Global Navigation Satellite Systems (GNSS) — Page 316, Lesson 313

Right, let's pick this up. We've just worked out how the receiver solves for its position using four satellites. Now I want to show you exactly what happens inside that mathematical process, and then we'll move on to the errors that degrade the whole system. So, when the receiver has signals from four satellites, it has to correct four things: the X, Y, and Z coordinates of its position, and time. Each satellite provides one element of the solution, so the receiver sets up four linear simultaneous equations, each with four unknown quantities: X, Y, Z, and T. It solves these by iteration — that means it keeps refining its guesses, plugging the results back in, until the numbers converge. This process removes the receiver time error, and because time error directly creates range error, it removes that too. That's the key point: using four satellites gives you a three-dimensional position fix and an accurate time reference — so we call it a four-dimensional fix, a 4D fix. Once the receiver has the X, Y, Z coordinates, it can transpose them into latitude and longitude, or any other earth reference system — for example, the UK Ordnance Survey grid — and altitude. Now, there's an important note here. Some receivers can produce a three-dimensional position using only three satellites, if you give them an input of altitude. In that case, the altitude input simulates a fourth satellite positioned at the centre of the earth. But be clear on this: the position produced that way will not be as accurate as the full 4D fix. Now let's move on to GPS errors. And I want you to note something right away: all the error values we're about to discuss are quoted at the 95% probability level. That's a statistical statement — it means that 95% of the time, the error will be within the stated value. First, ephemeris errors. These are errors in the satellite's calculation of its own position, caused by the gravitational effects of the sun, the moon, the planets, and solar radiation. The satellite's position is checked every 12 hours and, where necessary, updated. The maximum error here is 2.5 metres. Second, satellite clock error. Just like the ephemeris, the satellite clock is checked at least every 12 hours, and any error is passed to the satellite to be included in the broadcast. The maximum error here is 1.5 metres. Third — and this is the big one — ionospheric propagation error. When the radio energy passes through the ionosphere, it interacts with the ionized particles there, and this causes the radio energy to be slowed down. We call this the ionospheric delay. The delay depends on two things: the level of ionization and the frequency of the radio waves. The higher the frequency, the smaller the delay. The higher the levels of ionization, the greater the delay. Now, how does the receiver cope with this? The receiver contains an average model of the ionosphere, which it uses to make time corrections to the measured time interval. The state of the ionosphere is continuously checked at the monitoring stations, and the required modifications to the model are regularly updated to the satellites, and from there to the receivers. But here's the limitation: the propagation path from the satellite to the monitoring station will be very different to the path to your receiver. So this is only a partial solution. Here's the crucial relationship: the ionospheric delay is inversely proportional to the square of the frequencies. That's a precise mathematical statement — if you double the frequency, the delay drops by a factor of four. Because two different frequencies experience different delays, we can measure the difference in arrival time of the two signals. From that difference, we can deduce the total delay experienced, thereby minimising the error and calculating a very accurate range. And I want to stress this: this ionospheric error is the most significant of all the GPS errors. That figure shows you exactly why the two-frequency approach matters — only the C/A code is available to civilian users, and the reason two frequencies are so important is precisely this ionospheric correction we just discussed.