GPS – Less is better

In my previous posting on the issue of GPS track log accuracy, I made a case that the 500-600 points produced by recording every 6 meters (20 feet) or so should be plenty to give an accurate picture of the average 4 mile hike and that the additional 9000 or so points proffered by Garmin’s preferred (to put it mildly!) “1-second distance accumulation” method should not, assuming that the points are accurate, change the track length by any significant amount.

But we have the simple fact that my friends and I have gotten a substantial discrepancy on every single mountain hike we’ve taken with my new Garmin 62stc device, errors ranging from 10-40% – which I would call significant! So which one is right? Or, I should say, which is more accurate? Because nothing is perfect, after all. This is all about the accuracy.

Accuracy

I want to make it clear that I am not meaning to “dis” the accuracy of the GPS unit itself. The reason I didn’t take the thing back to REI after its first hike was that I could tell that it was handling the challenging conditions of hiking in the North Cascades quite well. It acquires a signal very quickly and holds it even under deep forest cover. It does a good job of tracking the switchbacks, reacting to a sudden direction change much faster than my other models. Each individual reading from the device is quite good, given the conditions.

But there will always be errors. Garmin only claims accuracy down to 2-3 meters (7-10 feet) in the very best conditions. If you are a few feet away from where you thought you were it is no big deal. The problems arise when you combine a whole bunch of slightly inaccurate readings:

• Just how big are these errors?
• Will they tend to cancel out?
• Or will they tend to accumulate?

Looking at the last point first: if the errors were even as small as 3-4 inches, 9000 of them would accumulate to about a half mile! So this is not a trivial question.

Error analysis is difficult, by its very nature, especially when you have no independent means of verification. Nice straight lines just don’t exist in the forests on the sides of mountains. The trails tend to be hidden under the trees and thus not visible on aerial or satellite photos, and I don’t have a ball of twine long enough to stretch along the trail. So there will have to be a lot of guesswork and some hand-waving, but I will try to make the estimates very conservative and give the benefit of every doubt.

Assumptions

• It is very, very conservative to say that you might expect any recorded point along the track to be off by a foot or so from the true path.
• Because of the steepness and roughness of the trails, we are often traveling at only 1 or 2 feet per second; we hardly ever average better than 3 fps (2 mph).

So the  “1-second distance accumulation” method will be recording points only a couple of feet apart, about one for each step we take. (I ask you to meditate for a moment on the wisdom of recording points a foot apart when we only know the position of each point to within ten feet… Now back to the conservative assumptions.)

• So it safe to assume that the errors in these points could easily be on the same order of magnitude as the distances being measured

Here come the numbers

Let me compare this to the situation you have when riding in a vehicle at, say, 20 mph (or 30 fps). In one second you travel about 30 feet and record a point that is off by, say 1 foot.

Sometimes the error in that point may be in the direction of travel, so the estimated distance might be as little as 29 feet or as much as 31 feet; but that sort of error will tend to cancel itself out – if this reading is a foot short, the next will likely be a foot long. This is not a problem since we are just interested in the total.

The sort of error that will NOT cancel is when the estimated point is off the true path to one side or the other. If this point is a foot off to the left, then the next might be off to the right, compounding the error rather than canceling. If the next point is off to the left also, then it might not add to the error, but there is no reason to expect that it would cancel it out. And when you go off the track you don’t have to go off to the other side on the next reading, but you do have to get back to the true path eventually if the track is to be worth anything.

What sort of error are we talking about here? The worst case would be where the estimated point is a foot to one side of the true path – I’m picturing a a right triangle with a 30 foot leg (the true distance) and a 1 foot leg (the error). The GPS will be measuring the straight line distance from the starting point to the estimated point, i.e., the hypotenuse of that triangle. Mr. Euclid assures us that the length of that hypotenuse is the square root of the sum of the squares of 30 and 1, that is, the square root of 901 or 30.0166. So the error is that 0.0166 foot or about 0.2 inch.

Which ain’t much. In terms of percentages an error of 0.2 inch in a measurement of 30 feet works out to about 0.057%. This is a very tiny error that explains why Garmin’s strategy works quite well when you use the device in a car, or even a bike.

But this all depends strongly on the fact that the thing being measured is thirty times larger than the error, more than an order of magnitude. If that ratio came down to 10 (one order of magnitude) the percentage error would still be small – about 0.5% – not enough to bother with.

Things go nuts, though, when the error is on the same order as the thing being measured. If the ratio comes down to 2, we get

And it gets quite silly when the ratio is 1:

This is enough to bother a hiker – if you were training for a trip down into the Grand Canyon by making “10 mile” hikes that were in fact only 6 miles long, you could get a nasty surprise when you get to Arizona!

Now I’m not saying that you will always get 40% error on a real life hike. There are lots of reasons why you might do better than this. But back when I was using the default settings on the Garmin I did in fact record some errors of that size. And while the errors are not always that large, they are ALWAYS ON THE HIGH SIDE, just as this analysis would predict. (Again, if an error takes you off the track, you have to make another error to get back on it.)

(By the way, the same discussion applies to the odometer. Assuming that they are using the same method to stoke the odometer, it would explain why I long ago gave up on its consistently, and often laughably, high readings. On one hike the odometer gained a quarter mile while we were sitting down having lunch.)

Conclusion

I know that this does not prove anything. The thing I hope you take away is this: the small errors that are unavoidable in each point of a track will tend to accumulate, which means that sometimes less is better. And given our slow rate of travel and difficult sky conditions, it would be very reasonable to expect the “1-second distance accumulation” method to generate errors on the scale that I’ve observed on the trail.

Ironic Epilogue

The cruel irony is that Garmin’s decision to paste the one-per-second track length into the track log makes almost no difference in most situations, such as in a vehicle. Or when one is power-walking along an urban trail with a clear view of the sky in all directions. So it may not do harm in those situations, it never does any good! This is the easiest kind of cost-benefit analysis:

• Substantial Cost.
• No Benefit. Ever.

In my last note to Garmin, I also made an appeal to their sense of fairness:

If a user believes that the “1 second distance accumulation” is a good idea and gives her good results, then she can choose that option in the track log setup. But if she has a valid reason NOT to sample that many points and wants to make a track with the points spaced further out FOR WHATEVER REASON (but her reason, not yours), then there is currently no way for her to measure the length of that track in the field. You may not like her reason, you may snicker behind your hand that she would want to do something you think is foolish, but it is (pardon the expression) none of your freakin’ business!