Showing papers on "Fractal dimension published in 1992"

Chaos and Fractals: New Frontiers of Science

Abstract: Causality Principle, Deterministic Laws and Chaos.- The Backbone of Fractals: Feedback and the Iterator.- Classical Fractals and Self-Similarity.- Lim and Self-Similarity.- Length, Area and Dimension: Measuring Complexity and Scaling Properties.- Encoding Images by Simple Transformations.- The Chaos Game: How Randomness Creates Deterministic Shapes.- Recursive Structures: Growing Fractals and Plants.- Pascal's Triangle: Cellular Automata and Attractors.- Irregular Shapes: Randomness in Fractal Constructions.- Deterministic Chaos: Sensitivity, Mixing, and Periodic Points.- Order and Chaos: Period-Doubling and Its Chaotic Mirror.- Strange Attractors: The Locus of Chaos.- Julia Sets: Fractal Basin Boundaries.- The Mandelbrot Set: Ordering the Julia Sets.

Fractal scaling of soil particle-size distributions: analysis and limitations

Abstract: Fractal scaling has recently been proposed as a model for soil particle-size distribution (PSD). In this work, the cumulative number of soil grains greater than a characteristic size, N(R > r), and the cumulative mass distribution, M(r < R), are developed and shown to be proportional to R³⁻ᴰ and R³⁻ᴰ, respectively, where r is the grain size, R is a specific measuring scale, and D is the fractal dimension. The cumulative-number approach to estimate D is shown to be sensitive to the assumed grain density and characteristic size, while the mass distribution is less sensitive to the assumed grain density and characteristic size, and therefore more appropriate for the analysis of field soils. These two models of fractal PSD behavior also constrain the fractal dimension to lie between 0 and 3 for field soils. With constraints on the fractal dimension, soils displaying strict fractal scaling in grain-size distribution are shown to be a rather small subset of those soils commonly encountered in the field. Earlier work has shown fractal scaling in many soil PSDs with fractal dimensions exceeding 3.0 using the number-based analysis. The fractal scaling and magnitude of the fractal dimensions found in previous work are shown to be an artifact of the plotting algorithms and assumptions on grain density and size. Although fractal scaling plays an important role in soil water retention and porosity, PSD data alone are not sufficient to fully characterize this scaling.

Fractal models in the earth sciences

Abstract: Preface. 1. What on Earth fractal? The self-similarity of rivers. How long is the Vistula river? The paradox of tortuosity: permeability of kaolinite-bearing sandstones. The permeability of shaly sandstones. Basic concepts of percolation theory. Percolation models of rock permeability. Deadly quarrels and coastlines. The coastline of Britain. A Fractal model of coastal erosion. Bifractal coastlines. The perimeter-area rule of Mandelbrot. Islands and lakes. The fractal shape of clouds. The "slit island" analysis of fracture surfaces. Mathematical appendix. The functional equations of similarity. Fractal curves are hot. References. 2. Fractals in Flatland: a romance of < 2 dimensions. The paradox of sedimentation rate. Stratigraphic hiatuses and sedimentation rate. From the Cantor dust to the Devil's staircase. A fractal model for stratigraphic hiatuses. Sadler's model of unsteady sedimentation and its fractal generalisation. Fractal analysis along a line: slip lines and fractures. Strange attractors, aggregates and geophysical networks. Fractal characterisation of geophysical measuring networks. Fractals in the plane: fractures - earthquakes - volcanoes. Cellular structures. Fracture networks, faults and earthquakes. Mathematical appendix. Different kinds of fractal dimensions and their numerical determination. References. 3. Korcak's law and fragmentation theory. The size-frequency relation for islands, lakes and caves. Fragmentation: from broken sea ice to the distribution of galaxics. The fractal theory of fragmentation. The Renormalization Group (RNG) model of rock fragmentation. Maximum-entropy people and fractal people. References. 4. Fractal surfaces. Fractal surfaces everywhere. Simple geometrical models of fractal surfaces. Analytical treatment of fractal surfaces. Wave scattering from fractal surfaces. Fractal models of porous rocks. Why are the pores fractal rather than smooth? Multifractal measures - not for the squeamish. References. 5. Of time and change. Paradoxes of time. The puzzle called the Hurst phenomenon. Paradoxes of the 1/f noise. On growth and form. References. Author index. Subject index.

Shear-induced aggregation and breakup of polystyrene latex particles

Abstract: The structure of aggregates formed by diffusion-limited aggregation can be characterized by a fractal dimension of df ≈ 1.75. Further, fluid shear changes the trajectories of the colliding particles and applies a force on the aggregates, so that both the shear and the resulting force can lead to a different structure. Shear forces also cause breakup of the aggregates, finally leading to a balance between growth and breakup. Monodispersed polystyrene latex spheres of 2.17 μm diameter in a highly electrolyte environment were sheared in a Couette-flow system. An increasing shear rate accelerated the growth and decreased the stable size of the aggregates. The density of the aggregates in the initial growth phase was characterized by a fractal dimension of df ≈ 2.1, changing to df ≈ 2.5 in later stages of their growth. The initial fractal dimension can be explained by accelerated restructuring. A change in the particle trajectories (relative to Brownian motion) has recently been shown (1) not to alter the fractal dimension. The further increase in the fractal dimension is correlated to the fragmentation and can be explained by a selective breakup that removes preferably the most porous parts from an aggregate.

The coexistence of species in fractal landscapes

Abstract: The degree of spatial dependence in a landscape can be modeled by varying the landscape's fractal dimension, D. I simulated landscapes in which D varied independently of environmental variability. Individuals of 10 species with modes placed evenly along the dominant environmental gradient were randomly placed on these landscapes and were allowed to reproduce, disperse, and compete. The results demonstrate that the effect of a landscape's environmental variability on species coexistence is affected by the degree of spatial dependence in a complex manner. Increasing the fractal dimension (decreasing spatial dependence) allows more species to exist per microsite and per landscape. However, extremely high fractal dimensions cause fewer species to coexist on the landscape scale. Observed habitat breadths are increased (and beta diversity is decreased) as a function of D. I suggest that these patterns can be explained by habitat area, the mass effect, ecological equivalency, and global biological constraints.

Growth of a self-assembled monolayer by fractal aggregation.

Abstract: Atomic force microscope images show that self-assembled monolayers of octadecyltrichlorosilane form on mica by nucleating isolated, self-similar domains. With increasing coverage, the fractal dimension of the growing domains evolves from 1.6 to 1.8. At higher coverage, continued growth is limited by adsorption from solution. Monte Carlo simulations that include, for the first time, adsorption as well as surface diffusion qualitatively reproduce both the growth kinetics and evolution of fractal structure, much better than a two-dimensional diffusion-limited-aggregation model.

Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames.

Abstract: A new method for the in situ optical determination of the soot-cluster monomer particle radius a, the number of monomers per cluster N, and the fractal dimension D is presented. The method makes use of a comparison of the volume-equivalent sphere radius determined from scattering-extinction measurements RSe and the radius of gyration Rg, which is determined from the optical structure factor. The combination of these data with the measured turbidity permits for a novel measurement of D. The parameters a and N are obtained from a graphical network-analysis scheme that compares R(se) and Rg. Corrections for cluster polydispersity are presented. The effects of uncertainty in various input parameters and assumptions are discussed. The method is illustrated by an application to data obtained from a premixed methane-oxygen flame, and reasonable values of a, N, andD are obtained.

Four methods to estimate the fractal dimension from self-affine signals (medical application)

Abstract: Four methods for estimating the fractal dimension, namely, relative dispersion, correlation, rescaled range, and Fourier (spectral) analysis, are described. Modifications of these methods for use on self-similar or self-affine signals are presented. It is found that correlation analysis and rescaled range analysis yield seriously biased results under many circumstances. Relative dispersion analysis is well suited for long signals. Spectral analysis gives the least biased results, and also has lowest variance in the estimates of the fractal dimension. >

Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors.

Abstract: Fractal dimensions and Lyapunov exponents can be estimated faster and more accurately and efficiently with the knowledge of the optimal embedding dimension and delay time. Two methods are presented to calculate the optimal embedding parameters for Takens's delay-time coordinates. The first, called the fill factor, is a procedure that yields a global measure of phase-space utilization for (quasi) periodic and strange attractors and leads to a maximum separation of trajectories within the phase space. The second, which we call local deformation, is complementary to the fill factor

A stochastic view of lower crustal fabric based on evidence from the Ivrea Zone

Abstract: Despite its complicated history the Ivrea Zone is considered to be a representative surface exposure of extended continental crust. We have digitized two standard 1:25,000 geological maps from this area and evaluated their structural statistics. Because of the subvertical orientation of the Ivrea Zone these maps can be considered as small-scale cross sections through the lower continental crust. The autocorrelation functions of the digitized lithologies measured from these maps show a clear self-similar or fractal rather than a Gaussian or deterministic trend. We found that an anisotropic von Karman correlation function with an aspect ratio around 4 and a Hurst number of 0.3, corresponding to a fractal dimension of 2.7, matches the observed data. Our results represent an explicit confirmation of previous indirect evidence for the fractal nature of lithospheric heterogeneities and provide the means to construct realistic crustal-scale seismic models of Ivrea-type lower continental crust.

A new approach to the determination of the surface fractal dimension of porous solids

Abstract: A new approach to the determination of surface fractal dimension is proposed. It is based on the approximation of a given surface by a set of inscribed equicurvature surfaces. The surface fractal dimension, d fs , is determined from the relationship between the area, S c , and the mean radius of curvature, a c , of these surfaces, S c ∼ a 2− d fs . It is the common relationship for the area of a fractal surface measured by a yardstick of varying size, whose role here is played by a c . The equicurvature surfaces can be realized in practice as the interfaces between fluids at the conditions of capillary equilibrium in the vicinity of a given surface. The area and the mean radius of curvature of equilibrium interfaces can be calculated on the basis of experimental data by using general thermodynamic relationships. Corresponding thermodynamic methods for calculating the surface fractal dimension are developed for capillary condensation and intrusion of a nonwetting fluid.

Surface measurement and fractal characterization of naturally fractured rocks

Abstract: A method of stylus measurement of fractured rock surfaces over relatively long sample lengths is described, together with methods of characterizing the fracture structure. Surface topography results are presented and analysed in several ways, including the use of 'fractal' methods, which show that naturally fractured rock can be classified as self-affine fractals and require two parameters, fractal dimension and topothesy for characterizing these naturally fractured rocks.

Global fractal dimension of human DNA sequences treated as pseudorandom walks.

Abstract: We used a pseudorandom-walk representation in a four-dimensional embedding to estimate the global fractal dimension D of 164 sequences from GenBank and generated length-matched control sequences of three types: random, matched in base content, and matched in dimer content. The mean D of the sequence was 1.631±0.137. This D was significantly ower than the D's for all three control types, indicating the presence of significant information content in DNA sequences not explained by base or dimer frequencies. This variation was due largerly to nonuiform distribution of bases and dimers within DNA sequences

Fractal dimension in the analysis of medical images

Abstract: The analysis of cardiac magnetic resonance (MR) images and X-rays of bone is considered. Each image type is approached using a different form of fractal parameterization. For the MR images, the goal of the study is segmentation, and to this end small regions of the image are assigned a local value of fractal dimension. For the bone X-rays, rather than segmentation, the large-scale structure is parameterized by its fractal dimension. In both cases, the use of fractals leads to the classification of the parameters of interest. When applied to segmentation, this analysis yields boundary discrimination unavailable through previous methods. For the X-rays, texture changes are quantified and correlated with physical changes in the subject. In both cases, the parameterizations are robust with regard to noise present in the images, as well as to variable contrast and brightness. >

Fractal Dimensions of Soil Aggregate-size Distributions Calculated by Number and Mass

Abstract: The fractal dimension, D, has been used to characterize soil aggregate-size distributions. However, D is based on the number-size relationship. In most soils applications, it is the mass-size relationship that is determined. A number-size distribution can be generated from the mass-size distribution, assuming scale-invariant shape and density. Variation in shape and density as a function of size may introduce errors in the calculation of D. We compared the fractal dimensions of soil aggregates estimated from mass-size distribution data (Dₘ), with those computed from actual number-size distribution data determined by counting (Dₙ). The fractal dimension ranged from 0.67 to 3.92 for Dₙ, and from 0.79 to 4.06 for Dₘ. A significant linear relation was found between Dₘ and Dₙ, with R² = 0.935. The resulting intercept and slope were not significantly different from zero and one, respectively, indicating a 1:1 relationship. This implies that the assumption of scale-invariant shape and density was valid across the range of aggregate sizes studied (5.0 × 10⁻¹ to 3.2 × 10¹ mm). Thus, the fractal dimension can be estimated from mass-distribution data within this range.

Fractal dimension of plants and body size distribution in spiders.

Abstract: 1. The fractal dimension of natural habitats may influence both numbers and body size distribution of arthropods. Plants with highly complicated and interrupted leaf shapes, i.e. whose fractal dimension is high, may have more space available for small animals than for large ones. 2. Morse et al. (1985) argued that scaling of body size with number of individuals may be predicted by combining assumptions of population densities as proportional to metabolic rate −1 and vegetation fractals. 3. The hypothesis by Morse et al. (1985) was tested in a field experiment in which spiders were allowed to invade artificial plants placed in oak and spruce habitats

Network properties of a two-dimensional natural fracture pattern

Abstract: The influence that fractures exert on the permeability of a fractured rock is, to a large extent, controlled by the nature of the network formed by the fracture system. Here, the network properties of a two-dimensional natural pattern, mapped from the surface of a sandstone layer, are investigated and compared to those of realizations of spatially randomly distributed line segments with similar orientation and length distributions and line segment density (line length per unit area) to the natural pattern. These patterns are composed of clusters of varying size and shape, made up of interconnected fracture traces or line segments. Comparing the natural pattern with the realizations, the natural pattern was found to contain roughly half the number of clusters while the mass (total line length) of the largest cluster is approximately double that of the realizations. The size of the largest cluster controls the connectivity of the patterns, as can be seen by comparing the largest cluster of the natural pattern, which connects all four sides of the region, with those of the realizations, which are unconnected or connect only two sides. Cluster scaling characteristics were found to be similar in the natural pattern and the realizations and show a crossover from a dimension of one (their topological dimension) to two (the dimension of the embedding medium) at a point that corresponds to the fracture spacing. An investigation of the self-similarity dimension, using the box-counting method, showed similar characteristics with a broad transition zone between one- and two-dimensional behaviour at smaller box sizes. The patterns are therefore found to be non-fractal. The effect of the spatial distribution shown by the natural pattern is thus to modify the manner in which fractures are distributed among clusters, increasing connectivity (and permeability in the case of open fractures), but does not affect the cluster scaling characteristics or the self-similarity dimension of the fracture patterns.

Test of static structure factors for describing light scattering from fractal soot aggregates

Abstract: The authors compare several static structure factors for light scattering from fractal aggregates. The variation between these structure factors is a result of different cutoff functions for the cluster density correlation function. Light scattering data obtained from soot aggregates in a premixed CH{sub 4}/O{sub 2} flame are fit with three respresentative structure factors. Fractal dimensions determined from these fits under the assumptions of a monodisperse size distribution are all unsatisfactory. Inclusion of a scaling cluster size distribution yields good fits to structure factors derived from correlation functions with cutoffs faster than exponential, the exponential structure factor yielding a poor fit. A Gaussian cutoff was found to work the best. 18 refs., 7 figs., 1 tab.