From Fractal Geometry to Fractal Analysis

Proper understanding of offshore wind speed variability is of essential importance in practice, which provides useful information to a wide range of coastal and marine activities. In this paper, long-term wind speed data recorded at various offshore stations are analyzed in the framework of fractal dimension analysis. Fractal analysis is a well-established data analysis tool, which is particularly suitable to determine the complexity in time series from a quantitative point of view. The fractal dimension is estimated using the conventional box-counting method. The results suggest that the wind speed data are generally fractals, which are likely to exhibit a persistent nature. The mean fractal dimension varies from 1.31 at an offshore weather station to 1.43 at an urban station, which is mainly associated with surface roughness condition. Monthly variability of fractal dimension at offshore stations is well-defined, which often possess larger values during hotter months and lower values during winter. This is partly attributed to the effect of thermal instability. In addition, with an increase in measurement interval, the mean and minimum fractal dimension decrease, whereas the maximum and coefficient of variation increase in parallel.

Urban skylines have a dynamic structure because they reflect changing conditions in time and space. This dynamic structure also has the potential to be an urban information tool for evaluating the visual qualities and complexities of cities. This article presents a new methodological perspective on urban skyline studies through fractal analysis in Istanbul. In the last 30 years, several computational techniques have been developed to measure the morphological characteristics of cities. Fractal analysis, the most well-known and widespread of these, is used in this study to analyze the complex structure of urban skylines obtained through visual data. The study aims to first expose the existing urban skyline of Istanbul and then assess the influence of the factors influencing this skyline on its appearance by using fractal geometry. The fractal dimension, measured by the box count is used for this purpose and that is a metric used to describe the complexity of urban skylines. In this study, in which Beyazıt Fire Tower is accepted as the reference point, the fractal dimensions of the skyline obtained from urban panoramas are calculated using the ImageJ Program and FracLac plugin. As a result of the fractal analysis of the skyline, fractal dimensions in the range of 1,0731-1,2171 were obtained. The skylines and their trends are presented through the rose diagram, and fractal analysis shows that two main elements complicate the appearance of the city the most which are: mosque and skyscrapers.

Characterization of pore structure is one of the key problems for fabrication and application research on porous materials. But, complexity of pore structure makes it difficult to characterize pore structure by Euclidean geometry and traditional experimental methods. Fractal theory has been proved effective to characterize the complex pore structure. The box dimension method based on fractal theory was applied to characterizing the pore structure of fiber porous materials by analyzing the electronic scanning microscope (SEM) images of the porous materials in this paper. The influences of image resolution, threshold value, and image magnification on fractal analysis were investigated. The results indicate that such factors greatly affect fractal analysis process and results. The appropriate magnification threshold and fractal analysis are necessary for fractal analysis.

Surface morphologies of supported polyethylene (PE) catalysts are investigated by an approach combining fractal with wavelet. The multiscale edge (detail) pictures of catalyst surface are extracted by wavelet transform modulus maxima (WTMM) method. And, the distribution of edge points on the edge image at every scale is studied with fractal and multifractal method. Furthermore, the singularity intensity distribution of edge points in the PE catalyst is analyzed by multifractal spectrum based on WTMM. The results reveal that the fractal dimension values and multifractal spectrums of edge images at small scales have a good relation with the activity and surface morphology of PE catalyst. Meanwhile the catalyst exhibiting the higher activity shows the wider singular strength span of multifractal spectrum based on WTMM, as well as the more edge points with the higher singular intensity. The research on catalyst surface morphology with hybrid fractal and wavelet method exerts the superiorities of wavelet and fractal theories and offers a thought for studying solid surfaces morphologies.

The fractal dimension is quantitatively to define the irregular characteristic of the shape in natural. It can be useful in describing morphological characteristics of various wear particles. This paper was undertaken to diagnose failure condition for sliding members in lubrication by fractal dimension. It will be possible to diagnose wear mechanism, friction and damage state of machines through analysis of shape characteristics for wear particle on driving condition by fractal parameters. In this study, the calculating and analyzing methods of fractal dimensions were constructed for the condition monitoring and wear particle analysis in lubricant condition. So, we carried out the Friction and wear test with the ball on disk type tester, and the fractal parameters of wear particle in lubricated conditions were calculated. Fractal parameters were defined as texture fractal dimension (), structure fractal dimension () and total fractal dimension (D).

<div class="htmlview paragraph">The physical and morphological properties of the particulate matter emitted from a 1.7-liter light-duty diesel engine were characterized by observing its evolution in size and fractal geometry along the exhaust system. A common-rail direct-injection diesel engine, the exhaust system of which was equipped with a turbocharger, EGR, and two oxidation catalysts, was powered with a California low-sulfur diesel fuel at various engine-operating conditions. A unique thermophoretic sampling system, a high-resolution transmission electron microscope (TEM), and customized image processing/data acquisition systems were key instruments that were used for the collection of particulate matter, subsequent imaging of particle morphology, and detailed analysis of particle dimensions and fractal geometry, respectively. The measurements were carried out at four different positions along the exhaust pipe. The rapid insertion of sampling probe into the exhaust stream, whose residence time resolves as short as 10 msec, provided near interference-free sampling of diesel particulates. From analyses for the soot samples collected at 2500-rpm and 25%-load, it was revealed that the primary particle diameter decreased from 28.6 nm to 19.1 nm and the corresponding radius of gyration of aggregate particles also decreased from 101.6 nm to 49.6 nm along the exhaust pipe. This reduction of particle sizes implies that particle sizes are significantly affected by aftertreatment components in the exhaust system. The detailed fractal analysis supported this finding; a fractal dimension of particles was slightly higher right after the first catalyst than those evaluated at other sampling positions, which indicates that particles became more spherical in shape during passing through the first catalyst. The fractal dimensions varied in a range of 1.5 to 1.7 along the exhaust pipe. It is speculated that the catalytic oxidation and aerodynamics of exhaust streams were major parameters influencing the distributions of measured sizes and fractal geometry. A chemical analysis using energy dispersion spectroscopy revealed that diesel particulates consist of elemental carbons in most portions.</div>

Three-dimensional modeling and analysis of fractal characteristics of rupture source combined acoustic emission and fractal theory

Fractal analysis and ultrasound techniques are pivotal in assessing osteoporosis by examining the complex structure of cancellous bone. Osteoporosis, marked by reduced bone mass and the degradation of bone microarchitecture, results in greater fragility and a higher risk of fractures. The poroelastodynamic model proposed in this study incorporates the Navier-Stokes equations of linear elasticity and Biot's theory of porous media, allowing the investigation of osteoporosis versus fractal dimension. Some mechanical properties of cancellous bone, such as porosity, density, elasticity, Poisson's ratio, and viscosity of bone marrow within the porous medium, are taken into account in this analysis. Our findings suggest that the fractal dimension of cancellous bone is a critical factor in predicting osteoporosis. Our study incorporates data from 84 women and 49 men to explore how fractal dimension can predict osteoporosis. We found that individuals with osteoporosis consistently exhibit lower fractal dimensions compared to those with normal bone, irrespective of gender or age. Analyzing rib data from three donors, we observed that an increase in fractal dimension correlates with a decrease in Young's modulus. Furthermore, examining lumbar spine data from 22 cadavers, we observed that increased bone mineral density is associated with higher fractal dimensions, while higher bone porosity correlates with reduced bone mineral density. Additionally, considering the attenuation of cancellous bone from 61 women, including 32 with osteoporotic bone and 29 with normal bone, shows that attenuation in normal bone is higher than in osteoporotic bone when accounting for porosity. Therefore, it can be concluded that increased porosity leads to a reduction in fractal dimension. From this study, the fractal dimension emerges as a critical metric in predicting osteoporosis, offering insights into the structural changes within cancellous bone that contribute to its fragility and susceptibility to fractures.

The aims of this study are to 4D modeling the distribution and concentration of soils magnetic susceptibility, that reflecting deposited magnetic particles regimes as contaminants along Tehran-Karaj highway carrying appreciable traffic. In this research, magnetometery surveys are carried out in an area of 10,000 m2 in the vicinity of this highway. Also, this study is based on; processing and integrated analysis of geophysical data and fractal geometry analysis of 9 sub and topsoil samples. In particular, in the January 2014 and March 2015, time-lapse magnetic susceptibility data has been carried out on 10 north-southern profiles. The interpretation of geophysical data made it possible to define the extent and the boundaries of anthropogenic contaminants. Also, as an innovation fractal geometry is studied and applied for preparing of the source of soil pollution. The SEM gray images of two soil samples are binarized by Otsu’s method in MATLAB. Fractal dimension (FD) =1.732 modified by enriched samples from anthropogenic contaminant, exhibits more fractality and regularities in comparison with the naturally contaminant samples with FD=1.405. Fractal dimension of contaminant sources allowed specifying 4D modeling results along with, the effects of the terrific as an anthropogenic agent on soil pollution has accumulated in this area.

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Determination of the initial saturation of a hydrocarbon reservoir requires resistivity log data and saturation exponent. A number of experimental investigations has shown that reservoir wettability affect the exponent value. A remarkable divergence conclusions, however, still appear in the literature. The objective of the present study was therefore to investigate the effect of rock wettability on saturation exponent. Fractal concepts have been applied to the image of thin-sections of a core sample and used to derive the resistivity equation. The advantage of this approach is that it is independent of rock-fluids equilibrium problems that commonly influence laboratory measurements. A general equation of electrical resistivity has been developed in this study. Electrical tortuosity, clay content, and rock wettability are incorporated in the equation. The present work employed twenty thin-sections of limestone and sandstones. It was found that the lowest exponent of 1.61 was obtained for strongly water-wet shaly sandstone and the highest value of 4.99 is for strongly oil-wet. The exponent consistently increases as the wetting condition is shifted from strongly water-wet toward oil-wet. It is close to 2.0 for clean sandstones at strongly water-wet, supporting the empirical formula of Archie. Introduction Estimation of a hydrocarbon reserve is strongly dependent of electric log data and the value of saturation exponent used. This exponent is usually either assumed to be 2.0 regardless of the reservoir wettability or is derived from laboratory measurements of electrical properties of the cores. Saturation exponent has been the subject to many laboratory investigations. The researchers arrived at considerably different quantitative results, particularly for wetting conditions at oil-wet side. Detailed review of this matter has been made by Anderson and Sondenaa et al. A lack of analytical investigation has been conducted to study the effect of wettability on electrical properties of rocks containing fluids. Fractal concepts have been applied to a wide range of research area related to porous media at various scales, from microscale (pore level) to megascale (field scale). A detailed review of fractal concepts applications has been presented by Sahimi and Yortsos. Briefly, a fractal object is an object that has a geometric dimension greater than its topological (Euclidian) dimension. Fractal geometry may describe the irregurality and fragmentation of natural objects and patterns. To characterize a fractal object or set, the fractal dimension D must be determined. A simple method is by "box counting". This is simply done by drawing a mesh of squares of side dimension that covers the set. The fractal dimension is determined from the slope of the relation between count and square side dimension. which is the fractal dimension that can yield any real value. To handle the complexity of the problems, the present study was directed to applying concepts of fractal analysis to determination of the electrical properties and thus saturation exponent. Pore structures and geometries can be quite complex in nature. The complexity stems from the irregularity of pore wall surfaces. The presence of clays may increase the complexity of pore structures. As a matter of fact fluids distribution within the pores is influenced by the wetting condition. This in turn would affect the fluids connectivity. Therefore, electric tortuosity depends not only on pores and fluids connectivity but also on the irregurality of main grain surfaces and volume and distribution of clays within the pores. As pores of a porous medium can be magnified by employing microscope, identification of fine details of the pores can be performed through the use of thin sections made. Ultimately, mathematical analysis coupled with fractal analysis is thus possible to study electrical properties of porous media. P. 143

<p>Fractal geometry methods offer a quantitative description of complex landscapes and can effectively elucidate the behavior of fluvial and coastal geomorphological systems. The bidimensional (2D) fractal geometry of a shoreline, obtainable with Koch’s curve method, may be expressed by a fractional number between 1.1 and 1.4. In this case, the former shoreline is less rugged than the latter, which tends to a Peano’s curve with a fractal dimension D = 2, that is a plan. These sizes, under initial analysis, may not supply direct information on the main processes responsible for modeling the coast. Actually, either erosion, sediment deposition, and tectonics could generate shapes with about the same fractal dimension, independently of their intensity, age, and time during which they acted, as a result of morphological convergence. From the results of research carried out along Mediterranean-type and Brazilian coastlands, it is possible to derive, from the fractal dimension, the primary and secondary processes behind modeling. Here distinguishing tectonic events from littoral dynamics, both responsible for coastal morphology and identifying which is predominant. Calculation and comparison of the 2D-fractal analysis over time along the selected littorals are useful for assessing sea storm impacts and tsunami flash remodeling, e.g., as a relationship between dimension and energy, and the post-event shoreline resilience time comparing the fractal dimensions before and after the main events related to the morphosedimentary features. The same principles can be applied to other geomorphological systems, such as dunes, river drainage networks, islands, and lagoons, representing an important tool for understanding and classifying these environments.</p>

Cake layer morphology in microfiltration of activated sludge wastewater based on fractal analysis

This paper proposes the novel method to assess Thai speech based on fractal analysis. The fractal algorithm, namely, Higuchi's method was selected to evaluate the fractal dimension (FD) of segmented speech signals. To show the FD changes in waveform over time, the time-dependent FD (TDFD) was proposed. Probability distribution of TDFDs using kernel density estimation was used as an additional parameter. The proposed method achieves the average percentages of assessment with successful rates of 90.0% (+8.9%) and 88.1% (+12.5%) for single and double syllables of Thai speech, respectively.

Characterization of interfacial interactions in bamboo pulp fiber/high-density polyethylene composites treated by nano CaCO3 impregnation modification using fractal theory and dynamic mechanical analysis