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

References

1. Tobler, Waldo (February 1993). "Three presentations on geographical analysis and modeling: Non-isotropic geographic modeling speculations on the geometry of geography global spatial analysis" (PDF). Technical Report. 93 (1). National center for geographic information and analysis. Retrieved 21 March 2013. Available also in HTML format.
2. Magyari-Sáska, Zsolt; Dombay, Ştefan (2012). "Determining minimum hiking time using DEM" (PDF). Geographia Napocensis. Anul VI (2). Academia Romana − Filiala Cluj Colectivul de Geografie: 124–129. Retrieved 21 March 2013.
3. Kondo, Yasuhisa; Seino, Yoichi (2010). "GPS-aided Walking Experiments and Data-driven Travel Cost Modeling on the Historical Road of Nakasendō-Kisoji (Central Highland Japan)". In Frischer, Bernard (ed.). Making history interactive: computer applications and quantitative methods in archaeology (CAA); proceedings of the 37th international conference, Williamsburg, Virginia, United States of America, March 22−26, 2009. BAR International Series. Oxford u.a.: Archaeopress. pp. 158–165. Retrieved 21 March 2013.
4. Imhof, Eduard (1950). Gelaende und Karte. Rentsch, Zurich.
5. Analyzing Tobler's Hiking Function and Naismith's Rule Using Crowd-Sourced GPS Data. Erik Irtenkauf. The Pennsylvania State University. May 2014
6. Kay, A. (2012). "Route Choice in Hilly Terrain" (PDF). Geogr Anal. 44 (2): 87–108. CiteSeerX 10.1.1.391.1203. doi:10.1111/j.1538-4632.2012.00838.x. Archived from the original (PDF) on 2012-11-14. Retrieved 19 January 2017.
7. Kay, A. (November 2012). "Pace and critical gradient for hill runners: an analysis of race records" (PDF). Journal of Quantitative Analysis in Sports. 8 (4). doi:10.1515/1559-0410.1456. ISSN 1559-0410. Retrieved 19 January 2017.