It's time to talk about the accuracy of positioning solutions and how it's measured. The best method is to plot all the measurement results on a map and mark a circle where a portion of these measurements fall. Depending on the portion chosen, it will be called different terms. For example, if we choose a circle where 50% of the points fall, it's called CEP (Circular Error Probable) or CEP50.
Unlike measurement times, we can distinguish between short-term solution stability (how closely points cluster within 3-5-10-30 minutes) and long-term stability. Satellites orbit the Earth in approximately 11 hours and 56 minutes, and after this period, satellite conditions repeat. Long-term stability is usually measured over 24 hours, or two orbits.
If you measure very stable solutions for a few hours, you may encounter solid Earth tides that can affect height by up to 30 centimeters and horizontal measurements by up to 5 centimeters. When measuring over a year or more (and very accurately), you can even detect the drift of continents (or the movement of tectonic plates) and the slow sliding of mountains towards the sea. These are millimeters and centimeters but can be measured. So, when you're given very accurate coordinates (accurate to millimeters), ask when and during which moon phase they were measured.
It's essential to distinguish between solution stability and accuracy. Stability refers to the size of the circle that most solutions fit into. Accuracy, on the other hand, is the distance from the center of this circle to the true coordinates.
In urban canyons, there are situations where accuracy can be worse than stability. Imagine a tall, massive concrete building to the north, like the Great Wall of China, with an angle of about 80 degrees from the horizon to the roof. To the south, there's a smaller building with a 40-degree angle to the roof. You won't be able to receive signals from satellites to the north due to the building blocking them. You also won't directly receive signals from the southern satellites, but you can receive their reflected signals off the northern building. As a result, the southern satellite signals will travel a longer path, causing the positioning solution to shift to the north.
Stability and accuracy largely depend on the parameter "elevation mask angle," which limits the reception of low-elevation satellites. On the one hand, low satellites provide more coordinate information (while high satellites provide more altitude and time information). On the other hand, low satellite signals are noisy due to the troposphere, and often not only direct but also reflected signals are received. The mask varies from 5 to 15 or more degrees for different conditions, and for timing solutions, it's recommended to be even higher, up to 20-30 degrees.
What we've discussed refers to post-measurement (aposteriori) evaluations. The information the receiver provides as solution characteristics is prior data (apriori). These are calculated from residual deviations in the matrix. In other words, the solution is computed using the least squares method, and the discrepancies in measurements from individual satellites compared to the average solution are transformed into an accuracy parameter.
© Eltehs SIA 2023
