## Saturday, July 10, 2010

### Rating Climbs: Introduction

There's been a number of formulas proposed for rating the difficulty of cycling climbs. The Tour de France, as do essentially all stage races held anywhere roads go up and down, rates climbs. However, these ratings depend on many factors other than the intrinsic nature of the road itself, for example position in a stage, position of the stage within the overall race, and on the number and nature of other climbs both within the stage and in other stages. The goal of rating climbs isn't to reproduce the ratings used in the Tour de France or any other race. Rather it's to come up with an intrinsic rating which describes how challenging that climb might be, by virtue of its climbing, for a typical fit rider with typical road gears without stopping to rest. Obviously a 200 mile flat road in the heat is challenging, but it has a climb rating of zero, while a 100 meter road gaining 5 meters is substantially easier, but has a climb rating larger than zero. So we're talking here about simply the climbing component of difficulty.

Everyone who rates climbs basically agrees on two things:
1. longer climbs rate higher than shorter climbs, for the same steepness
2. steeper climbs rate higher than more gradual climbs, for the same distance
Okay, so far so good. But after this simple point, there's less agreement.

Another nice feature of ratings is they are linear. In other words, if I have a long climb and I divide it into two segments, the rating of the total climb is the sum of the ratings of the individual climb segments. This may seem obvious, but it actually presents major difficulties. For example, I may know that a climb gains 1000 meters at an average grade of 5%. However, if I divide the climb into two, it isn't necessarily the case each segment also averages 5%. Maybe the first segment averages 6% and the second averages 4%. If the rating of a climb is a linear function of aggregate variables for that climb, for example a sum of coefficients multiplied by total climbing, total descending, and total distance, linearity will apply. But we may want more than this, so linearity may be tough to preserve.

But do you even want to preserve it? Climbing 1000 vertical meters at 10% may be more difficult than ten climbs at 10% of 100 meters each. Legs fatigue, and fatigue makes additional climbing tougher. This effect can be captured with a linear formula, of course, but this adds complexity. So for the purposes of this work, and in general for most of the rating schemes generally used, linearity is abandoned.

Then there's the issue of how descending during a climb should affect its rating. If descending is ignored, then the rating is based on gross climbing. On the other hand, if descending segments cancel commensurate climbing segments, then the rating is based on net climbing. We all agree that climbing up 1000 meters then descending 1000 meters is generally more difficult than riding without gaining or losing any altitude along the way. But a climb rating isn't necessarily rating the difficulty of riding the road. It's rating something different: the merits of that road as a climb. Descents provide a mental and physical break from the effort. So there is justification in using net climbing as opposed to gross climbing. This is what John Summerson does in his Guide to Climbing by Bike series. An additional benefit of net climbing is it is much easier to determine. All you need to know is the starting and ending altitudes of the climb, whereas for gross climbing, you need a detailed profile, preferably with points spaced no further apart than 50 meters or so, to capture even small dips in the road. Determining gross climbing is worth a whole blog post in itself. So virtually all the climb rating schemes use net climbing: it's simply easier to compile a list of ratings from available data, and in the end there's justification for that approach anyway.

Then there's the issue of gearing. Obviously with low enough gears and a bike set up with suitable geometry so the front wheel doesn't lift off the ground during pedaling, even the steepest roads lose their challenge. So for typical ratings, it is assumed a rider on a relatively standard bike with a relatively standard 2×10 drivetrain. Dan Guttierez made a wonderful bike specifically designed to climb Fargo Street in Los Angeles, a road which averages over 30%. Here's a video of him in action with his machine.

I'll discuss some formula options in my next post on the subject.