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If a GPS unit reports accuracy to 1 meter, does that mean that the receiver's absolute location at any time could be up to 1 meter incorrect in any direction, -or- does it mean that the initial location of the GPS is inaccurate to 1m but every relative movement of the GPS receiver is accurate to sub-millimeter level, relative to the initial position?

In other words, in scenario 1 above, your GPS could show your initial coordinates, and any movement of up to 1 meter in any direction may NOT be reflected as a change in coordinates. In scenario 2, every movement (down to the millimeter) would be reflected, but would all be offset by the initial error value.

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    \$\begingroup\$ The error is absolute \$\endgroup\$ – Scott Seidman May 13 '16 at 15:57
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    \$\begingroup\$ Why would GPS care about it's initial position? \$\endgroup\$ – Eugene Sh. May 13 '16 at 15:59
  • \$\begingroup\$ What's your initial position? The place you were born? :) \$\endgroup\$ – pipe May 13 '16 at 16:02
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Before you ask a question, it would be nice if you performed even a small amount of research. It shows a certain minimum of respect, you know? Googling "gps accuracy" will produce this government document and page 22 will give you a good idea of what you can expect with a good receiver.

With that said, what your gps unit is giving you is resolution, not accuracy. The link suggests that you should expect better than 4 meters accuracy more than 95% of the time, although there will be small number of readings with worse accuracy.

If you take a number of readings at the same location, spread out over several hours, you will see the reported location change as the positions of the gps satellites changes. So the first reading is not (necessarily) the most accurate. Separate readings are not necessarily related to each other, since the locations of the available satellites changes with time. What is related is the accuracy of all gps receivers which are relatively close to each other. This allows for augmentation. A gps receiver at a precisely known location can determine the gps location error (at that specific time) and broadcast this error to any other receivers in the area. These other receivers can use this error to compensate their own readings, and produce accuracies in the cm range.

Read the link.

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    \$\begingroup\$ Differential GPS was much more important when the military was dithering GPS. I'm not quite sure that differential GPS buys you all that much today \$\endgroup\$ – Scott Seidman May 13 '16 at 17:10
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    \$\begingroup\$ One of the main errors sources with GPS is due to atmospheric conditions which change over time but give a fairly consistent offset for an area, having local correction data is still useful for high accuracy. \$\endgroup\$ – Gorloth May 13 '16 at 22:46
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    \$\begingroup\$ @ScottSeidman A local differential GPS references achieves accuracy orders of magnitude beyond what the military-grade satellite-only system was designed to provide to authorized users, as it lets you factor out so many sources of error that are essentially constant in the local area... and it did that even while selective availability was still turned on. Sub-meter accuracy was never a design goal, but differential reference can make it a reality. \$\endgroup\$ – Chris Stratton May 14 '16 at 1:31
  • \$\begingroup\$ Thank you for your answer. My question was not meant to insult or disrespect anyone. It's something I've been pondering for a long time without getting a clear understanding of the situation. \$\endgroup\$ – Ryan Griggs May 14 '16 at 1:53

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