Friday, May 30, 2008

Race Revisions, part I: Uphill Battles

In this fan's opinion, the International Ski Federation has done a good job of developing new race formats for the elite tier of cross-country skiing - the World Cup, World Championships, and Winter Olympic Games. The now-common mass start to races of all distances is perhaps the most important innovation, but the mass start complements two "new" race formats: the individual sprints, run in both techniques over distances that vary from less than a thousand meters up to about 1500 meters, and the pursuit (a.k.a. skiathlon), using both techniques during a fairly long race, most commonly, two 7.5km courses for women or 15km courses for men.*

With these two race formats now reliable - and, I think, exciting - parts of the racing calendar, the time was ripe in 2005 for the FIS to do a bit more experimenting by developing the "Tour de Ski" (so far held in the 2006-2007 and 2007-2008 World Cup seasons) and, at the end of last season, the "mini tour" or "Grand Final."

As run so far, the Tour and Grand Final are, in essence, just radically compressed sets of races: respectively, seven or so discrete events staged over a week and a half and three races in three days. The TdS and GF have included a few twists such as "hunter-style" pursuits in which winning one day's race gives a skier a head start in the next day's race and, even more interestingly, the Tour's "Final Climb" - a colossally demanding distance race up a serious alpine slope, the Alpe Cermis, in Val di Fiemme, Italy.

Though it's still early days for the Final Climb as a tradition, the race has already started to acquire an aura that is matched - or, honestly, exceeded - only by the annual Holmenkollen marathon in Oslo. And no wonder: the Final Climb makes for compelling television, imposes heavy physical demands on racers (in 2007-2008, they had raced seven times in nine days), and promises to shake up the final standings of the Tour de Ski. In January, the women's climb in fact decided the Tour de Ski champion, when the young Swede Charlotte Kalla chased down and then skied away from the then-leader, Virpi Kuitunen of Finland.

Given all this, I think the FIS - and national and regional skiing organizations - ought to capitalize on the hillclimb idea. At the top level, adding a few more hillclimbs to the World Cup calendar, and maybe even to a World Championships or Olympics (if local geography permits it), would go a long way toward making XC skiing stand out as a uniquely grueling sport.

Happily, there seems to be mounting (ha!) interest in hillclimbs as a form of racing distinct from other distance events, such as, say, plain old hilly courses like those at Torino or Canmore. The 10km Horgi Opp race in Norway attracts some big names each April: the 2007 race saw Kristen Skjeldal take the win and wondergirl Therese Johaug finish seventh ahead of some top-notch men, while Anders Soedergren and Marthe Kristoffersen won their respective races in 2008.

There are probably a few other hillclimbs in Europe, but North America saw at least two such events last winter. Out west, the University of Utah staged the "Ski Up" race at the Snowbird resort in April, and out east, the Great Glen Trails in New Hampshire held the 10km "Ski to the Clouds" race in March. Elite skiers won both events: Josh Smullin and Wendy Wagner took the men's and women's Ski Up and Justin Freeman (like Wagner, an ex-Olympian) and Kelsey Allen won the men's and women's Ski to the Clouds. These two winter events followed on the "Climb to the Castle” rollerski race in New York last fall, which included just about every top-tier American ski racer. (Here in the Midwest, we're probably just looking for a big enough hill. We need someone in northern Minnesota or one of the river valleys to stage something like these races...)

Sure, these hillclimbs are few in number - so far. But with more buzz around the Final Climb of the Tour de Ski, and some luck in having that particular race actually matter to the Tour, we could soon see more of these races. In fact, the "father of the Tour de Ski," Vegard Ulvang, recently announced a plan to hold a "Tour of Barents" stage race in northernmost Europe. Among the planned events? A pursuit which will include two ascents of a 520-meter ski slope with a vertical difference of 140 meters.

This is good stuff, and suggestive of possible ways to vary the hillclimb. Mass starts versus interval starts or time trials, perhaps? Climbs followed by descents? Descents followed by climbs? Races that start at the very bottom of the climb, versus races that include some preliminary flats? A hillclimb championship each April? How about a pursuit that requires racers to swap equipment halfway up the slope? (Herringbone heaven!) Or a race up another fabled slope - L'Alpe d'Huez, anyone?

Perhaps - or certainly - the FIS need not go to any of those extremes in order to capitalize on the unique opportunity that I think the hillclimb format offers. Hell, just making sure to stage it at the end of each Tour de Ski - and maybe to plug one more such race into the regular World Cup calendar - would go a long way toward building more interest in and enthusiasm for the special character of our sport.

* Team sprints, a third new-ish race format, have so far been used too infrequently to really matter as a uniquely demanding event. According to the FIS calendar, the TSP has only been run 15 times on the World Cup since 1996, and has only been part of the big events for three scant years, having been staged at the '05 Worlds in Oberstdorf, the '06 Olympics in Torino, and the '07 Worlds in Sapporo. In other words, I think the jury is still out as to whether the TSP is still an experiment or should be considered on a par with the individual sprint or pursuit - much less the traditional races like the 10km/15km individual starts.

Friday, May 2, 2008

Anders Södergren vs Gravity: Mordarbacken Power Output

February 23, 2008, Falun, Sweden. There are just over two kilometers remaining in the Men's 30k Double Pursuit, and Anders Södergren has a problem.

Södergren is the best distance skier in Sweden, and the course is lined with thousands of screaming Swedes, hoping that can win one for the home team. The pursuit has played out like most World Cup mass starts do -- there is a lead group of twenty or so skiers still in contention. Many of them are barely hanging on -- but if they can make it to the last 500m, anything can happen in the final sprint. And that's where Södergren's problem is. He's one of the worst sprinters on the World Cup circuit, and if he enters the stadium with any of his 19 traveling companions, he is likely to disappoint the screaming masses.

Södergren's only chance is to get away on field's last trip over the Mordarbacken (literally, "Murder Hill"), one of the steepest extended climbs on the circuit. He leads the field as they reach the bottom of the climb. He has no choice but to attack from the front, and ski it as hard as he dares after an hour of racing.

[ Photo by oskarlin ]

Södergren punishes the field mightily. Over the top of the climb he has a 5.1 second lead over the World Cup overall leader, Lukas Bauer, with the rest of the field is disarray behind him. On the descent, his five second lead means he has over 50 meters of free space over Bauer. To the casual observer, he has won the race.

But Södergren has skied too hard. Despite the brief recovery afforded by the descent, he's unable to keep his work rate high enough to hold his lead. He's caught first by Bauer, and then by Norway's Tore Asle Gjerdalen. He enters the stadium with these two men, and as the sprint starts it is clear Södergren has nothing left, his muscles saturated with lactic acid and his spirit broken after being caught. He coasts in well behind the other two for third place.

While Södergren ultimately failed to win the pursuit, his ascent of the Mordarbacken was truly impressive, as he had been leading the field for several kilometers before the climb and, despite this, managed to best all challengers by five seconds in under two minutes of climbing.

The video from the race can be seen (WCSN subscription required) here, around 70 minutes into the broadcast.

Mordarbacken rises 72m in about 500m of skiing, (course profile), giving it an average grade of 14.4%. Södergren scaled it in 1:48, an average speed of 16.6 kph (4.61 meters per second).

But, just how fast is this? If you started racing Södergren at the base of the climb, completely fresh, could you have made it to the top with him? After all, he had just skied for an hour, and had to leave enough in reserve to make it to the finish. So just how fast is World Cup Attack Pace?

The best tool for measuring this is power output, a subject that should be familiar to all cyclists. We shall determine how many watts Södergren averaged during this effort -- and then we have a single number to describe how hard he was skiing, that can be compared to any other physical activity.

Södergren's first opponent during the climb is gravity. 72 meters is roughly 22 stories, which means he was climbing a story every five seconds. Imagine running up a stairwell, hitting a new floor every five seconds, for 22 stories -- that's the effort he's putting out.

Since Södergren weighs 78 kilos according to his webpage. He's also wearing top of line skis, boots, and poles, and wearing a full race suit (and the corresponding clothing underneath). Assuming these accessories weigh about 2kg, Södergren moves 80 kg total from the top of the hill to the bottom.

We can use the potential energy equation to see how much work this is:
PE = mgh.

As mentioned before, m = 80kg, g = 9.8 m/s/s, and h = 72m. With these numbers, we find that moving 80 kg to the top of Mordarbacken is worth 56448 joules of potential energy.

However, an 80kg spectator can also walk up the hill in ten minutes, and generate the same potential energy, which is clearly no athletic accomplishment. We need to account for time to get a true picture. Thus we invoke the power equation:

Power = Energy/Time

With Energy = 56448 J and Time = 108 seconds, we get a power output of 522.7 watts.

Any cyclist who owns a power meter will immediately recognize that to be a very big power output.

Unfortunately for Södergren, gravity is not the only problem he must contend with -- snow is far from being a frictionless surface! He undoubtedly has the best wax money can buy on his skis, but it is a warm day (5 C), and no wax can completely remove friction. Thus he has energy sapped from him with every stride by the very medium that allows his sport to exist -- the snow!

To find out how much work he must do against friction, we can use the equation for kinetic friction:

Friction Force = (mu)N, where N is the normal force between a body and the surface friction is being calculated with, and mu is coefficient of friction.

While Södergren is on an incline, he is also digging his ski edges in to prevent slipping backward. We can assume that his normal force is directly onto the snow and thus is equal to mg, or 80 kg * 9.8 m/s/s. This gives a normal force of 784 N.

Next we must calculate the coefficient of friction for a ski waxed with pure flouro on those snow conditions. Unfortunately, there is no way to know what this number is. The most quoted number online for the coefficient of a waxed ski is 0.05 -- but this number comes from a 1976 study. Surely ski waxes have improved since then -- we will tentatively assume that modern pure flouros are twice as good as that, and give him mu = 0.025.

Plugging this in, we find a friction force of (784 * 0.025) 19.6 N. Multiplying this by his rate of speed (4.61 m/s) gives 90.3 W -- thus, he is losing 90 W to the snow he is gliding on.

As he is overcoming snow resistance and gravity simultaneously, we can say that he is putting out (90.3 + 522.7 W) 623 W as he climbs the hill.

But wait! Södergren is not exercising in a vacuum. In exchange for not dying of asphyxiation, he must also overcome air resistance as he skis. At high speeds, this can be a huge limiter, but Södergren is only skiing at a little over 16.6 kph.

To calculate his aerodynamic drag, we can use this equation:

Drag Force = 1/2 r Cd A V^2

Where r = viscosity of the substance being penetrated, Cd = coefficient of drag, A = frontal area, and V = speed.

For air at sea level and 0 C, r = 1.293. It's a bit warmer than that in Falun, and slightly above the ocean, but 1.293 is probably very close.

For the coefficient of drag, we'll rate Södergren at 0.34, which is comparable to a Ferrari F360 Modena (or a Ford Sierra!). In any case, the human body is probably not more aerodynamic than a vehicle specifically engineered against drag.

For his frontal area, we will estimate him at 1.5m high and 0.4m wide. 1.5m is obviously a bit short, but his head is considerably narrower than the rest of his body, so we'll model him as a 1.5m x 0.4m rectangle (arms at his sides) an assume the area of his head makes up for the fact that his legs aren't as wide and torso + arms. In any case, a frontal area of 0.6 square meters is reasonable.

Plugging all these numbers in gives us
(0.5)*(1.293)*(0.6)*(0.34)*(4.61^2) = 2.8 N of aerodynamic drag. Compared to everything else, air resistance is pretty negligible. Nevertheless, multiplying this force by his speed gives us an extra (2.8 * 4.61 =) 12.9 W of power that must be generated to cancel out wind resistance.

Adding all these terms together gives us a average power output of 635.9 W, which Södergren sustained for 108 seconds. This might only be 0.85 of a horsepower, but it's enough to power 12-25 laptops. And more importantly, it gives us the final number that we can compare with other athletic accomplishments.

The sport that has the most power data available for it is cycling, so it is what we shall compare against. Lance Armstrong is often quoted as putting out 500 watts for 20 minutes during climbs in the Tour de France, however, this seems ridiculously high. Bradley McGee's recent 4000m pursuit world record (3 minutes and 30 seconds) was quoted at being 530+ watts.

Both of these comparisons have lower wattage numbers, but are for longer durations. Another resource we can compare with is Dr. Andrew Coggan's watts/kg profile table (here), which shows the watts/kg that athletes of varying fitness levels can sustain for various durations. Södergren's watts/kg come out to be 8.15 -- the highest level on that chart shows a world class cyclist can theoretically put out 11.5 w/kg for 60 seconds and 7.6 w/kg for 5 minutes. Södergren's achievement of 8.15 w/kg when not rested and continuing after clearly put him in the "world-class" range of athlete.

But we already knew this. Looking at the chart linked, we can run down the "1 minute" column to find the type of person that can sustain 8.15 w/kg for 60 seconds. This person could ski with Södergren for a minute (starting fully rested, at the bottom of Mordarbacken) and would crack just over halfway up -- just before the steepest part. Sound like you? The chart rates this guy as a "Jersey Rider" -- that is, someone who is a local bike racer. I believe the analogous class of ski racer would be "citizen racer" -- so if you fancy yourself as better than a decent citizen racer, congratulations, you could hang at least a minute with Anders Södergren!

To make it to the top with him (starting fresh, and collapsing at the line), you'll have to be better still. We see that a mid level international rider can put out 9.7 W/kg for 60 seconds and 6 w/kg for 5 minutes. Interpolating and recognizing the non-linear nature of sustaining power, I'd wager this guy can probably hold 8.15 w/kg for almost 2 minutes. This person compares well with the average college skier -- an exceptional athlete, trains year round, but still a few rungs below world class. So a college skier can make it up Mordarbacken with Södergren, as long as he doesn't ski 28k first and doesn't have to make it to the finish line.

And who could actually make it 28k with the lead pack, then put out 8.15 w/kg for 2 minutes, and then make it to the finish? We know the answer to that -- Anders Södergren.

Of course, Södergren's charge ultimately left him short of the finish. I think it's safe to say that his effort was too hard, and the price he paid for such power output was a corresponding loss of power over the final kilometer.

But the man can't sprint. He had to try. It was good enough to drop all but two of the best skiers in the world -- and it would have killed you or me.