## Wednesday, February 23, 2011

### adjusted VAM for hillclimb performance comparison

Strava reports the VAM, or rate of vertical altitude gain, for riders on different climbs. VAM allows comparison of climb efforts on climbs gaining different altitudes. It's correlated relatively strongly with power-to-mass ratio, and is considered a decent gauge of fitness. For example, I may climb Old La Honda while someone in France may climb Col de la Madone. We obviously can't simply compare times, or even average speeds, but the rider with the better VAM is probably the better climber.

However, one issue with VAM is that it is easier to produce a higher VAM on shorter climbs. It is well understood riders can produce a higher average power for shorter durations than for longer ones. So you certainly do not want to provide an unfair advantage to those riding shorter climbs.

Well, whenever one brings up the subject of the time-dependence of power, the CP model is the first thing to pop into mind. The CP model says:

maximum power = critical power + anaerobic work capacity / duration

or

Pmax = CP + AWC / t

A "typical" number for AWC is 90 seconds × CP, so assuming this, I get:

Pmax = CP ( 1 + 90 seconds / t )

The critical power model is very simplistic. It's based on the assumption there is a fixed reservoir of energy which can be fully converted into work over the duration of the effort. Additionally it assumes the "critical power" can be maintained indefinitely. Neither of these assumptions is particularly good: for times shorter than maybe 5 minutes, it becomes difficult to do all of AWC more work than CP times the duration. And for times much over a half hour it becomes difficult to sustain CP. But for climbing durations from 5 minutes to 30 minutes it works fairly well.

If instead of "critical power" (CP) I'm interested in "critical VAM" (cVAM), and I assume VAM is proportional to power (for fixed mass) I can then write:

VAM = cVAM ( 1 + 90 seconds / t )

or since I measure VAM and t :

cVAM = VAM / ( 1 + 90 seconds / t )

This is what I propose you use to rank climb efforts. Instead of "critical VAM" I'll call it "adjusted VAM":

adjusted VAM = VAM / ( 1 + 90 seconds / t )

For example, supposed someone does a climb lasting 6 minutes and climbs at 1800 meters/hour. Then I get

adjusted VAM = VAM / ( 1 + 1/4 ) = 80% VAM = 1440 meters/hour.

Someone else climbed for an hour with a VAM of 1500 meters/hour... his adjusted VAM would be (40 / 41) (1500) meters / hr = 1463 meters / hr.

So the 1500 meters in one hour beats the 1800 meters/hr for 6 minutes.

Considering the limitations of the CP model, getting an especially high adjusted VAM from a very short (less than 5 minutes) or very long (more than 30 minute) climb will be challenging. But it's better to undervalue short efforts than to overvalue them, as you want to encourage riders to make their efforts on substantive climbs, not little bumps which might not even qualify cat 4 in the Tour. And while the challenge may be tougher on very long climbs, with raw, naked VAM it's harder still.

added: It is true that steeper hills produce higher VAMs than more gradual hills due to the increased rolling resistance and wind resistance. However, I explicitly ignore that here. The point isn't to estimate power, the point is to give credit for climbing, and climbing is about increasing altitude quickly. After all, some people can produce enormous power on a dead-flat road, yet that is not an example of "climbing". So no extra points for pushing the wind aside or deforming rubber. If this means riders of steeper hills get more credit, that's not such a bad thing.