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

Let's pick up right where the mathematics of the fix leaves off. We've just seen that the receiver needs four satellites to solve for its X, Y, Z coordinates and the time error T. Now I want to walk you through exactly how that solution is computed, and then we'll move into the errors that degrade the whole system. So, the receiver has to correct the X, Y, Z coordinates and time to produce the fix. Since each satellite provides each of these elements, the receiver can set up four linear simultaneous equations, each with four unknown quantities — X, Y, Z, and T. It solves these by iteration, which simply means it makes a guess, checks the result, refines the guess, and repeats until the numbers converge. This iterative process removes the receiver time error, and hence the range errors that time error would cause. Here's the key consequence: the use of four satellites provides a three-dimensional fix — your position in space — and an accurate time reference. So together, that's a four-dimensional fix at the receiver. The X, Y, and Z coordinates can now be transposed into latitude and longitude, or any other earth reference system, such as 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, but they need an input of altitude to do it. The altitude simulates a fourth satellite positioned at the centre of the earth. However — and this is the limitation you must remember — the position produced this way will not be as accurate as the full four-dimensional fix. So three satellites plus an assumed altitude is a degraded fallback, not a primary mode. Now let's move to the errors. And I want you to note up front: all the error figures I'm about to give you are at the 95% probability level. That's a statistical statement — it means that 95% of the time, the error will be at or below 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, moon, 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, SV Clock Error. Just as with 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 is 1.5 metres. Third, and this is the big one — Ionospheric Propagation Error. The interaction of the radio energy with the ionized particles in the ionosphere causes the radio energy to be slowed down as it traverses the ionosphere. This is known as 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? The receiver contains an average model of the ionosphere, which is used 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 thence to the receivers. But here's the catch — the propagation path from the satellite to the monitoring station will be very different to that to the receiver. So this is only a partial solution. And here's the critical relationship you need to hold onto: the ionospheric delay is inversely proportional to the square of the frequencies. That means if you double the frequency, the delay drops to a quarter. Because two different frequencies will experience different delays, by measuring the difference in arrival time of the two signals, we can deduce the total delay experienced. That minimises the error and lets us calculate a very accurate range. This is the most significant of the errors. So the whole reason a dual-frequency receiver matters — and you'll see this in the figures — is that it can directly measure and remove the ionospheric delay, which is the dominant error source in the system.