6,209 research outputs found
Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular Boundaries
Irregular boundary lines can be characterized by fractal dimension, which
provides important information for spatial analysis of complex geographical
phenomena such as cities. However, it is difficult to calculate fractal
dimension of boundaries systematically when image data is limited. An
approximation estimation formulae of boundary dimension based on square is
widely applied in urban and ecological studies. However, the boundary dimension
is sometimes overestimated. This paper is devoted to developing a series of
practicable formulae for boundary dimension estimation using ideas from
fractals. A number of regular figures are employed as reference shapes, from
which the corresponding geometric measure relations are constructed; from these
measure relations, two sets of fractal dimension estimation formulae are
derived for describing fractal-like boundaries. Correspondingly, a group of
shape indexes can be defined. A finding is that different formulae have
different merits and spheres of application, and the second set of boundary
dimensions is a function of the shape indexes. Under condition of data
shortage, these formulae can be utilized to estimate boundary dimension values
rapidly. Moreover, the relationships between boundary dimension and shape
indexes are instructive to understand the association and differences between
characteristic scales and scaling. The formulae may be useful for the
pre-fractal studies in geography, geomorphology, ecology, landscape science,
and especially, urban science.Comment: 28 pages, 2 figures, 9 table
Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry
A change in the environmental conditions across space—for example, altitude or latitude—can cause significant changes in the density of a vegetation type and, consequently, in spatial connectivity. We use spatially explicit simulations to study the transition from connected to fragmented vegetation. A static (gradient percolation) model is compared to dynamic (gradient contact process) models. Connectivity is characterized from the perspective of various species that use this vegetation type for habitat and differ in dispersal or migration range, that is, “step length” across the landscape. The boundary of connected vegetation delineated by a particular step length is termed the “ hull edge.” We found that for every step length and for every gradient, the hull edge is a fractal with dimension 7/4. The result is the same for different spatial models, suggesting that there are universal laws in ecotone geometry. To demonstrate that the model is applicable to real data, a hull edge of fractal dimension 7/4 is shown on a satellite image of a piñon‐juniper woodland on a hillside. We propose to use the hull edge to define the boundary of a vegetation type unambiguously. This offers a new tool for detecting a shift of the boundary due to a climate change
A universal approach for drainage basins
Drainage basins are essential to Geohydrology and Biodiversity. Defining
those regions in a simple, robust and efficient way is a constant challenge in
Earth Science. Here, we introduce a model to delineate multiple drainage basins
through an extension of the Invasion Percolation-Based Algorithm (IPBA). In
order to prove the potential of our approach, we apply it to real and
artificial datasets. We observe that the perimeter and area distributions of
basins and anti-basins display long tails extending over several orders of
magnitude and following approximately power-law behaviors. Moreover, the
exponents of these power laws depend on spatial correlations and are invariant
under the landscape orientation, not only for terrestrial, but lunar and
martian landscapes. The terrestrial and martian results are statistically
identical, which suggests that a hypothetical martian river would present
similarity to the terrestrial rivers. Finally, we propose a theoretical value
for the Hack's exponent based on the fractal dimension of watersheds,
. We measure for Earth, which is close to
our estimation of . Our study suggests that Hack's law can
have its origin purely in the maximum and minimum lines of the landscapes.Comment: 20 pages, 6 Figures, and 1 Tabl
Fracturing ranked surfaces
Discretized landscapes can be mapped onto ranked surfaces, where every
element (site or bond) has a unique rank associated with its corresponding
relative height. By sequentially allocating these elements according to their
ranks and systematically preventing the occupation of bridges, namely elements
that, if occupied, would provide global connectivity, we disclose that bridges
hide a new tricritical point at an occupation fraction , where
is the percolation threshold of random percolation. For any value of in the
interval , our results show that the set of bridges has a
fractal dimension in two dimensions. In the limit , a self-similar fracture is revealed as a singly connected line
that divides the system in two domains. We then unveil how several seemingly
unrelated physical models tumble into the same universality class and also
present results for higher dimensions
Percolation with long-range correlated disorder
Long-range power-law correlated percolation is investigated using Monte Carlo
simulations. We obtain several static and dynamic critical exponents as
function of the Hurst exponent which characterizes the degree of spatial
correlation among the occupation of sites. In particular, we study the fractal
dimension of the largest cluster and the scaling behavior of the second moment
of the cluster size distribution, as well as the complete and accessible
perimeters of the largest cluster. Concerning the inner structure and transport
properties of the largest cluster, we analyze its shortest path, backbone, red
sites, and conductivity. Finally, bridge site growth is also considered. We
propose expressions for the functional dependence of the critical exponents on
Analyzing urban sprawl patterns through fractal geometry: the case of Istanbul metropolitan area
Over the last decade, there has been a rapid increase in the amount of literature on the measurement of urban sprawl. Density gradients, sprawl indexes which are based on a series of measurable indicators and certain simulation techniques are some quantitative approaches used in previous studies. Recently, fractal analysis has been used in analyzing urban areas and a fractal theory of cities has been proposed. This study attempts to measure urban sprawl using a sprawl index and analyses urban form through fractal analysis for characterizing urban sprawl in Istanbul which has not been measured or characterized yet.
In this study, measures of sprawl were calculated at each neighborhood level and then integrated within sprawl index through “density” and “proximity” factors. This identifies the pattern of urban sprawl during six periods from 1975 to 2005, and then the urban form of Istanbul is quantified through fractal analysis in given periods in the context of sprawl dynamics. Our findings suggest that the fractal dimension of urban form is positively correlated with the urban sprawl index score when urban growth pattern is more likely “concentrated”. However, a negative relationship has been observed between fractal dimension and sprawl index score when the urban growth pattern changes from the concentrated to the semi-linear form
- …