Tobler Hiking Function Calculator

Tobler Hiking Function

Geographer Waldo Tobler proposed this empirical formula in 1993 to estimate off-path walking speed as a function of terrain slope. It is widely used in GIS, search-and-rescue planning, and wargame movement models because it captures a key observation: humans walk fastest on a slight downhill, not on flat ground.

Formula

W = 6 × exp(−3.5 × |dh/dx + 0.05|) km/h

Where:

• W = walking speed (km/h)
• dh/dx = vertical change divided by horizontal distance (rise/run, signed)

The signed slope means positive = uphill, negative = downhill. The shift of +0.05 inside the absolute value places maximum speed at a 5% downhill grade.

Speed at Common Slopes

Every figure there comes straight out of the formula above, so you can check any of them on a phone calculator.

The asymmetry is real and well documented in physiology: uphill costs more than downhill at the same gradient. Notice the pairing in the table, though. A 10% descent and flat ground come out identical, and so do +5% and −15%, because the curve is symmetric about its −5% peak rather than about zero.

Worked Example: Mountain Day Hike

An 8-km trail with 600 m of total ascent (75 m/km average grade = 7.5%):

• Average slope: +0.075
• W = 6 × exp(−3.5 × |0.075 + 0.05|) = 6 × exp(−0.4375) = 3.87 km/h
• Time: 8 / 3.87 ≈ 2.07 hours = 2h 4m

Add ~10–15% for stops, navigation, and downhill descent if symmetric.

Off-Path Correction Factor

Tobler’s original paper recommends multiplying by 0.6 for off-trail (bushwhacking) terrain. Trail surface, vegetation, weather, snow, and pack weight all introduce further reductions.

Real-World Tuning

Tobler is average-fit hiker on dry trail. Adjust empirically for:

• Pack weight: subtract ~10% per 10 kg
• Trail condition: ~70–90% for rocky / uneven
• Snow / mud: 50–70% depending on depth
• Group hiking: pace of slowest member

Comparison to Naismith’s Rule

Naismith’s rule: 5 km/h on flat, plus 1 hour per 600 m of ascent. Run the same trail through both and you will not get the same answer, so it is worth knowing why.

Take 12 km with 900 m of ascent. Naismith gives 2.4 + 1.5 = 3.9 hours. Feed Tobler the average grade (900 m over 12 km, so 7.5%) and you get 3.87 km/h, which is 3.1 hours. A 26% gap on the same hike.

Naismith is not the one at fault. The problem is the average grade. A trail that climbs 900 m and comes back down is not a uniform 7.5% slope; it is roughly 6 km at +15% and 6 km at −15%. Work those legs separately and Tobler says 2.0 hours up plus 1.4 hours down, so 3.4 hours - much closer to Naismith, and from the same formula.

This happens because speed falls off faster than linearly as the ground steepens. Averaging the slope first and then looking up one speed always flatters the route. The steeper and more rolling the terrain, the worse the flattery gets.

So: use the rolling mode in the calculator above whenever the route goes up and comes back down, which is most of them. Single-slope mode is for one sustained grade, a climb to a hut or a valley walk out. For a route with several distinct sections, run each one and add them up.

Caveats

The function gives outbound speed. Descents add time but not necessarily proportionally. For dangerous descents (loose rock, exposure), real speed is much slower than Tobler predicts. Always pad your time estimate with breaks and contingency, especially in alpine terrain.