Slope is anisotropic, and therefore a slope raster map does not reflect the costs of crossing a raster cell in all directions (Conolly and Lake 2006, 217–218): The slope map normally records only the slope value of the direction of steepest descent, but the slope of a path traversing the surface is dependent on the direction of the path relative to the underlying surface.

According to van Leusen's view (2002, chapter 16, 12), basing the LCP analysis on the non-directional slope map has a minor effect on the quality of the model. Several authors of archaeological LCP studies share this view or are not able to apply anisotropic software (Vermeulen *et al*. 2001; Bellavia 2002; Kantner and Hobgood 2003; Batten 2007; Howey 2007; Chataigner and Barge 2008; Polla 2009). On the basis of an isotropic slope cost model, switchbacks are ineffective, because traversing a cell in the direction of the switchback road is as costly as climbing the steep slope. So isotropic cost models never create hairpin curves, which can be tested easily by LCP calculations on an artificial roof-shaped landscape (Herzog 2013b). Moreover, LCP reconstructions for the Zeitstraße in the hilly part of the Rhineland, Germany, suggest that LCPs based on effective slope outperform LCPs derived from an isotropic slope-dependent cost grid, when compared to the historic route (Herzog 2009b).

Several inadequate approaches dealing with effective slope have been proposed; an overview can be found in Herzog (2013b). These include IDRISI's anisotropic cost modules (Eastman 2006, 261–5), which construct an anisotropic function in such a way that for all parameter choices, downhill directions accumulate less expenditure than level ground. The IDRISI approach was also implemented in SAGA GIS (Olaya 2004, 161–2) and the SAGA code formed the basis of the SEXTANTE tools included in gvSIG (Olaya 2009).

ESRI's Path Distance procedure implements a fairly complex model for anisotropic cost calculations, which is controlled by eight parameters including the isotropic friction grid, the elevation raster, and two raster grids defining for each cell the horizontal and vertical directions respectively. Tripcevich (2009) published a tutorial including an example on how this procedure can be used for effective slope calculations with the popular Tobler cost function. Subsequently this approach has been used by Parslow (2009, 61) in her archaeological LCP study, but she does not check the resulting LCPs by comparing them with archaeological evidence.

A straightforward implementation of effective slope is described by Yu *et al*. (2003). The slope of a move between two neighbouring grid cells is derived directly from the differences in elevation and the distance between the two cell centres. The r.walk procedure of GRASS GIS is based on this approach. This algorithm can be readily extended so that it allows the subdivision of long moves into submoves, by estimating the elevation values of the intermediate stops from the two nearest neighbours (Figure 4). Any path in this model can be compared to walking on a tight wire that connects poles of different height. The poles are erected at the grid cell centres of the DEM and their heights are equivalent to the DEM altitudes.

The comparison to tight wire walking illustrates a drawback of the straightforward approach: the resulting path does not take the surrounding terrain into account. Any path needs a minimum breadth, and a contour line path along a steep rock requires construction work. So the straightforward approach assumes that whenever the LCP follows the contour line, this path was built, regardless of the effort required. But if construction work is an option, tunnels, bridges and the removal of rock outcrops should be considered as well. The extended straightforward approach published by Yu *et al*. (2003) allows the reconstruction of tunnels and bridges, but this extension has not been implemented in standard LCP software.

In fact, two different models should be considered: (i) roads involving a significant amount of construction effort such as Roman main roads (e.g. Czysz 2004) and (ii) natural paths, i.e. roads that 'made and maintained themselves' such as medieval roads in Britain (Hindle 2002, 6). There is no clear-cut division between these two types of roads, e.g. the construction effort for Roman minor roads was lower than for main roads, and some bridges are recorded for English medieval roads. Roads involving construction effort often modified the landscape so that their remains can be detected more easily by archaeological survey methods than those of natural paths. However, some sections of medieval routes in Britain also gradually imprinted traces on the surface in the form of sunken roads.

When hardly any effort was expended for road construction, the method proposed by Gonçalves (2010) for calculating least-cost wide paths addresses the problem of contour line paths. Currently, this approach does not seem widely applicable due to the necessity of a high resolution DEM combined with a computationally expensive algorithm. A less demanding alternative algorithm has been proposed though it has not yet been implemented (Herzog 2013b).

Rowe and Ross (1990) define zones on a steep slope that should be avoided by an agent on a natural path: the overturn range along the contour line, and steep up- or downhill movements for which the agent's force of propulsion or braking is not sufficient. With this model, the remaining zones on the slope still allow moving up or down a steep slope diagonally. So the recommendation of Olaya (2004, 161) of discarding cells with slopes exceeding a certain threshold is only appropriate for extremely steep slopes. This discussion shows that a perfect model for including effective slope in LCP calculations is not yet available.