## Wednesday, June 14, 2017

### Hiking Pace and Naismith's Rule

This is a guest post by Professor Paul Pharoah, University of Cambridge.

William W Naismith was a Scottish mountaineer who regularly wrote for the Scottish Mountaineering Club Journal in the 19th Century – a hiking blogger of his day. In 1892 he wrote a paragraph about a hike on Cruach Ardran, Stobinian, and Ben More in the Scottish Highlands. He concluded with: “Distance, ten miles; total climb, 6,300 feet; time, six and a half hours (including short halts). This tallies exactly with a simple formula, that may be found useful in estimating what time men in fair condition should allow for easy expeditions, namely, an hour for every three miles on the map, with an additional hour for every 2,000 feet of ascent.”

Naismith's rule was intended to estimate the total duration of a walk, and as such walks usually start and end at the same point the total ascent will be the same as the total descent. It is generally assumed that the rule should apply to walking uphill and that average walking pace downhill is the same as walking on the flat, even though this is clearly not realistic. The rule is widely used in the mountain walking community in the UK, but it has never been evaluated in conditions or terrain outside the typical fells and mountains of England and Scotland.

Tobler was a Swiss mountaineer and used data from walking in the Swiss Alps to come up with a much more complicated formula that predicts walking pace that will vary depending on the uphill and the downhill gradient. His formula is:
Speed (km per hour) = 6*e {-3.5*abs(S + 0.05)}
where S is the slope of the climb which is negative for a downhill. It predicts a maximum speed of 6 km per hour or 10 minutes per km on a down slope of 5 per cent. For those who prefer miles to kilometers a 10 minute kilometer is a 16 minute mile. This seems to be more sensible as it is easier to walk fast on a gentle down slope, but then gets harder as the slope gets steeper.

The graph shows the predicted walking pace in minutes per kilometer predicted by the Naismith rule and the Tobler function depending on the slope or gradient.

The Naismith rule is widely used in the mountain walking community in the UK, but it has never been evaluated in conditions or terrain outside the typical fells and mountains of England and Scotland. Nor are there studies validating the Tobler function. The availability of GPS recordings of hikes makes it possible to test out the accuracy of the Naismith and Tobler functions using real data from typical hikers from around the world. Three sources of data were used: 49 hikes done in various places worldwide by PP (green), 19 hikes done by Iron Hiker on some of California's most well known peaks (blue) and a pseudo-random set of 98 recordings downloaded from the www.wikiloc.com GPS track sharing website (red). The location of these hikes are shown on the map below.

Latitude, longitude, elevation and time were extracted from each .gpx file and used to calculate the distance and duration of each recorded segment. Segments were then combined to give “chunks” of about 100 metres in length. The net elevation change for each chunk was then calculated as the elevation difference between the first and last point of the chunk. And so for each chunk the walking pace could be calculated and compared with the slope of the chunk. A total of 166 tracks comprising 22,343 chunks of about 100m and 2,242 km of hiking were included in the analysis. The average length of the hikes was 13.9km and the average ascent was 810m with 833m of descent. The next graph shows the walking pace against slope for the 22,343 chunks. Superimposed on the graph are the Naismith and Tobler functions together with a line of best fit (a multi-variable fractional polynomial for the record) based on the Wikiloc data.

The line of best fit confirms the U-shaped relationship between pace and gradient, with the fastest pace being for slopes of -7%. [emphasis - Ed.] This U-shaped relationship between slope and walking pace is more easily seen for all three data sets when the chunks are grouped together according to slope.

The next three graphs show the total walking time for each of the 166 hikes plotted against the total walking time predicted by the best fit function, by Naismith’s rule and by the Tobler function (blue dots are Wikiloc hikes, green are Iron Hiker hikes and pink are PP hikes. A perfect prediction would lie on the dashed line. The times are, of course, well predicted by the best fit function. The Naismith function does fairly well, tending to slightly over estimate time taken (most of the points are below the dashed line). On the other hand, the Tobler function significantly over estimates the hiking time.

We also investigated the effect of altitude on hiking pace, and found that every kilometer of altitude from sea level cost about 65 seconds per kilometer of travel, or for every 1000' of altitude the cost is about 32 seconds per mile. [emphasis - Ed.] Fatigue might also be expected slow us down, but in this data set there was little difference in walking pace at the start and end of the hikes. Of course many other factors will affect walking time including individual fitness, load carried, terrain, conditions underfoot (wet or snow and ice), weather (wind, temperature and humidity). But these could not be taken into account as the data were not available.

This analysis confirms what all hikers know. The speed at which you walk is affected by the slope of the hill and both steep uphill gradients and steep downhill gradients slow you down. We also know that hiking pace is affected by multiple other factors, but the Naismith rule developed over 100 years ago is a useful rule-of-thumb that gives a fairly accurate estimate of total hiking time for a wide range of conditions and terrains for typical hikers.