The time required to complete an expedition or part thereof depends on the mode of travel, the slope, terrain and navigation difficulties along the route, and the age, fitness, experience and level of fatigue of the expedition members.
This section reviews some of the traditional models used for calculating the time for hiking expeditions, before presenting a new Extended Generic model which is adaptable to more expedition types, and which addresses other deficiencies of the traditional models.
Timing Models
Three Traditional Models
One of the most well-known methods used in hiking for estimating route time is the Naismith Rule, which takes the following form,
Naismith assumed a Level Speed value equivalent to 5km/hr, and an Elevation Factor of 1 hour per 600m of climb or descent.
There have been many variations of Naismith Rule, including speed adjustments for navigation and negotiation difficulty of the terrain and for fitness of the expedition members, and added time for fatigue over longer journeys. The Naismith-like template in the Downloads and Templates page uses the more generic relationship,
where TF is a Terrain Factor which reduces the effective speed. The speed is reduced by 40% if walking on a bush track with easy navigation, by 70% on a bush track but with difficult navigation, and by 90% if walking through uncharted, thick bush.
The time added for elevation changes is calculated separately for climb and descent, at a rate of C=1 hour per 600m for climbing, and D=1 hour per 1200m for descending. If the expedition members are fit and experienced, the time is reduced by one-third (Experience=⅔,) and then for routes of durations greater than 5 hours, an extra hour is added (Fatigue=1 if Route Time>5 hours.)
The average slope 𝜎 for a height change H over a distance D is given by
Langmuir maintained the level speed for shallow descents of up to 5° slope. For slopes between 5° and 12°, 10 min is subtracted per 300m. For steeper descents 10 min is added per 300m. Langmuir must have enjoyed running downhill, because this rule forecasts more than double the level speed for descents on slopes of 10° to 12°.
Tobler proposed the following formula for the walking speed W, as a direct function of the average slope,
The effect of the slope dependence for each of the above models is shown here, in which the ratio of the effective speed to that on level ground is plotted against the slope.
The region for which they have the most striking differences is when descending on slopes between 0° and -12°, and all exhibit rapid changes in speed over this range. This behaviour conflicts with experience. Walking speed will change very little for small changes in slope.
A New Speed Formula
The function S(𝜎) describes a curve which varies smoothly from 0 to 1 as the slope 𝜎 increases. The parameters s and w alter the shape of the curve. An increase in s shifts the whole curve to the right, stretching out the region where the curve remains close to zero. An increase in w increases the steepness of the curve, reducing the range of 𝜎 over which the curve increases to one.
Suggested values for the three parameters are r=0.8 (for a minimum speed 20% of the level speed,) s=0.05 (ascent), s=1 (descent) and w=10.
Comparison to real-world data
Irtenkauf has analyzed the Naismith and Tobler models against crowd-sourced GPS tracking data extracted from a global web repository. The figure shows the proposed model overlaid on Figure 6 from the reference.
The scatter in the data precludes an obvious distinction between models, except perhaps to eliminate the Langmuir approach.
Expedition Type Extension
Adventurous expeditions can be undertaken using a variety of modes of travel, of which hiking, or bushwalking, is just one example. They can also involve cycling, trail (horse) riding, ski touring, and canoeing or kayaking. For some types, such as cycling and ski touring, downhill travel will generally be faster than level ground.
This section describes how the S curve model for speed can be adapted to almost any type of expedition. The new model will be referred to as the Extended Generic (EG) model. It includes factors for obstacle avoidance and navigation difficulties, individual fitness and experience, and fatigue.
It is included as one of the timing models in the Excel Route Plan.
Pace
The first step is to change the basis of route segment timing to pace instead of speed. Pace is the reciprocal of speed, and when calculated using,
yields the time in minutes to cover a unit distance. The default speed of 5km/hr in the Naismith Rule is equivalent to a pace of 12 min/km. The advantage of using pace instead of speed is that it is in the numerator of the time equation and so is more intuitively related to the journey time. It is also easy to determine using some simple trials before the expedition.
The Extended Generic Model
The complete equation for a route segment time calculated using the Extended Generic (EG) model is,
The effective pace Peff incorporates the slope impacts by using the inverse of the relationship presented previously for speed,
where Plevel is the pace on level ground.
The pace will increase over the course of the day, as the fatigue of the members increases. In place of the single addition of a whole hour as used in the Naismith Rule, this model applies a linear increase in pace throughout the journey, based on the total travel time up to and including the route segment being evaluated.
The rate F is the time in hours to be added at the five-hour mark.
Excel Route Plan spreadsheet
The EG model is implemented in an Expedition Route Plan spreadsheet, described in the Excel Route Plan page. The tables of parameters for the EG model are described more fully in that page.
References
Irtenkauf, Eric (2014) “Analyzing Tobler’s Hiking Function and Naismith’s Rule Using Crowd-Sourced GPS Data” Retrieved from doc link
Langmuir, Eric (2013) “Mountaincraft and Leadership; A Handbook for Mountaineers and Hillwalking Leaders in the British Isles” 4th Edition ISBN 978-0-9568869-0-3 (book reference)
Naismith, W. W. (1892) “Excursions. Cruach Ardran, Stobinian, and Ben More”, Glasgow Digital Library. Retrieved from doc link
Tobler, Waldo (1993) “Three Presentations on Geographical Analysis and Modeling: Non- Isotropic Geographic Modeling; Speculations on the Geometry of Geography; and Global Spatial Analysis” Retrieved from permalink