Paths & distance


Cost surface analysis (van Leusen 2002, ch. 6) is a generic name for a series of GIS techniques based on the ability to assign a cost to each cell in a raster map, and to accumulate these costs by travelling over the map. This technique is rooted in traditional site catchment analysis or in the calculation of Thiessen (Voronoi) polygons.
A step further is allowing the simple “flat” geographical space to be supplanted by a set of complex cost surfaces incorporating many relevant properties of the terrain, like slopes, rivers to cross, or others. One may also replace the distance- and gravity based rules for defining the catchment or territory boundaries by a time- or energy expenditure based rule for accumulating costs.
A further refinement is the assignment of different weights to the sites or foci of the catchments, thus creating Weighted Voronoi diagrams.
A combination of the two (cost surface + site weights) is, to my knowledge, at the moment only possible with the Grass extension r.xtent (see map below).


Weighted Voronoi diagram taking into account also the landscape characteristics (slopes, and major rivers as obstacles), executed with the Grass extension r.xtent. The relative weights are indicated near the sites (red dots).

The cost distance function determines the accumulated cost for moving from a source over a cost surface.

This technique was experimented in the Cinigiano (GR, Tuscany) area.
The output units are arbitrary, and have to be calibrated in the field. In this case, each 2500-units slice turned out to correspond to 15-20 minutes of walking.


Cost distance function: the effort to move over the landscape starting from a single source (in the centre), taking into account slopes, and rivers as obstacles. The calculated “effort” has been sliced in 2500-size units, indicated by coloured bands. In red the modern roads, in blue the rivers.

Once performed the cost distance function, one can output the least-cost paths from a source point to one or more chosen destinations.

Cost surface analysis The area of Cinigiano (GR) on the topgraphical map

The area of Cinigiano (GR) on the topographical map, with in blue squares the source and destination points; a red dot indicates Roselle, a green triangle the top of the Mt. Amiata.

We determined the least cost path for walking or moving by cart between Monticello (to the east; main access to the area of Amiata) and Paganico (“hub” for access to Grosseto/San Martino and Roselle); distance as the crow flies about 20 km. The landscape is rolling to hilly, and crossed by rivers and streams of various size, running in flat to deeply incised valleys.

The typical landscape of the Cinigiano area.

The typical hilly-rolling landscape of the Cinigiano area.

The criteria considered were:
- slope (at first classified, later applying simply the percentage as weight)
- major and minor watercourses, more or less difficult to cross (their weight expressed through the width of a buffer)
- later also specific morphological elements such as ridges (easy to follow) and steeply incised valleys (very difficult to cross).

With the variation of internal weights and between factors, the result was always essentially two paths: a first along the Orcia valley in the north (hereafter named the “terrace road”) and a second to the south (the “ridge road”). Archaeological sites of Roman age (republican to late antique) are present along both paths.
Curious is that an in-between path was never produced !

The two least cost paths produced between Monticello and Paganico.

In yellow the two least cost paths produced between Monticello and Paganico, in red the modern roads.

Sections of the various routes were tested in the field, and moreover confronted with the roads traced on the Catasto Leopoldino of the early XIXth century.

It is evident that by varying the weights one can manipulate the results.
But it seems that this applies only up to a certain point, since in this case the possible solution was always one of these two, with small variations, and with a well defined tipping point between the values of the weights.

One should realize that least cost path analysis (like all modelling) does not provide definitive answers, but rather suggestions and tools for verification in the field or otherwise.