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Global Positioning System

The Global Positioning System (GPS) is the only fully functional Global Navigation Satellite System (GNSS) in the world. The GPS uses a constellation of between 24 and 32 Medium Earth Orbit satellites that transmit precise microwave signals, that enable GPS receivers to determine their current location, the time, and their velocity (including direction). GPS was developed by the United States Department of Defense. Its official name is NAVSTAR-GPS. Although NAVSTAR-GPS is not an acronym, a few backronyms have been created for it. The GPS satellite constellation is managed by the United States Air Force 50th Space Wing.

Similar satellite navigation systems include the Russian GLONASS (incomplete as of 2008), the upcoming European Galileo positioning system, the proposed COMPASS navigation system of China, and IRNSS of India.

After Korean Air Lines Flight 007 was shot down in 1983 after straying into the USSR's prohibited airspace, President Ronald Reagan issued a directive making GPS freely available for civilian use as a common good. Since then, GPS has become a widely used aid to navigation worldwide, and a useful tool for map-making, land surveying, commerce, scientific uses, and hobbies such as geocaching. Also, the precise time reference is used in many applications including scientific study of earthquakes, and synchronization of telecommunications networks.

The first satellite navigation system, Transit, used by the United States Navy, was first successfully tested in 1960. Using a constellation of five satellites, it could provide a navigational fix approximately once per hour. In 1967, the U.S. Navy developed the Timation satellite which proved the ability to place accurate clocks in space, a technology the GPS relies upon. In the 1970s, the ground-based Omega Navigation System, based on signal phase comparison, became the first worldwide radio navigation system. The design of GPS is based partly on similar ground-based radio navigation systems, such as LORAN and the Decca Navigator developed in the early 1940s, and used during World War II. Additional inspiration for the GPS came when the Soviet Union launched the first Sputnik in 1957. A team of U.S. scientists led by Dr. Richard B. Kershner were monitoring Sputnik's radio transmissions. They discovered that, because of the Doppler effect, the frequency of the signal being transmitted by Sputnik was higher as the satellite approached, and lower as it continued away from them. They realized that since they knew their exact location on the globe, they could pinpoint where the satellite was along its orbit by measuring the Doppler distortion.

Basic concept of GPS operation

A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS satellites high above the Earth. Each satellite continually transmits messages containing the time the message was sent, a precise orbit for the satellite sending the message (the ephemeris), and the general system health and rough orbits of all GPS satellites (the almanac). These signals travel at the speed of light (which varies between vacuum and the atmosphere). The receiver uses the arrival time of each message to measure the distance to each satellite, from which it determines the position of the receiver (conceptually the intersection of spheres - see trilateration). The resulting coordinates are converted to more user-friendly forms such as latitude and longitude, or location on a map, then displayed to the user.

It might seem that three satellites would be enough to solve for a position, since space has three dimensions. However, a three satellite solution requires the time be known to a nanosecond or so, far better than any non-laboratory clock can provide. Using four or more satellites allows the receiver to solve for time as well as geographical position, eliminating the need for a very accurate clock. In other words, the receiver uses four measurements to solve for four variables: x, y, z, and t. While most GPS applications use the computed location and not the (very accurate) computed time, the time is used in some GPS applications such as time transfer and traffic signal timing.

Although four satellites are required for normal operation, fewer may be needed in some special cases. If one variable is already known (for example, a ship or plane may already know its altitude), a receiver can determine its position using only three satellites. Some GPS receivers may use additional clues or assumptions (such as re-using the last known altitude, dead reckoning, inertial navigation, or including information from the vehicle computer) to give degraded answers when fewer than four satellites are visible.

Position calculation introduction

To provide an introductory description of how a GPS receiver works, errors will be ignored in this section. Using messages received from a minimum of four visible satellites, a GPS receiver is able to determine the satellite positions and time sent. The x, y, and z components of position and the time sent are designated as left [x_i, y_i, z_i, t_iright ] where the subscript i denotes the satellite number and has the value 1, 2, 3, or 4. Knowing the indicated time the message was received tr_i, the GPS receiver can compute the indicated transit time, left (tr_i-t_iright ) . of the message. Assuming the message traveled at the speed of light, c, the distance travelled, p_i can be computed as left (tr_i-t_iright )c . Knowing the distance from GPS receiver to a satellite and the position of a satellite implies that the GPS receiver is on the surface of a sphere centered at the position of a satellite. Thus we know that the indicated position of the GPS receiver is at or near the intersection of the surfaces of four spheres. In the ideal case of no errors, the GPS receiver will be at an intersection of the surfaces of four spheres. The surfaces of two spheres if they intersect in more than one point intersect in a circle. A figure, two sphere surfaces intersecting in a circle, is shown below. Two points at which the surfaces of the spheres intersect are clearly shown in the figure. The distance between these two points is the diameter of the circle of intersection.

The article, trilateration, shows mathematically that two spheres intersecting in more than one point intersect in a circle.

A circle and sphere surface in most cases of practical interest intersect at two points, although it is conceivable that they could intersect in 0 or 1 point. Another figure, Surface of Sphere Intersecting a Circle (not disk) at Two Points, is shown to aid in visualizing this intersection. Again trilateration clearly shows this mathematically. The correct position of the GPS receiver is the intersection that is closest to the surface of the earth for automobiles and other near earth vehicles. The correct position of the GPS receiver is also the intersection which is closest to the surface of the sphere corresponding to the fourth satellite. (The two intersections are symmetrical with respect to the plane containing the three satellites. If the three satellites are not in the same orbital plane, the plane containing the three satellites will not be a vertical plane passing through the center of the earth. In this case one of the intersections will be closer to the earth than the other. The near-earth intersection will be the correct position for the case of a near-earth vehicle. The intersection which is farthest from earth, may be the correct position for the case of global positioning systems in deep space probes or other vehicles.)

Correcting a GPS receiver's clock

The method of calculating position for the case of no errors has been explained. One of the most important errors is the error in the GPS receiver clock. Because of the very large value of c, the speed of light, the estimated distances from the GPS receiver to the satellites, the pseudoranges, are very sensitive to errors in the GPS receiver clock. This seems to suggest that an extremely accurate and expensive clock is required for the GPS receiver to work. On the other hand, manufacturers would like to make an inexpensive GPS receiver which can be mass marketed. The manufacturers were thus faced with a difficult design problem. The technique that solves this problem is based on the way sphere surfaces intersect in the GPS problem.

It is likely the surfaces of the three spheres intersect since the circle of intersection of the first two spheres is normally quite large and thus the third sphere surface is likely to intersect this large circle. It is very unlikely that the surface of the sphere corresponding to the fourth satellite will intersect either of the two points of intersection of the first three since any clock error could cause it to miss intersecting a point. However the distance from the valid estimate of GPS receiver position to the surface of the sphere corresponding to the fourth satellite can be used to compute a clock correction. Let r_4 denote the distance from the valid estimate of GPS receiver position to the fourth satellite and let p_4 denote the pseudorange of the fourth satellite. Let da = r_4 - p_4. Note that da is the distance from the computed GPS receiver position to the surface of the sphere corresponding to the fourth satellite. Thus the quotient, b = da / c , provides an estimate of

(correct time) - (time indicated by the receiver's on-board clock), and the GPS receiver clock can be advanced if b is positive or delayed if b is negative.

System detail

System segmentation

The current GPS consists of three major segments. These are the space segment (SS), a control segment (CS), and a user segment (US).

Space segment

The space segment (SS) comprises the orbiting GPS satellites, or Space Vehicles (SV) in GPS parlance. The GPS design originally called for 24 SVs, eight each in three circular orbital planes, but this was modified to six planes with four satellites each. The orbital planes are centered on the Earth, not rotating with respect to the distant stars. The six planes have approximately 55° inclination (tilt relative to Earth's equator) and are separated by 60° right ascension of the ascending node (angle along the equator from a reference point to the orbit's intersection). The orbits are arranged so that at least six satellites are always within line of sight from almost everywhere on Earth's surface.

Orbiting at an altitude of approximately 20,200 kilometers (12,600 miles or 10,900 nautical miles; orbital radius of 26,600 km (16,500 mi or 14,400 NM)), each SV makes two complete orbits each sidereal day. The ground track of each satellite therefore repeats each (sidereal) day. This was very helpful during development, since even with just four satellites, correct alignment means all four are visible from one spot for a few hours each day. For military operations, the ground track repeat can be used to ensure good coverage in combat zones.

As of March 2008, there are 31 actively broadcasting satellites in the GPS constellation. The additional satellites improve the precision of GPS receiver calculations by providing redundant measurements. With the increased number of satellites, the constellation was changed to a nonuniform arrangement. Such an arrangement was shown to improve reliability and availability of the system, relative to a uniform system, when multiple satellites fail.

Some reports in 2008 indicated that the 32nd satellite was causing difficulties for some GPS receivers.

Control segment

The flight paths of the satellites are tracked by US Air Force monitoring stations in Hawaii, Kwajalein, Ascension Island, Diego Garcia, and Colorado Springs, Colorado, along with monitor stations operated by the National Geospatial-Intelligence Agency (NGA). The tracking information is sent to the Air Force Space Command's master control station at Schriever Air Force Base in Colorado Springs, which is operated by the 2nd Space Operations Squadron (2 SOPS) of the United States Air Force (USAF). Then 2 SOPS contacts each GPS satellite regularly with a navigational update (using the ground antennas at Ascension Island, Diego Garcia, Kwajalein, and Colorado Springs). These updates synchronize the atomic clocks on board the satellites to within a few nanoseconds of each other, and adjust the ephemeris of each satellite's internal orbital model. The updates are created by a Kalman filter which uses inputs from the ground monitoring stations, space weather information, and various other inputs.

Satellite maneuvers are not precise by GPS standards. So to change the orbit of a satellite, the satellite must be marked 'unhealthy', so receivers will not use it in their calculation. Then the maneuver can be carried out, and the resulting orbit tracked from the ground. Then the new ephemeris is uploaded and the satellite marked healthy again.

User segment

The user's GPS receiver is the user segment (US) of the GPS. In general, GPS receivers are composed of an antenna, tuned to the frequencies transmitted by the satellites, receiver-processors, and a highly-stable clock (often a crystal oscillator). They may also include a display for providing location and speed information to the user. A receiver is often described by its number of channels: this signifies how many satellites it can monitor simultaneously. Originally limited to four or five, this has progressively increased over the years so that, as of 2007, receivers typically have between 12 and 20 channels.

GPS receivers may include an input for differential corrections, using the RTCM SC-104 format. This is typically in the form of a RS-232 port at 4,800 bit/s speed. Data is actually sent at a much lower rate, which limits the accuracy of the signal sent using RTCM. Receivers with internal DGPS receivers can outperform those using external RTCM data. As of 2006, even low-cost units commonly include Wide Area Augmentation System (WAAS) receivers.

Many GPS receivers can relay position data to a PC or other device using the NMEA 0183 protocol, or the newer and less widely used NMEA 2000. Although these protocols are officially defined by the NMEA, references to the these protocols have been compiled from public records, allowing open source tools like gpsd to read the protocol without violating intellectual property laws. Other proprietary protocols exist as well, such as the SiRF and MTK protocols. Receivers can interface with other devices using methods including a serial connection, USB or Bluetooth.

Navigation signals

Each GPS satellite continuously broadcasts a Navigation Message at 50 bit/s giving the time-of-week, GPS week number and satellite health information (all transmitted in the first part of the message), an ephemeris (transmitted in the second part of the message) and an almanac (later part of the message). The messages are sent in frames, each taking 30 seconds to transmit 1500 bits.

The first 6 seconds of every frame contains data describing the satellite clock and its relationship to GPS time. The next 12 seconds contain the ephemeris data, giving the satellite's own precise orbit. The ephemeris is updated every 2 hours and is generally valid for 4 hours, with provisions for updates every 6 hours or longer in non-nominal conditions. The time needed to acquire the ephemeris is becoming a significant element of the delay to first position fix, because, as the hardware becomes more capable, the time to lock onto the satellite signals shrinks, but the ephemeris data requires 30 seconds (worst case) before it is received, due to the low data transmission rate.

The almanac consists of coarse orbit and status information for each satellite in the constellation, an ionospheric model, and information to relate GPS derived time to Coordinated Universal Time (UTC). A new part of the almanac is received for the last 12 seconds in each 30 second frame. Each frame contains 1/25th of the almanac, so 12.5 minutes are required to receive the entire almanac from a single satellite. The almanac serves several purposes. The first is to assist in the acquisition of satellites at power-up by allowing the receiver to generate a list of visible satellites based on stored position and time, while an ephemeris from each satellite is needed to compute position fixes using that satellite. In older hardware, lack of an almanac in a new receiver would cause long delays before providing a valid position, because the search for each satellite was a slow process. Advances in hardware have made the acquisition process much faster, so not having an almanac is no longer an issue. The second purpose is for relating time derived from the GPS (called GPS time) to the international time standard of UTC. Finally, the almanac allows a single frequency receiver to correct for ionospheric error by using a global ionospheric model. The corrections are not as accurate as augmentation systems like WAAS or dual frequency receivers. However it is often better than no correction since ionospheric error is the largest error source for a single frequency GPS receiver. An important thing to note about navigation data is that each satellite transmits only its own ephemeris, but transmits an almanac for all satellites.

All satellites broadcast at the same two frequencies, 1.57542 GHz (L1 signal) and 1.2276 GHz (L2 signal). The receiver can distinguish the signals from different satellites because GPS uses a code division multiple access (CDMA) spread-spectrum technique where the low-bitrate message data is encoded with a high-rate pseudo-random (PRN) sequence that is different for each satellite. The receiver knows the PRN codes for each satellite and can use this to reconstruct the actual message data. The message data is transmitted at 50 bits per second. Two distinct CDMA encodings are used: the coarse/acquisition (C/A) code (a so-called Gold code) at 1.023 million chips per second, and the precise (P) code at 10.23 million chips per second. The L1 carrier is modulated by both the C/A and P codes, while the L2 carrier is only modulated by the P code. The C/A code is public and used by civilian GPS receivers, while the P code can be encrypted as a so-called P(Y) code which is only available to military equipment with a proper decryption key. Both the C/A and P(Y) codes impart the precise time-of-day to the user.

Satellite frequencies

  • L1 (1575.42 MHz): Mix of Navigation Message, coarse-acquisition (C/A) code and encrypted precision P(Y) code, plus the new L1C on future Block III satellites.
  • L2 (1227.60 MHz): P(Y) code, plus the new L2C code on the Block IIR-M and newer satellites.
  • L3 (1381.05 MHz): Used by the Nuclear Detonation (NUDET) Detection System Payload (NDS) to signal detection of nuclear detonations and other high-energy infrared events. Used to enforce nuclear test ban treaties.
  • L4 (1379.913 MHz): Being studied for additional ionospheric correction.
  • L5 (1176.45 MHz): Proposed for use as a civilian safety-of-life (SoL) signal (see GPS modernization). This frequency falls into an internationally protected range for aeronautical navigation, promising little or no interference under all circumstances. The first Block IIF satellite that would provide this signal is set to be launched in 2009.

C/A code

Demodulation and decoding

Since all of the satellite signals are modulated onto the same L1 carrier frequency, there is a need to separate the signals after demodulation. This is done by assigning each satellite a unique pseudorandom sequence known as a Gold code, and the signals are decoded, after demodulation, using modulo 2 addition of the Gold codes corresponding to satellites n1 through nk, where k is the number of channels in the GPS receiver and n1 through nk are the pseudorandom numbers associated with the satellites. The result of these modulo 2 additions are the 50 bps navigation messages from satellites n1 through nk. The Gold codes used in GPS are a sequence of 1023 bits with a period of one millisecond. These Gold codes are highly mutually orthogonal, so that it is unlikely that one satellite signal will be misinterpreted as another. As well, the Gold codes have good auto-correlation properties.

There are 1025 different Gold codes of length 1023 bits, but only 32 are used. These Gold codes are quite often referred to as "pseudo-random noise" since they contain no data. However, this may be misleading since they are actually deterministic sequences.

If the almanac information has previously been acquired, the receiver picks which satellites to listen for by their PRN numbers. If the almanac information is not in memory, the receiver enters a search mode and cycles through the PRN numbers until a lock is obtained on one of the satellites. To obtain a lock, it is necessary that there be an unobstructed line of sight from the receiver to the satellite. The receiver can then acquire the almanac and determine the satellites it should listen for. As it detects each satellite's signal, it identifies it by its distinct C/A code pattern, then measures the received time for each satellite. To do this, the receiver produces an identical C/A sequence using the same PRN number as depicted in the diagram, referenced to its local clock, starting at the same time the satellite sent it. It then computes the offset to the local clock that generates the maximum correlation. This offset is the time delay from the satellite to the receiver, as told by the receiver's clock. Since the PRN repeats every millisecond, this offset is precise but ambiguous, and the ambiguity is resolved by looking at the data bits, which are sent at 50 Hz (20 ms/bit) and aligned with the PRN code.

Next, the orbital position data, or ephemeris, from the Navigation Message is then downloaded to calculate precisely where the satellite was at the start of the message. A more-sensitive receiver will potentially acquire the ephemeris data more quickly than a less-sensitive receiver, especially in a noisy environment.

Position calculation advanced

Before providing a more mathematical description of position calculation, the introductory material on this topics is reviewed. To describe the basic concept of how a GPS receiver works, the errors are at first ignored. Using messages received from four satellites, the GPS receiver is able to determine the satellite positions and time sent. The x, y, and z components of position and the time sent are designated as left [x_i, y_i, z_i, t_iright ] where the subscript i denotes which satellite and has the value 1, 2, 3, or 4. Knowing the indicated time the message was received tr_i, the GPS receiver can compute the indicated transit time, left (tr_i-t_iright ) . of the message. Assuming the message traveled at the speed of light, c, the distance traveled, p_i can be computed as left (tr_i-t_iright )c . Knowing the distance from GPS receiver to a satellite and the position of a satellite implies that the GPS receiver is on the surface of a sphere centered at the position of a satellite. Thus we know that the indicated position of the GPS receiver is at or near the intersection of the surfaces of four spheres. In the ideal case of no errors, the GPS receiver will be at an intersection of the surfaces of four spheres. The surfaces of two spheres if they intersect in more than one point intersect in a circle. A figure, Two Sphere Surfaces Intersecting in a Circle, is shown below depicting this which hopefully will aid the reader in visualizing this intersection. Two points at which the surfaces of the spheres intersect are clearly shown in the figure. The distance between these two points is the diameter of the circle of intersection. If you are not convinced of this, consider how a side view of the intersecting spheres would look. This view would look exactly the same as the figure because of the symmetry of the spheres. And in fact a view from any horizontal direction would look exactly the same. This should make it clear to the reader that the surfaces of the two spheres actually do intersect in a circle. The article, trilateration, shows mathematically how the equation for a circle is determined. A circle and sphere surface in most cases of practical interest intersect at two points, although it is conceivable that they could intersect in 0, 1 or infinite points. Another figure, Surface of Sphere Intersecting a Circle (not disk) at Two Points, is shown below to aid in visualizing this intersection. Again trilateration clearly show this mathematically. The correct position of the GPS receiver is the one that is closest to the fourth sphere. This paragraph has described the basic concept of GPS while ignoring errors. The next problem is how to process the messages when errors are present. Let b denote the clock error or bias, the amount by which the receiver's clock is slow. The GPS receiver has four unknowns, the three components of GPS receiver position and the clock bias left [x, y, z, bright ]. The equation of the sphere surfaces are given by (x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = bigl((tr_i + b - t_i)cbigr)^2, ; i=1,2,3,4. Another useful form of these equations is in terms of the pseudoranges, which are simply the ranges approximated based on GPS receiver clock's indicated (i.e. uncorrected) time so that p_i = left (tr_i - t_i right )c. Then the equations becomes:

p_i = sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}- bc,; i=1,2,3,4.. Two of the most important methods of computing GPS receiver position and clock bias are (1) trilateration followed by one dimensional numerical root finding and (2) multidimensional Newton-Raphson calculations. These two methods along with their advantages are discussed.

  • The receiver can solve by trilateration followed by one dimensional numerical root finding. This method involves using Trilateration to determine the intersection of the surfaces of three spheres. It is clearly shown in trilateration that the surfaces of three spheres intersect in 0, 1, or 2 points. In the usual case of two intersections, the solution which is nearest the surface of the sphere corresponding to the fourth satellite is chosen. The surface of the earth can also sometimes be used instead, especially in the case of civilian GPS receivers since it is illegal in the United States to track vehicles of more than 60,000 feet in altitude. The bias, b is then computed based on the distance from the solution to the surface of the sphere corresponding to the fourth satellite. Using an updated received time based on this bias, new spheres are computed and the process is repeated. One advantage of this method is that it involves one dimensional as opposed to multidimensional numerical root finding.
  • The receiver can utilize multidimensional Newton-Raphson calculations. Linearize around an approximate solution, say left [x^{(k)}, y^{(k)}, z^{(k)}, b^{(k)}right ] from iteration k, then solve four linear equations derived from the quadratic equations above to obtain left [x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)}right ]. The radii are large and so the sphere surfaces are close to flat. This near flatness may cause the iterative procedure to converge rapidly in the case where b is near the correct value and the primary change is in the values of x, y,; and; z, since in this case the problem is merely to find the intersection of nearly flat surfaces and thus close to a linear problem. However when b is changing significantly, this near flatness does not appear to be advantageous in producing rapid convergence, since in this case these near flat surfaces will be moving as the spheres expand and contract. This possible fast convergence is an advantage of this method. Also it has been claimed that this method is the "typical" method used by GPS receivers. A disadvantage of this method is that according to , "There are no good general methods for solving systems of more than one nonlinear equations." For a more detailed description of the mathematics see Multidimensional Newton Raphson.
  • Other methods include:
  • Solving for the intersection of the expanding signals form light cones in 4-space cones
  • Solving for the intersection of hyperboloids determined by the time difference of signals received from satellites utilizing multilateration,
  • Solving the equations in accordance with .

  • More than four satellites should be used, if available. This results in an over-determined system of equations with no unique solution, which must be solved by least-squares or a similar technique. If all visible satellites are used, the results are always at least as good as using the four best, and usually better. Also the errors in results can be estimated through the residuals. With each combination of four or more satellites, a geometric dilution of precision (GDOP) vector can be calculated, based on the relative sky positions of the satellites used. As more satellites are picked up, pseudoranges from more combinations of four satellites can be processed to add more estimates to the location and clock offset. The receiver then determines which combinations to use and how to calculate the estimated position by determining the weighted average of these positions and clock offsets. After the final location and time are calculated, the location is expressed in a specific coordinate system such as latitude and longitude, using the WGS 84 geodetic datum or a local system specific to a country.

Finally, results from other positioning systems such as GLONASS or the upcoming Galileo can be used in the fit, or used to double check the result. (By design, these systems use the same bands, so much of the receiver circuitry can be shared, though the decoding is different.)

P(Y) code

Calculating a position with the P(Y) signal is generally similar in concept, assuming one can decrypt it. The encryption is essentially a safety mechanism: if a signal can be successfully decrypted, it is reasonable to assume it is a real signal being sent by a GPS satellite. In comparison, civil receivers are highly vulnerable to spoofing since correctly formatted C/A signals can be generated using readily available signal generators. RAIM features do not protect against spoofing, since RAIM only checks the signals from a navigational perspective.

Error sources

Sources of User Equivalent Range Errors (UERE)
Source Effect
Ionospheric effects ± 5 m
Ephemeris errors ± 2.5 m
Satellite clock errors ± 2 m
Multipath distortion ± 1 m
Tropospheric effects ± 0.5 m
Numerical errors ± 1 m
The position calculated by a GPS receiver requires the current time, the position of the satellite and the measured delay of the received signal. The position accuracy is primarily dependent on the satellite position and signal delay.

To measure the delay, the receiver compares the bit sequence received from the satellite with an internally generated version. By comparing the rising and trailing edges of the bit transitions, modern electronics can measure signal offset to within about one percent of a bit time, or approximately 10 nanoseconds for the C/A code. Since GPS signals propagate at the speed of light, this represents an error of about 3 meters.

Position accuracy can be improved by using the higher-chiprate P(Y) signal. Assuming the same one percent bit time accuracy, the high frequency P(Y) signal results in an accuracy of about 30 centimeters.

Electronics errors are one of several accuracy-degrading effects outlined in the table above. When taken together, autonomous civilian GPS horizontal position fixes are typically accurate to about 15 meters (50 ft). These effects also reduce the more precise P(Y) code's accuracy.

Atmospheric effects

Inconsistencies of atmospheric conditions affect the speed of the GPS signals as they pass through the Earth's atmosphere, especially the ionosphere. Correcting these errors is a significant challenge to improving GPS position accuracy. These effects are smallest when the satellite is directly overhead and become greater for satellites nearer the horizon since the path through the atmosphere is longer (see airmass). Once the receiver's approximate location is known, a mathematical model can be used to estimate and compensate for these errors.

Because ionospheric delay affects the speed of microwave signals differently depending on their frequency — a characteristic known as dispersion - delays measured on two or more frequency bands can be used to measure dispersion, and this measurement can then be used to estimate the delay at each frequency. Some military and expensive survey-grade civilian receivers measure the different delays in the L1 and L2 frequencies to measure atmospheric dispersion, and apply a more precise correction. This can be done in civilian receivers without decrypting the P(Y) signal carried on L2, by tracking the carrier wave instead of the modulated code. To facilitate this on lower cost receivers, a new civilian code signal on L2, called L2C, was added to the Block IIR-M satellites, which was first launched in 2005. It allows a direct comparison of the L1 and L2 signals using the coded signal instead of the carrier wave.

The effects of the ionosphere generally change slowly, and can be averaged over time. The effects for any particular geographical area can be easily calculated by comparing the GPS-measured position to a known surveyed location. This correction is also valid for other receivers in the same general location. Several systems send this information over radio or other links to allow L1-only receivers to make ionospheric corrections. The ionospheric data are transmitted via satellite in Satellite Based Augmentation Systems (SBAS) such as WAAS (available in North America and Hawaii), EGNOS (Europe and Asia) or MSAS (Japan), which transmits it on the GPS frequency using a special pseudo-random noise sequence (PRN), so only one receiver and antenna are required.

Humidity also causes a variable delay, resulting in errors similar to ionospheric delay, but occurring in the troposphere. This effect both is more localized and changes more quickly than ionospheric effects, and is not frequency dependent. These traits make precise measurement and compensation of humidity errors more difficult than ionospheric effects.

Changes in receiver altitude also change the amount of delay, due to the signal passing through less of the atmosphere at higher elevations. Since the GPS receiver computes its approximate altitude, this error is relatively simple to correct, either by applying a function regression or correlating margin of atmospheric error to ambient pressure using a barometric altimeter.

Multipath effects

GPS signals can also be affected by multipath issues, where the radio signals reflect off surrounding terrain; buildings, canyon walls, hard ground, etc. These delayed signals can cause inaccuracy. A variety of techniques, most notably narrow correlator spacing, have been developed to mitigate multipath errors. For long delay multipath, the receiver itself can recognize the wayward signal and discard it. To address shorter delay multipath from the signal reflecting off the ground, specialized antennas (e.g. a choke ring antenna) may be used to reduce the signal power as received by the antenna. Short delay reflections are harder to filter out because they interfere with the true signal, causing effects almost indistinguishable from routine fluctuations in atmospheric delay.

Multipath effects are much less severe in moving vehicles. When the GPS antenna is moving, the false solutions using reflected signals quickly fail to converge and only the direct signals result in stable solutions.

Ephemeris and clock errors

While the ephemeris data is transmitted every 30 seconds, the information itself may be up to two hours old. Data up to four hours old is considered valid for calculating positions, but may not indicate the satellites actual position. If a fast TTFF is needed, it is possible to upload valid ephemeris to a receiver, and in addition to setting the time, a position fix can be obtained in under ten seconds. It is feasible to put such ephemeris data on the web so it can be loaded into mobile GPS devices. See also Assisted GPS.

The satellite's atomic clocks experience noise and clock drift errors. The navigation message contains corrections for these errors and estimates of the accuracy of the atomic clock. However, they are based on observations and may not indicate the clock's current state.

These problems tend to be very small, but may add up to a few meters (10s of feet) of inaccuracy.

Selective availability

GPS includes a (currently disabled) feature called Selective Availability (SA) that can introduce intentional, slowly changing random errors of up to a hundred meters (328 ft) into the publicly available navigation signals to confound, for example, the guidance of long range missiles to precise targets. When enabled, the accuracy is still available in the signal, but in an encrypted form that is only available to the United States military, its allies and a few others, mostly government users. Even those who have managed to acquire military GPS receivers would still need to obtain the daily key, whose dissemination is tightly controlled.

Prior to being turned off, SA typically added signal errors of up to about 10 meters (32 ft) horizontally and 30 meters (98 ft) vertically. The inaccuracy of the civilian signal was deliberately encoded so as not to change very quickly. For instance, the entire eastern U.S. area might read 30 m off, but 30 m off everywhere and in the same direction. To improve the usefulness of GPS for civilian navigation, Differential GPS was used by many civilian GPS receivers to greatly improve accuracy.

During the Gulf War, the shortage of military GPS units and the ready availability of civilian ones caused many troops to buy their own civilian GPS units: their wide use among personnel resulted in a decision to disable Selective Availability. This was ironic, as SA had been introduced specifically for these situations, allowing friendly troops to use the signal for accurate navigation, while at the same time denying it to the enemy—but the assumption underlying this policy was that all U.S. troops and enemy troops would have military-specification GPS receivers and that civilian receivers would not exist in war zones. But since many American soldiers were using civilian devices, SA was also denying the same accuracy to thousands of friendly troops; turning it off (by removing the added-in error) presented a clear benefit to friendly troops.

In the 1990s, the FAA started pressuring the military to turn off SA permanently. This would save the FAA millions of dollars every year in maintenance of their own radio navigation systems. The amount of error added was "set to zero" at midnight on May 12000 following an announcement by U.S. President Bill Clinton, allowing users access to the error-free L1 signal. Per the directive, the induced error of SA was changed to add no error to the public signals (C/A code). Clinton's executive order required SA to be set to zero by 2006; it happened in 2000 once the US military developed a new system that provides the ability to deny GPS (and other navigation services) to hostile forces in a specific area of crisis without affecting the rest of the world or its own military systems.

Selective Availability is still a system capability of GPS, and error could, in theory, be reintroduced at any time. In practice, in view of the hazards and costs this would induce for US and foreign shipping, it is unlikely to be reintroduced, and various government agencies, including the FAA, have stated that it is not intended to be reintroduced.

One interesting side effect of the Selective Availability hardware is the capability to correct the frequency of the GPS cesium and rubidium atomic clocks to an accuracy of approximately 2 × 10-13 (one in five trillion). This represented a significant improvement over the raw accuracy of the clocks.

On 19 September 2007, the United States Department of Defense announced that future GPS III satellites will not be capable of implementing SA, eventually making the policy permanent.

Relativity

According to the theory of relativity, due to their constant movement and height relative to the Earth-centered inertial reference frame, the clocks on the satellites are affected by their speed (special relativity) as well as their gravitational potential (general relativity). For the GPS satellites, general relativity predicts that the atomic clocks at GPS orbital altitudes will tick more rapidly, by about 45.9 microseconds (μs) per day, because they are in a weaker gravitational field than atomic clocks on Earth's surface. Special relativity predicts that atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by about 7.2 μs per day. When combined, the discrepancy is about 38 microseconds per day; a difference of 4.465 parts in 1010.. To account for this, the frequency standard onboard each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz. Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.

Sagnac distortion

GPS observation processing must also compensate for the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. A Lorentz transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.

Interference

Natural sources

Since GPS signals at terrestrial receivers tend to be relatively weak, it is easy for other sources of electromagnetic radiation to desensitize the receiver, making acquiring and tracking the satellite signals difficult or impossible.

Solar flares are one such naturally occurring emission with the potential to degrade GPS reception, and their impact can affect reception over the half of the Earth facing the sun. GPS signals can also be interfered with by naturally occurring geomagnetic storms, predominantly found near the poles of the Earth's magnetic field. GPS signals are also subjected to interference from Van Allen Belt radiation when the satellites pass through the South Atlantic Anomaly.

Artificial sources

In automotive GPS receivers, metallic features in windshields, such as defrosters, or car window tinting films can act as a Faraday cage, degrading reception just inside the car.

Man-made EMI (electromagnetic interference) can also disrupt, or jam, GPS signals. In one well documented case, the entire harbor of Moss Landing was unable to receive GPS signals due to unintentional jamming caused by malfunctioning TV antenna preamplifiers. Intentional jamming is also possible. Generally, stronger signals can interfere with GPS receivers when they are within radio range, or line of sight. In 2002, a detailed description of how to build a short range GPS L1 C/A jammer was published in the online magazine Phrack.

The U.S. government believes that such jammers were used occasionally during the 2001 war in Afghanistan and the U.S. military claimed to destroy six GPS jammers during the Iraq War, including one that was destroyed ironically with a GPS-guided bomb. Such a jammer is relatively easy to detect and locate, making it an attractive target for anti-radiation missiles. The UK Ministry of Defence tested a jamming system in the UK's West Country on 7 and 8 June 2007.

Some countries allow the use of GPS repeaters to allow for the reception of GPS signals indoors and in obscured locations, however, under EU and UK laws, the use of these is prohibited as the signals can cause interference to other GPS receivers that may receive data from both GPS satellites and the repeater.

Due to the potential for both natural and man-made noise, numerous techniques continue to be developed to deal with the interference. The first is to not rely on GPS as a sole source. According to John Ruley, "IFR pilots should have a fallback plan in case of a GPS malfunction". Receiver Autonomous Integrity Monitoring (RAIM) is a feature now included in some receivers, which is designed to provide a warning to the user if jamming or another problem is detected. The U.S. military has also deployed their Selective Availability / Anti-Spoofing Module (SAASM) in the Defense Advanced GPS Receiver (DAGR). In demonstration videos, the DAGR is able to detect jamming and maintain its lock on the encrypted GPS signals during interference which causes civilian receivers to lose lock.

Accuracy enhancement

Augmentation

Augmentation methods of improving accuracy rely on external information being integrated into the calculation process. There are many such systems in place and they are generally named or described based on how the GPS sensor receives the information. Some systems transmit additional information about sources of error (such as clock drift, ephemeris, or ionospheric delay), others provide direct measurements of how much the signal was off in the past, while a third group provide additional navigational or vehicle information to be integrated in the calculation process.

Examples of augmentation systems include the Wide Area Augmentation System, Differential GPS, Inertial Navigation Systems and Assisted GPS.

Precise monitoring

The accuracy of a calculation can also be improved through precise monitoring and measuring of the existing GPS signals in additional or alternate ways.

After SA, which has been turned off, the largest error in GPS is usually the unpredictable delay through the ionosphere. The spacecraft broadcast ionospheric model parameters, but errors remain. This is one reason the GPS spacecraft transmit on at least two frequencies, L1 and L2. Ionospheric delay is a well-defined function of frequency and the total electron content (TEC) along the path, so measuring the arrival time difference between the frequencies determines TEC and thus the precise ionospheric delay at each frequency.

Receivers with decryption keys can decode the P(Y)-code transmitted on both L1 and L2. However, these keys are reserved for the military and "authorized" agencies and are not available to the public. Without keys, it is still possible to use a codeless technique to compare the P(Y) codes on L1 and L2 to gain much of the same error information. However, this technique is slow, so it is currently limited to specialized surveying equipment. In the future, additional civilian codes are expected to be transmitted on the L2 and L5 frequencies (see GPS modernization, below). Then all users will be able to perform dual-frequency measurements and directly compute ionospheric delay errors.

A second form of precise monitoring is called Carrier-Phase Enhancement (CPGPS). The error, which this corrects, arises because the pulse transition of the PRN is not instantaneous, and thus the correlation (satellite-receiver sequence matching) operation is imperfect. The CPGPS approach utilizes the L1 carrier wave, which has a period 1000 times smaller than that of the C/A bit period, to act as an additional clock signal and resolve the uncertainty. The phase difference error in the normal GPS amounts to between 2 and 3 meters (6 to 10 ft) of ambiguity. CPGPS working to within 1% of perfect transition reduces this error to 3 centimeters (1 inch) of ambiguity. By eliminating this source of error, CPGPS coupled with DGPS normally realizes between 20 and 30 centimeters (8 to 12 inches) of absolute accuracy.

Relative Kinematic Positioning (RKP) is another approach for a precise GPS-based positioning system. In this approach, determination of range signal can be resolved to a precision of less than 10 centimeters (4 in). This is done by resolving the number of cycles in which the signal is transmitted and received by the receiver. This can be accomplished by using a combination of differential GPS (DGPS) correction data, transmitting GPS signal phase information and ambiguity resolution techniques via statistical tests—possibly with processing in real-time (real-time kinematic positioning, RTK).

Timekeeping

While most clocks are synchronized to Coordinated Universal Time (UTC), the atomic clocks on the satellites are set to GPS time. The difference is that GPS time is not corrected to match the rotation of the Earth, so it does not contain leap seconds or other corrections which are periodically added to UTC. GPS time was set to match Coordinated Universal Time (UTC) in 1980, but has since diverged. The lack of corrections means that GPS time remains at a constant offset (19 seconds) with International Atomic Time (TAI). Periodic corrections are performed on the on-board clocks to correct relativistic effects and keep them synchronized with ground clocks.

The GPS navigation message includes the difference between GPS time and UTC, which as of 2006 is 14 seconds due to the leap second added to UTC December 31 2005. Receivers subtract this offset from GPS time to calculate UTC and specific timezone values. New GPS units may not show the correct UTC time until after receiving the UTC offset message. The GPS-UTC offset field can accommodate 255 leap seconds (eight bits) which, at the current rate of change of the Earth's rotation, is sufficient to last until the year 2330.

As opposed to the year, month, and day format of the Gregorian calendar, the GPS date is expressed as a week number and a day-of-week number. The week number is transmitted as a ten-bit field in the C/A and P(Y) navigation messages, and so it becomes zero again every 1,024 weeks (19.6 years). GPS week zero started at 00:00:00 UTC (00:00:19 TAI) on January 6 1980, and the week number became zero again for the first time at 23:59:47 UTC on August 21 1999 (00:00:19 TAI on August 22 1999). To determine the current Gregorian date, a GPS receiver must be provided with the approximate date (to within 3,584 days) to correctly translate the GPS date signal. To address this concern the modernized GPS navigation message uses a 13-bit field, which only repeats every 8,192 weeks (157 years), and will not return to zero until near the year 2137.

Modernization

Having reached the program's requirements for Full Operational Capability (FOC) on July 17, 1995, the GPS completed its original design goals. However, additional advances in technology and new demands on the existing system led to the effort to modernize the GPS. Announcements from the U.S. Vice President and the White House in 1998 initiated these changes, and in 2000 the U.S. Congress authorized the effort, referring to it as GPS III.

The project aims to improve the accuracy and availability for all users and involves new ground stations, new satellites, and four additional navigation signals. New civilian signals are called L2C, L5 and L1C; the new military code is called M-Code. Initial Operational Capability (IOC) of the L2C code is expected in 2008. A goal of 2013 has been established for the entire program, with incentives offered to the contractors if they can complete it by 2011 (See GPS signals).

Applications

The Global Positioning System, while originally a military project, is considered a dual-use technology, meaning it has significant applications for both the military and the civilian industry.

Military

The military applications of GPS span many purposes:

  • Navigation: GPS allows soldiers to find objectives in the dark or in unfamiliar territory, and to coordinate the movement of troops and supplies. The GPS-receivers commanders and soldiers use are respectively called the Commanders Digital Assistant and the Soldier Digital Assistant.
  • Target tracking: Various military weapons systems use GPS to track potential ground and air targets before they are flagged as hostile. These weapon systems pass GPS co-ordinates of targets to precision-guided munitions to allow them to engage the targets accurately. Military aircraft, particularly those used in air-to-ground roles use GPS to find targets (for example, gun camera video from AH-1 Cobras in Iraq show GPS co-ordinates that can be looked up in Google Earth).
  • Missile and projectile guidance: GPS allows accurate targeting of various military weapons including ICBMs, cruise missiles and precision-guided munitions. Artillery projectiles with embedded GPS receivers able to withstand accelerations of 12,000G have been developed for use in 155 mm howitzers.
  • Search and Rescue: Downed pilots can be located faster if they have a GPS receiver.
  • Reconnaissance and Map Creation: The military use GPS extensively to aid mapping and reconnaissance.
  • The GPS satellites also carry a set of nuclear detonation detectors consisting of an optical sensor (Y-sensor), an X-ray sensor, a dosimeter, and an Electro-Magnetic Pulse (EMP) sensor (W-sensor) which form a major portion of the United States Nuclear Detonation Detection System.

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Civilian

Many civilian applications benefit from GPS signals, using one or more of three basic components of the GPS: absolute location, relative movement, and time transfer.

The ability to determine the receiver's absolute location allows GPS receivers to perform as a surveying tool or as an aid to navigation. The capacity to determine relative movement enables a receiver to calculate local velocity and orientation, useful in vessels or observations of the Earth. Being able to synchronize clocks to exacting standards enables time transfer, which is critical in large communication and observation systems. An example is CDMA digital cellular. Each base station has a GPS timing receiver to synchronize its spreading codes with other base stations to facilitate inter-cell hand off and support hybrid GPS/CDMA positioning of mobiles for emergency calls and other applications. Finally, GPS enables researchers to explore the Earth environment including the atmosphere, ionosphere and gravity field. GPS survey equipment has revolutionized tectonics by directly measuring the motion of faults in earthquakes.

To help prevent civilian GPS guidance from being used in an enemy's military or improvised weaponry, the US Government controls the export of civilian receivers. A US-based manufacturer cannot generally export a GPS receiver unless the receiver contains limits restricting it from functioning when it is simultaneously (1) at an altitude above 18 kilometers (60,000 ft) and (2) traveling at over 515 m/s (1,000 knots). These parameters are well above the operating characteristics of the typical cruise missile, but would be characteristic of the reentry vehicle from a ballistic missile.

GPS tours are also an example of civilian use. The GPS is used to determine which content to display. For instance, when approaching a monument it would tell you about the monument.

GPS functionality has now started to move into mobile phones en masse. The first handsets with integrated GPS were launched already in the late 1990’s, and were available for broader consumer availability on networks such as those run by Nextel, Sprint and Verizon in 2002 in response to US FCC mandates for handset positioning in emergency calls. Capabilities for access by third party software developers to these features were slower in coming, with Nextel opening those APIs up upon launch to any developer, Sprint following in 2006, and Verizon soon thereafter.

Timeline

  • In 1972, the US Air Force Central Inertial Guidance Test Facility (Holloman AFB) conducted developmental flight tests of two prototype GPS receivers over White Sands Missile Range, using ground-based pseudo-satellites.
  • In 1978 the first experimental Block-I GPS satellite was launched.
  • In 1983, after Soviet interceptor aircraft shot down the civilian airliner KAL 007 that strayed into prohibited airspace due to navigational errors, killing all 269 people on board, U.S. President Ronald Reagan announced that the GPS would be made available for civilian uses once it was completed.
  • By 1985, ten more experimental Block-I satellites had been launched to validate the concept.
  • On February 14, 1989, the first modern Block-II satellite was launched.
  • In 1992, the 2nd Space Wing, which originally managed the system, was de-activated and replaced by the 50th Space Wing.
  • By December 1993 the GPS achieved initial operational capability.
  • By January 17, 1994 a complete constellation of 24 satellites was in orbit.
  • Full Operational Capability was declared by NAVSTAR in April 1995.
  • In 1996, recognizing the importance of GPS to civilian users as well as military users, U.S. President Bill Clinton issued a policy directive declaring GPS to be a dual-use system and establishing an Interagency GPS Executive Board to manage it as a national asset.
  • In 1998, U.S. Vice President Al Gore announced plans to upgrade GPS with two new civilian signals for enhanced user accuracy and reliability, particularly with respect to aviation safety.
  • On May 2, 2000 "Selective Availability" was discontinued as a result of the 1996 executive order, allowing users to receive a non-degraded signal globally.
  • In 2004, the United States Government signed a historic agreement with the European Community establishing cooperation related to GPS and Europe's planned Galileo system.
  • In 2004, U.S. President George W. Bush updated the national policy, replacing the executive board with the National Space-Based Positioning, Navigation, and Timing Executive Committee.
  • November 2004, QUALCOMM announced successful tests of Assisted-GPS for mobile phones.
  • In 2005, the first modernized GPS satellite was launched and began transmitting a second civilian signal (L2C) for enhanced user performance.
  • On September 14 2007, the aging mainframe-based Ground Segment Control System was transitioned to the new Architecture Evolution Plan.
  • The most recent launch was on March 15, 2008. The oldest GPS satellite still in operation was launched on July 4 1991, and became operational on August 30 1991.

Satellite numbers
Block Launch Period Satellites launched Currently in service
I 1978–1985 10+11 0
II 1985–1990 9 0
IIA 1990–1997 19 13
IIR 1997–2004 12+11 12
IIR-M 2005–2008 6+22 6
IIF 2009– 0+102 0
Total 58+21+122 31
1Failed
2In preparation.
(Last update: 12 April 2008)

Awards

Two GPS developers received the National Academy of Engineering Charles Stark Draper Prize for 2003:

One GPS developer, Roger L. Easton, received the National Medal of Technology on February 13 2006 at the White House.

On February 10, 1993, the National Aeronautic Association selected the Global Positioning System Team as winners of the 1992 Robert J. Collier Trophy, the most prestigious aviation award in the United States. This team consists of researchers from the Naval Research Laboratory, the U.S. Air Force, the Aerospace Corporation, Rockwell International Corporation, and IBM Federal Systems Company. The citation accompanying the presentation of the trophy honors the GPS Team "for the most significant development for safe and efficient navigation and surveillance of air and spacecraft since the introduction of radio navigation 50 years ago."

Other systems

Other satellite navigation systems in use or various states of development include:

Multidimensional Newton-Raphson for GPS

This section provides a more detailed discussion of the equations used in the second method described in Position calculation advanced. The linearized equations are developed using the appropriate partial derivatives and the algorithm is described. In the same method is discussed but the equations are not shown. Let x, y; and z; denote the true coordinates of GPS receiver position at time, t. Let b denote the unknown clock error or bias, the amount by which the receiver's clock is slow. Let the coordinates of each satellite, and the time the message was sent, be left [x_i, y_i, z_i, t_iright ], let the GPS clock's indicated received time be tr_i; mbox{for} i=1,2,3,4 and c be the speed of light. The pseudorange is computed as p_i = left (tr_i - t_i right )c. Assume the message travels at the speed of light, then the pseudorange satisfies the equation,

p_i = sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}- bc, ;; mbox{for} ; i=1,2,3,4. qquad (1)

When an approximate solution, left [x^{(k)}, y^{(k)}, z^{(k)}, b^{(k)}right ] rather than the exact solution, left [x, y, z, bright ] is used in equation 1, there is a residual, f_i^{(k)}. Transforming p_i to the right hand side of the equation there results,

f_i^{(k)} = sqrt{(x^{(k)}-x_i)^2 + (y^{(k)}-y_i)^2 + (z^{(k)}-z_i)^2}- b^{(k)}c - p_i, ;; mbox{for} ; i=1,2,3,4. qquad (2)

A solution will have been found when f_i^{(k)} is zero or sufficiently close to zero for i=1,2,3,4.

In order to linearize equation 2, the partial derivatives are computed as

frac{partial f_i^{(k)}} {partial x^{(k)}} = frac {(x^{(k)} - x_i)} {R_i^{(k)} }, frac{partial f_i^{(k)}} {partial y^{(k)}} = frac {(y^{(k)}-y_i)} {R_i^{(k)}}, frac{partial f_i^{(k)}} {partial z^{(k)}} = frac {(z^{(k)} -z_i)} {R_i^{(k)} }, frac{partial f_i^{(k)}} {partial b^{(k)}} = -c qquad (3)

where

R_i^{(k)} = sqrt{(x^{(k)}-x_i)^2 + (y^{(k)}-y_i)^2 + (z^{(k)}-z_i)^2}.

Linearizing the right hand side of equation 2 about the approximate solution, left [x^{(k)}, y^{(k)}, z^{(k)}, b^{(k)}right ] there results

f_i^{(k)} = frac {(x^{(k)}-x_i)} {R_i^{(k)}} x^{(k)} + frac {(y^{(k)}-y_i)} {R_i^{(k)}} y^{(k)} + frac {(z^{(k)}-z_i)} {R_i^{(k)}} z^{(k)} - b^{(k)}c - p_i + epsilon_i ;; mbox{for} ; i=1,2,3,4 qquad (4)

where epsilon_i is the residual due to linearization which is in addition to the residual, f_i^{(k)} , due to an approximate solution.

In order to drive f_i^{(k+1)} closer to zero choose the values left [x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)}right ] such that

- f_i^{(k)} = frac {(x^{(k)}-x_i)} {p_i} (x^{(k+1)}- x^{(k)}) + frac {(y^{(k)}-y_i)} {p_i} (y^{(k+1)}- y^{(k)});+ frac {(z^{(k)}-z_i)} {p_i} (z^{(k+1)}- z^{(k)}) - (b^{(k+1)} - b^{(k)})c - p_i. qquad (5)

That is choose the values

left [x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)}right ]

such that the residual in equation 2 changes by approximately - f_i^{(k)}.

Let

Delta x^{(k+1)} = (x^{(k+1)}- x^{(k)}), Delta y^{(k+1)} = (y^{(k+1)}- y^{(k)}), Delta z^{(k+1)} = (z^{(k+1)}- z^{(k)}),
and Delta b^{(k+1)} = (b^{(k+1)}- b^{(k)}).

Substituting and transposing p_i to the left hand side of the equation, there results

- f_i^{(k)} + p_i = frac {(x^{(k)}-x_i)} {p_i} Delta x^{(k+1)} + frac {(y^{(k)}-y_i)} {p_i} Delta y^{(k+1)} ;+ frac {(z^{(k)}-z_i)} {p_i} Delta z^{(k+1)} - Delta b^{(k+1)} c. qquad (6)
Equations 6 provide a set of four linear equations in four unknowns, the delta terms. They are in a form for solution. Using the values of ; Delta x^{(k+1)}, Delta y^{(k+1)}, Delta z^{(k+1)} and ; Delta b^{(k+1)} determined by this linear equation solution,
left [x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)}right ]

is evaluated using

x^{(k+1)} = x^{(k)} + Delta x^{(k+1)}, y^{(k+1)} = y^{(k)} + Delta y^{(k+1)}, z^{(k+1)} = z^{(k)} + Delta z^{(k+1)},

and b^{(k+1)} = b^{(k)} + Delta b^{(k+1)}. qquad (7)

Then set k = k + 1 in equations 2 through 6, plug the terms

left { x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)} right }

from equations 7 into equations 2, set k = k + 1 in equations 7, and reevaluate the residuals in equations 2. This procedure is repeated until the residuals are sufficiently small in magnitude.

See also

References

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