Case Study: Global Positioning System

A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community. Comments are welcome by e-mail to the project directors: Susan L. Forman or Lynn A. Steen.

Global Positioning System (GPS) is a satellite-based system of signals that enables specially designed receivers to calculate precise positions on the surface of the earth. The satellite system, operated by the U.S. Department of Defense, became operational in 1995. Four satellites are located in six circular high-altitute (20,200 km) orbits spaced 60 degrees apart and inclined at 55 degrees from the Equator. Each satellite has a period of 12 hours and transmits on two radio frequencies its position and the exact time. Typically, four or five satellites are visible at any time from any point on the surface of the inhabited part of the earth.

Hand-held battery-powered receivers, now widely available, use signals from these satellites to calculate the receiver's three coordinates of latitude, longitude, and altitude. A marvel of engineering and mathematics, GPS is now widely used for business (in agriculture, surveying, transportation), emergencies (for 911 calls and other rescue operations), driving (in cars), and recreation (by motorists, boaters, and hikers).

Some uses of GPS require merely the data from one location, which can then be transfered to a map to identify the position of the receiver. Other uses (for example, in Precision Agriculture) involve the collection of data from many different points in order to construct a map or feed information to a Geographic Information System where the position data are related to other information. Typical receivers also indicate the positions (altitude, azimuth, and identification number) of the satellites from which signals are being received. Thus GPS receivers can also be used to identify and track the satellites as they traverse their orbits.

Examples of Use

GPS systems are being widely used to track anything that moves--from freight trains to airplanes, from tractors to yachts. Farmers use GPS to link geographic information with soil analysis and crop yield; airlines are beginning to use it to guide planes; and freight companies use it to track cargo across the country. Examples:

GPS data relayed by satellites from moving trains and trucks help dispatchers track freight as it moves across the country, thereby enabling customers to plan for deliveries with precision. This system also greatly helps reduce theft of valuable freight.
Wilderness hikers now regularly use GPS receivers to monitor their positions, which they can then plot on a map. Before GPS, they would determine location by taking compass readings to distant landmarks and marking these directions on a map to find the point of intersection.
Since GPS receivers are capable of recording a limited amount of data associated with each position, surveyors who wants to mark locations of trees, bushes, and other natural features on a plot of land can record these items while carrying a GPS receiver around the property.

GPS receivers provide a natural means of displaying latitude and longitude, information that is not normally evident. Experience with GPS reveals the changes in latitude and longitude that correspond to more routine measurements such as kilometers and miles (or feet and yards). Data recorded from a stationary GPS receiver quickly reveal a pattern of random errors that can be used to determine the reliability of the information.

Underlying Mathematics

All GPS activities--from decoding satellite signals to locating the satellites themselves, from interpreting GPS data to creating specialized maps--build on a substantial base of mathematics, especially geometry and statistics. The system itself involves the three-dimensional geometry of the surface of the earth and the orbits of the GPS satellites. Translation of three-dimensional data into two-dimensional maps involves many choices of projection and interpolation. Wise use of the system also requires a thorough understanding of the limitations of its accuracy and of strategies for reducing variability.

If there were no sources of error, the receiver's position could be calculated by simple algebra and spherical trigonometry from two satellite signals. The distance of the receiver from each satellite is determined by the time the signal takes in transit from the satellite to the receiver. Two such distances (measured from different satellites) are sufficient to determine the position of the receiver on the two-dimensional surface of the earth; three satellites give enough data to calculate altitude as well.

This calculation, called trilateration, is akin to triangulation except that it uses the lengths of three lines (rather than the angles from three points) to locate the receiver. Mathematically, it involves solving three simultaneous equations that are based on the Pythagorean formula in three dimensions. It also requires knowledge of conversion factors to transform the satellite information into actual latitude, longitude, and altitude. These calculations are within the scope of standard algebra and geometry courses--and also within the capabilities of a computer chip.

Correcting Errors

Unfortunately, none of the data involved in this ideal trilaterization is perfect. The receiver's clock, although regularly corrected from satellite signals, is not likely to be perfectly aligned with the clocks on the satellites. Moreover, atmospheric conditions will cause slight variations in the speed of the signal as it approaches the surface of the earth. And the satellite may not be precisely where the data it transmits about its position say it is.

GPS receivers use various strategies for reducing these inevitable errors. To correct for discrepencies between the receiver's and the satellites' clocks, all GPS receivers employ data from four satellites instead of three. The additional data make it possible to solve a system of four simultaneous equations in which the unknowns are the three coordinates of position (latitude, longitude, and altitude) and the error in the receiver's clock. (If only three satellites are visible, the receiver omits calculation of altitude and reports only sea-level data for latitude and longitude.)

Errors due to atmospheric conditions and satellite position are not unique to the receiver, but affect everyone in the same region the same way. So in certain areas (e.g., airports, harbors, farming country), ground stations are built whose positions are known precisely. Each station receives GPS data from the overhead satellites, calculates position from them, and compares the results with its known correct position. It then broadcasts a signal that conveys this error to nearby GPS receivers designed to receive and utilize this second, error-correcting signal.

Under normal circumstances, without error-correcting efforts, the GPS system produces position information that is accurate to approximatley 100 meters. The error-correcting systems now in use reduce errors to under 10 meters--quite adequate for most civilian uses.

References

Thompson, Richard B. "Global Positioning Systems: The Mathematics of GPS Receivers." Mathematics Magazine 71:4 (1998) 260-269.

Kopytoff, Verne G. "18 Wheels, G.P.S., and Radar." New York Times, March 4, 1999, D1, D7.

Trimble GPS Information

GIS, GPS, and Remote Sensing (Florida)

Associate of Technology in GIS and GPS (Kansas)

Terrain Analysis (New York)

Supported by the Advanced Technological Educaiton (ATE) program at the National Science Foundation. Opinions and information on this site are those of the authors and do not represent the views of either the ATE program or the National Science Foundation.