202
Applications of Fractal Geometry and Percolation Theory to Landscape Analysis and Assessments
all lead to the same value for the index, which we
then call the dimension of the set.
Fractals are characterized by so-called symmetries or pansymmetries (Li, 1986; Li et aI., 1992),
which are invariance under dilations or contractions. Hence the best fractals are those that exhibit
the maximum of invariance. A fractal invariant under ordinary geometric similarity is called self-similar (Mandelbrot, 1983). Self-similar has two
meanings. We can understand similar as a loose,
everyday synonym of analogous. But there is also
the strict textbook sense of contracting similarity.
It expresses that each part is a linear geometric reduction of the whole, with the same reduction ratios in all directions. Self-similarity cannot be compatible with analyticity. Random fractals are
self-similar only in a statistical sense, and to describe them it is more appropriate to use the term
scale invariance than self-similarity. More recent
developments have extended, in particular, to include self-affine, in which the reductions are still
linear but the reduction ratios change in different
directions.
Fractals have been used to study nonlinear spatial and temporal phenomena (such as D as a measure of complexity), but they can also be extended
to abstract objects developing in a phase space,
such as models of dynamic complex systems. Sizefrequency distributions describing structured systems can also have a fractal dimension. There are
several methods of measuring the fractal dimension, which include changing coarse-graining level
(i.e., box-counting methods, information fractals,
etc.), using the fractal measure relations (i.e.,
perimeter-area-volume methods), using the correlation function (i.e., autocorrelations, semivariograms, etc.), using the distribution function (i.e.,
hyperbolic distribution), and using the power spectrum (i.e., Fourier transformation, filters, wavelets,
etc.) (Li et al., 1992). Although the theoretical origins of fractals in measure theory may seem abstruse, the basic ideas of fractal geometry are extremely simple and intuitive, and we can begin to
work with them very quickly.
Fractal dimensions can be positive, negative
(Mandelbrot, 1990), complex (Pietronero and
Tosatti, 1986), fuzzy (Feng et aI., 1991), and multifractals (Mandelbrot, 1989). Generally, there are
three properties of fractal forms: heterogeneity,
self-similarity, and the absence of a characteristic
scale of length. These geometric features are also
characteristic of patch patterns in landscape. The
fractal dimension D has been shown to be a useful
way to characterize the geometric structure of a
number of these patchy spatial patterns (Milne,
1988).
The fractal concept is also useful for characterizing certain aspects of landscape patch dynamics.
Consider a complex process of landscape patch
change that cannot be expressed in terms of a simple characteristic rate, but instead is regulated by a
self-similar or self-affine mechanism in time. The
multiplicity in time scales will be reflected in a
power spectrum with a broad profile of responses.
The fractal scaling between variations on different
time scales will lead to a frequency spectrum having an inverse power-law distribution. The fractal
analysis of cluster-phase dynamics in southern
Texas savanna landscape in Section 14.5 is an example.
14.3 Current Fractal Applications
in Landscape Ecology
Recently, Milne (1990), Sugihara and May (1990),
and Li (2000) reviewed fractal applications in ecological research and landscapes. Some interesting
aspects associated with spatial patterns and landscape dynamics, combined with the author's current research in this field, are introduced.
14.3.1 Perimeter Versus
Area Relationships
Krummel et aI. (1987) used fractal models to show
that patch shape varies with patch size. The relationship is generalized to fractal patches by the
equality between area and patch length, A = {3LD,
where A = patch area, L = patch perimeter, {3 =
constant, and D = fractal dimension. By using data
of patch area and patch perimeter, and relationships
of log A = log {3 + D log L, we can easily estimate
the fractal dimension D. They found a marked (p <
0.001) discontinuity in D, with D = 1.20 ± 0.02 at
small scales and D = 1.52 ± 0.02 at large scales.
The discontinuity occurred at areas of around 60 to
70 hectares. Their result is interpreted to indicate
that human disturbances predominate at small
scales, making for smoother geometry and lower
D, while natural processes (e.g., geology, distributions of soil types) continue to predominate at
larger scales.
In general, the area of fractal patches can be expressed as a function of perimeter raised to an exponent. Similar studies can be found in DeCola
(1989), Rex and Malanson (1990), and Haslett
(1994).
Applications of Fractal Geometry and Percolation Theory to Landscape Analysis and Assessments
all lead to the same value for the index, which we
then call the dimension of the set.
Fractals are characterized by so-called symmetries or pansymmetries (Li, 1986; Li et aI., 1992),
which are invariance under dilations or contractions. Hence the best fractals are those that exhibit
the maximum of invariance. A fractal invariant under ordinary geometric similarity is called self-similar (Mandelbrot, 1983). Self-similar has two
meanings. We can understand similar as a loose,
everyday synonym of analogous. But there is also
the strict textbook sense of contracting similarity.
It expresses that each part is a linear geometric reduction of the whole, with the same reduction ratios in all directions. Self-similarity cannot be compatible with analyticity. Random fractals are
self-similar only in a statistical sense, and to describe them it is more appropriate to use the term
scale invariance than self-similarity. More recent
developments have extended, in particular, to include self-affine, in which the reductions are still
linear but the reduction ratios change in different
directions.
Fractals have been used to study nonlinear spatial and temporal phenomena (such as D as a measure of complexity), but they can also be extended
to abstract objects developing in a phase space,
such as models of dynamic complex systems. Sizefrequency distributions describing structured systems can also have a fractal dimension. There are
several methods of measuring the fractal dimension, which include changing coarse-graining level
(i.e., box-counting methods, information fractals,
etc.), using the fractal measure relations (i.e.,
perimeter-area-volume methods), using the correlation function (i.e., autocorrelations, semivariograms, etc.), using the distribution function (i.e.,
hyperbolic distribution), and using the power spectrum (i.e., Fourier transformation, filters, wavelets,
etc.) (Li et al., 1992). Although the theoretical origins of fractals in measure theory may seem abstruse, the basic ideas of fractal geometry are extremely simple and intuitive, and we can begin to
work with them very quickly.
Fractal dimensions can be positive, negative
(Mandelbrot, 1990), complex (Pietronero and
Tosatti, 1986), fuzzy (Feng et aI., 1991), and multifractals (Mandelbrot, 1989). Generally, there are
three properties of fractal forms: heterogeneity,
self-similarity, and the absence of a characteristic
scale of length. These geometric features are also
characteristic of patch patterns in landscape. The
fractal dimension D has been shown to be a useful
way to characterize the geometric structure of a
number of these patchy spatial patterns (Milne,
1988).
The fractal concept is also useful for characterizing certain aspects of landscape patch dynamics.
Consider a complex process of landscape patch
change that cannot be expressed in terms of a simple characteristic rate, but instead is regulated by a
self-similar or self-affine mechanism in time. The
multiplicity in time scales will be reflected in a
power spectrum with a broad profile of responses.
The fractal scaling between variations on different
time scales will lead to a frequency spectrum having an inverse power-law distribution. The fractal
analysis of cluster-phase dynamics in southern
Texas savanna landscape in Section 14.5 is an example.
14.3 Current Fractal Applications
in Landscape Ecology
Recently, Milne (1990), Sugihara and May (1990),
and Li (2000) reviewed fractal applications in ecological research and landscapes. Some interesting
aspects associated with spatial patterns and landscape dynamics, combined with the author's current research in this field, are introduced.
14.3.1 Perimeter Versus
Area Relationships
Krummel et aI. (1987) used fractal models to show
that patch shape varies with patch size. The relationship is generalized to fractal patches by the
equality between area and patch length, A = {3LD,
where A = patch area, L = patch perimeter, {3 =
constant, and D = fractal dimension. By using data
of patch area and patch perimeter, and relationships
of log A = log {3 + D log L, we can easily estimate
the fractal dimension D. They found a marked (p <
0.001) discontinuity in D, with D = 1.20 ± 0.02 at
small scales and D = 1.52 ± 0.02 at large scales.
The discontinuity occurred at areas of around 60 to
70 hectares. Their result is interpreted to indicate
that human disturbances predominate at small
scales, making for smoother geometry and lower
D, while natural processes (e.g., geology, distributions of soil types) continue to predominate at
larger scales.
In general, the area of fractal patches can be expressed as a function of perimeter raised to an exponent. Similar studies can be found in DeCola
(1989), Rex and Malanson (1990), and Haslett
(1994).