## Friday, October 22, 2010

### grade histogram comparison of climbs

It occurred to me one way to compare climbs would be to compare the distribution of grades. But a simple histogram wouldn't do, for example how much of a climb is between 10% and 10.1%. What I really care about is how much of a climb is at least 10%.

Then there's the question of what is meant by "how much". You could count up the distance, but for steep roads, distance isn't the real story. 100 meters of 20% road isn't half "as much" as 200 meters of 10%: they climb about the same vertical. The more relevant question is how much altitude is gained at a given grade or steeper.

Then how do you assess grade? If I measure the route profile to the resolution of a pebble, then the grade may tend to be extremely high. So you need to apply numerical smoothing. So I did exactly what I did in rating climbs:
1. Interpolate the profile to a 10 meter mesh
2. Smooth the profile with a characteristic length of 50 meters
3. Estimate how much time it takes "a typical rider" to ride each section of the course
4. Apply a smoothing function with time constant 15 seconds, since steep sections which can be ridden in 15 seconds or less are easier than something of the same grade which is part of a more extended steep segment due to momentum and stored muscular energy.
After this, I added the altitude gained by each small 10-meter piece of the climb in a "bin" determined by its gradient, then accumulated the altitude climbed in these gradient "bins", going from steep gradients to flat. I didn't consider descending segments.

Here I plot some results for climbs from the last two years of the Low-Key Hillclimbs. These are the most highly-rated (using the Low-Key rating formula) climbs the series has done so far: Alba Road, Bohlman-Norton-Quickert-On Orbit-Bohlman, and Welch Creek Road. Each of these climbs is nasty, that nobody would deny. I also add in this Saturday's Sierra Road, and week 2's Old La Honda Road, which is the 100-point reference for the rating system, being the world's canonical cycling climb.

First, Old La Honda is small potatoes compared to these other guys. On the logarithmic scale, the content starts dropping off a serious cliff at around 9%: of the 393 vertical meters gained, only 50 meters exceeds 10%

Sierra Road is a lot tougher than Old La Honda. It gains around 170 meters at 10% or more. still, it's not quite in the league of the remaining three.

Alba Road appears next, a lot tougher than Sierra, but not quite delivering the super-steepness in the same quantity as Welch or Bohlman-Norton. But hey -- how's that? It ranks higher using the Low-Key Hillclimb rating scheme. To tell why this is requires a closer look.

Consider Welch Creek first. Alba gains more vertical (grade at least 0%) and holds that advantage out to grades of at least 8%. Only beyond this point does Welch Creek win. Since total climbing matters, even if there is a bonus given to steeper portions, Alba's increased net climbing wins the day even if Welch Creek has the edge in steep. And Alba certainly is no slouch on steep.

But what about Bohlman-Norton? That an interesting one: Bohlman-Norton gains more altitude starting with almost every grade. How can it therefore rank lower?

The answer is On Orbit Lane, which has the nastiest grades of that climb, contains a descent before returning back to Bohlman, which continues the climb. The rating system deducts points for descents along the way, partally offsetting some of the climbing. The histogram doesn't show the descending portion, nor for that matter does it show perfectly flat sections. it just shoes portions which contribute positively to total climbing. Were that descent not there the Bohlman-On Orbit climbs would win over Alba, hands down.

So Alba wins on the points because it is relatively relentless. And that's how it should be. That little descent at the end of On Orbit Lane helps a lot. But despite it, Bohlman-Norton is still a formidable foe. Many would say it is considerably harder than Alba, and this plot shows why: around 130 vertical meters at at least 15% grade. My legs hurt just looking at the plot. It all comes down to individual preference.

Now I'll compare with two plots from my recent trip to Italy.

In black, I have Alba, the highest rated of the Low-Key climbs. From Italy, first the legendary Ghisallo in orange. You can see we in the San Francisco Bay area can be proud of our climbs here: the Alba, while not a world-class climb of the standards of the Dolomites or the Alps, perhaps, still shows its clear superiority to the climb to the Madonna, at least in terms of raw numbers (sorry: the Italian climb wins by an order of magnitude or more on style, and Alba has its share of style).

Then I plot in red the climb I rode, until I walked, from Dongo to the Rifuge Giovo. Wow! That's some serious grade action there. Add in the rough-hewn gravel surface on the steep portion, and you have a climb which would have been challenging on a mountain bike, let alone a skinny-tired road bike.

Plots like these go a lot deeper than simple ratings at lending insight into what a climb has to offer. But they really require detailed profile data. The profiles typically published in Europe, for example with segments 500 meters or 1 km long, aren't enough. The devil's in the details.