 years it has made the transition from concept to reality, representing today an operational system on which the entire world has become dependent. Both technical improvements and an enlightened national policy will be necessary to ensure its continued growth into the 21st century.

Dr. Robert A. Nelson, P.E. is president of Satellite Engineering Research Corporation, a satellite engineering consulting firm in Bethesda, MD. He is Via Satellite’s Technical Editor.

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