Technical note 3: Collaborative Research Center 1070 – Geoscientific and archaeological research
Field methods
Until the beginning of the 20th century the determination of SET was performed manually (see Valde-Nowak 2002: 65). In the 1970s, pedometers and maps were used (Jarman 1972: 712). Depending on the location of the site, four or more transects in different directions were used to analyse possible exploitation territory by analysing the walking distance (experimental study) within in specific time frame. Based on the experiences and notes from the field survey, ‘time-contour lines’ or ‘isochronic distances’ were drawn on a map (Jarman 1972: 713; Higgs 1975: appendix A; Bailey and Davidson 1983: 93). Obviously, this approach is very time consuming and expensive. In addition, the SET that were defined using this approach are subjective and no longer reproducible today. Bailey and Davidson summarized some of the major difficulties in determining the ‚isochronic distances‘: „In practice the walks were often carried out by students who were unfamiliar with the terrain, unused to walking long distances, and whose transects were influenced one way or another by modern roads and footpaths, barbed wire fences, bulls, unfriendly dogs or landowners, and the location of bars! The original Mt. Carmel study also had to allow for minefields and military manoeuvres“ (Bailey and Davidson 1983: 93). In order to deal with some of these hurdles, Bailey and Davidson combined field surveys with the analysis of topographic maps. In order to do so, they used rules developed by William W. Naismith (1856–1935), which were used by mountaineers to calculate time-distances. In principle Naismith assumes that in two hours on flat ground a distance of 10 km can be covered on foot, for each 300 meters altitude difference an additional half hour is added: „On a map at scale 1:25.000 with contours at 50 m intervals, isochronic limits may be calculated with a pair of compasses. With the compasses set at 1 cm, each unit of distance on the map is equivalent to 3 min. on the ground, and each contour is equivalent to an extra 5 min“ (Bailey and Davidson 1983: 94).
The form of a SET depends on the terrain surrounding a site. In landscapes with a balanced and flat relief SET often have an almost circular shape. As one might expect, in mountainous regions this is not the case. Due to strong relief differences SET tend to have a distorted form (Higgs and Vita-Finzi 1972: 33; Jarman 1972: 710, 713; Bailey and Davidson 1983: 93, 96; Valde-Nowak 2002: 65). Because of that, SET based on time-distance factors provide a more realistic picture of the potentially used catchment area of a site in mountainous regions.
The concept and the development of site catchment analysis had a huge impact in archaeological research5. In Germany, site catchment is often used in order to perform comparative studies on soil preferences in prehistoric time periods (Gringmuth-Dallmer and Altermann 1985; Paetzold 1992; Fries 2005). However, only a few examples can be found, in which archaeologists use SET based on time-distance factors and topography (Valde-Nowak 2002; Uthmeier et al. 2008; Roubis et al. 2011; Cappenberg 2014).
Computational methods
The increasing availability of spatial data and fast developments in computing technologies as well as geographical information systems (GIS) enables implementing time-cost-functions in various ways. Well known commercial GIS software products offer different functions to compute cost surfaces and cost distances to estimate the effort needed to cross a certain landscape (Rogers et al. 2014a; Rogers et al. 2014b). Especially with the increased availability of high resolution as well as large-scale digital elevation models as provided by the Shuttle Radar Topographic Mission (Rodriguez et al. 2005; Farret al. 2007; Jarvis et al. 2008) the methods and their results become more and more interesting for the scientific community to analyse societies, functions and resources. In general, the common workflow for time-cost analysis comprises four steps (Fig. 1). The first step is to create a cost surface based on an input dataset and an arbitrary cost function, where cost is measured in time. The second step is a neighbourhood analysis based on a set of multiple transition layers. The number of these layers depends on the number of moving directions from the centre cell to the neighbouring ones. The chosen cell is the surrounding one being reachable by the smallest expenditure of time. Thirdly, accumulating the time along the fastest path provides the final time-cost raster. Finally, the visualization of the spatial expansion of the moving patterns results from isochrones.
However, for many purposes using commercial software is cost and time intensive. In addition, using different software implementations of time-cost analysis on the same data produces dissimilar results that are incommensurable (Herzog 2013). Therefore we implemented one of the most famous time-distance functions in an open source environment (Programming Language R) to address a wide-range of scientists and to enable a potential use in analytical questions.
Fig. 1: General workflow for time-cost analysis in four steps.
Tobler Hiking Function
As mentioned before, there are various ways to apply and/or implement least-cost analysis within a
wide-range of archaeology research. Numerous studies use the hiking function by Waldo R. Tobler (1961; see also Herzog 2013 and Herzog 2014) first implemented by Gorenflo and Gale (1990). Tobler (1993) developed an empirical model based on the empirical marching data of the Swiss military given by Imhof (1950). Marching time depends on multiple factors, such as length and quality of path, altitude difference, weather conditions and darkness as well as marching competence and luggage. In addition, hiking speed is greater for short distances than for long-lasting marches and small groups cover distances faster than columns (Imhof 1950). The Tobler Hiking Function is the empirical quantification of the walking velocity to cross a certain terrain by using a digital elevation model (DEM) as well as the first derivative (dh/dx):
V = 6e { -3.5 abs ( s + 0.05 ) }
where V is the walking velocity in km/h, e is the base of the natural logarithms, and s is dh/dx [dh and dx must be measured in the same unit; slope] (Tobler 1993). This formula calculates a maximum velocity of around 6 km/h on gently downslope direction from -5 to -2 degrees and on flat terrain around 5 km/h (Fig. 2). Fig. 2 also shows a decreasing speed of hiking with an increasing slope gradient because overcoming steeper slopes is time-consuming and exhausting. The empirical data of Imhof (1950) is limited to small groups hiking on defined paths and ways with average speed of 4.5 to 5 km/h on flat terrain. To address off-path traveling, reducing mean hiking speed to 3 km/h (Imhof 1950), Tobler (1993) argued to include an off-path multiplier of 0.6.
Fig. 2: The Tobler Hiking Function (Tobler 1993).
Main cost factor: slope
According to Tobler (1993), slope is the foundation of time-cost analysis. Therefore, almost all archaeological studies use slope as a cost factor (Herzog 2010; Herzog 2014). Slope is an anisotropic cost factor depending on the directions of movement (Herzog 2013). However, there is a huge number of different slope algorithms which use pixel-based analysis, each addressing particular questions and certain landscape conditions or data quality. Slope is the first derivative of the terrain representing the vertical change of the elevation (Behrens 2003). The calculation of the slope gradient bases on a neighbourhood analysis using a moving window (Fig. 3).
Fig. 3: Moving Window approach for deviating terrain attributes (e.g. slope) from a digital elevation model (see Behrens 2003).
The ultimate principle of deriving slope from an elevation model is calculating the difference of height between the centre cell and its surroundings. Depending on the slope algorithm chosen, the number and combination of the cells nearby varies. The most common slope algorithms are the mean slope gradient (Zevenbergen and Thorne 1987) for smooth terrains as well as the maximum slope gradient (e.g. Guth 1995) for identifying streamlines (Behrens 2003). Both approaches are using a moving window technique to calculate a slope angle between the centre cell and its neighbourhood. Using Zevenbergen and Thorne (1987), the slope angle bases on accounting the cardinal cells only (C2, C4, C6, C8, Fig. 3), whereas the maximum slope algorithm by Guth (1995) uses diagonal neighbours additionally. Using cardinal neighbours has the advantage of including the nearest pixels only, resulting in a high local accuracy, but tends to get noisy if the terrain is very heterogeneous or the quality of the DEM is low. This slope angle is the average slope gradient of the neighbourhood. In contrast, the maximum slope angle results from the pixel showing the maximal difference in altitude to the centre of the moving window (Behrens 2003). Besides these two widespread approaches, there are other popular algorithms as Fleming and Hoffer (1979) as well as Ritter (1987) for smooth surfaces and Horn (1981) for rough surfaces, both being included in the r-package “raster” (Hijmans 2016).
Other cost factors
Besides slope as an anisotropic cost factor, there are isotropic cost factors including topographic, social and cultural factors which influence crossing the landscape (Herzog 2013). Particular types of land cover or water bodies complicate traversing any region. The water volume of mountain streams can vary between passable and impassable during the day. Besides, vegetation age, stand diversity and density influence the hiking speed and energy effort. Moreover, substrate, bedrock, subsoil and general underground cause tough sledding (Imhof 1950). In addition, the terrain includes areas hardly passable for human beings. Pixels of steep slopes represent areas with lower velocity. Allocating these zones as impassable barriers leads to their exclusion from the time-cost analysis. In addition to anisotropic factors, the time-cost analysis uses friction layers to include isotropic cost factors that influence spatial moving patterns.
Time-cost calculation
As mentioned above, slope and the Tobler Hiking Function are a reliable foundation of numerous time-cost analysis (Herzog 2013; Herzog 2014). Therefore, we use the Tobler Hiking Function to calculate the velocity to cross each pixel cell using a slope raster dataset. The Tobler Hiking Function is best suited for flat terrain over gently to moderate slopes (Herzog 2014). Thus, for reducing errors at steep slopes (e.g. >16–20°) we implemented an optional damping cost factor lowering the hiking velocity tremendously at these areas. The (damped) velocity raster is the final cost surface.
The final step computing the time-cost surface by a stepwise or cell-by-cell-based approach to account for traversing the landscape is the most time and computational intensive part. As the number of moving directions from one cell to a neighbouring cell is relevant, four, eight and 16 directions are differentiated. The naming of moving characteristics originates from chess moves. Rook move (four directions) means just following cardinal directions, queen move (eight directions) additionally enables diagonally shifting and knight move (16 directions) respects a combination of cardinal and diagonal movements (Fig. 4). An increasing number of directions results in a growing computing time but also in moving patterns of humans being more realistic.
To address the spatial resolution of each individual DEM dataset we implemented a geo-correction of the time-cost raster considering the cardinal and diagonal movement through pixel cells. Finally, each pixel of the time-cost raster contains the accumulated time needed to reach it from the initial location.
Fig. 5 exemplifies the SETs of a one-hour and a two-hour hike starting from a neolithic test site in the Black Forest disregarding (1) and respecting (2, 3) the local terrain. Hence, the scaling down of the potential exploitation territory is 23.3% considering the local terrain. Including a damping factor of 16° (3) effects another reduction of 4% from original 254 km² to 186.5 km². The spatial restriction enables the purposive focus on particular questions.
Fig. 4: Moving characteristics using four (a, rook move), eight (b, queen move) and 16 directions (c, knight move).
Fig. 5: Three plots of time-cost-analysis using different terrain information. Excluding the terrain results in circular isochrones around the initial point (1). By including the slope gradient, the covered distances per time interval continue along the terrain (2). The exaggeration of steep slopes creates moving patterns continuing strictly along high-angle hillsides (3). The contour lines represent the covered distance per hour.