Global Positioning System

Some Graphics Below have been borrowed from the references

PHY 312

5/6/98

The Global Positioning System (GPS), which was developed to meet military needs of the Department of Defense but now is used in every day life, is a radio based navigation system that gives three dimensional coverage of Earth 24 hours a day in any weather condition. It is the only system today able to show your exact position on the Earth anytime, anywhere and it is one of history’s most exciting and revolutionary developments. The GPS system can tell location anywhere on or above the Earth within about 300 feet. Greater accuracy could be achieved (within less then three feet) with corrections calculated by a GPS receiver at a known fixed location. The Department of Defense developed and maintains GPS. Even though the system has been completed only recently it has already proved to be a most valuable aid to U.S. military forces. Operation Desert Storm was performed using GPS. Soldiers were able to go places and maneuver in sandstorms or at night even when he troops who lived there couldn’t. Navy ships use GPS for rendezvous, minesweeping, and aircraft operation. GPS is also used for navigation in planes, ships, cars, and trucks. GPS has become important in almost all military operations and weapon systems.

GPS consists of three parts: the space segment, the control segment and user segment.

The space segment consists of satellites, 24 in all: 21 navigational SVs and 3 active spares orbit at 11,000 nautical miles above the Earth. There are six orbital planes (with 4 SVs in each), equal spaced (60 degrees apart), and inclined about 55 degrees with respect to the equatorial plane. This constellation provide the user with the ability to receive the signal form five to eight SVs nearly 100% of the time at any position on Earth. It takes12 hours each to go around the Earth once (one orbit). Clocks in satellites keep accurate time to within three nanoseconds - that’s 0.000000003 of a second. "Clock-driven transmitters send out synchronous time signals, tagged with the position and time of the transmission event, so that a receivers near the Earth can determine its position and then by decoding navigation messages from four satellites to find the transmission event coordinates, and then solving four simultaneous one-way signal propagation equations" (Ashby, Neil). Conversely, gamma-ray detectors on the satellites could determine space-time coordinates of a nuclear event by measuring signal arrival times and solving four one-way propagation delay equations. The time is very important because the receiver must determine exactly how long it takes for signal to travel from each GPS satellite. Satellites transmit the signals that can be detected by anyone with GPS receiver. Satellites paths are monitored by ground stations. The first GPS satellite was launched in 1978 and first 10 satellites were developmental satellites, called Block I. From 1989 to 1993 23 additional production satellites, called Block II, were launched. The launch of 24^{th} satellite in 1994 completed the system.

The GPS control segment consists of a system of monitor stations located around the world (Hawaii and Kwajalein in the Pacific Ocean; Diego Garcia in the India Ocean; Ascension Island in the Atlantic Ocean; and Colorado Springs, Colorado) a master ground station at Falcon Air Force Base in Colorado Springs, Colorado; and four large ground antenna stations that broadcast signals to the satellites.

Monitor stations measure signal from SVs which are incorporated into orbital model of each satellites. The models compute orbital data and clock correction for each satellite. Master Control station uploads orbital and clock data to SVs which send subsets of the data to GPS receivers over radio signals.

GPS user segment consists of receivers which could be hand carried or installed on aircraft, ships, tanks, submarines, cars, and trucks. Receivers detect, decode, and process GPS satellite signals. The signal is converted into position, velocity, and time estimates. In total there are five pieces of data that GPS receiver can take measurements on. Receiver records positions in Latitude and Longitude which can be translated in various Datums and coordinate systems for mapping.

There are two GPS Positioning Services specified in the Federal Radionavigation Plan:

- Precise Positioning Service (PPS)

Authorized users with specially equipped receivers use PPS. Accuracy:

22 meter horizontal accuracy

- meter vertical accuracy

100 nanosecond time accuracy

- Standard Positioning Service (SPS)

Most receivers are capable of receiving and using SPS signal. Accuracy:

100 meter Horizontal accuracy

156 meter Vertical accuracy

340 nanoseconds time accuracy

The principle behind GPS is the measurement of distance or "range" between receiver and the satellites. The satellites also tell us exactly where they are in their orbits above the Earth. The problem is that by measuring time, Doppler shifts, gravitational frequency shifts and propagation delays has to be taken into account so the user could determine position accurately. An error of a billionth of a second in time corresponds to an error in location of 1 foot.

GPS is one of the first operating systems, excluding high-energy accelerators, that has important effects from relativity. When the first satellite was launched in 1997 there was still a lot of controversy about the Einstein’s theory of Special and General Relativity. First satellite contained the first Cesium clock to be placed in the orbit and there was a lot of people who doubted that relativity effects are real. A frequency synthesizer was build into satellite clock system so that after launch, if it proved to be that clock would run at the rate predicted by the General Relativity, it could be turned on to bring clocks in the coordinate rate necessary for operation. The clock on board was operated for about 20 days without turning on the synthesizer and the frequency measure during that interval was 442.5 parts in 10^{12} faster then the clocks on the ground. If no correction would of took place this would resulted in error of about 38,000 nanoseconds per day. The frequency predicted by General Relativity was only within 3.97 parts of 10^{12} which was within accuracy capabilities of orbiting clock. It is not easy right now to perform relativity tests using GPS because satellite clocks are actively corrected to within 1 microsecond of Universal Coordinated Time.

There are several reasons that relativity is very important in GPS: GPS satellites have a large velocity, there is large gravitational potential differences between that of the satellites and that of the users, and there is significant Earth rotation effects. These effects themselves might not be that important but because GPS satellites are equipped with atomic clocks relativistic effect should be taken into account.

"There are three primary consequences of relativity effects:

- There is a fixed frequency offset in the satellite’s clock rate when observed from Earth. Most of the effect is purposely removed by slightly offsetting the satellite clocks in frequency prior to launch, the so-called "factory offset" of the clock.
- The slight eccentricity of each satellite orbit causes an additional periodic clock error effect that varies with the satellite’s position in its orbit plane.
- There is also effect (Sagnac delay) caused by the Earth’s rotation during the time of transit of the satellite signal from satellite to the ground" (Parkinson, Bradford. pg. 623-634).

Moving users on the Earth surface or near it or fixed users at some altitude about the Earth surface have to make additional corrections caused the their velocity and the height above the ground.

The net effect of relativity for a zero eccentricity GPS satellite is a combination of effects caused by satellites velocity (Special Relativity effect) and Earth gravitational field (General Relativity effect). This produces small fixed frequency offset in addition to classical Doppler shift.

One of the effects of Einstein’s Special Relativity theory is time dilation. Clocks moving with high velocity run slower then clocks with the smaller relative velocity. Therefore, clocks in the GPS satellites will run slower compare to the clocks on the Earth because GPS satellites have a large velocity. In more details the rates that clocks tick compare to the static clock are given by the formulas:

Te^{2} = (1 - Ve^{2}/c^{2})Ts^{2}; and Tg^{2} = (1 - Vg^{2}/c^{2})Ts^{2}, where Te is proper time of the clock on the surface of the Earth, Ve is the clock’s linear velocity due to the rotation of Earth, Ts is proper time of static clock, Tg is proper time of the clock in the GPS satellite, Vg is linear velocity of the GPS satellite in the orbit. Therefor the ratio at which clock will tick could be derived from the formulas above: Te^{2}/Tg^{2} = (1 - Ve^{2}/c^{2})/ (1 - Vg^{2}/c^{2}). This effect has a fixed value. Clocks in the GPS satellites run slow by 6 millionth of a second per day.

The effect of General Relativity is that clocks at the higher altitude above the Earth run faster then the clocks on the surface of the Earth. From the Schwarzschild metric we could calculate the rates at which clocks of particular interest tick compare to the clocks at infinite distance from source of gravitational field. Lets compare the rates of two static clocks: one on the surface of the Earth and other in the GPS satellite. For a static clock:

dT^{2} = -dS^{2} = (1 - R/r)dTi^{2}, where T is proper time of the clock we interested in, Ti is proper time of the clock infinite distance away, and R = 2MG/c^{2}, where M is mass of the Earth, G is Newtonian gravitation constant and c is speed of light. Therefore the ratio of rates at which clock on the surface of Earth and in GPS satellite will tick is given by formula:

dTe^{2}/dTg^{2} = (1 - R/re)/(1 - R/rg), where re is radius of Earth and rg is distance from the center of the Earth to GPS satellite. This effect also has a fixed value. The clocks in the GPS satellites run fast by 45 millionth of second per day. Therefore combining Special and General Relativity effects we conclude that clock in GPS satellites run fast by 39 millionth of a second per day which if not taken into account could produce position error of 12 kilometers.

GPS satellites do not follow strict circular orbit around the Earth. There exist slight orbit eccentricity and therefore it causes periodic clock error effect that varies with satellites position in the orbit. Consider two positions of the satellite in the orbit (Position1 and Position2 in the diagram). As we could see, distance d1 from the Earth to the satellite in Position1 is not equal to distance d2 from the Earth to the satellite in Position2. Also there is differences in velocities at which satellite travels in those two positions. Velocity of satellite is Position2 will be bigger then velocity of satellite in Position1. Therefore there is going to be different Special and General relativity effect when satellite orbits around the Earth. GPS orbit around the Earth is very close to the circle, so periodic errors are very small but have a fixed values and could be included in the system.

Simultaneity is very important concept in GPS. For users to determine position and time clock in the GPS satellites have to be synchronized. If we use inertial frame of reference, clocks could be synchronized using Einstein’s synchronization procedures (if light rays emitted from two different points arrive at the midpoint of those points at the same time, then the events or transmitting the light ray from two original points occurred simultaneous). However users of GPS are moving (Earth is rotating) and noninertial effect take place. Consider situation illustrated in diagram above. Clocks A, B, C, and D are rotating around some fixed clock E. Velocities of the clocks are given by Va, Vb, Vc, Vd. Furthermore all velocities are equal and there exist centripetal (??) acceleration which causes clocks to move around clock E and therefore direction of velocities are changing with respect to coordinate axis x (represented in the diagram). Sagnac effect plays important role in the system. It would be desirable to synchronize clocks in the rotating frame fixed to the Earth because most of GPS users are at the rest or nearly so on the Earth’s surface. Because Earth rotates, Sagnac effect is large enough in the GPS and the clocks can’t be synchronized in the rotating frame and there is necessity for different approach to synchronize the clocks. Lets try to draw space-time diagram in the reference frame of clock E from diagram above. Lets choose a particular moment in time when velocity of clock A is in x direction, velocity of clock C is in direction opposite to x direction and velocities of clocks B and D in x direction is equal to zero. This situation is illustrated in the previous diagram. So space-time diagram looks like this:

Notice that velocities of clocks D and B are zero so their world lines are parallel to the world line of cock E and lines of simultaneity are the same as for clock E. What does it meat for clocks A, B, C, and D to be synchronized? At any point in time all of these clock should show the same time coordinate. Say we use reference frame of clock A. Lets look at the diagram. Consider clocks A some line of simultaneity (ta = 0 in our diagram). Points J and L represent crossings with clock B and D world lines. If clocks are synchronized, clock B and D at the points J and L should read the same time as clock A (ta = 0). We could clearly see from the picture that when clock A reads time ta = 0, clock B reads time te>0 and clock D reads time te<0. Same analysis apply in the reference frame of the clock C (points M and K on world lines of clocks D and B). Therefore we can’t synchronize clock using any rotating frame of reference. In the GPS synchronization is performed in the Earth-Centered Inertial frame using constancy of speed of light.

In addition to relativity effects explained above there exist effects caused by the user velocity and height of the user above geoid. Some of these effect might cancel or partially cancel in estimation of the position. However these effects might be significant if the user is another satellite in orbit.

There also exist secondary effects that are smaller then the accuracy level required by the user. Tidal potential effect on the clocks is result of the rotating Earth revolving around the Sun and therefore exposed to Sun’s gravitational field. However, both the GPS satellites and the user are in orbit around the Sun at almost the same position so a lot of this effect cancels.

The Earth’s gravity potential is not spherical because of the ellipsoidal shape of the Earth, which cause effect which is not modeled into the system (quadrupole field effect). However this effect is also very small. If higher accuracy is desirable for GPS system then this effect should be taken into account. If this effect is not taken into account gravitation potential of the Earth (F ) is given by the formula:

F
= -G*Me/r, where G is Newtonian gravitational constant, Me is the mass of the Earth, and r is distance from the observation point to the center of the Earth. This formula is consistent with spherical shape of the Earth. If we want to account for ellipsoidal shape of the Earth we extend our formula to account the factor 1/r^{2}. Therefore,

F
= (GMe/r) * [ 1 - J2(Re/r)^{2} P2(cos Q
)], where J2 is Earth quadrupole moment coefficient, Re is the Earth’s equatorial radius and P2(x) = (3x^{2} - 1)/2 is Legendre polynomial of second degree. This extension is sufficient for desirable accuracy of GPS but if military applications will require higher and higher accuracy from GPS system Gravitation Potential might need to be extended to account for Earth’s not perfect ellipsoidal shape (this means that reciprocals of the higher powers of r will have to be included in the formula).

Another effect of the clocks in the GPS satellites is so called Shapiro delay. The cause of this delay is that light’s velocity changes when it is exposed to Earth gravitational field. This delay was measured experimentally and was proved to be negligible.

Another effect on clocks is cause by Earth’s mass rotating on its axis (Lense-Thirring effect - frame dragging). This effect slightly modifies solution to Einstein’s equations and generates slightly different metric. However these effects are negligible for the purposes of GPS.

Relativistic effects in GPS system is very important. If higher accuracy will be desirable secondary effects of relativity will have to be implemented in the improved systems together with primary effects. Currently uncertainty of position using Precise Positioning Code is now around 2.4 meters and a lot of people are interested in reducing the error to the millimeter level. Even though the system is not at that high accuracy, it still provides a lot of examples for the applications in relativity.

"New and surprising applications of positioning determination and time transfer based on GPS are continually being invented. Civilain applications include for example, tracking elephants in Africa, studies of crustal plate movements, surveying, mapping, exploration, salvage in the open ocean, vehicle fleet tracking, search and rescue, power line fault location, and synchronization of telecommunications nodes" (Ashby, Neil).

Bibliography

Ashby, Neil. "General Relativity in the Global Positioning System".

http://vishnu.nirvana.phys.psu.edu/mog/mog9/node9.html

Dana, Peter. "Global Positioning System Overview".

http://www.utexas.edu/depts/grg/gcraft/notes/gps/gps.html

Ashby, Niel in "Global Positioning System: Theory and

Applications" by Parkinson, Bradford and Spilker, James.