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

Right, let's pick this up with the heart of how a GNSS receiver actually measures distance. We've talked about the satellites and the signals, but now we're getting into the geometry and the timing that turns those signals into a position. The very first thing to understand is that the initial measurement of range is called pseudo-range. The word "pseudo" is crucial here. It means "false" or "not genuine." It's called pseudo-range because it has not yet been corrected for receiver clock error. So, when the receiver first calculates how far away a satellite is, that number is slightly wrong, and we call that raw, uncorrected number the pseudo-range. Now, let's think about how the receiver uses these measurements. It uses four SVs — that's four Space Vehicles, which is just the technical term for the satellites. It constructs a three-dimensional fix using the pseudo-ranges from those four SVs. Each range corresponds to a position somewhere on the surface of a sphere with a radius in excess of 10,900 NM. So, if the receiver knows it is a certain distance from one satellite, it knows it is somewhere on the surface of an imaginary sphere centered on that satellite, and that sphere has a radius of more than 10,900 nautical miles. Let's build this up step by step. If you have just one measurement, you're somewhere on the surface of one big sphere. Now, if you take a second measurement from a second satellite, you get a second sphere. The intersection of two range spheres will give a circular position line. So, with two measurements, you're not at a single point — you're somewhere on a circle, which is where those two spheres cross each other. Now, introduce a third range sphere. The intersection of three spheres will produce two positions that are several thousand miles apart. One of those positions will be on or close to the surface of the earth, and the other position will be out in space. So, theoretically, it would be possible to use just three pseudo-ranges to produce a position, by simply rejecting the space position and keeping the one on the earth. But here's the catch, and this is the key point. A fourth range position line is needed because of the way the receiver compensates for receiver time errors. Let me explain why. The receiver has an accurate crystal oscillator to provide time. However, the accuracy of that crystal oscillator does not compare with the accuracy of the SV clocks — the satellites carry atomic clocks, which are vastly more precise. So, there will always be an error in the time measurement, and hence in the computation of range. Furthermore, and this is a clever detail, the receiver clock is deliberately kept in error by a small factor to ensure that the correction process can only go in one direction. This is why the initial calculated range is known as a pseudo-range. Because of this deliberate error, the position lines will not meet in a single point but will form what's called a 'cocked hat' — that's the triangle of error you get when three lines don't intersect perfectly. Let me give you a concrete example of how this error works. If the receiver clock is permanently 1 millisecond fast, then the receiver will overestimate each range by about 162 NM. Think about that: a tiny timing error of one millisecond translates into a huge distance error of 162 nautical miles. So, when the receiver sets about calculating the correct ranges, it knows that it must reduce the pseudo-ranges. It knows the error is always in one direction, so it knows which way to correct. So, to tie it all together: the receiver needs four satellites. Three would give you two possible positions, and you could reject the one in space. But because the receiver's clock isn't perfect, the fourth measurement is what allows the receiver to solve for that clock error and produce a clean, accurate three-dimensional fix. That's the fundamental reason GNSS needs four satellites in view.