The Global Positioning System (GPS) consists of 24 operating satellites and several spares. Some of these spare satellites are also in use or can readily be activated once an operating satellite becomes dysfunctional. Taking the specific orbits of this many satellites into consideration, an observer can see at least four satellites at any time from any location on the earth.
Methods to use GPS are more or less complicated depending on the desired accuracy and speed of positioning. For a simple example, assume that the orbital positions of the satellites can be accurately computed with respect to the earth at any time. Further assume that a GPS receiver on the ground can measure the distance between a receiver and a satellite for at least three satellites at the same time. By defining the receiver location with three coordinates, such as latitude, longitude and height, one can readily write three equations that relate the three distance observations to the known coordinates of the satellites and the unknown coordinates of the receiver. These three equations can be solved for the three unknowns.
The distance to the satellites is measured by timing signals transmitted by the satellites that travel with the speed of light toward a receiver on the ground. Because of the high speed of light, it is necessary that the instant of signal transmission at the satellite and the instant of signal reception at the receivers antenna be accurately registered in order for the distance and, consequently, the position calculation to be accurate. Satellites in fact carry atomic clocks. Receivers, in contrast, contain inexpensive and therefore less accurate clocks. As a result, we must allow for a timing error to occur as the arriving satellite signal is timed at the receiver. Because the signals arriving at a receiver from all satellites are measured at the same time, the distance measurements are all falsified by the same receiver clock error, which must be calculated in order to determine an accurate position. The complete position determination of the receiver consequently requires four unknowns: the receiver clock error and the three receiver coordinates. Measuring distances to at least four satellites allows one to set up four equations that can be solved for these four unknowns. This leads to the fundamental requirement for a truly global positioning system that at least four satellites be visible at any time from any location on the earth. In practice, receivers observe all visible satellites to determine the best estimates of both the receiver clock and location.
The solution described above is usually called the navigation solution. This type of calculation is implemented in virtually all receivers, including inexpensive handheld devices. In addition to determining position, the time is available to an accuracy of better than one microsecond. GPS receivers are therefore very valuable for time synchronization between remotely located clocks. Time synchronization is an important element in modern communication by Internet, telephone, TV broadcast and many other means.
A closer look at the underlying theories and techniques readily reveals that GPS positioning is not at all simple. This may be of little interest to hikers and motorists who use GPS to get from here to there. But for scientists, GPS is a utility with an endless list of applications, ranging from ionospheric and tropospheric studies to earth crust deformations. The list is equally long for engineering applications. GPS has truly become a national utility.
Originally published on June 3, 2002.