1. (x1 - X)² + (y1 - Y)² + (z1 - Z)² = c²(t1 - T-d1)²
2. (x2 - X)² + (y2 - Y)² + (z2 - Z)² = c²(t2 - T-d2)²
3. (x3 - X)² + (y3 - Y)² + (z3 - Z)² = c²(t3 - T-d3)²
4. (x3 - X)² + (y3 - Y)² + (z4 - Z)² = c²(t4 - T-d4)²

GPS Satellite Data Packets
As indicated elsewhere, GPS satellites send out radio signals in packets of digital data, each paket indicating where and when it set out on its journey to the receiver. How does a GPS satellite "know"this information? The main reason that this information is available to the computer on-board the GPS satellite is that there is on-board a very accurate atomic clock, which keeps time to an impressive accuracy, which is regularly compared (and corrected against) even more accurate atomic clocks at groundstations. Thus all the clocks on the GPS satellites are both highly accurate and almost precisely synchronised, essential for accurate position fixes by receivers.
As to the satellites position. Each satellite is accurately tracked by groundstations, so a computer model of the orbit is constantly updated. Regular updates of details of the next orbit are given to each satellite by the groundstations, so that it can compute its actual position to within centimetres at each instant.

Ionospheric Effects on GPS
Light, and radio wave travels at a constant speed in empty space, the velocity of light. Signals from GPS satellites are most effected by their passage through the ionosphere, the region above the atmosphere starting about 50 km up, where most of the very scant gas molecules present are ionised. Thus for instant, rather than oxygen molecules floating around, what's in the ionosphere are oxygen atoms that have lost one electron, called oxygen ions, together with electrons. Overall the ionosphere is neutral, but this mix of ions and elctrons is called a plasma and has several effects on light and radio waves, of which one is simply to delay passage. In order to compensate for the ionosphere, GPS signals are sent at two frequencies, so that for precise navigation, the ionospheric error can be roughly corrected. Nevertheless, even for the best precision GPS location, there's an error of about 30 m due to ionospheric uncertainies. If this figure appears large in comparison to various claims as to GPS accuracy, please note that our discussion is about ordinary GPS, using only a single receiver. Much higher accuracy in relative distances is obtained by Differential GPS.

Operational Details re GPS
The GPS Operational Constellation consists of 24 satellites: 21 navigational SVs and 3 active spares orbit the earth in 12 hour orbits. These orbits repeat 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. The Master Control facility is located at Falcon Air Force Base 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.

Precision of GPS

SPS Standard Positioning System	PPS Precise Positioning Service
SPS Predictable Accuracy 100 meter horizontal accuracy 156 meter vertical accuracy 340 nanoseconds time accuracy	PPS Predictable Accuracy 22 meter Horizontal accuracy 27.7 meter vertical accuracy 100 nanosecond time accuracy

Notes
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.
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.
t is anticipated that the GPS receiver to be used on the Australian scientific satellite FedSat 1, will have PPS capabilities. Thus the GPS system will supply position data required for the magnetometer experiment (NEWMAG) on board. However, more notably, the GPS signals themselves will be used for ionospheric studies, especially of the Southern Ocean region, where no effective models exist for the ionosphere,
The figures for the precision of SPS and PPS given above are credible for periods of low solar activity, but are overly generous for the Southern Pacific region. Errors due to poor modelling of the ionosphere can exceed 30 m.

References

• Australian Ionospheric Prediction Service (IPS) Index Page
• Australian Ionospheric Prediction Service (IPS) Educational material.
• ATMOS-GPS GPS Signals Used for meteorology.
• Navstar Global Positioning System Fact sheet from USAF unit responsible for GPS satellite launches.
• Types of GPS Receivers. From web site of Lowrance Electronicss, GPS receiver manufacturer, but useful.
• Global Positioning System Overview by Peter H. Dana Department of Geography, University of Texas at Austin. Excellent technical review of GPS -- Tertiary level

Prepared by Dr Harvey Cohen, H.Cohen@latrobe.edu.au

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