Joe,

I just ran across your April article in GeoWorld at http://www.geoplace.com/gw/2001/0501/0501bmp.asp.

Congratulations on keeping a sustained focus on the modeling and analytical aspects of GIS--more people need to see and appreciate how it is done.

The caption indicating that hiking times could be adjusted for terrain slope caught my eye, because it's an interesting problem. What makes it interesting is that a simple grid operation alone is inadequate for this analysis: the local direction of the path matters. For instance, a technique that does not account for path direction would assign the same "slope weight" to a trail winding horizontally along a steep mountain cliff side as it would to the trail that drives straight up the cliff. It seems this is what your article advocates; I hope that I have just mis-read it.

If you capture both slope and aspect in grids, then there are simple and obvious ways to use map algebra to combine a direction grid with slope/aspect to compute directional slope: this is the traditional "directional gradient" calculation of multivariate calculus. A better way, I find, is to extract elevations from the vector representation of a trail, use those to compute local slopes along the trail, and apply those slopes to evaluate different trails.

Designing a trail that accounts for along-trail slopes, scenery, and other factors can usually be formulated as a constrained global optimization problem. For instance, you might want to find a trail from mountaintop to valley that minimizes the maximum slope, minimizes total curvature, minimizes length, and maximizes scenic value, subject to the constraints that it begins at the start, ends at the finish, and never wanders outside a feasible region (such as a national park, or a parcel bounded by interstate highways and other barriers). You cannot achieve all these, but some reasonable objective function that combines these attributes could capture a stakeholder's preference structure for trading off among them. Some solutions might yield to classical calculus of variations attacks, but more likely simulated annealing would be the best approach.

Lest the idea of applying such a technological cannon to hiking trail design strike some as frivolous, consider that the same techniques can be brought fruitfully to bear on highway design (minimize time, gas mileage, construction costs, environmental impact, and balance cut with fill), transmission line siting, pipeline siting, selection of air and sea routes, and many other applications. I am pleased that commonly available GIS software provides an adequate computational environment to formulate, attack, and solve such problems.

Yours truly,

Bill Huber

Quantitative Decisions
[whuber@quantdec.com]