The formula proposed by Givoni and Goldstein (1971) suggests that multiplying the slope costs and the terrain factor is the most appropriate combination of cost components. However, experiments by Santee et al. (2003) indicate that a constant terrain factor might not be appropriate for modelling the energy expenditure of walking on all slopes. Their experiments suggested that the terrain factor on steep slopes (8.6 and 12% in the experiments) is higher than on level ground.
The disadvantages of adding cost components have already been discussed elsewhere (Herzog 2013a). The main problem is that some sort of standardisation is required so that the cost components considered are comparable. The standardisation based on minimum and maximum values is not on statistically safe ground and the results depend on the selection of study area boundaries. Another approach tries to modify the main cost function by adding costs in case of difficult terrain and using negative values to model facilitating features, and in this case the risk of an accumulated negative cost value cannot be ruled out. Unfortunately, the r.walk function of GRASS GIS currently only supports adding costs.
Analysing modern accessibility, Nelson (2000) presents another alternative for combining costs:
Cost = Slope_Cost * Precedence(Barriers, Roads, Rivers,..., Land_Cover)
In this model, barriers such as borders will slow down the progress independent of all the other factors involved. For example, a fast car on a motorway has to stop at the border. If no barrier is present, fast roads will speed up progress, and in the absence of roads and rivers, land cover plays an important role. An appropriate raster GIS procedure (called Merge in ESRI products) is required to combine a set of raster grids into one single raster where the order of the input defines the order of precedence, i.e. borders overlay roads which in turn overlay rivers which in turn overlay land cover. This approach is quite intuitive and could be applied to archaeological LCP calculations, especially if three or more cost components are to be combined.