A Modified Fractal Dimension as a Measure of Landscape Diversity

Fractals hove been used recently to describe spatial patterns in mony iandscape-leve1 applicotions. One such appiication has been to measure the geometnk complexity of landscape features. This paper describes a modified fractal dimension to be used as a measure of distribution of Iandscape diversity in a classified GIS image. The resulting modified fractal dimension calculation consisientiy describes diversity for the landscape, accounting not only for patch shape, but also for patch juxtaposition ond evenness. Recently, researchers have turned to fractals to describe spatia1 pattems in a variety of landscape-leve1 applications (Burrough, 1986; O'Neill et 01.. 1988; De Cola, 1989; Lam, 1990; Milne, 1990; Rex and Malason, 1990; Tumer, 1990; Polidori et ai., 1991; Ripple et al., 1991; Baker and Cai, 1992). Most of this work used fractals to characterize the shape of features within a landscape. Generally, fractal dimension was calculated by computing the slope of a regression line between the natural logarithm of perimeter and area pairs calculated from the feature(s) of interest. For instance, Burrough (1986) used the natural log of one-fourth the perimeter against the natural log of the area. This technique requires the presence of a fairly large number of landscape patches in order to generate enough perimeter and area pairs to accurately calculate the regression. In other work. the fractal dimension of individual landscape features was calculated as an aid in land-cover classification (De Cola, 1989). Lam (1990) used fractals to discuss the spatial complexity of three land types. Milne (1990) applied fractals to estimate the probability of locating a landscape patch within a larger landscape. These researchers used fractals to estimate landscape complexity as a function of patch shape; in most cases, a single fractal value was determined for the entire landscape for selected cover types or for all cover types (Turner, 1990). Natural resource managers and researchers working at the landscape leve1 need to understand the spatial dynamics of diversity within a given landscape, not just the overall diversity of a landscape. Our research attempts to address this need by examining how to evaluate the diversity of landscapes at various scales (e.g., with various changes of extent, and/or grain size) and how to evaluate the distribution of diversity over a landscape (i.e., what areas within a given landscape are more diverse than others). As shown by past work in the remote sensing, GIS, and landscape ecology literature (Mandelbrot, 1983; O'Neill et al., 1988; Peitgen and Saupe, 1988; De Cola, 1989; Turner et al., 1989; Lam, 1990; Milne, 1990; Rex and Malason, 1990; Introduction Ripple, 1991). fractals can be applied to a variety of landscape ecology problems because they convenientiy describe many of the irregular, fragmented patterns found in nature (Mandelbrot, 1983). W h i l e the applications and calculation of fractals vary. our use of fractals is limited to describing the degree to which the area of a landscape patch (a continuous grouping of grid cells representing the same landscape feature) is related to its edge and how this measure can be modified to address diversity. By determining the fractal relationship of patch area to patch edge for a given landscape, measures of the geometric diversity of that landscape and, therefore, the complexity of patch interaction within it can be determined. However, as Rex and Malason (1990) have shown, the shape of a landscape patch is not the only factor which affects ecological processes within a landscape; they show that the juxtaposition of a patch to other patches can also have significant effects. Furthermore, a complete definition of diversity needs to include patch richness (e.g., number of different patches) and patch evenness (distribution of patches across the landscape) as well (Shannon and Weaver, 1962). Past research has also demonstrated that as the extent of a given landscape changes, so do the various landscape indices, including diversity (Tumer et ai., 1989). Given this, the estimation of landscape diversity is dependent on defining the extent of the landscape. in order to determine the distribution of diversity within a landscape, reduced areas need to be examined. Unfortunately, using the regression method for computing a fractal index for small landscapes is particularly problematic. As the landscape extent is reduced, the number of patches present in the landscape is also reduced. With the lower number of patches, fewer perimeter and area pairs are generated for computing the slope of the regression line, meaning that the computation is based on too few data points and is, therefore, inaccurate. The diversity index proposed here combines patch complexity. as calculated by fractals, with richness and evenness of patches within a landscape. This index combinas work by Patton (1975), who addressed patch edge diversity, and work by Shannon and Weaver (1962), who evaluated species richness and evenness. By combining the definitions of diversity of Patton (1975). Shannon and Weaver (1962), and Rex and Malason (1990). a definition for landscape diversity which is a function of the number and types of patches, their distribution (juxtaposition), and their shape was obtained. Fractal indices have been used in many cases as a diversity index, but unfortunately can only account for diversity arising from patch shape; therefore, a pure fractal index [based solely on patch shape) is only a partial measure of landscape diversity. The methodology presented here modifies the fractal dimension measured by accounting for both the geometry PhotogrammeMc Engineering & Remote Sensing, Vol. 59, No. 10, October 1993, pp. 1517-1520. 01993 American Society for Photogrammets, and Remote Sensing 0099-1112/93/5910-1517$03.00/0 Eric R. Olsen R. Douglas Ramsey David S. Winn Department of Geography and Earth Resources. Utah State University, Logan, UT 84322-5240.