Saturday, July 17, 2010

Rating Climbs: comparison of my formula with Fiets/Summerson formula

The magazine Fiets has a formula for climb rating which is proportional to:

rating = net climbing² / distance + (peak altitude - 1000 meters) / 100,

where the altitude term is set to zero if the altitude is less than 1000 meters. I've already noted this altitude part is silly: it allows a flat road to have a positive rating. For the rest of this blog I'll neglect the altitude part. I already mentioned Summerson uses the same climbing and grade dependence as Fiets, although his altitude adjustment is superior.  The magazine actually normalizes this rating by 10 meters, so a climb averaging 10% for 1000 vertical meters, which my representation ranks as 100 meters, would be rated 10 (no units).

Recall my formula is:

rating = net climbing × ( 1 + [10 × net climbing / distance]² )

Now suppose one climb rates 1% higher in the Fiets rating than another. How would the climbs compare in my rating? Well, if they have the same grade and differ by distance, the answer is they will also differ by 1% in my rating: both ratings scale in proportion to distance if the road grade is the same.

But now consider the case where the net climbing is the same, but the road grade differs. For this, it's useful to differentiate the equations. To simplify the writing, I'll make the following definitions:

c : total climbing
g : total climbing / total distance
r : rating

What I'm interested in here is "if I change the grade by 1%, by what percent does the rating change?" This is written numerically as (g / r) ∂ r / ∂ g .

For the Fiets formula, it's obvious:

(g / r) ∂ r / ∂ g = 1.

Since the r is proportional to g, a 1% change in g (with c fixed) results in a 1% change in r.

For my formula, it's a bit more complex:

(g / r) ∂ r / ∂ g = 2 ×(10 g)² / [ 1 + (10 g)²]

The break-even point, at which a change in grade has the same relative effect in both the Fiets formula and my formula, is found by setting this to one (as it is in the Fiets formula). The solution is trivial:

g = 1 / 10 .

So if the climb averages more than 100 meters gained per kilometer traveled, then my formula places greater priority on grade differences. If it averages less than this, then the Fiets formula places greater priority on grade changes.

This is also evident in a numerical comparison. The following plot shows the ratio of the two ratings versus climbing / distance. At 10%, the ratio is flat, as I just described.

A quick numerical comparison from this list: the Mortirolo (1450 meters), one of the most feared steep climbs in the Tour of Italy ranks slightly easier (136.3 meters) than the Stelvio (137.2 meters), perhaps the longest climb used in that same race. The Mortirolo has a higher average grade (10.5% to 7.5%) but the Stelvio climbs substantially more (1841 m to 1300 m). Note I'm again ignoring the altitude term.

With my formula, it's not as close, with the Stelvio a clear leader, 2863 meters versus 2729 meters for the Mortirolo. With the Fiets formula, the ratings differ by only 0.6%, while the advantage to the longer climb grows to 4.9% with my formula. This is consistent with the analysis I presented, however: since most of the difference in the grades is on the low side of 10%, the Fiets formula is placing more emphasis on steepness.

Another example. Here's the climb ratings versus distance yielding the maximum rating for each of the two rating systems. The Fiets formula places more emphasis on grade, so the steeper climbs tend to be ranked relatively higher, the gradual climbs relatively lower, with Fiets than with my formula. This is most notable with Mount Hamilton, which ranks relatively lowly with the Fiets formula. With Hamilton, the climbing is at a relatively modest grade, but the average is reduced even further by the intermediate descents. Another consequence of this strong emphasis on grade is that it is more likely a climb can rank higher by eliminating part of the beginning and/or end to increase the peak grade at the expense of total climbing.

Fiets rating versus climbing
Fiets rating versus climbing

Of course, my choice of 10% as the cross-over point was somewhat arbitrary, based on my personal experience. For someone living in Fort Lauderdale Florida instead of San Francisco, the comfort level with steep roads might be less, and "crossover grade" might be lower. Or if you ride a fixed gear bike, unable to shift to accomodate steep grades, it might also be lower. But I feel 10% is a generally fair choice assuming riders accustomed to the hills with geared bikes. I know in the examples from the Low-Key Hillclimbs I fear Diablo more than shorter and steeper Alba and Soda Springs, which rank comparable to Diablo under Fiets but which rank lower with my formula. So I give the nod, for my own experience, to my formula.

Still, for an ensemble of riders over roads where the grade may vary considerably about the mean, maybe the Fiets formula's simplicity makes sense. Sure, for me a 4% grade may be no tougher than a 2% grade, and for me a 20% grade may be much, much tougher than 15%, but for a relatively untrained rider or one on a fixie, that 4% grade may be much more challenging than the 2%, or for a rider with a broad-span triple chainring drivetrain, the 20% may not be a big deal. So I'm not convinced my formula is generally better, even if for a specific rider (for example, me) on a specific bike (for example, with a 36-50/11-26) it may better capture the realities of how grade affects the challenge of a climb.

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