scispace - formally typeset
Search or ask a question

Showing papers on "Fractal dimension published in 2001"


Book
27 Sep 2001
TL;DR: In this article, the authors present a mathematical model for chaotic multidimensional flows and fractal dimension calculation based on the Lyapunov exponents and the Hamiltonian chaos.
Abstract: Preface 1. Introduction 2. One-dimensional maps 3. Nonchaotic multidimensional flows 4. Dynamical systems theory 5. Lyapunov exponents 6. Strange attractors 7. Bifurcations 8. Hamiltonian chaos 9. Time-series properties 10. Nonlinear prediction and noise reduction 11. Fractals 12. Calculation of fractal dimension 13. Fractal measure and multifractals 14. Nonchaotic fractal sets 15. Spatiotemporal chaos and complexity A. Common chaotic systems B. Useful mathematical formulas C. Journals with chaos and related papers Bibliography Index

1,687 citations


Journal ArticleDOI
TL;DR: In this paper, the authors provide guidelines for the accurate and practical estimation of exponents and fractal dimensions of natural fracture systems, including length, displacement and aperture power law exponents.
Abstract: Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

1,153 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a review of scattering and absorption of light by fractal aggregates, which are typically diffusion limited cluster aggregates with fractal dimensions of D.
Abstract: This paper presents a review of scattering and absorption of light by fractal aggregates. The aggregates are typically diffusion limited cluster aggregates (DLCA) with fractal dimensions of D

1,027 citations


Journal ArticleDOI
TL;DR: This study demonstrates that a careful selection of fractal dimension algorithm is required for specific applications, and the most common methods of estimating the fractaldimension of biomedical signals directly in the time domain are analyzed and compared.
Abstract: The fractal dimension of a waveform represents a powerful tool for transient detection. In particular, in analysis of electroencephalograms and electrocardiograms, this feature has been used to identify and distinguish specific states of physiologic function. A variety of algorithms are available for the computation of fractal dimension. In this study, the most common methods of estimating the fractal dimension of biomedical signals directly in the time domain (considering the time series as a geometric object) are analyzed and compared. The analysis is performed over both synthetic data and intracranial electroencephalogram data recorded during presurgical evaluation of individuals with epileptic seizures. The advantages and drawbacks of each technique are highlighted. The effects of window size, number of overlapping points, and signal-to-noise ratio are evaluated for each method. This study demonstrates that a careful selection of fractal dimension algorithm is required for specific applications.

444 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce simple stochastic models which allow for any combination of fractal dimension and Hurst exponent, and synthesize images from these models, with arbitrary fractal properties and power-law correlations.
Abstract: Fractal behavior and long-range dependence have been observed in an astonishing number of physical systems. Either phenomenon has been modeled by self-similar random functions, thereby implying a linear relationship between fractal dimension, a measure of roughness, and Hurst coefficient, a measure of long-memory dependence. This letter introduces simple stochastic models which allow for any combination of fractal dimension and Hurst exponent. We synthesize images from these models, with arbitrary fractal properties and power-law correlations, and propose a test for self-similarity.

265 citations


Journal ArticleDOI
TL;DR: In this article, the influence of surface roughness on the adhesion of elastic solids was studied and a partial detachment transition preceding a full detachment transition was found in the case of a self-affine fractal.
Abstract: We study the influence of surface roughness on the adhesion of elastic solids. Most real surfaces have roughness on many different length scales, and this fact is taken into account in our analysis. We consider in detail the case when the surface roughness can be described as a self-affine fractal, and show that when the fractal dimension Df>2.5, the adhesion force may vanish, or be at least strongly reduced. We consider the block-substrate pull-off force as a function of roughness, and find a partial detachment transition preceding a full detachment one. The theory is in good qualitative agreement with experimental data.

253 citations


Journal ArticleDOI
TL;DR: In this article, the effect of scale on the surface roughness of rock joints was investigated using a 3-D laser scanner with high accuracy and resolution on a silicon rubber replica of a natural rock joint surface.

217 citations


Journal ArticleDOI
01 Aug 2001-Methods
TL;DR: Practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns, including fractal dimensionality are reviewed.

194 citations


Journal ArticleDOI
TL;DR: In this paper, the porosity and permeability by scale dissolution and precipitation in porous media is described based on fractal attributes of the pores, realization of flow channels as a bundle of uniformly distributed mean-size cylindrical and tortuous hydraulic flow tubes, a permeability-porosity relationship conforming to Civan's power law flow units equation, and the pore surface scale precipitation and dissolution kinetics.
Abstract: Variation of porosity and permeability by scale dissolution and precipitation in porous media is described based on fractal attributes of the pores, realization of flow channels as a bundle of uniformly distributed mean-size cylindrical and tortuous hydraulic flow tubes, a permeability-porosity relationship conforming to Civan's power law flow units equation, and the pore surface scale precipitation and dissolution kinetics. Practical analytical solutions, considering the conditions of typical laboratory core tests and relating the lumped and phenomenological parameters, were derived and verified by experimental data. Deviations of the empirically determined exponents of the pore-to-matrix volume ratio compared to the Kozeny-Carman equation were due to the relative fractal dimensions of pore attributes of random porous media. The formulations provide useful insights into the mechanism of porosity and permeability variation by surface processes and accurate representation of the effect of scale on porosity and permeability by simpler lumped-parameter models.

184 citations


Journal ArticleDOI
TL;DR: In this article, the authors applied multifractal techniques to characterize contrasting PSDs and to identify multifracfal parameters potentially useful for classification and prediction, which indicated that multifractals are promising descriptors of PSDs.
Abstract: A particle-size distribution (PSD) constitutes a fundamental soil property correlated to many other soil properties. Accurate representations of PSDs are, therefore, needed for soil characterization and prediction purposes. A power-law dependence of particle mass on particle diameter has been used to model soil PSDs, and such power-law dependence has been interpreted as being the result of a fractal distribution of particle sizes characterized with a single fractal dimension. However, recent studies have shown that a single fractal dimension is not sufficient to characterize a distribution for the entire range of particle sizes. The objective of this study was to apply multifractal techniques to characterize contrasting PSDs and to identify multifracfal parameters potentially useful for classification and prediction. The multifractal spectra of 30 PSDs covering a wide range of soil textural classes were analyzed. Parameters calculated from each multifractal spectrum were: (i) the Hausdorff dimension, f(α); (ii) the singularities of strength, α; (iii) the generalized fractal dimension, D q ; and (iv) their conjugate parameter the mass exponent, τ (q), calculated in the range of moment orders (q) of between -10 and -10 taken at 0.5 lag increments. Multifractal scaling was evident by an increase in the difference between the capacity D 0 and the entropy D 1 dimensions for soils with more than 10% clay content, Soils with <10% clay content exhibited single scaling. Our results indicate that multifractal parameters are promising descriptors of PSDs. Differences in scaling properties of PSDs should be considered in future studies.

176 citations


Journal ArticleDOI
01 Sep 2001-Geoderma
TL;DR: In this paper, the surface fractal dimension of the pore-solid interface was measured by fitting two straight lines to the log-log plot and finding a crossover point at a scale of about 14 μm, forming the border between textural and structural fractality.


Journal ArticleDOI
01 Jan 2001-Langmuir
TL;DR: In this article, small-angle neutron scattering and dynamic and static light scattering measurements were used to probe the structures of aqueous and organic-solvent-based magnetic fluids comprising dispersed magnetite nanoparticles (∼10 nm in diameter) stabilized against flocculation by adsorbed alkanoic acid layers.
Abstract: Small-angle neutron scattering and dynamic and static light scattering measurements were used to probe the structures of aqueous and organic-solvent-based magnetic fluids comprising dispersed magnetite nanoparticles (∼10 nm in diameter) stabilized against flocculation by adsorbed alkanoic acid layers. A core−shell model fitted to a set of neutron scattering spectra obtained from contrast variation experiments allowed the determination of the iron oxide core size and size distribution, the thicknesses of the surfactant shells, and the spatial arrangement of the individual particles. The magnetic colloidal particles appear to form compact fractal clusters with a fractal dimension of 2.52 and a correlation length of ∼350 A in aqueous magnetic fluids, consistent with the structures of clusters observed directly using cryo-TEM (transmission electron microscopy), whereas chainlike clusters with a fractal dimension of 1.22 and a correlation length of ∼400 A were found for organic-solvent-based magnetic fluids. T...

Journal ArticleDOI
TL;DR: The fractal structure of star formation on large scales in disk galaxies is studied using the size distribution function of stellar aggregates in kiloparsec-scale star fields as mentioned in this paper.
Abstract: The fractal structure of star formation on large scales in disk galaxies is studied using the size distribution function of stellar aggregates in kiloparsec-scale star fields. Archival Hubble Space Telescope images of 10 galaxies are Gaussian-smoothed to define the aggregates, and a count of these aggregates versus smoothing scale gives the fractal dimension. Fractal and Poisson models confirm the procedure. The fractal dimension of star formation in all of the galaxies is ~2.3. This is the same as the fractal dimension of interstellar gas in the Milky Way and nearby galaxies, suggesting that star formation is a passive tracer of gas structure defined by self-gravity and turbulence. Dense clusters such as the Pleiades form at the bottom of the hierarchy of structures, where the protostellar gas is densest. If most stars form in such clusters, then the fractal arises from the spatial distribution of their positions, giving dispersed star fields from continuous cluster disruption. Dense clusters should have an upper mass limit that increases with pressure, from ~103 M⊙ in regions like the solar neighborhood to ~106 M⊙ in starbursts.

Journal ArticleDOI
TL;DR: A lower bound of the box size is found and the reason for having it is provided and indicates the need for limiting the box sizes within certain bounds.

Journal ArticleDOI
TL;DR: In this paper, a framework for the mechanics of solids, deformable over fractal subsets, is outlined, and an extension of the Gauss-Green theorem to fractional operators is proposed to demonstrate the duality principle for fractal media.

Journal ArticleDOI
TL;DR: In this paper, the authors used two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension and found that fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation by increasing surface irregularity and hence roughness.
Abstract: This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.

Journal ArticleDOI
15 Sep 2001
TL;DR: In this article, the power law behavior of scattered light intensity as a function of scattering wave vector is observed in all cases and is suggestive of fractal structure, and the fractal dimensions obtained fall within the expected range of 1.8 to 2.3 observed for colloidal aggregates.
Abstract: Suspensions of a variety of different aluminum oxides have previously been shown to require very high concentrations of chloride and nitrate anions (>0.5 M) to induce rapid aggregation. This high stability has been accredited to the presence of surface forces considered to be due to the formation of highly charged Al13 polymeric species at slightly acidic pH's and aluminum oxyhydroxide gel formation under alkaline conditions. The effect of this stability on the structure of the resulting aggregates is investigated here using well-established static light-scattering techniques. Power law behavior of scattered light intensity as a function of scattering wave vector is observed in all cases and is suggestive of fractal structure. The fractal dimensions obtained fall within the expected range of 1.8 to 2.3 observed for colloidal aggregates but do not appear to follow the typical observations for colloids destabilized by indifferent electrolytes where lower fractal dimensions are associated with rapid (diffusion-limited) aggregation and higher fractal dimensions with slower (reaction-limited) aggregation. Indeed, relatively constant fractal dimensions (2.10 to 2.25) are observed over the range of salt concentrations at which the slow to rapid aggregation rate transformation occurs with, if anything, a slightly higher fractal dimension observed for higher aggregation rates. The presence of specifically binding sulfate anions appears to negate the strong near-distance repulsive forces leading to rapid aggregation at low (1 to 2 mM) sulfate concentrations. Significantly lower fractal dimensions (1.85 to 1.91) are observed for aggregates formed by destabilization using sulfate ions than obtained when chloride or nitrate are used with, again, an apparent slight increase in fractal dimension upon increasing aggregation rate.

Journal ArticleDOI
TL;DR: The presence of multifractility is confirmed, the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.

Journal ArticleDOI
TL;DR: In this paper, the structures of cluster-cluster and particlecluster fractal-like aggregates were investigated and the morphological properties of aggregates undergoing partial sintering and restructuring were also investigated.

Journal ArticleDOI
TL;DR: In this paper, the authors estimate Levy-stable (fractal) distributions that can accurately account for skewness, kurtosis, and fat tails of the returns.
Abstract: It is argued that the study of the correct specification of returns distributions has attractive implications in financial economics. This study estimates Levy-stable (fractal) distributions that can accurately account for skewness, kurtosis, and fat tails. The Levy-stable family distributions are parametrized by the Levy index (α), 0 < (α), ≤ 2, and include the normal distribution as a special case (α = 2). The Levy index, α, is the fractal dimension of the probability space. The unique feature of Levy-stable family distributions is the existence of a relationship between the fractal dimension of the probability space andthe fractal dimensionof the time series. This relationshipis simply expressed in terms of Hurst exponent (H), i.e. α = 1/ H. In addition, Hurst exponent is related to long-memory effects. Thus, estimating the Levy index allows us to determine long-memory effects. The immediate practical implication of the present work is that on the one hand we estimate the shape of returns distributions...

Journal ArticleDOI
TL;DR: The fractal signatures of small regions of interest (32x32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images.

Journal ArticleDOI
TL;DR: This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski–Bouligand, and mass‐radius fractal dimensions.
Abstract: Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated Bone biopsies were obtained from 148 osteoporotic male patients Bone volume (BV/TV), trabecular characteristics (TbN, TbSp, TbTh), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincare number were computed The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L) Logarithmic relationships were observed between BV/TV and the fractal dimensions The best correlation was obtained with D(MR) and the lowest with D(MB) Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, TbSp) Linear relationships were observed among the three fractal techniques which appeared highly correlated A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (TbTh, strut); for marrow cavities (Euler, ICI, TbSp, star volume, L); and for the complexity of the network (TbN and the three D's) A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae Computation time is also an important determinant when choosing one of these methods

Journal ArticleDOI
TL;DR: In this paper, a simple constitutive law is derived for determining the fractal dimension of an aggregate, resulting from a coagulation event between aggregates with different fractal dimensions.

Journal ArticleDOI
TL;DR: In this article, a stereoscopic scanning electron microscopic (SEM) method based on surface areas tallied over a much wider range of measurement scales was used to evaluate the fractal characteristics of fracture surfaces of cement pastes and mortars.

Journal ArticleDOI
TL;DR: In this article, a new pattern recognition algorithm based on multiple intensity clips was developed, which assures an optimal adaptation to the solar structure under study, and is demonstrated by application to the intensity structure of solar granulation near the disk center, both speckle reconstructed and not.
Abstract: We have developed a new pattern-recognition algorithm based on multiple intensity clips which assures an optimal adaptation to the solar structure under study. The shapes found at higher clip levels are repeatedly extended to lower levels, thus filling more and more of the observed intensity contours. Additionally, at each intensity threshold new shapes, exceeding the level, are integrated. The number and height of the levels can be optimized making this `multiple level tracking' algorithm (MLT) superior to commonly used Fourier-based recognition techniques (FBR). The capability of MLT is demonstrated by application to the intensity structure of solar granulation near the disk center, both speckle reconstructed and not. Comparisons with Doppler maps prove its reliability. The granular pattern recognized by MLT differs essentially from that obtained with FBR. Elongated `snake-like' granules do not occur with MLT and, consequently, the perimeter-area distribution exhibits only a marginal `second branch' of higher fractal dimension, which dramatically diminishes the better the MLT pattern matches the granular structure. The final distribution obtained with optimized parameters has a single fractal dimension near 1.1, making the question of a `critical size', a `second branch', and the often discussed dimension of 4/3; highly questionable. This result is equally obtained from application of MLT to the corresponding Doppler velocity map of granular up-flows. In contrast, the pattern of down-flows contains some elongated `snake-like' structures with higher fractal dimension. We also use the new algorithm to recognize speckle-reconstructed limb faculae, which MLT separates from their granular surroundings, and compare both, granules and faculae, using large statistical samples. The facular grains near cosθ=57° exhibit a slightly different ellipticity than the (geometrically foreshortened) adjacent granules. However, small facular grains are more round than small granules and larger grains are more similar to granules.

Journal ArticleDOI
01 Nov 2001
TL;DR: In this paper, different collision frequency kernel calculation schemes were evaluated to simulate experimentally determined latex particle aggregation kinetics, and the intermediate collision scheme proposed by Veerepanneni and Wiesner (J. Colloid Interface Sci. 177, 45-57 (1996)) is in agreement with experimental aggregation kinetic if considered along with experimentally derived fractal dimension evolution.
Abstract: Aggregate structure, i.e., fractal dimension, has a coupled geometric and hydrodynamic impact on the aggregate collision frequencies. In this work, we evaluated different collision frequency kernel calculation schemes to simulate experimentally determined latex particle aggregation kinetics. Light scattering was used to simultaneously measure aggregate size and fractal dimension throughout the latex aggregation experiments. The intermediate collision scheme proposed by Veerepanneni and Wiesner ( J. Colloid Interface Sci. 177, 45–57 (1996)) is in agreement with experimental aggregation kinetics if considered along with experimentally determined fractal dimension evolution. Aggregate restructuring was found to be an important mechanism in explaining aggregation kinetics.

Journal ArticleDOI
TL;DR: In this paper, different fractal models of porous solid are described and compared, and fractal dimension values calculated from the different models are consistent between them, which can be used as stone pore system descriptor and, in addition, allow the differentiation between weathered and unweathered stones.

Journal ArticleDOI
TL;DR: Water-sorption measurements confirm that interface smoothing is due predominantly to the water condensing in the most strongly curved asperities rather than covering the surface with a wetting film of uniform thickness, where a Porod behavior typical of smooth interfaces is observed instead.
Abstract: Small-angle x-ray and neutron scattering are used to characterize the surface roughness and porosity of a natural rock which are described over three decades in length scales and over nine decades in scattered intensities by a surface fractal dimension D=2.68{+-}0.03 . When this porous medium is exposed to a vapor of a contrast-matched water, neutron scattering reveals that surface roughness disappears at small scales, where a Porod behavior typical of smooth interfaces is observed instead. Water-sorption measurements confirm that such interface smoothing is due predominantly to the water condensing in the most strongly curved asperities rather than covering the surface with a wetting film of uniform thickness.

Journal ArticleDOI
01 Nov 2001-Polymer
TL;DR: In this article, the authors analyzed carbon black polymer composites in terms of their fractal geometry and found that the fractal dimension of these composites increases with the number of filler points.