Department of Geography, University of Texas at Austin. These materials may be used for study, research, and education in not-for-profit applications. All commercial rights are reserved. Please credit the author, Peter H. Dana, The Geographer's Craft Project, Department of Geography, The University of Texas at Austin.

• GPS is funded by and controlled by the U. S. Department of Defense (DOD). While there are many thousands of civil users of GPS world-wide, the system was designed for and is operated by the U. S. military.

• GPS provides specially coded satellite signals that can be processed in a GPS receiver, enabling the receiver to compute position, velocity and time.

• Four GPS satellite signals are used to compute positions in three dimensions and the time offset in the receiver clock.

• Position and Time from Four GPS Satellite Signals

• The Space Segment of the system consists of the GPS satellites. These space vehicles (SVs) send radio signals from space.

• GPS Satellite

• The nominal GPS Operational Constellation consists of 24 satellites that orbit the earth in 12 hours. There are often more than 24 operational satellites as new ones are launched to replace older satellites. The satellite orbits repeat almost the same ground track (as the earth turns beneath them) once each day. The orbit altitude is such that the satellites repeat the same track and configuration over any point approximately each 24 hours (4 minutes earlier each day). There are six orbital planes (with nominally four SVs in each), equally spaced (60 degrees apart), and inclined at about fifty-five degrees with respect to the equatorial plane. This constellation provides the user with between five and eight SVs visible from any point on the earth.

• GPS Constellation

• GPS Satellites and Ground Tracks

• GPS Nominal Orbit Planes

Control Segment

• The Control Segment consists of a system of tracking stations located around the world.

• GPS Master Control and Monitor Network

• The Master Control facility is located at Schriever Air Force Base (formerly Falcon AFB) in Colorado. These monitor stations measure signals from the SVs which are incorporated into orbital models for each satellites. The models compute precise orbital data (ephemeris) and SV clock corrections for each satellite. The Master Control station uploads ephemeris and clock data to the SVs. The SVs then send subsets of the orbital ephemeris data to GPS receivers over radio signals.

• GPS Control Monitor

User Segment

• The GPS User Segment consists of the GPS receivers and the user community. GPS receivers convert SV signals into position, velocity, and time estimates. Four satellites are required to compute the four dimensions of X, Y, Z (position) and Time. GPS receivers are used for navigation, positioning, time dissemination, and other research.

  • Navigation in three dimensions is the primary function of GPS. Navigation receivers are made for aircraft, ships, ground vehicles, and for hand carrying by individuals.

  • GPS Navigation

  • Precise positioning is possible using GPS receivers at reference locations providing corrections and relative positioning data for remote receivers. Surveying, geodetic control, and plate tectonic studies are examples.

  • Time and frequency dissemination, based on the precise clocks on board the SVs and controlled by the monitor stations, is another use for GPS. Astronomical observatories, telecommunications facilities, and laboratory standards can be set to precise time signals or controlled to accurate frequencies by special purpose GPS receivers.

  • Research projects have used GPS signals to measure atmospheric parameters.

GPS Positioning Services Specified In The Federal Radionavigation Plan

Precise Positioning Service (PPS)

• Authorized users with cryptographic equipment and keys and specially equipped receivers use the Precise Positioning System. U. S. and Allied military, certain U. S. Government agencies, and selected civil users specifically approved by the U. S. Government, can use the PPS.

• PPS Predictable Accuracy

  • 22 meter Horizontal accuracy

  • 27.7 meter vertical accuracy

  • 100 nanosecond time accuracy

Standard Positioning Service (SPS)

• Civil users worldwide use the SPS without charge or restrictions. Most receivers are capable of receiving and using the SPS signal. The SPS accuracy is intentionally degraded by the DOD by the use of Selective Availability.

• SPS Predictable Accuracy

  • 100 meter horizontal accuracy

  • 156 meter vertical accuracy

  • 340 nanoseconds time accuracy

• These GPS accuracy figures are from the 1994 Federal Radionavigation Plan. The figures are 95% accuracies, and express the value of two standard deviations of radial error from the actual antenna position to an ensemble of position estimates made under specified satellite elevation angle (five degrees) and PDOP (less than six) conditions.

• For horizontal accuracy figures 95% is the equivalent of 2drms (two-distance root-mean-squared), or twice the radial error standard deviation. For vertical and time errors 95% is the value of two-standard deviations of vertical error or time error.

• Receiver manufacturers may use other accuracy measures. Root-mean-square (RMS) error is the value of one standard deviation (68%) of the error in one, two or three dimensions. Circular Error Probable (CEP) is the value of the radius of a circle, centered at the actual position that contains 50% of the position estimates. Spherical Error Probable (SEP) is the spherical equivalent of CEP, that is the radius of a sphere, centered at the actual position, that contains 50% of the three dimension position estimates. As opposed to 2drms, drms, or RMS figures, CEP and SEP are not affected by large blunder errors making them an overly optimistic accuracy measure

• Some receiver specification sheets list horizontal accuracy in RMS or CEP and without Selective Availability, making those receivers appear more accurate than those specified by more responsible vendors using more conservative error measures.

• The SVs transmit two microwave carrier signals. The L1 frequency (1575.42 MHz) carries the navigation message and the SPS code signals. The L2 frequency (1227.60 MHz) is used to measure the ionospheric delay by PPS equipped receivers.

• Three binary codes shift the L1 and/or L2 carrier phase.

  • The C/A Code (Coarse Acquisition) modulates the L1 carrier phase. The C/A code is a repeating 1 MHz Pseudo Random Noise (PRN) Code. This noise-like code modulates the L1 carrier signal, "spreading" the spectrum over a 1 MHz bandwidth. The C/A code repeats every 1023 bits (one millisecond). There is a different C/A code PRN for each SV. GPS satellites are often identified by their PRN number, the unique identifier for each pseudo-random-noise code. The C/A code that modulates the L1 carrier is the basis for the civil SPS.

  • The P-Code (Precise) modulates both the L1 and L2 carrier phases. The P-Code is a very long (seven days) 10 MHz PRN code. In the Anti-Spoofing (AS) mode of operation, the P-Code is encrypted into the Y-Code. The encrypted Y-Code requires a classified AS Module for each receiver channel and is for use only by authorized users with cryptographic keys. The P (Y)-Code is the basis for the PPS.

  • The Navigation Message also modulates the L1-C/A code signal. The Navigation Message is a 50 Hz signal consisting of data bits that describe the GPS satellite orbits, clock corrections, and other system parameters.

• GPS Signals

• The GPS Navigation Message consists of time-tagged data bits marking the time of transmission of each subframe at the time they are transmitted by the SV. A data bit frame consists of 1500 bits divided into five 300-bit subframes. A data frame is transmitted every thirty seconds. Three six-second subframes contain orbital and clock data. SV Clock corrections are sent in subframe one and precise SV orbital data sets (ephemeris data parameters) for the transmitting SV are sent in subframes two and three. Subframes four and five are used to transmit different pages of system data. An entire set of twenty-five frames (125 subframes) makes up the complete Navigation Message that is sent over a 12.5 minute period.

• Data frames (1500 bits) are sent every thirty seconds. Each frame consists of five subframes.

• Data bit subframes (300 bits transmitted over six seconds) contain parity bits that allow for data checking and limited error correction.

• Navigation Data Bits

• Clock data parameters describe the SV clock and its relationship to GPS time.

• Ephemeris data parameters describe SV orbits for short sections of the satellite orbits. Normally, a receiver gathers new ephemeris data each hour, but can use old data for up to four hours without much error. The ephemeris parameters are used with an algorithm that computes the SV position for any time within the period of the orbit described by the ephemeris parameter set.

• Sample Ephemeris and Clock Data Parameters

• SV Ephemeris Parameter to SV Position Algorithm

• SV Clock Parameter to SV Clock Correction Algorithm

• Almanacs are approximate orbital data parameters for all SVs. The ten-parameter almanacs describe SV orbits over extended periods of time (useful for months in some cases) and a set for all SVs is sent by each SV over a period of 12.5 minutes (at least). Signal acquisition time on receiver start-up can be significantly aided by the availability of current almanacs. The approximate orbital data is used to preset the receiver with the approximate position and carrier Doppler frequency (the frequency shift caused by the rate of change in range to the moving SV) of each SV in the constellation.

• Sample Almanac Parameters

• Each complete SV data set includes an ionospheric model that is used in the receiver to approximates the phase delay through the ionosphere at any location and time.

• Sample Ionospheric Parameters

• Each SV sends the amount to which GPS Time is offset from Universal Coordinated Time. This correction can be used by the receiver to set UTC to within 100 ns.

• Sample UTC Parameters

• Other system parameters and flags are sent that characterize details of the system.

• Code Phase Tracking (Navigation)

• The GPS receiver produces replicas of the C/A and/or P (Y)-Code. Each PRN code is a noise-like, but pre-determined, unique series of bits.

• The receiver produces the C/A code sequence for a specific SV with some form of a C/A code generator. Modern receivers usually store a complete set of precomputed C/A code chips in memory, but a hardware, shift register, implementation can also be used.

• C/A Code Generator

• The C/A code generator produces a different 1023 chip sequence for each phase tap setting. In a shift register implementation the code chips are shifted in time by slewing the clock that controls the shift registers. In a memory lookup scheme the required code chips are retrieved from memory.

• C/A Code Phase Assignments

• The C/A code generator repeats the same 1023-chip PRN-code sequence every millisecond. PRN codes are defined for 32 satellite identification numbers.

• C/A Code PRN Chips

• The receiver slides a replica of the code in time until there is correlation with the SV code.

• Correlation Animation (250k)

• Short PRN Code Segment

• If the receiver applies a different PRN code to an SV signal there is no correlation.

• No PRN Correlation

• When the receiver uses the same code as the SV and the codes begin to line up, some signal power is detected.

• Partial PRN Correlation

• As the SV and receiver codes line up completely, the spread-spectrum carrier signal is de-spread and full signal power is detected.

• Full PRN Correlation

• A GPS receiver uses the detected signal power in the correlated signal to align the C/A code in the receiver with the code in the SV signal. Usually a late version of the code is compared with an early version to insure that the correlation peak is tracked.

• Simplified GPS Receiver Block Diagram

• A phase locked loop that can lock to either a positive or negative half-cycle (a bi-phase lock loop) is used to demodulate the 50 HZ navigation message from the GPS carrier signal. The same loop can be used to measure and track the carrier frequency (Doppler shift) and by keeping track of the changes to the numerically controlled oscillator, carrier frequency phase can be tracked and measured.

• Data Bit Demodulation and C/A Code Control

• The receiver PRN code start position at the time of full correlation is the time of arrival (TOA) of the SV PRN at receiver. This TOA is a measure of the range to SV offset by the amount to which the receiver clock is offset from GPS time. This TOA is called the pseudo-range.

• Pseudo-Range Navigation

• The position of the receiver is where the pseudo-ranges from a set of SVs intersect.

• Intersection of Range Spheres

• Position is determined from multiple pseudo-range measurements at a single measurement epoch. The pseudo range measurements are used together with SV position estimates based on the precise orbital elements (the ephemeris data) sent by each SV. This orbital data allows the receiver to compute the SV positions in three dimensions at the instant that they sent their respective signals.

• Four satellites (normal navigation) can be used to determine three position dimensions and time. Position dimensions are computed by the receiver in Earth-Centered, Earth-Fixed X, Y, Z (ECEF XYZ) coordinates.

• ECEF X, Y, and Z

• Time is used to correct the offset in the receiver clock, allowing the use of an inexpensive receiver clock.

• SV Position in XYZ is computed from four SV pseudo-ranges and the clock correction and ephemeris data.

• GPS SV and Receiver XYZ

• Receiver position is computed from the SV positions, the measured pseudo-ranges (corrected for SV clock offsets, ionospheric delays, and relativistic effects), and a receiver position estimate (usually the last computed receiver position).

• Pseudo-Range Navigation Solution Example

• Ephemeris Data Set Used in Pseudo-Range Navigation Solution Example

• Three satellites could be used determine three position dimensions with a perfect receiver clock. In practice this is rarely possible and three SVs are used to compute a two-dimensional, horizontal fix (in latitude and longitude) given an assumed height. This is often possible at sea or in altimeter equipped aircraft.

• Five or more satellites can provide position, time and redundancy. More SVs can provide extra position fix certainty and can allow detection of out-of-tolerance signals under certain circumstances.

• Receiver Position, Velocity, and Time

• Position in XYZ is converted within the receiver to geodetic latitude, longitude and height above the ellipsoid.

• Geodetic Coordinates

• ECEF XYZ to Geodetic Coordinate Conversion

• Geodetic to ECEF XYZ Coordinate Conversion

• Latitude and longitude are usually provided in the geodetic datum on which GPS is based (WGS-84). Receivers can often be set to convert to other user-required datums. Position offsets of hundreds of meters can result from using the wrong datum.

• Geodetic Datum Overview, Department of Geography, University of Texas at Austin

• Velocity is computed from change in position over time, the SV Doppler frequencies, or both.

• Time is computed in SV Time, GPS Time, and UTC.

• SV Time is the time maintained by each satellite. Each SV contains four atomic clocks (two cesium and two rubidium). SV clocks are monitored by ground control stations and occasionally reset to maintain time to within one-millisecond of GPS time. Clock correction data bits reflect the offset of each SV from GPS time.

• SV Time is set in the receiver from the GPS signals. Data bit subframes occur every six seconds and contain bits that resolve the Time of Week to within six seconds. The 50 Hz data bit stream is aligned with the C/A code transitions so that the arrival time of a data bit edge (on a 20 millisecond interval) resolves the pseudo-range to the nearest millisecond. Approximate range to the SV resolves the twenty millisecond ambiguity, and the C/A code measurement represents time to fractional milliseconds. Multiple SVs and a navigation solution (or a known position for a timing receiver) permit SV Time to be set to an accuracy limited by the position error and the pseudo-range error for each SV.

• SV Time is converted to GPS Time in the receiver.

• SV Time to GPS Time Data Bits

• GPS Time is a "paper clock" ensemble of the Master Control Clock and the SV clocks. GPS Time is measured in weeks and seconds from 24:00:00, January 5, 1980 and is steered to within one microsecond of UTC. GPS Time has no leap seconds and is ahead of UTC by several seconds.

• Time in Universal Coordinated Time (UTC) is computed from GPS Time using the UTC correction parameters sent as part of the navigation data bits.

• At the transition between 23:59:59 UTC on December 31, 1998 and 00:00:00 UTC on January 1, 1999, UTC was retarded by one-second. GPS Time is now ahead of UTC by 13 seconds.

• UTC from GPS Time

• Sample UTC Parameters

• Carrier Phase Tracking (Surveying)

• Carrier-phase tracking of GPS signals has resulted in a revolution in land surveying. A line of sight along the ground is no longer necessary for precise positioning. Positions can be measured up to 30 km from reference point without intermediate points. This use of GPS requires specially equipped carrier tracking receivers.

• The L1 and/or L2 carrier signals are used in carrier phase surveying. L1 carrier cycles have a wavelength of 19 centimeters. If tracked and measured these carrier signals can provide ranging measurements with relative accuracies of millimeters under special circumstances.

• Tracking carrier phase signals provides no time of transmission information. The carrier signals, while modulated with time tagged binary codes, carry no time-tags that distinguish one cycle from another. The measurements used in carrier phase tracking are differences in carrier phase cycles and fractions of cycles over time. At least two receivers track carrier signals at the same time. Ionospheric delay differences at the two receivers must be small enough to insure that carrier phase cycles are properly accounted for. This usually requires that the two receivers be within about 30 km of each other.

• Carrier phase is tracked at both receivers and the changes in tracked phase are recorded over time in both receivers.

• Carrier Phase Tracking

• All carrier-phase tracking is differential, requiring both a reference and remote receiver tracking carrier phases at the same time.

• Unless the reference and remote receivers use L1-L2 differences to measure the ionospheric delay,  they must be close enough to insure that the ionospheric delay difference is less than a carrier wavelength.

• Using L1-L2 ionospheric measurements and long measurement averaging periods, relative positions of fixed sites can be determined over baselines of hundreds of kilometers.

• Phase difference changes in the two receivers are reduced using software to differences in three position dimensions between the reference station and the remote receiver. High accuracy range difference measurements with sub-centimeter accuracy are possible. Problems result from the difficulty of tracking carrier signals in noise or while the receiver moves.

• Two receivers and one SV over time result in single differences.

• Single Difference Survey

• Two receivers and two SVs over time provide double differences.

• Post processed static carrier-phase surveying can provide 1-5 cm relative positioning within 30 km of the reference receiver with measurement time of 15 minutes for short baselines (10 km) and one hour for long baselines (30 km).

• Rapid static or fast static surveying can provide 4-10 cm accuracies with 1 kilometer baselines and 15 minutes of recording time.

• Real-Time-Kinematic (RTK) surveying techniques can provide centimeter measurements in real time over 10 km baselines tracking five or more satellites and  real-time radio links between the reference and remote receivers.

GPS Error Sources

• GPS errors are a combination of noise, bias, blunders.

• Noise, Bias, and Blunders

  • Noise errors are the combined effect of PRN code noise (around 1 meter) and noise within the receiver noise (around 1 meter).

• Bias errors result from Selective Availability and other factors

  • Selective Availability (SA)

    • SA is the intentional degradation of the SPS signals by a time varying bias. SA is controlled by the DOD to limit accuracy for non-U. S. military and government users. The potential accuracy of the C/A code of around 30 meters is reduced to 100 meters (two standard deviations).

    • The SA bias on each satellite signal is different, and so the resulting position solution is a function of the combined SA bias from each SV used in the navigation solution. Because SA is a changing bias with low frequency terms in excess of a few hours, position solutions or individual SV pseudo-ranges cannot be effectively averaged over periods shorter than a few hours. Differential corrections must be updated at a rate less than the correlation time of SA (and other bias errors).

  • Other Bias Error sources;

    • SV clock errors uncorrected by Control Segment can result in one meter errors.

    • Ephemeris data errors: 1 meter

    • Tropospheric delays: 1 meter. The troposphere is the lower part (ground level to from 8 to 13 km) of the atmosphere that experiences the changes in temperature, pressure, and humidity associated with weather changes. Complex models of tropospheric delay require estimates or measurements of these parameters.

    • Unmodeled ionosphere delays: 10 meters. The ionosphere is the layer of the atmosphere from 50 to 500 km that consists of ionized air. The transmitted model can only remove about half of the possible 70 ns of delay leaving a ten meter un-modeled residual.

    • Multipath: 0.5 meters. Multipath is caused by reflected signals from surfaces near the receiver that can either interfere with or be mistaken for the signal that follows the straight line path from the satellite. Multipath is difficult to detect and sometime hard to avoid.

• Blunders can result in errors of hundred of kilometers.

  • Control segment mistakes due to computer or human error can cause errors from one meter to hundreds of kilometers.

  • User mistakes, including incorrect geodetic datum selection, can cause errors from 1 to hundreds of meters.

  • Receiver errors from software or hardware failures can cause blunder errors of any size.

• Noise and bias errors combine, resulting in typical ranging errors of around fifteen meters for each satellite used in the position solution.

• Geometric Dilution of Precision (GDOP) and Visibility

  • GPS ranging errors are magnified by the range vector differences between the receiver and the SVs. The volume of the shape described by the unit-vectors from the receiver to the SVs used in a position fix is inversely proportional to GDOP.

    • Poor GDOP, a large value representing a small unit vector-volume, results when angles from receiver to the set of SVs used are similar.

    • Poor GDOP
      Good GDOP, a small value representing a large unit-vector-volume, results when angles from receiver to SVs are different.

    • Good GDOP

  • GDOP is computed from the geometric relationships between the receiver position and the positions of the satellites the receiver is using for navigation. For planning purposes GDOP is often computed from Almanacs and an estimated position. Estimated GDOP does not take into account obstacles that block the line-of-sight from the position to the satellites. Estimated GDOP may not be realizable in the field.

    • Good Computed GDOP and Bad Visibility Equals Poor GDOP

  • GDOP terms are usually computed using parameters from the navigation solution process.

  • Pseudo-Range Navigation Solution Example

  • GDOP Computation Example

  • In general, ranging errors from the SV signals are multiplied by the appropriate GDOP term to estimate the resulting position or time error. Various GDOP terms can be computed from the navigation covariance matrix. ECEF XYZ DOP terms can be rotated into a North-East Down (NED) system to produce local horizontal and vertical DOP terms.

  • GDOP Components

    • PDOP = Position Dilution of Precision (3-D), sometimes the Spherical DOP.

    • HDOP = Horizontal Dilution of Precision (Latitude, Longitude).

    • VDOP = Vertical Dilution of Precision (Height).

    • TDOP = Time Dilution of Precision (Time).

  • While each of these GDOP terms can be individually computed, they are formed from covariances and so are not independent of each other. A high TDOP (time dilution of precision), for example, will cause receiver clock errors which will eventually result in increased position errors.

• The idea behind all differential positioning is to correct bias errors at one location with measured bias errors at a known position. A reference receiver, or base station, computes corrections for each satellite signal.

• Because individual pseudo-ranges must be corrected prior to the formation of a navigation solution, DGPS implementations require software in the reference receiver that can track all SVs in view and form individual pseudo-range corrections for each SV. These corrections are passed to the remote, or rover, receiver which must be capable of applying these individual pseudo-range corrections to each SV used in the navigation solution. Applying a simple position correction from the reference receiver to the remote receiver has limited effect at useful ranges because both receivers would have to be using the same set of SVs in their navigation solutions and have identical GDOP terms (not possible at different locations) to be identically affected by bias errors.

• Differential Code GPS (Navigation)

  • Differential corrections may be used in real-time or later, with post-processing techniques.

    • Real-time corrections can be transmitted by radio link. The U. S. Coast Guard maintains a network of differential monitors and transmits DGPS corrections over radiobeacons covering much of the U. S. coastline. DGPS corrections are often transmitted in a standard format specified by the Radio Technical Commission Marine (RTCM).

    • Corrections can be recorded for post processing. Many public and private agencies record DGPS corrections for distribution by electronic means.

    • Private DGPS services use leased FM sub-carrier broadcasts, satellite links, or private radio-beacons for real-time applications.

    • To remove Selective Availability (and other bias errors), differential corrections should be computed at the reference station and applied at the remote receiver at an update rate that is less than the correlation time of SA. Suggested DGPS update rates are usually less than twenty seconds.

  • DGPS removes common-mode errors, those errors common to both the reference and remote receivers (not multipath or receiver noise). Errors are more often common when receivers are close together (less than 100 km). Differential position accuracies of 1-10 meters are possible with DGPS based on C/A code SPS signals.

  • Differential Code-Phase Navigation

  • Errors Reduced by Differential Corrections

• Differential Carrier GPS (Survey)

  • All carrier-phase tracking is differential, requiring both a reference and remote receiver tracking carrier phases at the same time.

  • In order to correctly estimate the number of carrier wavelengths at the reference and remote receivers, they must be close enough to insure that the ionospheric delay difference is less than a carrier wavelength. This usually means that carrier-phase GPS measurements must be taken with a remote and reference station within about 30 kilometers of each other.

  • Special software is required to process carrier-phase differential measurements. Newer techniques such as Real-Time-Kinematic (RTK) processing allow for centimeter relative positioning with a moving remote receiver.

  • Differential Carrier-Phase Positioning

• Common Mode Time Transfer

  • When time information is transferred from one site to another, differential techniques can result in time transfers of around 10 ns over baselines as long as 2000 km.

  • Common-Mode Time Transfer

• Receiver costs vary depending on capabilities. Small civil SPS receivers can be purchased for under $200, some can accept differential corrections. Receivers that can store files for post-procesing with base station files cost more ($2000-5000). Receivers that can act as DGPS reference receivers (computing and providing correction data) and carrier phase tracking receivers (and two are often required) can cost many thousands of dollars ($5,000 to $40,000). Military PPS receivers may cost more or be difficult to obtain.

• Other costs include the cost of multiple receivers when needed, post-processing software, and the cost of specially trained personnel.

• Project tasks can often be categorized by required accuracies which will determine equipment cost.

  • Low-cost, single-receiver SPS projects (100 meter accuracy)

  • Medium-cost, differential SPS code Positioning (1-10 meter accuracy)

  • High-cost, single-receiver PPS projects (20 meter accuracy)

  • High-cost, differential carrier phase surveys (1 mm to 1 cm accuracy)

• GPS Applications, Costs, and Signals

• Global Positioning System Standard Positioning Service Specification, 2nd Edition, June2, 1995. Available on line from United States Coast Guard Navigation Center

• NAVSTAR GPS User Equipment Introduction. 1996. Available on line from United States Coast Guard Navigation Center

• GPS Joint Program Office. 1997. ICD-GPS-200: GPS Interface Control Document. ARINC Research.Available on line from United States Coast Guard Navigation Center

• Hoffmann-Wellenhof, B. H. Lichtenegger, and J. Collins. 1994. GPS: Theory and Practice. 3rd ed.New York: Springer-Verlag.

• Institute of Navigation. 1980, 1884, 1986, 1993. Global Positioning System monographs. Washington, DC: The Institute of Navigation.

• Kaplan, Elliott D. ed. 1996. Understanding GPS: Principles and Applications. Boston: Artech House Publishers.

• Leick, Alfred. 1995. GPS Satellite Surveying. 2nd. ed. New York: John Wiley & Sons.

• National Imagery and Mapping Agency. 1997. Department of Defense World Geodetic System 1984: Its Definition and Relationship with Local Geodetic Systems. NIMA TR8350.2 Third Edition. 4 July 1997. Bethesda, MD: National Imagery and Mapping Agency. Available on line from  National Imagery and Mapping Agency

• Parkinson, Bradford W. and James J. Spilker. eds. 1996. Global Positioning System: Theory and Practice. Volumes I and II. Washington, DC: American Institute of Aeronautics and Astronautics, Inc.

• Wells, David, ed. 1989. Guide to GPS positioning. Fredericton, NB, Canada: Canadian GPS Associates.