Deriving New Minimum Cost Pathways from Existing Paths
Dean, Denis J., Cartography and Geographic Information Science
Geographic Information System (GIS) "based cost spreading techniques are widely used to find minimum cost paths from specified starting point(s) to specified ending point(s). These raster-based techniques assume that the costs of traversing each raster cell within the area of interest are known. In general, cost spreading algorithms require as inputs:
* The locations of the starting and ending point(s);
* A raster database that describes the cost of traversing each raster cell in the study area; and in the case of anisotropic costs;
* Additional databases and/or mechanisms that allow the cell traversing costs to be modified to account for the direction of travel.
The output of cost-spreading analysis is typically a new raster database showing the minimum-cost path connecting the starting point/ending point pair that produces the smallest total traversing cost of all possible pairs of starting and ending points. The cost-spreading algorithms that generate these outputs employ a form of dynamic programming, which offers a reliable and relatively efficient way of solving what on its surface appears to be a fairly formidable problem (Huriot et al. 1989; Smith 1989).
Cost-spreading techniques have been used in applications as diverse as finding routes for forest access roads (Dean 1997) to analyzing networks of blood vessels in medical imagery (Olabarriaga et al. 2003). However despite their apparent differences, all previous applications of cost spreading share a common assumption; specifically, they all assume that traversing costs are known a priori. This reliance on a priori cost definitions has prevented cost-spreading techniques from addressing at least one common set of problems. These unaddressable problems require deducing traversing costs from existing minimum cost pathways and then using these deduced costs to find new pathways. For example, consider a situation commonly encountered by natural resource managers. These managers are frequently required (often by law but always by the dictates of good management practices) to develop and evaluate a range of alternative management plans before selecting a final plan to implement within the area under their control (Cutter and Renwick 2004). Each of these alternative plans may involve the development of new access roads and/or trails. If conventional cost-spreading techniques are to be used to find minimum cost routes for these new paths, it is necessary to conduct in-depth studies to determine path building costs throughout the area of interest. This is often financially impractical (Dean 1997; Liu and Sessions 1993). However, the study area often includes existing roads and/or trails. Clearly, managers in these situations could benefit from a mechanism whereby they could use existing roads and/or paths to deduce traversing costs, and then use these deduced costs in standard cost-spreading procedures to identify minimal cost paths for new proposed roads and/or paths.
This study presents a technique for estimating traversing costs from an existing minimum cost path. It is assumed that an existing minimum cost path is present and that the components of traversing costs are known (e.g., road-building costs may be a function of slope, gradient and soil type), and that traversing costs are a linear combination of their component costs (e.g., road-building costs from the previous example could be computed as a linear function of slope, gradient, and soil type). Unknown will be the parameters of the linear equation that constructs traversing costs from its components. I will start by presenting techniques for estimating these parameters in the isotropic case. I will then extend these techniques to include the anisotropic case. Both cases will be illustrated using example problems.
Case 1: Isotropic Traversing Costs
Assume that a minimum cost path exists from starting point [P. …