This Christmas I got a little handheld GPS, which I’ve been using mostly for geocaching. As the device acquires signals from the various satellites dutifully orbiting overhead, it displays your position coordinates and a figure indicating the estimated uncertainty. At the beginning of the acquisition or if the view of the sky is poor, it might be something like 50 feet. If you have a clear view of the sky and are receiving signals from many satellites, it might be as low as 11 or 12 feet.*
This figure is called the circular error probable. In essence, there is a 50% probability that your GPS-reported position is within one CEP distance of the actual position. The terminology originates in military bombing of ground targets, and was used in the context of “50% of bombs will land within one CEP of the target”. The assumption is that the position error obeys a Gaussian distribution, which is not such a bad assumption for GPS, which is subject to a lot of small error sources and thus more or less subject to the central limit theorem.
However, we’re measuring a distance on a 2-d plane. Distances can’t be negative, so the standard bell curve doesn’t work. We have to use the 2-dimensional Gaussian function, which quoting Wikipedia is:
This is more complicated than we need. The uncertainties are the same in both directions, and since we have that symmetry we don’t need to bother with x and y anyway. We’ll just assume the function is centered at the origin and use x^2 + y^2 = r^2 and write the function in terms of r. This gives:
We’ll go ahead and call this our Sunday Function. This is positive everywhere and decays smoothly out from the origin, as it should. I’ll graph it, picking A to normalize the function and σ so that the CEP is 1:
The ordinary 1-d Gaussian distribution has its width given by its standard deviation, but we could give the width in terms of something like a CEP, where 50% of the population is within a distance D of the mean. If we did this, we could calculate that 82% are within 2D of the mean, 95.6% are within 3D of the mean, and so forth. But it turns out that the 2-d Gaussian function that characterizes GPS error doesn’t generate the same values, (Because there’s an extra factor of r inside the integral, due to the Jacobian. Might make a good though somewhat technical post later.)
As it happens, if the CEP is 1, there’s already a 93.7% chance you’re within twice the CEP, and a 99.8% chance you’re within three times the CEP. If we graph both the cumulative probability distributions for being within x of the mean for both the 1-d and 2-d Gaussians, they look like this:
Here I’ve picked the widths for each so that both cumulative distributions are equal to 1/2 at x = 1. The 2-d function approaches 1 considerably faster. This is pretty convenient for GPS users, for instance. Even if there’s only a 50% chance you’re within (say) 20 feet of the reported position, it’s highly likely that you won’t be all that much farther away even in the worst case.