If you’ve ever tried mountain biking, you’ll know that many factors influence the amount of work you do in the saddle, such as riding speed, terrain and weather. But perhaps most important is the size of the hills you tackle.

The question that Dennis Rapaport, from Bar Ilan University in Israel, tackles today is how to quantify this factor. And it turns out that it’s not so easy to work out.

The obvious way to start is to find some way of determining the cumulative vertical distance travelled during a bike ride. So Rapaport used both GPS and barometric measurements during a lengthy ride to find out how they compare.

His results are a little surprising. While both methods produce altitude measurements at specific waypoints during the ride, he says that the GPS data is much noisier with all kinds of spikes and troughs that are caused by momentary loss of satellite signals and other signal degradation. The effect of this noise is that the GPS method tends to overestimate the cumulative vertical distance travelled.

The barometric data deduces changes in height by measuring changes in atmospheric pressure (assuming that there is no change due to the weather). The device that Rapaport used is periodically calibrated using GPS but this happens only occasionally so the device is not so susceptible to noise. Rapaport believes that this gives a more accurate estimate.

There’s a sting in the tail, however. The final vertical distance estimation depends crucially on the number of waypoint measurements during the trip. Any distance travelled in between is simply averaged. But how many measurements are necessary?

It turns out there’s no easily definable answer, no standard length scale that ought to be used. “Estimating cumulative ascent is an ill-defined task,” concludes Rapaport. The question has no unique answer.

The problem, of course, is analogous to measuring a coastline or any fractal quantity. “At best, a range of estimates can be obtained, hopefully one that is comparatively narrow,” says Rapaport.

And that means that cyclists are destined to never know exactly how much vertical distance they travel.

Ref: arxiv.org/abs/1011.4778: Evaluating Cumulative Ascent: Mountain Biking Meets Mandelbrot