The model defines fragmentation categories in terms of the amount and connectivity of forest in fixed-area windows. There are geometric constraints such that, for example, it is not possible to obtain a low value of connectivity where there is a large amount of forest (O’Neill et al. 1996). Knowledge of the feasible parameter space is not critical in a classification model. However, model sensitivity and robustness to real pattern differences could partly depend on how much of the feasible parameter space is realized when the model is applied. In this appendix, we describe the feasible parameter space of the model and illustrate the realized portion for the analysis of South America with 81-km² windows.

Let Pff (connectivity) and Pf (amount) define the x and y axes, respectively, of the parameter space. The upper part of the feasible parameter space does not extend to the corner at [0.0, 1.0]. It is anchored on the left [0.5, 0.0] for a checkerboard pattern and on the right [1.0, 1.0] for complete forest cover. The constraining upper curve between those points is monotonic and convex. It could be drawn by plotting the values of [Pff, Pf] obtained by starting with a checkerboard pattern and adding forest pixels one at a time until the window contained only forest pixels.

The parameter space is also constrained on the right boundary because the maximum connectivity cannot be obtained unless the analysis window is fully forested, and on the bottom boundary because connectivity is undefined when there is no forest. The shape of the lower curve joining [0.0, 0.0] and [1.0, 1.0] is not necessarily monotonic and concave (see below), but in principle it could be drawn.

The realized parameter space, that is, the set of [Pff, Pf] values actually obtained in a particular analysis, will depend on actual forest patterns as well as the details of model implementation. The realized parameter space for South America (Fig. 72) corresponds to the continental fragmentation map (Fig. 13). The figure shows the range of observed values but not the relative frequencies; most of the values are close to the main diagonal.

The left part of the parameter space illustrates that actual pattern (at this scale) is seldom like a checkerboard. Any convex, monotonic curve that connects the values [0.0, 0.5] and [1.0, 1.0] leaves a large amount of feasible space unfilled under the left part of the curve. A simulation study (not shown) suggests that the upper limit of the realized parameter space may represent an asymptote for random forest loss superimposed upon different initial amounts and patterns of forest.

The visually striking and periodic pattern of fine-scale detail on the bottom margin is an artifact because it disappears when larger windows are used. At the scale used here, there are constraints on the number of ways that a fixed amount of forest can be arranged in a finite window; some combinations are impossible and some others are possible only if there is enough forest. The rounding of continuous [0.0, 1.0] values to integer [1,255] values (see text) probably accentuates the effect; rounding by itself is not sufficient because the left margin has no similar pattern. For these reasons, the bottom margin of the parameter space (Fig. 72) only approximates the feasible curve.