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Tobler's hiking function

Formula for estimating hiking speed

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

Tobler's hiking function is an exponential function determining the hiking speed, taking into account the slope angle. It was formulated by Waldo Tobler. This function was estimated from empirical data of the Swiss cartography professor Eduard Imhof.

Formula

Walking velocity:

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

: \frac{dh}{dx}=S=\tan\theta

where

: W = walking velocity [km/h] : 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°.

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.

Pace

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

: p=0.6e^{\displaystyle3.5\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

Slope(deg)	Gradient(dh/dx)	Speed	Pace	km / h	mi / h	min / km
-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

References

Tobler, Waldo. (February 1993). "Three presentations on geographical analysis and modeling: Non-isotropic geographic modeling speculations on the geometry of geography global spatial analysis". National center for geographic information and analysis.
(2012). "Determining minimum hiking time using DEM". Academia Romana − Filiala Cluj Colectivul de Geografie.
(2010). "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". Archaeopress.
(1950). "Gelaende und Karte". Rentsch, Zurich.
[https://gis.e-education.psu.edu/sites/default/files/capstone/Irtenkauf_596B_20140430.docx Analyzing Tobler's Hiking Function and Naismith's Rule Using Crowd-Sourced GPS Data]. Erik Irtenkauf. The Pennsylvania State University. May 2014
Kay, A.. (2012). "Route Choice in Hilly Terrain". Geogr Anal.
Kay, A.. (November 2012). "Pace and critical gradient for hill runners: an analysis of race records". Journal of Quantitative Analysis in Sports.

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