16 April 2014

Is it possible to compare mountain ultramarathons? Calculations of the equivalent flat distance as a function of the elevation gain and comparisons of past course record times and speeds

This past weekend the largest Swedish 100 mile trail race, Täby Extreme Challenge (TEC) 100, was held just outside Stockholm. I followed the live race reports on the net and it was thrilling to see the competition fold out between Andreas Falk,   Elov Olsson and Johan Steene. The winner Andreas finished with a great time of 13 hours 34 minutes 31seconds. Still, it was more than one hour slower than Jonas Buud’s really great course record of 12:32 from 2010. Looking at this race record, giving a record speed of 12.9 km/h I started to think about if it would be possible to compare this with record times and speeds on mountain ultramarathons with a higher positive altitude difference (D+). I have previously in a blog post compared a number of mountain ultramarathons with regards to D+/km and noted that Barkley Marathons with a D+/km of 112 really stands out. Among the more famous 100 mile mountain ultramarathon races, most have a D+/km of between 55 to 65, with the exception of Western States Endurance Run (WSER) and Leadville 100, which both have a D+/km or around 30. TEC100 has an even more modest D+/km of 11.3.

Looking in the literature for ways to compare mountain trails with varying climbing distances I came across two different ways to adjust for the climbs and to calculate the equivalent flat distance of mountainous trails with varying elevation profiles. The first method (Method A in figures below) I found is described in Saugy’s paper entitled “Alterations of Neuromuscular Function after the World’s Most Challenging Mountain Ultra-Marathon” comparing Tor des Géants (TDG) with UTMB published in PLoS ONE 2103; 8: e65596. They used the formula flat-equivalent distance = distance (km)+  positive elevation change (meters)/100 and this yields a value of 570 km for the full TDG course and 264 km for UTMB. I have not found the reference for this formula and whether it has been validated or used before.

The second method (Method B in figures below) I found was the so called Naismith's rule, invented by the Scottish mountaineer William W. Naismith already in 1892. According to this rule 3 miles (=15,840 feet) of distance is equivalent in time terms to 2000 feet of climb. In other words, 1 meter of ascent is equivalent to 7.92 (=15840/2000) meters of horizontal distance so time would be a linear function of ascent and distance.  Obviously, Naismith’s rule has been widely used for long time and has been proven useful for walking in particular in hilly terrain with moderate slopes and not too technical terrain. A recent publication tested whether if it would be applicable in shorter fell running competitions and found a good correlation between time and flat equivalent distance using the correction (see figure below) (Scarf “Route choice in mountain navigation, Naismith’s rule, and the equivalence of distance and climb” Journal of Sports Sciences 2007; 25: 719 – 726).
From Scarf 2007
However, in Scarfs study there were only distances up until 80 kilometers included and for the longer races the correlation seemed less good. Another weakness was that the gradients of the races were not compared and that the difficulties with not only uphill, but also downhill, running at steep slopes were not considered. In Naismith’s rule running downhill is the same as running on level ground, something which is clearly not the case in most mountain environments. A third problem is the varying ground terrain and the technicality of the trails. The model has consequently been challenged and in an elegant paper entitled “Pace and critical gradient for hill runners: An analysis of race records” by Kay published in Journal of Quantitative Analysis in Sports. 2012; 8: 1559-0410 (Online) clearly shows that for steeper gradients the Naismith’s rule is clearly not applicable. Based on data from 82 uphill and 14 downhill races he proposes a quartic model where pace can be predicted based on gradient (see figure below). However, again, there are no longer races included in Kay’s analysis either.
From Kay 2012. Pace versus gradient

I have looked at some of the more popular and well-known mountain ultramarathon trail races over 100 miles in order to see whether the method found in Saugy’s paper (Method A) or Naismith’s rule (Method B) can be applied also in longer distances. Not surprisingly, the course record times of 100 mile races varied quite extensively from just under 15 hours to over 25 hours (see table below).

The variation in course records appeared to be at least partly a function of the amount of elevation gain (D+) and both methods to adjust for this appeared actually quite good (see figure below where blue denotes original distance, red distance adjusted to method A and green distance adjusted to method B [Naismith’s rule]).

Just looking at the original record speed as a function of the elevation gain (D+) shows surprisingly good correlation among the races around 100 miles (again, blue original speed, red speed adjusted by method A and green speed adjusted to method B [Naismith's rule]).

There are not many races even longer than 100 miles, but adding Tor des Géants (TDG) to the calculations appear to make the methods less predictable, but still quite good  (in the figures below the same color coding applies to WSER, UTMB, Hardrock100 and TDG). I have not added Barkley Marathons to any of the figures as that race with its extreme course profile and conditions really is an outlier.  
Comparison of WSER, UTMB, Hardrock100 & TDG

Comparison of WSER, UTMB, Hardrock100 & TDG
In summary, when comparing mountain ultramarathons positive elevation gain (D+) appears to be a very important factor in order to determine course record times and speeds. I have done some preliminary calculations of two methods to determine the equivalent flat distance of some of the well-known races and found that both methods are quite good albeit there is certainly room for improvement taking into account for instance average slope of climbs, technical passages and ground conditions, the average altitude over sea level etc. Furthermore, the course record time might not reflect how non-elite runners will perform and how the adjustment method would work for them. Nevertheless, regardless I think that this exercise shows the obvious that if you are planning to run a mountain ultramarathon it is very important to collect climbing distance.  


  1. For what it's worth, Barkley distance is most likely 125 to 130 miles, not the claimed 100 miles, so its D+/km would actually be a bit lower.

    Interesting analysis. Thanks.

    1. Thanks Jon! Interesting, assuming a distance of around 205 kilometers makes the D+/km for Barkley a more reasonable 87.9. The adjusted distances will be 385 km and 348 km with the methods above - and the race suddenly fits much better in the models. It is still somewhat of an outlier though and judging by the finish rate not everything is about D+