## Friday, July 16, 2010

### Rating Climbs: Low-Key Hillclimbs and my simple formula

My last post I showed how to apply a simple difficulty formula to a climb from the 2009 Low-Key Hillclimb series, Bohlman-Norton-Quickert-Kittridge-On Orbit-Bohlman. To summarize, the approach was to find the road segment which maximized the rating, and to use that to rate the net climb. Using the aggregate statistics from what we used for the Low-Key Hillclimb, 593 meters gained over 7.15 km, yielded a rating of 999 meters (the same rating as a very gradual climb gaining that altitude). However, by eliminating the first 160 meters (which are gradual) and the final 1100 meters (which gain only 20 meters altitude) the rating can be improved to 1083 meters, an 8.4% increase.

Here's how the formula rates the climbs from the 2008, 2009, and 2010 Low-Key Hillclimbs. I added in the climb from Kennedy Road, for which we'd applied for a permit last year but were denied. On the x-axis is the altitude gained over the segment used for the rating (which is sometimes less than the full climb length). On the y-axis is the rating calculated for that segment: further up on the plot is considered "more difficult". Ratings and altitude gained have been normalized to Old La Honda Road, the international unit standard for bicycle climbs.

The ratio of the rating to the altitude gained depends on the grade. For sufficiently shallow grades the rating depends only on the altitude gained, so there's a minimum line on the plot below which a rating is mathematically impossible. I've used Old La Honda as a partition between what is a "steep" and what is a "gradual" climb: less than Old La Honda's 7.3% and I consider the climb gradual, steeper and I consider the climb steep.

There are plenty of examples where the rating system ranks higher a climb with less vertical feet gained due to its steepness. The key question is whether these ratings actually reflect the feel of the climb. They key parameter in the rating which affects how steepness influences the rating is that coefficient, for which I chose 10. A higher coefficient and steepness matters more. I suspect no two people would agree on the proper ranking of these climbs, and except for the obvious simplicity of a coefficient of 10, different people would have differing ideas on what the optimal coefficient should be.

A key weakness with the method is it neglects the grade uniformity. Some climbs, like Alba, Quimby, Jamison Creek, Welch Creek, and the Bohlman climbs have exceptionally steep sections which are lost in simply considering the aggregate statistics. To give proper credit to this sort of case I'll propose a formula which uses the full profile data in a later post.

Back to the simple formula: I noted that with Bohlman-Norton the rating was improved by using a subset of the full section of road used by Low-Key. I calculated how for all of the climbs in the plot how the rating of the full climb compared to the rating of the optimal segment. The result was that on average the optimal rating came from using 93.4% of the distance of the road, including 98.1% of the climbing, with the full road length scoring 97.9% the rating of that optimal segment. So had I used only the aggregate statistics from the series, perhaps not having access to full profile data, I'd have scored the climbs on average 2.2% low. Climbs were the rating loss was at least 5% included Kennedy Fire Road (11.7%), Bear Gulch (9.5%), Bohlman-Norton (7.7%), and Metcalf Road (6.1%). On 6 of the 24 climbs the full route was optimal.

So that's enough on this matter. Next I'll look a bit more closely at comparing the Fiets/Summerson formula with my rating.