2x1=10

While plan­ning an eleven-day trekking trip through the Hardan­gervid­da in Nor­way, I came across the age old prob­lem of esti­mat­ing the walk­ing time for a giv­en path on the map. While one is eas­i­ly able to deter­mine the times for the main west-east and north-south routes from a trav­el guide, there sad­ly is no infor­ma­tion about those self-made prob­lems (i.e. cus­tom routes). Obvi­ous­ly, a sim­ple and cor­rect solu­tion needs to be found.

Of course, there is no such thing. When search­ing for hik­ing time rules, two can­di­dates pop up reg­u­lar­ly: Naismith’s rule (includ­ing Tranter’s cor­rec­tions), as well as Tobler’s hik­ing func­tion.

William W. Naismith’s rule — and I couldn’t find a sin­gle sci­en­tif­ic source — is more a rule of thumb than it is exact. It states:

For every 5 kilo­me­tres, allow one hour. For every 600 metres of ascend, add anoth­er hour.

which reads as

where \(|\vec{w}|\) is the walk­ing speed, \(\Delta s\) the length on the hor­i­zon­tal plane (i.e. “for­ward”), \(\Delta a\) the ascend (i.e. the dif­fer­ence in height) and \(\theta\) the slope.

function [w, t, slope] = naismith(length, ascend)
    slope = ascend/length;
    t = length*(1/5) + ascend*(1/0.6);
    w = length./t;
end

That looks like

Inter­est­ing­ly, this implies that if you climb a 3 km moun­tain straight up, it will take you 5 hours. By recog­nis­ing that \(5 \textrm{km} / 0.6 \textrm{km} \approx 8.3 \approx 8\) , the 8 to 1 rule can be employed, which allows the trans­for­ma­tion of any (Nai­smith-ish) track to a flat track by cal­cu­lat­ing

So a track of \(20 \textrm{km}\) in length with \(1 \textrm{km}\) of ascend would make for \(\mathrm{km} + 8 \cdot 1 \mathrm{km} = 28 \mathrm{km}\) of total track length. Assum­ing an aver­age walk­ing speed of \(5 \mathrm{km/h}\) , that route will take \(28 \mathrm{km} / 5 \mathrm{km/h} = 5.6 \mathrm{h}\) , or 5 hours and 36 min­utes. Although quite inac­cu­rate, some­body found this rule to be accu­rate enough when com­par­ing it against times of men run­ning down hills in Nor­way. Don’t quote me on that.

Robert Aitken assumed that 5 km/h might be too much and set­tled for 4 km/h on all off-track sur­faces. Unfor­tu­nate­ly the Nai­smith rule still didn’t state any­thing about descent or slopes in gen­er­al, so Eric Lang­muir added some refine­ments:

When walk­ing off-track, allow one hour for every 4 kilo­me­tres (instead of 5 km). When on a small decline of 5 to 12°, sub­tract 10 min­utes per 300 metres (1000 feet). For any steep­er decline (i.e. over 12°), add 10 min­utes per 300 metres of descent.

Now that’s the stuff won­der­ful­ly non-dif­fer­en­tiable func­tions are made of:

That is:

It should be clear that 12 km/h is an high­ly unlike­ly speed, even on roads.

function [w, t, slope] = naismith_al(length, ascend, base_speed)
    if ~exist('base_speed', 'var')
        base_speed = 4; % km/h
    end

    slope = ascend/length;
    
    t = length*(1/base_speed);
    if slope >= 0
        t = t + ascend*(1/0.6);
    elseif atand(slope) <= -5 && atand(slope) >= -12
        t = t - abs(ascend)*((10/60)/0.3);
    elseif atand(slope) < -12
        t = t + abs(ascend)*((10/60)/0.3);
    end
    
    w = length./t;
end

So Wal­do Tobler came along and devel­oped his “hik­ing func­tion”, an equa­tion that assumes a top speed of 6 km/h with an inter­est­ing fea­ture: It — though still indif­fer­en­tiable — adapts grace­ful­ly to the slope of the ground. That func­tion can be found in his 1993 report “Three pre­sen­ta­tions on geo­graph­i­cal analy­sis and mod­el­ing: Non-isotrop­ic geo­graph­ic mod­el­ing spec­u­la­tions on the geom­e­try of geog­ra­phy glob­al spa­tial analy­sis” and looks like the fol­low­ing:

It boils down to the fol­low­ing equa­tion of the walk­ing speed \(|\vec{w}|\) “on foot­paths in hilly ter­rain” (with \(s=1\) ) and “off-path trav­el” (with \(s=0.6\) ):

where \(\tan(\theta)\) is the tan­gent of the slope (i.e. ver­ti­cal dis­tance over hor­i­zon­tal dis­tance). By tak­ing into account the exact slope of the ter­rain, this func­tion is supe­ri­or to Naismith’s rule and a much bet­ter alter­na­tive to the Lang­muir bug­fix, espe­cial­ly when used on GIS data.

function [w] = tobler(slope, scaling)
    w = scaling*6*exp(-3.5 * abs(slope+0.05));
end

It how­ev­er lacks the one thing that makes the Nai­smith rule stand out: Tranter’s cor­rec­tions for fatigue and fit­ness. (Yes, I know it gets weird.) Sad­ly these cor­rec­tions seem to only exists in the form of a mys­ti­cal table that looks, basi­cal­ly, like that:

where the min­utes are a rather obscure mea­sure of how fast some­body is able to hike up 300 metres over a dis­tance of 800 metres ($20^\circ$). With that table the rule is: If you get into nas­ti­er ter­rain, drop one fit­ness lev­el. If you suck at walk­ing, drop a fit­ness lev­el. If you use a 20 kg back­pack, drop one lev­el. Sad­ly, there’s no equa­tion to be found, so I had to make up one myself.

hours   = [2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24];
fitness = [15, 20, 25, 30, 40, 50];
table   = [1,    1.5,  2,    2.75, 3.5, 4.5,  5.5,  6.75, 7.75,  10,   12.5, 14.5, 17,  19.5, 22, 24;
           1.25, 2.25, 3.25, 4.5,  5.5, 6.7,  7.75, 8.75, 10,    12.5, 15,   17.5, 20,  23,  NaN, NaN;
           1.5,  3,    4.25, 5.5,  7,   8.5,  10,   11.5, 13.25, 15,   17.5, NaN,  NaN, NaN, NaN, NaN;
           2,    3.5,  5,    6.75, 8.5, 10.5, 12.5, 14.5, NaN,   NaN,  NaN,  NaN,  NaN, NaN, NaN, NaN;
           2.75, 4.25, 5.75, 7.5,  9.5, 11.5, NaN,  NaN,  NaN,   NaN,  NaN,  NaN,  NaN, NaN, NaN, NaN;
           3.25, 4.75, 6.5,  8.5,  NaN, NaN,  NaN,  NaN,  NaN,   NaN,  NaN,  NaN,  NaN, NaN, NaN, NaN];

By look­ing at the table and the mesh plot it seems that each time axis for a giv­en fit­ness is log­a­rith­mic.

I did a log-log plot and it turns out that the series not only appear to be log­a­rith­mic in time, but also in fit­ness. By deriv­ing the (log-log-)linear regres­sion for each series, the fol­low­ing equa­tions can be found:

These ear­ly approx­i­ma­tions appear to be quite good, as can be seen in the fol­low­ing lin­ear plot. The last three lines \(t_{30}\) , \(t_{40}\) and \(t_{50}\) how­ev­er begin to drift away. That’s expect­ed for the last two ones due to the small num­ber of sam­ples, but the \(t_{30}\) line was irri­tat­ing.

My first assump­tion was that the \(t_{40}\) and \(t_{50}\) lines sim­ply are out­liers and that the real coef­fi­cient for the time vari­able is the (out­lier cor­rect­ed) mean of \(1.2215 \pm 0.11207\) . This would imply, that the inter­sect coef­fi­cient is the vari­able for fit­ness.

Unfor­tu­nate­ly, this only seems to make things bet­ter in the log-log plot, but makes them a lit­tle bit worse in the lin­ear world.

Equi-dis­tant inter­sect coef­fi­cients also did not do the trick. Well, well. In the end, I decid­ed to give the brute force method a chance and defined sev­er­al fit­ting func­tions for the use with genet­ic algo­rithm and pat­tern search solvers, includ­ing expo­nen­tial, third-order and sig­moidal forms. The best ver­sion I could come up with was

This func­tion results in a least squared error of about 21.35 hours over all data points. The fol­low­ing shows the orig­i­nal sur­face from the table and the syn­thet­ic sur­face from the func­tion.

A max­i­mum devi­a­tion of about 1 hour can be seen clear­ly in the fol­low­ing error plot for the $t_{30}$ line, which real­ly seems to be an out­lier.

For com­par­i­son (here’s the orig­i­nal table), this is the syn­thet­ic cor­rec­tion table: