Showing papers on "Fractal dimension published in 1995"

Simple mathematical model of a rough fracture

Abstract: A simple mathematical model of rough-walled fractures in rock is described which requires the specification of only three main parameters: the fractal dimension, the rms roughness at a reference length scale, and a length scale describing the degree of mismatch between the two fracture surfaces Fractured samples, collected from natural joints and laboratory specimens, have been profiled to determine the range of these three parameters in nature It is shown how this surface roughness model can be implemented on a computer, allowing future detailed study of the mechanical and transport properties of single fractures and the scale dependence of these properties

Applications of fractals in soil and tillage research: a review

Abstract: Fractals are spatial and temporal model systems generated using iterative algorithms with simple scaling rules. This paper reviews the literature on spatial fractals as it applies to soil and tillage research. Applications of fractals in this area can be grouped into three broad categories: (i) description of soil physical properties; (ii) modeling soil physical processes; (iii) quantification of soil spatial variability. In terms of physical properties, fractals have been used to describe bulk density, pore-size distribution, pore surface area, particle-size distribution, aggregate-size distribution, ped shape and soil microtopography. In terms of physical processes, fractals have been used to model adsorption, diffusion, transport of water and solutes, brittle fracture and fragmentation. In terms of spatial variability, fractals have been applied to quantify distributions of soil properties and processes using semivariograms, power spectra and multifractal spectra. Further research is needed to investigate the specificity of different fractal models, to collect data for testing these models, and to move from the current descriptive paradigm towards a more predictive one. Fractal theory offers the possibility of quantifying and integrating information on soil biological, chemical and physical phenomena measured at different spatial scales.

Determination of the Surface Fractal Dimension for Porous Media by Mercury Porosimetry

Abstract: A scaling relation is presented after a concrete analysis of the properties of topology and mercury porosimetry processes for porous media. Then a new method and the corresponding procedure using the above scaling relation are set up in order to obtain the surface fractal dimensions of porous media from the experimental data of mercury intrusion. In contrast with the existing methods of this kind exhibited in literature, the results simulated by means of the method for several kinds of porous media conform to the prerequisite for being surface fractals, which means the surface fractal dimensions should be in the very range: 2≤D<3. At the same time, this method gives almost the same surface fractal dimensions as the adsorption method. The evidence deduced from the results and analyses shows that this method may be a promising one to decide or at least estimate the surface fractal dimension for porous media

Fractal Morphology Analysis of Combustion-Generated Aggregates Using Angular Light Scattering and Electron Microscope Images

Abstract: Experimental studies of the fractal morphology of flame-generated aggregates are described here, considering not only the fractal dimension, D f , but also the fractal prefactor (lacunarity), k g , both of which are shown to be needed to fully characterize aggregates. Measurements were made using angular light scattering (ALS) and thermophoretic sampling followed by transmission electron microscopy (TEM) for soot aerosols found in laminar and turbulent flame environments. D f and the prefactor k g were simultaneously inferred from ALS measurements using the optical properties of aggregates composed of small primary particles. TEM-based inferences of these fractal properties involve analysis of aggregate-projected images from which the actual morphologies are obtained by correlating the radius of gyration to the outer radius (half of the maximum length) of an aggregate. Both of our procedures for determining the detailed morphology of aggregates yield D f = 1.7 ± 0.15 and k g = 2.4 ± 0.4 for carbonaceous soot, in good agreement with earlier TEM measurements involving multiple angle images. Furthermore, we show that these values also seem to be valid for other materials such as alumina aggregates, suggesting that not only the fractal dimension but also the fractal prefactor are universal properties of aggregates found in combustion environments due to similar mechanisms of aggregation. This universality for D f and k g is observed at various positions in four different flame types with nine various gaseous and liquid fuels for aggregates' with mean primary particle radii and number of primary particles in aggregates in the range 12-26 nm and 2-10 4 , respectively.

Multifractal scaling properties of a growing fault population

Abstract: SUMMARY A numerical rupture model, introduced in Cowie, Vanneste & Sornette (1993), is used to simulate the growth of faults in a tectonic plate driven by a constant plate boundary velocity. We find that the plate initially deforms by uncorrelated nucleation of small faults reflecting the distribution of material properties. With increasing strain, growth and coalescence of existing faults dominate over nucleation, a power-law distribution of fault sizes appears, and the fault pattern is fractal. Furthermore, the combined effect of fault clustering and the correlation between fault displacement and fault size leads to a strongly multifractal deformation pattern. We show theoretically that the multifractal spectrum depends explicitly on the exponent c, which defines the size distribution of the faults, as size and displacement are correlated. For different realizations of the numerical model, we calculate the exponent c, and fractal structure of the deformation through time as strain accumulates. We explore in detail the time evolution of the capacity (D0), information (D1), and correlation (D2) fractal dimensions. We relate these scaling parameters to the physical mechanisms of fault nucleation, growth and linkage during different phases of the deformation and discuss the factors that determine the values of the exponents. A consistently observed systematic decrease in the values of c, D1 and D2 through time indicates that the relative strain contribution of the smallest faults decreases as the total strain increases, a signature of the localization of faulting.

Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension

Abstract: We examine the estimation of selectivities for range and spatial join queries in real spatial databases. As we have shown earlier [FK94], real point sets: (a) violate consistently the “uniformity” and “independence” assumptions, (b) can often be described as “fractals”, with non-integer (fractal) dimension. In this paper we show that, among the infinite family of fractal dimensions, the so called “Correlation Dimension” Dz is the one that we need to predict the selectivity of spatial join. The main contribution is that, for all the real and synthetic point-sets we tried, the average number of neighbors for a given point of the point-set follows a power law, with LI& as the exponent. This immediately solves the selectivity estimation for spatial joins, as well as for “biased” range queries (i.e., queries whose centers prefer areas of high point density). We present the formulas to estimate the selectivity for the biased queries, in&ding an integration constant (KLshape,) for each query shape. Finally, we show results on real and synthetic point sets, where our formulas achieve very low relative errors (typically about lo%, versus 40%-100% of the uniformity assumption).

Fractals in the Earth Sciences

Abstract: Fractal Distributions in Geology, Scale Invariance, and Deterministic Chaos (D.L. Turcotte, J. Huang). Some Long Run Properties of Geophysical Records (B.B. Mandelbrot, J.R. Wallis). Some Remarks on the Numerical Estimation of Fractal Dimension (S.A. Pruess). Measuring the Dimension of Selfaffine Fractals: The Example of Rough Surfaces (S.R. Brown). A Review of the Fractal Character of Natural Fault Surfaces with Implications for Friction and the Evolution of Fault Zones (W.L. Power, T.E. Tullis). Fractals and Ocean Floor Topography: A Review and a Model (A. Malinverno). Fractal Transitions on Geological Surfaces (C.H. Scholz). Fractal Scaling of Fracture Networks (C.C. Barton). Fractal Fragmentation in Crustal Shear Zones (C.G. Sammis, S.J. Steacy). Fractal Distribution of Fault Length and Offsets: Implications on Brittle Deformation Evaluation: The Lorraine Coal Basin (T. Villemin et al.). Fractal Dynamics of Earthquakes (P. Bak, K. Chen). Mineral Crystallinity in Igneous Rocks: Fractal Method (A.D. Fowler). Fractal Structure of Electrum Dendrites in Bonanza Epithermal AuAg Deposits (J.A. Saunders, P.A. Schoenly). Appendix. Index.

The relationship between fractional calculus and fractals

Abstract: The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).

Grain-size distribution in deforming subglacial tills: Role of grain fracture

Abstract: New ways of looking at grain-size distributions may yield insights into sedimentary processes or environments. For example, during shearing of a granular material, alignments of grains, or bridges, develop with orientations such that compressive forces parallel to these alignments support most of the applied shear stress. If deformation is due to failure of such bridges by fracture, rather than by, say, dilation, the grain-size distribution will tend toward one that provides the maximum support for the grains. This size distribution is fractal and has a fractal dimension of 2.6. We analyzed the grain-size distribution of three deforming tills collected from beneath modern glaciers. The size distributions are fractal, and the mean fractal dimension is ∼2.9, suggesting an excess of fines. For comparison, grain-size distributions of samples from some other common sedimentary environments were also analyzed. Samples of dune sand and of glacial outwash were not fractal, but a debris-flow sample was, and had a fractal dimension of 2.8.

Fractal Analysis of Scaling and Spatial Clustering of Fractures

Abstract: Fractures exist over a wide range of scales, from the largest faults to microfractures, and this range is primarily responsible for scaling effects observed in fractured-rock hydrology and bulk mechanical properties of fractured rock (Witherspoon and others, 1979; Thorp and others, 1983; DeMarsily, 1985).

The perimeter-area fractal model and its application to geology

Abstract: Perimeters and areas of similarly shaped fractal geometries in two-dimensional space are related to one another by power-law relationships. The exponents obtained from these power laws are associated with, but do not necessarily provide, unbiased estimates of the fractal dimensions of the perimeters and areas. The exponent (DAL) obtained from perimeter-area analysis can be used only as a reliable estimate of the dimension of the perimeter (DL) if the dimension of the measured area is DA=2. If DA DL. Similar relations hold true for area and volumes of three-dimensional fractal geometries. The newly derived results are used for characterizing Au associated alteration zones in porphyry systems in the Mitchell-Sulphurets mineral district, northwestern British Columbia.

Estimation and simulation of fractal stochastic point processes

Abstract: We investigate the properties of fractal stochastic point processes (FSPPs). First, we define FSPPs and develop several mathematical formulations for these processes, showing that over a broad range of conditions they converge to a particular form of FSPP. We then provide examples of a wide variety of phenomena for which they serve as suitable models. We proceed to examine the analytical properties of two useful fractal dimension estimators for FSPPs, based on the second-order properties of the points. Finally, we simulate several FSPPs, each with three specified values of the fractal dimension. Analysis and simulation reveal that a variety of factors confound the estimate of the fractal dimension, including the finite length of the simulation, structure or type of FSPP employed, and fluctuations inherent in any FSPP. We conclude that for segments of FSPPs with as many as 106 points, the fractal dimension can be estimated only to within ±0.1.

Fractal character of cold-deposited silver films determined by low-temperature scanning tunneling microscopy

Abstract: High-resolution scanning-tunneling-microscope (STM) topographic images of vacuum-deposited Ag films are reported. Films were formed and imaged at 100 and 300 K. Images of films deposited at 300 K, annealed to 560 K, and then returned to 300 K are also presented. The topographic surfaces of the low-temperature films are found to be self-affine fractals with a local Hausdorff-Besicovitch dimension D=2.5. The low-temperature films exhibit intense surface-enhanced Raman spectra (SERS). Films deposited at 300 K do not possess significant fractal character and are not SERS active. We show that the apparent local fractal dimension obtained by analyzing STM topographic images depends critically on the algorithm used. Three such methods (cube counting, triangulation, and power spectrum analysis) are assessed. A method is proposed for obtaining reliable fractal dimensions by analyzing the experimental STM topographic images using several algorithms and comparing the results to a calibration curve generated by applying the same algorithms to simulated fractal surfaces of known Hausdorff-Besicovitch dimension.

Evaluation of the dispersional analysis method for fractal time series.

Abstract: Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is 0.5 for positively correlated series, and = 0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 - D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (> 0.8) and low H ( 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

Fractal dimension of electroencephalographic time series and underlying brain processes

Abstract: Fractal dimension has been proposed as a useful measure for the characterization of electrophysiological time series. This paper investigates what the pointwise dimension of electroencephalographic (EEG) time series can reveal about underlying neuronal generators. The following theoretical assumptions concerning brain function were made (i) within the cortex, strongly coupled neural assemblies exist which oscillate at certain frequencies when they are active, (ii) several such assemblies can oscillate at a time, and (iii) activity flow between assemblies is minimal. If these assumptions are made, cortical activity can be considered as the weighted sum of a finite number of oscillations (plus noise). It is shown that the correlation dimension of finite time series generated by multiple oscillators increases monotonically with the number of oscillators. Furthermore, it is shown that a reliable estimate of the pointwise dimension of the raw EEG signal can be calculated from a time series as short as a few seconds. These results indicate that (i) The pointwise dimension of the EEG allows conclusions regarding the number of independently oscillating networks in the cortex, and (ii) a reliable estimate of the pointwise dimension of the EEG is possible on the basis of short raw signals.

Multifractal nature of concrete fracture surfaces and size effects on nominal fracture energy

Abstract: Experimental evidence of the fractality of fracture surfaces has been widely recognized in the case of concrete, ceramics and other disordered materials. An investigationpost mortem on concrete fracture surfaces of specimens broken in direct tension has been carried out, yielding non-integer (fractal) dimensions of profiles, which are then related to the ‘renormalized fracture energy’ of the material. No unique value for the fractal dimension can be defined: the assumption of multifractality for the damaged, material microstructure produces a dimensional increment of the dissipation space with respect to the number 2, and represents the basis for the so-called multifractal scaling law. A transition from extreme Brownian disorder (slope 1/2) to extreme order (zero slope) may be evidenced in the bilogarithmic diagram: the nominal fracture energyGF increases with specimen size by following a nonlinear trend. Two extreme scaling regimes can be identified, namely the fractal (disordered) regime, corresponding to the smallest sizes, and the homogeneous (ordered) regime, corresponding to the largest sizes, for which an asymptotic constant value ofGF is reached.

Enhancement of physiological signals using fractal analysis

Abstract: A technique for the derivation of a pulse oximetry signal using fractal dimension analysis of detected light signals. First and second light sources transmit light through the patient's finger or reflect light off the blood vessels in the patient's finger. A light detector detects light from each of the light sources and generates a measured intensity signal. The measured intensity signal includes the true intensity of light transmitted from each of the light sources as well as noise introduced during the measurement process. A data sample from each of the light sources is digitized and a set of equations developed as a function of a ratio value indicative of oxygen saturation in the patient. The fractal dimension is determined for the set of signal functions over the normal physiological range for the ratio value. Maximum and/or minimum fractal dimension values are calculated to determine the desired ratio values which are possible indicatives of the ratio of true physiological signals or noise signals. The ratio values are subsequently processed to determine the oxygen saturation within the patient.

Can fractal features be used for recognizing 3-d partial discharge patterns

Abstract: Fractals have been used extensively to provide a description and to model mathematically many of the naturally occurring sampler shapes, such as coastlines, mountain ranges, clouds, etc., and have also received increased attention in the field of image processing, for purposes of segmentation and recognition of regions and objects present in natural scenes. Among the numerous fractal features that could be defined and used for an image surface, fractal dimension and lacunarity have been found to be useful for recognition purposes. Partial discharges (PD) occurring in all HV insulation systems is a very complex phenomenon, and more so are the shapes of the various 3-d patterns obtained during routine tests and measurements. It has been fairly well established that these pattern shapes and underlying defects causing PD have a 1:1 correspondence, and therefore methods to describe and quantify these pattern shapes must be explored, before recognition systems based on them could be developed. This contribution reports preliminary results of such a study, wherein the 3-d PD pattern surface was considered to be a fractal, and the computed fractal features (fractal dimension and lacunarity) were analyzed and found to possess fairly reasonable pattern discriminating abilities. This new approach appears promising, and further research is essential before any long-term predictions can be made. >

On the fractal structure of two-dimensional quantum gravity

Abstract: We provide evidence that the Hausdorff dimension is 4 and the spectral dimension is 2 for two-dimensional quantum gravity coupled the matter with a central charge $c \leq 1$. For $c > 1$ the Hausdorff dimension and the spectral dimension monotonously decreases to 2 and 1, respectively.