Tobler Hiking Function Calculator
Predict walking speed from terrain slope using the Tobler hiking function.
Plan trail times accurately on mixed up-and-down terrain for any route.
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
| Slope | Speed (km/h) | Notes |
|---|---|---|
| +30% (steep up) | 1.65 | Very slow climb |
| +20% | 2.36 | Moderate climb |
| +10% | 3.36 | Mild uphill |
| +5% | 4.01 | Gentle up |
| 0% (flat) | 5.04 | Standard pace |
| −5% | 6.00 | Maximum speed (Tobler peak) |
| −10% | 5.04 | Same as flat |
| −20% | 3.36 | Knees-burning descent |
| −30% | 2.36 | Hard descent (tendons) |
The asymmetry — uphill is harder than downhill at the same gradient — is real and well documented in physiology.
Worked Example — Mountain Day Hike
A 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 + 1 hour per 600 m of ascent. Tobler is more accurate for varied terrain because it integrates slope continuously. For long hikes with mixed up and down, run Tobler segment by segment for best results.
Caveats
The function gives outbound speed — descents back down 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.