Wednesday, February 24, 2010

How is Hiking Speed Affected by Steepness? A GPS Graphical Analysis

The short answer is we slow down as the trail gets steeper. Yes, I know, pretty obvious.

But how much? And what steepness / speeds are hiking a 10% grade fire road fast the same exertion as hiking up a 30 % grade mountain at 2 mph?

This is one of many things interesting to look at through the experimental data that a GPS device can collect. Take a look at the graph below, which shows my hiking speed versus the steepness of the trail while on a hike up Iron Mountain in the San Gabriels.

The grade is defined as (vertical distance / horizontal distance) at any given moment. Velocity in this case is defined as the horizontal velocity ('true velocity' would be in the diagonal direction - but this does not vary much from the horizontal). The measurement is really speed (velocity being direction dependent).

The left side of the graph is information about my movement downhill, and the right side information about uphill. We'll focus only on the uphill part, although it is interesting to see the symmetry created...

So over a whole hike with many samples, we get a picture of how my velocity changed as a function of the steepness of the trail. For this hike, I was going at a very strong, consistent pace, so we can assume my energy expenditure was similar at different grades. For instance, I was working equally as hard at (0.2, 3 mph) and (0.4, 2 mph). 

However, notice as the grade decrease from 0.2 to 0, my velocity increases, but not that much. It reaches a limit of about 4 mph. Well we are hiking, right? So I was keeping to actually walking and not running, even though it was probably possible that I could run at those speeds. 

And if I wanted to make sure that my exertion was the same at all grades, I'd have to run on the lower ones. So now I present you a similar graph for when I did a trail run in the Santa Monica Mountains. The pace I was running at was similar in exertion to the Iron mountain hike I did.

Now you can see my velocity increases a lot at the lower grades of steepness. Let's combine the two graphs.

Now we are starting to see a clearer picture. My top velocity around 0 grade is a bit above 8 mph, and it drops quickly as the grade increases. It keeps dropping quickly, but eventually the rate at which my speed decreases begins to slow. We notice that at grades above 0.2, my speed decreases, but not by as much.

Now, I want to find an equation that best relates my speed to the steepness, so I could determine that based on the information in this graph.

Such a line could be called an "isoEnergy" line. At every point on the line, I am exerting the same amount of energy. The position of this line will shift outward or inward depending on a person's fitness and the length of the hike. But given an individualized equation, any person could then estimate approximately how long a hike (and/or trail run) would take them.

For example, if a hike is 1000 ft / mile, that is about 0.19 grade and I'd be able to hike the uphill in about 3 mph. If I had a gps track from someone else, I could input the steepness at every point on the trail and determine even more accurately how long it would take.

You will also notice that if just looking at the hiking data, at grades under 0.15, there is a sort of hiking "deadspace". It is an area where I will not be working hard, unless I'm running it. But that's not hiking, and I don't like to do that on most hikes. So some time is lost.

Is this all really important? No, but maybe it justs get you thinking about this stuff. There's plenty more of this to look at too. Maybe you have some suggestions as well?


  1. Really cool stuff Ze!! I might have to invest in a GPS just to play with this kind of data!

    What would be really interesting is, given an individual's data, to determine their optimal hiking grade for a given altitude gain.

    For example, consider Mt Whitney with the Main Mt Whitney Trail vs the Mountaineer's Route.
    MMWT: 6000ft in 11mi, grade = 0.10
    MR: 6000ft in 5mi, grade = 0.23

    Estimated times, based on your data:
    MMWT: 11mi @ 4mph = 2.75 hours
    MR: 5mi @ 2.5mph = 2 hours

    I think it's clear those numbers should be higher due to elevation (next graph: altitude vs speed?), but it would be interesting to see where the "sweet spot" is, to help with route selection, and determine how steep to make switchbacks when in the backcountry.

  2. damn, everyone at SGMDF beat me to my idea. Still, really cool stuff!

  3. Have you looked into Geospatial Science at all? More commonly known as GIS? Spatial statistics use this type of information using linear regression to predict values (velocity, indices, etc.) Great post...keep it up.

  4. No I haven't...not sure how I would incorporate it. Can you give an example? Thanks

  5. Zé, please contact me. A publisher is interested in reprinting your material.

  6. They are almost symetrical, unlike resulting from Langmuir rule or even Tobler's hiking function. According to Langmuir, hiking speed downhill should increase between -5 and -12 deg (grade -0.087 to -0.21) and at -0.21 sharp threshold should appear.

  7. I have been looking for a method to obtain data from a gpx track that you must have used in the plots you have in this post. Did you do all the leg work yourself or did you use a software application/library?