Genghis,
(1) Are we certain that's how GPS determines velocity, and it's not using some quasi-doppler function? (Archae86 certainly believes so, and that was my general belief also).
That's how survey grade receivers do it. OTOH, those are not specifically designed for navigation. Now that you mention, it is possible that aviation Garmins & similar do use carrier-phase Doppler measurements. Your best bet is to ring the manufacturer of your instrument to find out (in fact, I will do so, out of curiousity)
(2) This assumes the same random variation in errors at each position report, and that the reported position at T2 isn't some function of the reported position at T1, as well as of the actual position?
Position errors can be assumed uncorrelated for this application, even though in fact they aren't. However, that could still be taken care of by using the formula for correlated error propagation if you feel so inclined (that will make your afternoon miserable though). For ET[12], as long as the elapsed period is not too big (i.e. long term clock drift not applicable), those are uncorrelated.
- I do agree with the error propogation method however, at-least assuming that the total errors are normally distributed, which seems a fair assumption.
As you say, it's a "fair" assumption, i.e., close enough to the truth for most practical purposes. In reality, the distribution of 3D autonomous GPS fixes looks a bit like a slightly flattened rugby ball.
Note that the PDF above deals with a different issue, as far as I can tell from the abstract. What they are doing is using raw GPS data (carrier wave measurements) for calculating speed. Whether your receiver works in a similar way or not will determine how applicable their research is to your question. Looks interest nonetheless, so that's my bedtime reading for tonight
Let me know how you get on. I needed to look into this a few years ago for a similar application, but unfortunately I don't think I keep a copy of my notes anymore.