Editing Global Positioning System (section)

==== Spheres ====
[[File:2D Trilat Scenario 2019-0116.jpg|thumb|2-D Cartesian true-range multilateration (trilateration) scenario]]

The measured ranges, called pseudoranges, contain clock errors. In a simplified idealization in which the ranges are synchronized, these true ranges represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at the intersection of the surfaces of these spheres; see [[trilateration]] (more generally, true-range multilateration). Signals from at minimum three satellites are required, and their three spheres would typically intersect at two points.<ref>{{cite web|url=http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0203-1-10-instruments/modern.htm|title=Modern navigation|work=math.nus.edu.sg|access-date=December 4, 2018|archive-url=https://web.archive.org/web/20171226024421/http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0203-1-10-instruments/modern.htm|archive-date=December 26, 2017}}</ref> One of the points is the location of the receiver, and the other moves rapidly in successive measurements and would not usually be on Earth's surface.

In practice, there are many sources of inaccuracy besides clock bias, including random errors as well as the potential for precision loss from subtracting numbers close to each other if the centers of the spheres are relatively close together. This means that the position calculated from three satellites alone is unlikely to be accurate enough. Data from more satellites can help because of the tendency for random errors to cancel out and also by giving a larger spread between the sphere centers. But at the same time, more spheres will not generally intersect at one point. Therefore, a near intersection gets computed, typically via least squares. The more signals available, the better the approximation is likely to be.