So, if you read part 1 of this topic, you know that we need to determine the navigation satellite's ephemeris and clock errors - because those somehow affect the accuracy of your receiver. As mentioned in the previous post, these errors are the same (mostly) as those broadcast by dgps systems - so how do dgps systems determine the errors? Its actually pretty cool. We know how a receiver works - it moves and it's position updates as it moves around - the position being calculated by the governing navigation equations. Suppose the receiver was fixed at a known (surveyed) location? It's location is no longer an unknown in the navigation equations - now we can take the satellite position as an unknown and solve for it instead. Since the satellite is broadcasting it's precise (predicted) position, we can difference the two and get the ephemeris error. Easy.
What about the clock error? Well, we'd need a 'surveyed' clock state to compare the satellite's clock state against. This is achieved at some monitoring stations (as these fixed stations are known) by using a local atomic clock to compare states against. Other techniques exist as well, such as syncing to the USNO time standard.
Two problems occur when calculating the difference in this manner:
- They difference data becomes spatially decorrelated
- The data is noisy
Let's look at the second problem first. Measured data is noisy by definition. Many types of measurement errors can be included in any measurement. One common resolution to this problem is to use a filter to smooth the measurements. In the case of GPS measurements, a Kalman filter is commonly used in practice. The output of the filter is a smoothed measurement, providing a better estimate of the ephemeris or clock error. We'll use this result to solve the first problem.
The first problem arises out of the fact that the differences are good for that location only. As you start to move away from the surveyed location where the difference was calculated, it becomes less and less effective. This is because the differences were determined using specific line-of-sight paths through the atmosphere - the farther you move away from the station, the more the line-of-sight paths change. How can you overcome this spatial decorrelation?
In practice, at the GPS Master Control Station, (and in other operational global differential networks) this is overcome by using the smoothed estimate for the clock and ephemeris error for any location on the globe. There is a chance that the estimated error will diverge from the observed error, but the more surveyed stations there are supplying observations, the smaller the chance. Judicious measurement weighting and Kalman filter tuning can also reduce that likelihood.
So what do these ephemeris and clock errors look like over time? Which one affects your receiver's accuracy more? Is there a plot?
Here are the errors for the entire constellation for a whole day:
Y axis in meters
Y axis in meters
Notice how some of these errors tend to run off. This is the estimate diverging from the predicted, broadcast ephemeris or clock state. When these divergences get too big, new ephemeris predictions are made and a new upload is sent to the satellite, bringing the age of data back to zero and the ephemeris and clock errors back to zero (or close to zero).
So how do these errors combine to provide the error your receiver will experience?
Ahhhh, now you have to wait until Part 3.
Smooth sailing! (...maybe smoothed sailing? )
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