Uncertainties in GPS Positioning

The Global Positioning System (GPS) is now so widely available that it can be hard to remember how (relatively) new it is, and how remarkable. As one of its early users I feel some edginess in reviewing a book whose title starts with the word “uncertainties”. It was the remarkable accuracy and precision that first struck me.

Uncertainties there are, however, and the current book describes what they are and how they arise. But the title is somewhat misleading because the book is really a technical introduction to GPS with engineering and mathematical details. The treatment of uncertainties and their causes is only one part.

The author first provides some background on the general positioning problem and how it has been approached in previous position fixing and navigation systems. He introduces GPS in this context and describes some of the ways it is being used. Map making and surveying are pretty natural applications. Not so obvious are applications to sports, geophysics, and wildlife management, not to mention the use as a worldwide time reference. GPS also clearly has a role in transportation from civil aviation to intelligent transportation systems and emergency services.

The baseline GPS system has 24 satellites in six earth-centered orbital planes. Each orbital plane has four satellites and at least one spare. The satellites transmit a signal that enables a receiver to compute time-of-flight values from four satellites and thereby find its position in three dimensions. While the geometry of the calculation is not complicated, the time synchronization is complex and begins with a broadcast pseudo-random signal code that is matched and compared with the receiver’s own clock and used to compute time of arrival of each signal. Corrections because of relativistic effects are incorporated to handle the different clock rates of the satellite and the receiver. The author describes the basic elements of this process, and provides some sample GPS computations for idealized problems.

Uncertainties in GPS occur largely because of measurement errors or interpolation errors related to sampling rates. Errors arise because: ionospheric effects cause a propagation delay, the satellite onboard clocks drift, ephemeris errors occur so satellite orbits are known imprecisely, satellite geometry is unfavorable, multipath propagation occurs with reflection from terrestrial objects, and hardware errors occur. A variety of remedies are available and they depend on the application. While GPS is adequate in many circumstances, it is not good enough to guide ships entering or leaving a harbor or for landing an aircraft. The author describes some augmentations of GPS that might make these possible.

One current complaint about GPS comes from runners who have learned that GPS consistently overestimates the distances they run. The statistics of measurement — not GPS itself — are responsible. (For anyone who might be interested, this paper describes the statistical issues and provides a formula that estimates how big the error will be.)

Although the author offers a good deal of information about the GPS system and its uses, the treatment is rather choppy and not particularly well integrated. The level of detail also varies considerably. Just about everything a reader might want to know is here, but it might take a lot of work to put the pieces together.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his