204
Applications of Fractal Geometry and Percolation Theory to Landscape Analysis and Assessments
properties such as permeability (Loehle and Li,
1996b), and uncertainty in ecological systems (Li,
unpublished manuscript).
14.3.3 Patch Hierarchical Scaling
Different observational scales capture different aspects of structure, and these transitions are signaled
by shifts in the apparent dimension of an object.
This latter fact suggests an interesting application
of fractals as a method for distinguishing hierarchical size scales of patches in nature, such as how
to determine boundaries between hierarchical levels and how to determine the scaling rules for extrapolating within each level.
Bradbury et al. (1984) examined the possibility
of hierarchical scaling in an Australian coral reef.
They used the dividers method in transects across
the reef to determine whether D depends on the
range of length scales. They found that three ranges
of scale correspond nicely with the scales of three
major reef structures: 10 cm corresponds to the size
of anatomical features within individual coral
colonies; 20 to 200 cm corresponds to the size
range of whole, adult living colonies; and 5 to 10
m is the size range of major geomorphologic structures. This showed that the shifts in D at different
scales appear to signal where the breakpoints occur in the hierarchical organization of reefs.
In our recent study on simple patch patterns,
change of fractal dimensions seems to predict hierarchical scales of patch size and structure in nature, both of different mean grain densities within
patch and spatial patterns between patches. Since
the change in the fractal dimension may tell us
something about the underlying physical and biological processes, D could be used as a scaling indicator of patch phase transition and may help us
decide on the appropriate scale of ecological experiment, assessment, and management.
14.3.4 Fractal Spatial Patterns and
Modified Brownian Dynamics
There are simple relationships between persistence
measured by the parameter H in modified Brownian diffusion models and fractal exponents D. Hastings et al. (1982) and Mandelbrot (1983) have discussed how fractal exponents may be incorporated
into diffusion processes, as a scaling factor for normalizing increments in space and time. They find
that D may be used as an index of succession in
circumstances where simple patch-extinction models are reasonable. Sugihara and May (1990) consider that it would be interesting to follow up these
provocative anecdotes with careful studies to determine to what extent D computed from snapshots
can be used as an index of physiological state or
persistence of patches in time and how such persistence may relate to the spatial scales involved.
Recent studies have shown that many of nature's
seemingly patch shapes can be effectively characterized and modeled as random fractals based on
generalizations of fractional Brownian motion
(Voss, 1988). The variance of the increments of a
process (i.e., the semivariogram) is required to be
independent of scale; the condition of independence of scale is satisfied by such generalizations
of fractional Brownian motion. Extending the ideas
in fractal correlation analysis of patchy systems is
addressed in Li (2000).
14.3.5 Multiple-scale Sampling and
Data Analysis
Biotic and metric scale-dependent structures exist
in landscapes. Biotic scale dependence originates
from the different responses of organisms to the
abundance of a resource. Metric scale dependence
results when physical processes produce statistically similar aggregations of abiotic quantities,
such as water, temperature, or minerals (Milne,
1990). From a fractal point of view (Mandelbrot,
1977, 1983), we consider a sampling space dimension Ds and a specific ecological phenomena
dimension Dp embedded in a space of dimension
E. If Ds + Dp > E, we will obtain a nonzero measure; that is, a specific sampling space dimension
Ds used in space of dimension E can only detect
such phenomena of different dimension Dp > E -
Ds. Lovejoy et al. (1986) argued that measuring network inhomogeneity by the fractal dimension raises
new problems concerning the detectability of
sparse phenomena. They suggested a new criterion
for evaluating measuring networks: to detect geophysical phenomena, not only must a network have
sufficient spatial resolution, but it must also have
sufficient dimensional resolution. Because our
sampling space is at multiple scales, of course, such
sampling is multiple-scale sampling.
Theoretically, data from different scales have
their corresponding probability distributions. Statistical fractals show us that the sum of a large number of identically distributed random variables has
self-similarity with the distribution as each of the
contributors to the sum. The class of distributions
having this property is called the Levy distributions
(Mandelbrot, 1977, 1983). This result implies that
Applications of Fractal Geometry and Percolation Theory to Landscape Analysis and Assessments
properties such as permeability (Loehle and Li,
1996b), and uncertainty in ecological systems (Li,
unpublished manuscript).
14.3.3 Patch Hierarchical Scaling
Different observational scales capture different aspects of structure, and these transitions are signaled
by shifts in the apparent dimension of an object.
This latter fact suggests an interesting application
of fractals as a method for distinguishing hierarchical size scales of patches in nature, such as how
to determine boundaries between hierarchical levels and how to determine the scaling rules for extrapolating within each level.
Bradbury et al. (1984) examined the possibility
of hierarchical scaling in an Australian coral reef.
They used the dividers method in transects across
the reef to determine whether D depends on the
range of length scales. They found that three ranges
of scale correspond nicely with the scales of three
major reef structures: 10 cm corresponds to the size
of anatomical features within individual coral
colonies; 20 to 200 cm corresponds to the size
range of whole, adult living colonies; and 5 to 10
m is the size range of major geomorphologic structures. This showed that the shifts in D at different
scales appear to signal where the breakpoints occur in the hierarchical organization of reefs.
In our recent study on simple patch patterns,
change of fractal dimensions seems to predict hierarchical scales of patch size and structure in nature, both of different mean grain densities within
patch and spatial patterns between patches. Since
the change in the fractal dimension may tell us
something about the underlying physical and biological processes, D could be used as a scaling indicator of patch phase transition and may help us
decide on the appropriate scale of ecological experiment, assessment, and management.
14.3.4 Fractal Spatial Patterns and
Modified Brownian Dynamics
There are simple relationships between persistence
measured by the parameter H in modified Brownian diffusion models and fractal exponents D. Hastings et al. (1982) and Mandelbrot (1983) have discussed how fractal exponents may be incorporated
into diffusion processes, as a scaling factor for normalizing increments in space and time. They find
that D may be used as an index of succession in
circumstances where simple patch-extinction models are reasonable. Sugihara and May (1990) consider that it would be interesting to follow up these
provocative anecdotes with careful studies to determine to what extent D computed from snapshots
can be used as an index of physiological state or
persistence of patches in time and how such persistence may relate to the spatial scales involved.
Recent studies have shown that many of nature's
seemingly patch shapes can be effectively characterized and modeled as random fractals based on
generalizations of fractional Brownian motion
(Voss, 1988). The variance of the increments of a
process (i.e., the semivariogram) is required to be
independent of scale; the condition of independence of scale is satisfied by such generalizations
of fractional Brownian motion. Extending the ideas
in fractal correlation analysis of patchy systems is
addressed in Li (2000).
14.3.5 Multiple-scale Sampling and
Data Analysis
Biotic and metric scale-dependent structures exist
in landscapes. Biotic scale dependence originates
from the different responses of organisms to the
abundance of a resource. Metric scale dependence
results when physical processes produce statistically similar aggregations of abiotic quantities,
such as water, temperature, or minerals (Milne,
1990). From a fractal point of view (Mandelbrot,
1977, 1983), we consider a sampling space dimension Ds and a specific ecological phenomena
dimension Dp embedded in a space of dimension
E. If Ds + Dp > E, we will obtain a nonzero measure; that is, a specific sampling space dimension
Ds used in space of dimension E can only detect
such phenomena of different dimension Dp > E -
Ds. Lovejoy et al. (1986) argued that measuring network inhomogeneity by the fractal dimension raises
new problems concerning the detectability of
sparse phenomena. They suggested a new criterion
for evaluating measuring networks: to detect geophysical phenomena, not only must a network have
sufficient spatial resolution, but it must also have
sufficient dimensional resolution. Because our
sampling space is at multiple scales, of course, such
sampling is multiple-scale sampling.
Theoretically, data from different scales have
their corresponding probability distributions. Statistical fractals show us that the sum of a large number of identically distributed random variables has
self-similarity with the distribution as each of the
contributors to the sum. The class of distributions
having this property is called the Levy distributions
(Mandelbrot, 1977, 1983). This result implies that