Experiments with GPS

The biggest mystery in the universe is clearly the male brain. What happens if you leave my husband alone with a GPS receiver? He'll spend several hours measuring the position of a table. For what I'm concerned the table is on the patio. Besides this, I'm every product developers nightmare since instead of reading manuals or tutorials I randomly click or push buttons till I've figured out what they're good for. That's a good procedure to find out every single way to crash the system, but usually not particularly efficient to actually use the device or software. Stefan instead goes and reads the manual!In this case, he's been playing around with a brand new Garmin etrex H GPS receiver. Among other things it allows you to store a time sequence of position measurements, a so-called track, which you can then upload on Google maps to see which part of the forest you've been straying around in. This cute high-tech gadget is shown in the photo. On the display (click to enlarge) you can see a schematic map of the sky with the GPS satellites in view, and the strengths of the signals received from these satellites indicated in the bars at the bottom of the screen. Moreover, the display says that the accuracy of the position measurement is 5 meters.Now, Stefan wanted to figure out the meaning of this accuracy, and if it can be improved by averaging over many repeated measurements. So, he put the GPS receiver on said patio table, and had the device measure the position every second over a duration of a bit less than 3 hours. Then he downloaded the measurement series with several thousand data points, plotted them, and computed the average value. Remarkably, this simple experiment delivered clear evidence that spacetime is discrete! Shown in the figure below is the longitude and latitude of the data points, transformed to metric UTM coordinates, as blue crosses. The yellow dot is the average value, and the ellipse has half-axes of 1 standard deviation. Several thousand measurements correspond to just 16 different positions.This 3-d figure shows the weights of the data points from the above figure:Of course, the position measurements in the time series are not really statistically independent, so one has to be careful when interpreting the result. If one repeats a position measurement after the short time interval of just one second, one expects a very similar result since the signals used likely come from the same satellites which haven't changed their position much. Over the course of time, however, the satellites whose signal one receives are likely to change. To see this effect, Stefan computed the autocorrelation function of the measurement series, shown in the figure below:The autocorrelation function, a function of the time delay τ between two measurements, tells you how long it takes till you can consider the measurements to be uncorrelated. The closer to zero the autocorrelation function, the less correlated the measurements. A positive value indicates the measurements are correlated on the same side of the average value, a negative value that they're correlated in opposite directions.How do we interpret these results? The origin of the discreteness of the measuring points is likely a result of rounding or some artificially imposed uncertainty. (The precision of commercial devices is usually limited as to disable their use for military purposes.) It remains somewhat unclear though whether the origin is in the device's algorithm or already in the signal received. The initial drop of the autocorrelation functions in the figure means that after roughly half an hour, the position measurements are statistically independent. But why the autocorrelation function does not simply fall to zero and instead indicates complete anticorrelation in the y coordinate (latitude) data for a delay of about 1.5 hours and seems to hint at a periodicity is not entirely clear to us - more data and a more sophisticated analysis clearly are necessary.Anyway, finally, to finish this little experiment, Stefan uploads the graphs to blogger. And then asks his wife to weave a story around it. The biggest mystery in the universe..."You do not really understand something unless you can explain it to your grandmother." ~ Albert Einstein