Global Navigation Satellite Systems (GNSS) — Page 316, Lesson 311

Let's pick this up right where the geometry of the fix gets interesting. We've got the pseudo-range concept, and now I want to walk you through exactly why the receiver needs four satellites instead of three, and what that means for the shape of your position. First, let's lock in the definition. The initial measurement of range is called a pseudo-range because it has not yet been corrected for receiver clock error. That's the precise term, and it's the foundation of everything here. The receiver uses four SVs — that's space vehicles, the satellites — and constructs a three-dimensional fix using the pseudo-ranges from those four SVs. Now, here's the geometry. Each range corresponds to a position somewhere on the surface of a sphere, and that sphere has a radius in excess of 10,900 nautical miles. So imagine each satellite measurement drawing a sphere around that satellite. The intersection of two range spheres will give you a circular position line — two measurements put you somewhere on that circle. That's fine, but it's not a fix yet. When you introduce a third range sphere, you get two positions, and they're several thousand miles apart. One of those positions will be on or close to the surface of the earth, and the other will be out in space. So in principle, you could use just three pseudo-ranges to produce a position, by rejecting the space position. That's a real possibility, and the receiver could do it. But here's the catch, and this is the core of why we need the fourth satellite. A fourth range position line is needed because of the way the receiver compensates for receiver time errors. Let me explain that carefully. The receiver has an accurate crystal oscillator to provide time. But that accuracy does not compare with the accuracy of the SV clocks — the satellites carry atomic clocks. So there will always be an error in the time measurement, and hence in the computation of range. Furthermore, the receiver clock is deliberately kept in error by a small factor, to ensure that the correction process can only go in one direction. That's a deliberate design choice, and it's why the initial calculated range is called a pseudo-range. So because of that clock error, the position lines will not meet in a point. Instead, they will form what's called a 'cocked hat' — that's the triangle of uncertainty where the three spheres don't intersect cleanly. Let me give you a concrete example to make this real. If the receiver clock is permanently 1 millisecond fast, then the receiver will overestimate each range by about 162 nautical miles. So when the receiver sets about calculating the correct ranges, it knows that it must reduce the pseudo-ranges. That's the whole point of the fourth satellite — it gives the receiver enough information to solve for that clock error and correct all the ranges down to their true values. So to tie it together: three spheres give you two candidate positions, one on earth and one in space. You could reject the space one. But because of the receiver clock error, the ranges are all off by the same unknown amount, and the position lines won't meet cleanly — they form a cocked hat. The fourth satellite lets the receiver solve for that clock error and collapse the cocked hat into a proper three-dimensional fix. That's why four SVs are the standard for a full GNSS position.