What is the length of a trail?

Last weekend, I went with the scouts to pine mountain.  We did a loop I’d done before . I expected about a twelve mile loop based on an old GPS track I’d made. The new GPS gave 10.1!  We stayed at the new Jenkins Spring campsite which was excellent.  One the chief volunteers in the Pine Mountain Trail association came by and said that there were a lot of Eagle projects to be done on the trail (which is entirely built and maintained with volunteer labor).

new map of the loop

new map of the loop

What’s going on?

The new GPS uses both the Russian and US satellite systems as well as having a more sensitive antenna and thus simply more satellites.  Therefore the distances are more accurate – with less wobble. Also the two systems have different and less correlated systematic errors so that the estimated precision is more accurate with the combination than with either. Thus the ruler used to measure the distance was smaller than before.

On the surface this is sort of an “anti-fractal”. It’s well known, or at least should be, that as rulers get smaller the distances measured gets larger. One simply measures more of the little in and outs on the curve and hence arrives at a longer distance. With the GPS estimates, which depend on point measurements, the idea is a little different. Here there is a swarm of (we can pretend in the limit of large numbers) normally distributed points drawn around a true track. Hence the calculated distance includes the sum of a bunch of random “wobble vectors”. The spread of the wobble is smaller with the new system and so the distance is more accurate. So the fractal measure in this case is actually in the statistics of the sampling and not the curve being measured.

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