Tobler's hiking function

Tobler's hiking function is an exponential function determining the hiking speed, taking into account the slope angle.^[1][2][3] It was formulated by Waldo Tobler. This function was estimated from empirical data of Eduard Imhof.^[4]

Tobler's hiking function – walking speed vs. slope angle chart.

Formula

Walking velocity:

${\displaystyle W=6e^{\displaystyle -3.5\left\vert {\frac {dh}{dx}}+0.05\right\vert }}$

${\displaystyle {\frac {dh}{dx}}=S=\tan \theta }$

where

W = walking velocity [km/h]^[2]

dh = elevation difference,

dx = distance,

S = slope,

θ = angle of slope (inclination).

The velocity on the flat terrain is 5 km / h, the maximum speed of 6 km / h is achieved roughly at -2.86°.^[5]

On flat terrain this formula works out to 5 km/h. For off-path travel, this value should be multiplied by 3/5, for horseback by 5/4.^[1]

Pace

Pace is the reciprocal of speed.^[6][7] For Tobler's hiking function it can be calculated from the following conversion:^[7]

${\displaystyle p=0.6e^{\displaystyle 3.5\left\vert m+0.05\right\vert }}$

where

p = pace [s/m]

m = gradient uphill or downhill (dh/dx = S in Tobler's formula),

Sample values

Pace in minutes per kilometer or mile vs. slope angle for Tobler's hiking function.

Slope (deg)	Gradient (dh/dx)	Speed		Pace
		km / h	mi / h	min / km	min / mi	s / m
-60	-1.73	0.02	0.01	3603.9	5799.9	216.23
-50	-1.19	0.11	0.07	543.9	875.3	32.63
-40	-0.84	0.38	0.24	158.3	254.7	9.50
-30	-0.58	0.95	0.59	63.3	101.9	3.80
-25	-0.47	1.40	0.87	42.9	69.1	2.58
-20	-0.36	2.00	1.24	30.0	48.3	1.80
-15	-0.27	2.80	1.74	21.4	34.5	1.29
-10	-0.18	3.86	2.40	15.6	25.0	0.93
-5	-0.09	5.26	3.27	11.4	18.3	0.68
-2.8624	-0.05	6.00	3.73	10.0	16.1	0.60
0	0	5.04	3.13	11.9	19.2	0.71
1	0.02	4.74	2.94	12.7	20.4	0.76
5	0.09	3.71	2.30	16.2	26.0	0.97
10	0.18	2.72	1.69	22.1	35.5	1.32
15	0.27	1.97	1.23	30.4	49.0	1.83
20	0.36	1.41	0.88	42.6	68.5	2.56
25	0.47	0.98	0.61	60.9	98.1	3.66
30	0.58	0.67	0.41	89.9	144.6	5.39
40	0.84	0.27	0.17	224.6	361.5	13.48
50	1.19	0.08	0.05	771.8	1242.1	46.31

See also

• Naismith's rule
• Preferred walking speed
• Tobler hyperelliptical projection
• Tobler's first law of geography
• Tobler's second law of geography
• Waldo Tobler bibliography