Tobler's hiking function

File:Tobler's hiking function.svg

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

File:Pace, Tobler's hiking function.svg in minutes per kilometer or mile vs. slope angle for Tobler's hiking function.]]

rowspan="2" \| Slope (deg)	rowspan="2" \| Gradient (dh/dx)	colspan = "2" \| Speed	colspan = "3" \| Pace
class="wikitable"
km / h	mi / h	min / km	min / mi	s / m
style="text-align: right;" \| -60	-1.73	0.02	0.01	3603.9	5799.9	216.23
style="text-align: right;" \| -50	-1.19	0.11	0.07	543.9	875.3	32.63
style="text-align: right;" \| -40	-0.84	0.38	0.24	158.3	254.7	9.50
style="text-align: right;" \| -30	-0.58	0.95	0.59	63.3	101.9	3.80
style="text-align: right;" \| -25	-0.47	1.40	0.87	42.9	69.1	2.58
style="text-align: right;" \| -20	-0.36	2.00	1.24	30.0	48.3	1.80
style="text-align: right;" \| -15	-0.27	2.80	1.74	21.4	34.5	1.29
style="text-align: right;" \| -10	-0.18	3.86	2.40	15.6	25.0	0.93
style="text-align: right;" \| -5	-0.09	5.26	3.27	11.4	18.3	0.68
style="text-align: right; font-weight: bold; background-color: #EAECF0;" \| -2.8624	-0.05	6.00	3.73	10.0	16.1	0.60
style="text-align: right; font-weight: bold; background-color: #EAECF0;" \| 0	0	5.04	3.13	11.9	19.2	0.71
style="text-align: right;" \| 1	0.02	4.74	2.94	12.7	20.4	0.76
style="text-align: right;" \| 5	0.09	3.71	2.30	16.2	26.0	0.97
style="text-align: right;" \| 10	0.18	2.72	1.69	22.1	35.5	1.32
style="text-align: right;" \| 15	0.27	1.97	1.23	30.4	49.0	1.83
style="text-align: right;" \| 20	0.36	1.41	0.88	42.6	68.5	2.56
style="text-align: right;" \| 25	0.47	0.98	0.61	60.9	98.1	3.66
style="text-align: right;" \| 30	0.58	0.67	0.41	89.9	144.6	5.39
style="text-align: right;" \| 40	0.84	0.27	0.17	224.6	361.5	13.48
style="text-align: right;" \| 50	1.19	0.08	0.05	771.8	1242.1	46.31

References

{{Reflist|refs=

{{cite journal |last=Tobler |first=Waldo |date=February 1993 |title=Three presentations on geographical analysis and modeling: Non-isotropic geographic modeling speculations on the geometry of geography global spatial analysis |journal=Technical Report |volume=93 |issue=1 |publisher=National center for geographic information and analysis |url=http://escholarship.org/uc/item/05r820mz.pdf |access-date=21 March 2013}} Available also in [http://www.geodyssey.com/papers/tobler93.html HTML] format.

{{cite journal |last1=Magyari-Sáska |first1=Zsolt |last2=Dombay |first2=Ştefan |year=2012 |title=Determining minimum hiking time using DEM |journal=Geographia Napocensis |volume=Anul VI |issue=2 |pages=124–129 |publisher=Academia Romana − Filiala Cluj Colectivul de Geografie |url=http://geographianapocensis.acad-cluj.ro/Revista/volume/nr_2_2012/pdf/Magyari_Dombay.pdf |access-date=21 March 2013}}

{{cite book |last1=Kondo |first1=Yasuhisa |last2=Seino |first2=Yoichi |editor-first=Bernard |editor-last=Frischer |series=BAR International Series |title=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 |publisher=Archaeopress |location=Oxford u.a. |year=2010 |pages=158–165 |chapter=GPS-aided Walking Experiments and Data-driven Travel Cost Modeling on the Historical Road of Nakasendō-Kisoji (Central Highland Japan)| chapter-url=https://www.academia.edu/670947 |access-date=21 March 2013 }}

{{cite book |last1=Imhof |first1=Eduard |year=1950 |title=Gelaende und Karte |publisher=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

{{cite journal |last=Kay |first=A. |title=Route Choice in Hilly Terrain |journal=Geogr Anal |volume=44 |issue=2 |pages=87–108 |date=2012 |url=http://www.lboro.ac.uk/microsites/maths/research/preprints/papers10/10-10.pdf |archive-url=https://wayback.archive-it.org/all/20121114225956/http://www.lboro.ac.uk/microsites/maths/research/preprints/papers10/10-10.pdf |url-status=dead |archive-date=2012-11-14 |doi=10.1111/j.1538-4632.2012.00838.x |access-date=19 January 2017 |citeseerx=10.1.1.391.1203 }}

{{cite journal |last=Kay |first=A. |title=Pace and critical gradient for hill runners: an analysis of race records |journal=Journal of Quantitative Analysis in Sports |volume=8 |issue=4 |date=November 2012 |url=https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/16478/1/PaceCG_published.pdf |doi=10.1515/1559-0410.1456 |issn=1559-0410 |access-date=19 January 2017}}

}}

Category:Hiking

Tobler's hiking function

Formula

Pace

Sample values

See also

References