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Continuously Operating Reference Stations (CORS) locations are often utilized as "substitute" GPS fixes and possess millimeter accuracy. Given enough time and the right operating conditions, most high end GPS receivers (the kind used for surveying, etc) can easily achieve cm accuracy. Military receivers utilize a second signal with shorter bit length to achieve higher accuracy, since the time of travel (satellite to earth) can be determined more accurately. Their signal is encrypted for obvious reasons.
The biggest issue to GPS positioning is the ionosphere - the solar wind causes the ionosphere to become charged, which alters the timing of the GPS signal path. Most GPS algorithms incorporate a standard ionospheric model to "correct" for this issue, but if the iono starts to deviate from the model, more error will insue. As we start to enter the 11 year sun spot cycle, more extensive solar storms = a wild(er) iono = degraded GPS. In some instances, the iono storms can become so large as to temporarily block the GPS signal. GPS will generally worse around 12-3 pm, and much better at midnight for this reason as well.
Other big problems are multi-path, where the signal bounces off things like buildings or trees before finally arriving at your receiver (lengthening the time of arrival), and the geometry of the satellites relative to your position - better to have them widely spread out than clustered together. Again, various algorithms have different ways of mitigating these isses, with varying results. In general, any decent receiver should locate you within 10 meters or less; in most cases, within 3 meters.
GPS is sampling at a rate of ~1 reading per second and determing position at point A and then at point B. The time of the two readings is also determined based upon an oscillating crystal. Velocity is therefore simple distance / time. The accuracy depends upon the quality of both the time estimate and the position estimate. If you are going straight at constant velocity, the average of many readings becomes pretty good. If you just drive around in a circle, the accuracy will be less. At stong accelerations/ decellerations the "action" is happening faster than the 1 second sampling rate and the unit will have trouble getting a good estimate. A decent GPS unit should give a pretty close result to the speedometer and give an indication that one (or the other) is WAY off, but I wouldn't utilize it to calibrate a speedometer or challenge a ticket! (Although with a good lawyer anything is possible :-/ )
