Inverse box-counting method: A fractal-based procedure to create biospheric landscape patterns

Abstract : Planners and designers are interested in replicating biospheric landscape patterns to reclaim surface mines to match existing natural landscape patterns. One approach that shows promise is the use of fractal geometry to generate biospheric landscape patterns. While the measurement of the actual fractal dimension of a landscape can be difficult, a box-counting method was developed at INHP, Angers, France which approximates the spatial patterns of biospheric landscapes. Essentially the procedure entails covering a natural object/pattern with a regular grid of size r and then one simply counts the number of grid boxes, N(r), that contain some part of the object. The boxes are subdivided and the value of r is progressively reduced and N(r) is similarly re-measured until some of the boxes become empty (containing no landscape objects of interest). Then the fractal dimension of the object is approximated to be the log(N(r))/Iog(l/r). We illustrate this procedure by measuring and replicating a stand of trees in the Upper Peninsula of Michigan. Our study revealed a fractal number of 1.0 17 (p < 0.01), with a mean of 77.4 trees per 100 in by 100 in stand, and a standard deviation of 34.87 trees per stand.