topographic area calculations

[DON EBERT]

Jim, 

Here's a simplistic and rather laborious way of calculating the surface
area of 3-dimensional polygons. If anyone knows of an easier way of doing
this, I would appreciate hearing about it. First use your elevation map to
calculate out a slope map using percent slope as the output. Then,
assuming your elevation map cells are square instead of rectangular, the
3-d area of the cell will be run * sqrt(run^2 + rise^2), where run = the
length of elevation cell and rise = the slope value * run. 
     
              /@\
            /@@@@@\
          /@@@@@@@@@\
        /@@@@@@@@@@@@/|
       \@@@@@@@@@@@/  |
       r \@@@@@@@/    | rise
        u  \@@@/      | (run * %_slope)
          n  /________|
                 run
               

In r.mapcalc the formula would be:

area = x * sqrt(x^2 + (x * slope)^2); substituting cell length for x

Next, determine which cells lie in the interior of your polygon and those 
that are on the periphery. There are a number of ways to do this. I 
typically import my polygon boundaries as line features, run v.to.rast, 
reclass the output map assigning 0 to the polygon boundary and 1 to 
everything else, run r.clump, then reclass the output map assigning 1 to 
0, 2 to the interior clump, and 0 to everything else. This results in a 
map where the periphery cells are coded 1 and the interior cells are 
coded 2. 

The final area of your 3-d polygon will be approximately the sum of the
areas of the interior cells plus 1/2 the total of the area of the
periphery cells. Since this methodology works remarkably well on flat
surfaces, I've made the assumption that it works equally well in the 3rd
dimension. I usually output the cell area values to do the final summation
in either a spreadsheet or database since the final area can be quite
large. To do so I run 2 commands: 

        r.stats -1zg in=polygon out=polygon.tmp
        r.what area < polygon.tmp > polygon.3d

This outputs an ascii file with one row for each cell consisting of its 
easting, northing, polygon code, and area value.

Hope this helps.

Don Ebert
Research Associate
National Biological Service
Cooperative Research Unit
University of Nevada - Las Vegas 

On Mon, 1 Apr 1996, James Taulman wrote:

>       I have UTM x,y coordinate contour files representing probability 
> boundaries for animal home range utilization distributions.  I can add 
> elevation data from digitized USGS quads.  I need to be able to calculate 
> accurate topographic areas within these contours in GRASS.  Any 
> information helpful to this project would be most appreciated.
>       Jim Taulman
>