Multifractal Distribution Research Articles

Multiscaling in the 3D critical site-diluted Ising ferromagnet

We study numerically the appearance of multiscaling behavior in the 3D ferromagnetic Ising site-diluted model, in the form of a multifractal distribution of the decay exponents for the spatial correlation functions at the critical temperature. We have computed the exponents of the long-distance decay of higher moments of the correlation function, up to the 10th power, by studying three different quantities: global susceptibilities, local susceptibilities and correlation functions. We have found very clear evidence of multiscaling behavior.

Multifractal scaling analyses of the spatial diffusion pattern of COVID-19 pandemic in Chinese mainland

Revealing spatio-temporal evolution regularity in the spatial diffusion of epidemics is helpful for preventing and controlling the spread of epidemics. Based on the real-time COVID-19 datasets by prefecture-level cities, this paper is devoted to exploring the multifractal scaling in spatial diffusion pattern of COVID-19 pandemic and its evolution characteristics in Chinese mainland. The ArcGIS technology and box-counting method are employed to extract spatial data and the least square regression based on rescaling probability (μ-weight method) is used to calculate fractal parameters. The results show multifractal distribution of COVID-19 pandemic in China. The generalized correlation dimension spectrums are inverse S-shaped curves, but the fractal dimension values significantly exceed the Euclidean dimension of embedding space when moment order q«0. The local singularity spectrums are asymmetric unimodal curves, which slant to right. The fractal dimension growth curves are shown as quasi S-shaped curves. From these spectrums and growth curves, the main conclusions can be drawn as follows: First, self-similar patterns developed in the process of COVID-19 pandemic, which seems to be dominated by multifractal scaling law. Second, the spatial pattern of COVID-19 across China can be characterized by global clustering with local disordered diffusion. Third, the spatial diffusion process of COVID-19 in China experienced four stages, i.e., initial stage, the rapid diffusion stage, the hierarchical diffusion stage, and finally the contraction stage. This study suggests that multifractal theory can be utilized to characterize spatio-temporal diffusion of COVID-19 pandemic, and the case analyses may be instructive for further exploring natural laws of spatial diffusion.

Thermal Control: Multifractal Distribution of Local Plastic Strains

Thermal Control: Multifractal Distribution of Local Plastic Strains

Thermal Control: Multifractal Distribution of Local Plastic Strains

Methods that make it possible to evaluate the heterogeneity of the distribution of local strains and stresses indirectly through the values of the temperature distribution in thermograms are presented. In this case, methods of fractal analysis in the form of multifractal spectra, image clustering, and multiscale analysis of discrete wavelet images decomposition are used. It is shown that, with increasing damage to the material, the width of the multifractal spectrum increases and local strains are redistributed along the plate length during loading. The use of the multiple-scale wavelet decomposition makes it possible to establish scale elements that have the greatest strain changes.

Geochemical multifractal modeling of soil and stream sediment data applied to gold prospectivity mapping of the Pitangui Greenstone Belt, northwest of Quadrilátero Ferrífero, Brazil

Geochemical mapping has become essential to prospective assessments in both brownfield and greenfield exploration targeting. During the last two decades, several techniques have been applied to the identification of mineralization's geochemical signature. Among these techniques, Hierarchical Clustering (HC) and Principal Component Analysis (PCA) have been widely used. Other techniques, like Multifractal Spectrum (MS) and Local Singularity Mapping (LSM), are efficient in the generation of models suitable to determine the type of database statistical distribution and mapping zones of geochemical enrichment and depletion. We combined these techniques for soil and stream sediment geochemistry and generated two prospective geochemical models for the gold deposits from the Pitangui Greenstone Belt (PGB), located at northwest of Quadrilátero Ferrífero – Brazil. The results of PCA and HC showed a geochemical association of As-Bi-Mo-Sb with gold mineralization in the PGB. According with descriptive statistical analysis, these elements do not display normal or log-normal statistical distribution, which allowed the application of multifractal models on the geochemical mapping. MS technique revealed that all elements associated with gold have a multifractal distribution. In addition, geochemistry data shows that high contents of elements measured in the stream sediment analysis displays greater dispersion when compared to elements of soil geochemistry data, due to their mobility in environmental conditions. We used LSM technique to map geochemical enrichment of the elements associated with gold. The maps were evaluated using the Ground Truth Analysis (GTA) and the results showed a good performance in the identification of prospective zones. The anomalies identified in both geochemical maps showed a strong correlation between geochemical enrichment zones and geological factors that favors gold mineralization. These results bring a new perspective on the distribution of geochemical anomalies using, for the first time, the multifractal geochemical modeling on the PGB. The geochemical maps point to new targets for mineral exploration and contribute significantly for the prospective models of the region.

Spatial Multifractal and b Value Characteristics of Mine Earthquakes under High Tectonic Stress

Mining-induced earthquake is one of the common risk factors affecting the safety of mining engineering. In order to reveal the information characteristics of mine seismicity and predict the high-energy mine seismicity, based on the microseismic monitoring data of 250105 working face in Huating Coal Mine, the spatial multifractal and magnitude-frequency distribution relationship of mine seismicity activity are analyzed. It was found that under high tectonic stress, the spatial multifractal characteristics of mine seismicity are obvious, and the magnitude-frequency distribution satisfies the Gutenberg-Richter law. There is a positive linear correlation between the fractal dimension D 2 and b value, and the spatial distribution of mine seismicity is correlated with the magnitude-frequency distribution. In terms of D ∞ and D 2 - D ∞ , the seismic aggregation degree and fractal structure uniformity are discussed, respectively, and the parameter variation is related to rock burst events. The variations of D 2 and b values are strongly correlated with the main earthquake. The descending gradient of b value and D 2 is the largest in the region where the main shock occurred, and the b value and D 2 take on lower values. The significant decrease of b value and D 2 can be regarded as the precursor feature of the main shock.

Estimating effective permeability using connectivity and branch length distribution of fracture network

In this study, a new approach to estimate the effective permeability of the fracture network is proposed. New functional expressions are derived considering laminar fluid flow through fracture networks. Effective permeability of the fracture network is expressed as a function of topological connectivity, statistical branch segment length distributions, maximum, and minimum branch segment lengths. It is observed that the statistical cumulative branch segment length distribution of fracture networks can follow both fractal and multi-fractal patterns. Hence, analytical expressions of effective permeability are proposed considering both fractal and multi-fractal distribution of branch segment lengths. Obtained expressions are validated using numerical simulation results from past studies. It is observed from derived expressions that, effective permeability of fracture networks increases non-linearly with increasing topological connectivity and fractal dimensions of statistical branch segment length distributions. However, the non-linearity in the relationship is mainly governed by the scaling exponent and scaling coefficient of the fracture network. With increasing scaling exponent and scaling coefficient, non-linearity in the relation between effective permeability and topological connectivity increases. It is observed from the derived relation that, effective permeability of a fracture network is more sensitive to longer branch segments compared to the shorter branch segments. Effective permeability of fracture networks increases rapidly with increase in length of the branch segments. Hence it is interpreted, longer branch segments in fracture networks play a critical role in determining the effective permeability of the fractured reservoir.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Disorder-Driven Multifractality Transition in Weyl Nodal Loops.

The effect of short-range disorder in nodal line semimetals is studied by numerically exact means. For arbitrary small disorder, a novel semimetallic phase is unveiled for which the momentum-space amplitude of the ground-state wave function is concentrated around the nodal line and follows a multifractal distribution. At a critical disorder strength, a semimetal to compressible metal transition occurs, coinciding with a multi- to single-fractality transition. The universality class of this critical point is characterized by the correlation length and dynamical exponents. At considerably higher disorder, an Anderson metal-insulator transition takes place. Our results show that the nature of the semimetallic phase in nonclean samples is fundamentally different from a clean nodal semimetal.

Multifractality and the distribution of the Kondo temperature at the Anderson transition

Using numerical simulations, we investigate the distribution of Kondo temperatures at the Anderson transition. In agreement with previous work, we find that the distribution has a long tail at small Kondo temperatures. Recently, an approximation for the tail of the distribution was derived analytically. This approximation takes into account the multifractal distribution of the wavefunction amplitudes (in the parabolic approximation), and power law correlations between wave function intensities, at the Anderson transition. It was predicted that the distribution of Kondo temperatures has a power law tail with a universal exponent. Here, we attempt to check that this prediction holds in a numerical simulation of Anderson's model of localisation in three dimensions.

Quantifying Fractal Dimension in Integrated Experimental Data of Tight Sandstones

Determining the microscopic pore structures of tight sandstones is becoming one of the most challenging efforts, and the strong heterogeneity makes the accurate assessment still a problem. In this research, we report a new criterion for pore structure typing based on the fractal geometry theory. The fractal dimension values were first accurately calculated through intrusion and nonintrusion methods. The results show that the pores in tight sandstones have multifractal distributions and different types of pore structure were divided based on various tests. The relationships between petrophysical properties and a series of multifractal parameters have been analyzed in detail. The fractal dimension values of type 1 sandstones derived from the throat curves in rate-controlled mercury intrusion methods can well characterize the porosity of the research area, while the others did not respond well. Finally, a multifractal criterion was proposed to analyze the petrological and pore structures by combined observations and experiments. The new criterion exhibits perfect performance in the prediction of the storage capacity. The multifractal model proposed in this research helps to assess the pore structures of tight sandstones and helps to characterize the reservoir quality in hydrocarbon exploration and development.

Density and tracer statistics in compressible turbulence: Phase transition to multifractality.

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state. This is demonstrated to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers; however, at finite Mach numbers, the assumption of mixing is needed, which we demonstrate to be not self-evident because of the possibility of self-organization.

Multi-fractal distribution analysis for pore structure characterization of tight sandstone—A case study of the Upper Paleozoic tight formations in the Longdong District, Ordos Basin

Abstract With proven gas potential, the gas charging and accumulation mechanisms for the Upper Paleozoic tight gas reservoir in the Longdong District of the Ordos Basin remain controversial. Pore structure is the most direct and critical factor that affects gas charging and accumulation in the reservoir. However, the pore structure characterization of tight sandstone is far more difficult than that for conventional reservoir rocks due to significant heterogeneity. The present work aims to introduce a new quantitative method for pore structure characterization that integrates multi-fractal distribution analysis, an effective and profitable method for quantitative heterogeneity analysis, and high-pressure mercury intrusion porosimetry (HPMIP). Twenty-one samples from targeted tight formations with different gas-bearing status were obtained for analysis. The results show that the best formations for gas charging and accumulation in the Upper Paleozoic tight sandstone formations in the Longdong District are the formations with more large pores, more homogeneous small pores, and well-developed micro-fractures. The existence of micro-fractures is the key factor that affects gas charging and accumulation. Tight sandstone samples were categorized into three types in accordance with the HPMIP curve configuration. Due to the high heterogeneity of pore structures, analyses based solely on HPMIP curve configuration has its shortcomings, which can be compensated by multi-fractal distribution analysis. The multi-fractal distribution of target tight formations could be interpreted as two linear segments, representing large pores and small pores. The separating points of the two segments vary from 0.1 μm to 0.5 μm in different samples due to different pore structures. Large pores are not ideal fractals in gas-bearing formations due to the existence of micro-fractures, whereas they are fractals in dry formations where micro-fractures are not developed. Small pores are all ideal fractals, with the highest heterogeneity in type γ samples and the lowest heterogeneity in type α samples. A conclusion was made in this work: if the slope of segment one in the multi-fractal distribution of mercury saturation (SHg) versus capillary pressure (PC) is larger than 1, this measure could be accepted as an empirical indicator for the existence of micro-fractures.

A new method for geochemical anomaly separation based on the distribution patterns of singularity indices

Singularity analysis is one of the most important models in the fractal/multifractal family that has been demonstrated as an efficient tool for identifying hybrid distribution patterns of geochemical data, such as normal and multifractal distributions. However, the question of how to appropriately separate these patterns using reasonable thresholds has not been well answered. In the present study, a new method termed singularity-quantile (S-Q) analysis was proposed to separate multiple geochemical anomaly populations based on integrating singularity analysis and quantile-quantile plot (QQ-plot) analysis. The new method provides excellent abilities for characterizing frequency distribution patterns of singularity indices by plotting singularity index quantiles vs. standard normal quantiles. From a perspective of geochemical element enrichment processes, distribution patterns of singularity indices can be evidently separated into three groups by means of the new method, corresponding to element enrichment, element generality and element depletion, respectively. A case study for chromitite exploration based on geochemical data in the western Junggar region (China), was employed to examine the potential application of the new method. The results revealed that the proposed method was very sensitive to the changes of singularity indices with three segments when it was applied to characterize geochemical element enrichment processes. And hence, the S-Q method can be considered as an efficient and powerful tool for separating hybrid geochemical anomalies on the basis of statistical and inherent fractal/multifractal properties.

Engineering slow light and mode crossover in a fractal-kagome waveguide network

We present an analytically exact scheme of unraveling a multitude of flat, dispersionless photonic bands in a kagome waveguide strip where each elementary plaquette hosts a deterministic fractal geometry of arbitrary size. The number of non-dispersive eigenmodes grows as higher and higher order fractal geometry is embedded in the kagome motif. Such eigenmodes are found to be localized with finite support in the kagome strip and exhibit a hierarchy of localization areas. The onset of localization can, in principle, be delayed in space by an appropriate choice of frequency of the incident wave. The length scale at which the onset of localization for each mode occurs, can be tuned at will as prescribed here using a real space renormalization method. Conventional methods of extracting the non-dispersive modes in such geometrically frustrated lattices fail as a non-translationally invariant fractal decorates the unit cells in the transverse direction. The scheme presented here circumvents this difficulty, and thus may inspire the experimentalists to design similar fractal incorporated kagome or Lieb class of lattices to observe a multifractal distribution of flat photonic bands.

Multifractal and Levy-stable statistics of soil surface moisture distribution derived from 2D image analysis

Soil moisture distribution usually presents extreme variation at multiple spatial scales. A Levy stable model can assist multifractal analysis for a complete statistical characterization of such extreme variability. The objectives of the present work were (i) to characterize superficial soil moisture distribution patterns using a multifractal approach and (ii) to propose a 2D/3D model for estimating the Levy stable index from the multifractal distribution. Photographs (KodakTM EasyShare C182, 12 Mpx resolutions) were taken at three different times of soil wetting process after soil irrigation (e.g., 5, 10 and 20 min.). ImageJ Software was used for image processing and multifractal analysis. Levy stable index, μ, was estimated from the f(α) curve using a 2D extension of a previous model. All gray scale images were approximately bimodal. Multifractal analysis showed that D0 was consistently <2, which gives evidence on the fractal structure of the support. All the investigated soil moisture distributions fitted well the multifractal log-Levy stable model with stability index within the range 1 < μ < 2 and Hurst exponent (H) 0.5 < H < 1. This implies that soil moisture pattern can be described by sub-Gaussian statistics (e.g., far from Central Limit Theorem) and it can arise as the outcome of persistence due to long memory effects. The results could be useful for improving the statistical characterization and the predictive capability of future models related to spatially variable soil processes.

Lacunarity analysis of atomic configurations: application to ethanol-water mixtures.

Lacunarity analysis is a scale-dependent method quantifying the translational invariance in patterns. In this work it is used to characterize the distribution of several subsets of atoms in molecular systems. Binary clusters and one-component (ethanol or water) hydrogen-bonded clusters of ethanol-water mixtures with 0-100mol% ethanol content were analyzed. Molecular dynamics simulations created the configurations, and all were in good agreement with the respective experimental x-ray diffraction data. Lacunarity analysis revealed that the placement of the one-component clusters at low concentration can be described by a multifractal distribution, especially in the case of ethanol. Most of the cases these clusters are not isolated entities, but form islands in binary clusters.

Computer simulation of the interplay between fractal structures and surrounding heterogeneous multifractal distributions. Applications

In a large number of physical, biological and environmental processes interfaces with high irregular geometry appear separating media (phases) in which the heterogeneity of constituents is present. In this work the quantification of the interplay between irregular structures and surrounding heterogeneous distributions in the plane is madeFor a geometric set A⊂R2 and a mass distribution (measure) μ supported in S⊂R2, being A⊂S, the mass μ(A(ε)) gives account of the interplay between the geometric structure and the surrounding distribution. A computation method is developed for the estimation and corresponding scaling analysis of μ(A(ε)), being A a fractal plane set of Minkowski dimension D and μ a multifractal measure produced by random multiplicative cascades. The method is applied to natural and mathematical fractal structures in order to study the influence of both, the irregularity of the geometric structure and the heterogeneity of the distribution, in the scaling of μ(A(ε)). Applications to the analysis and modeling of interplay of phases in environmental scenarios are given.

Multifractal distribution of geochemical elements in Dexing porphyry copper deposit, Jiangxi Province, China

Hydrothermal mineralization can cause enrichment or depletion of elements in ore bodies and surrounding country rocks. Local metal-enriched rocks can become an ore body that can be mined at an economic profit. Therefore, understanding the spatial distribution of elements caused by mineralization is crucial for mineral exploration and resource assessment. In this study, a total of 24 representative samples were collected from different veins within the Tongchang ore body of the Dexing porphyry Cu–Mo Deposit. The objectives of this work were firstly to identify different vein types and secondly to evaluate the potential of multifractal parameters, constructed using the so-called method of moments, to characterize the spatial and temporal distribution patterns of concentration of elements on mineral surfaces, which may provide insight into hydrothermal mineralization and its effect on spatial and temporal distribution of element concentration. Based on the analytical results and previous investigations, the vein systems within the deposit can be divided into four main types as follows: (1) early quartz veins (A-type veins) related to K-feldspar alteration; (2) quartz-pyrite-molybdenite-chalcopyrite veins (B-type veins) related to chlorite+illite alteration; (3) quartz-pyrite-chalcopyrite veins (D-type veins) related to late phyllic (muscovite+quartz) alteration; and (4) the latest carbonate-sulphate-oxide veins (H-type veins). The multifractal distributions of concentration of chemical elements on the surface of pyrite crystals selected from veins of different types were investigated using digital images generated by means of ESEM. To achieve our goal, the distribution patterns of element concentration values of eight elements (Cu, Fe, Mn, Ni, P, S, Si and Ti) on the pyrite surface selected from B-type, D-type and H-type veins and veinlets were represented by multifractal spectra. Several multifractal parameters were calculated; however, the most significant ones are ∆α and ∆αL. These parameters are found to be higher in the D-type veins related to late phyllic alteration and mineralization in comparison with other veins and vein systems, indicating that the ore minerals containing Cu and Fe sulphides are highly enriched in D-type veins. This result is consistent with previous research on the area as well as new field observations, chemical and petrographic analysis carried out for this study. Therefore, the multifractal parameters, particularly ∆α and ∆αL, appear to be promising to discriminate between mineralized veins developed during different stages of hydrothermal activity which led to development of the porphyry system at Tongchang ore body.

The geochemical multi-fractal characteristics and mineralization of the Dehelongwa copper-gold deposit

The spatial distribution of ore-forming elements is the result of interaction and influence among multiple factors in the process of mineralization, and is the specific embodiment of non-linearity and coupling of mineralization. The fractal theory has such functions as to characterize non-lineariry and coupling, and can find out the deterministic law from the complicated coupling process. The primary geochemical fractal distribution of metallogenic elements such as Cu and Au possesses the characteristics of multi-fractal distribution in the Dehelongwa copper-gold deposit. Copper follows 2-d fractal distribution, and Au follows 3- or 4-d fractal distribution. The primary geochemical fractal distribution has a certain similarity on the same prospecting line, and this similarity is not obvious among different prospecting lines. The secondary geochemical fractal distribution of Cu and Au is higher in similarity, and simpler than the primary geochemical fractal distribution. These results show that the mineralization of Cu and Au is out of sync and also is inconsistent in this deposit, and the mineralization of Cu mainly experienced two mineralization stages, while the mineralization of Au experienced three to four mineralization stages. Besides the mineralization of Cu and Au shows a certain direction and along the strike of vertical orebodies it is relatively balanced. While along the strike of the ore bodies the mineralization of Cu and Au shows a significant difference. The hypergenic geologic process has a certain effect on the balance of spatial distribution of Cu and Au. From these it can be seen that the mineralization process of this deposit is well characterized by the geochemical fractal distribution of Cu and Au.