Local times of linear multifractional stable sheets

Let X^H(u)(u)={X^H(u)(u);u∈R^N+}be linear multifractional stable sheets with index functional H(u),where H(u)=(H1(u),…,HN(u))is a function with values in(0;1)N.Based on some assumptions of H(u),we obtain the existence of the local times of X^H(u)(u)and establish its joint continuity and the Holder regularity.These results generalize the corresponding results about fractional stable sheets to multifractional stable sheets.

A note on approximation to multifractional Brownian motion

In this paper,we prove approximations of multifractional Brownian motions with moving-average representations and of those with harmonizable representations in the space of continuous functions on [0,1]. These approximations are constructed by Poisson processes.

How big are the increments of a multifractional Brownian motion?

In this paper, we study globle path behavior of a multifractional Brownian motion, which is ageneralization of the fractional Brownian motion.

Hitting Probabilities and Fractal Dimensions of Multiparameter Multifractional Brownian Motion

The main goal of this paper is to study the sample path properties for the harmonisable-type N-parameter multifractional Brownian motion, whose local regularities change as time evolves. We provide the upper and lower bounds on the hitting probabilities of an （N, d）-multifractional Brownian motion. Moreover, we determine the Hausdorff dimension of its inverse images, and the Hausdorff and packing dimensions of its level sets.

Correlation analysis between the Aral Sea shrinkage and the Amu Darya River

The shrinkage of the Aral Sea,which is closely related to the Amu Darya River,strongly affects the sustainability of the local natural ecosystem,agricultural production,and human well-being.In this study,we used the Bayesian Estimator of Abrupt change,Seasonal change,and Trend(BEAST)model to detect the historical change points in the variation of the Aral Sea and the Amu Darya River and analyse the causes of the Aral Sea shrinkage during the 1950–2016 period.Further,we applied multifractal detrend cross-correlation analysis(MF-DCCA)and quantitative analysis to investigate the responses of the Aral Sea to the runoff in the Amu Darya River,which is the main source of recharge to the Aral Sea.Our results showed that two significant trend change points in the water volume change of the Aral Sea occurred,in 1961 and 1974.Before 1961,the water volume in the Aral Sea was stable,after which it began to shrink,with a shrinkage rate fluctuating around 15.21 km3/a.After 1974,the water volume of the Aral Sea decreased substantially at a rate of up to 48.97 km3/a,which was the highest value recorded in this study.In addition,although the response of the Aral Sea's water volume to its recharge runoff demonstrated a complex non-linear relationship,the replenishment of the Aral Sea by the runoff in the lower reaches of the Amu Darya River was identified as the dominant factor affecting the Aral Sea shrinkage.Based on the scenario analyses,we concluded that it is possible to slow down the retreat of the Aral Sea and restore its ecosystem by increasing the efficiency of agricultural water use,decreasing agricultural water use in the middle and lower reaches,reducing ineffective evaporation from reservoirs and wetlands,and increasing the water coming from the lower reaches of the Amu Darya River to the 1961–1973 level.These measures would maintain and stabilise the water area and water volume of the Aral Sea in a state of ecological restoration.Therefore,this study focuses on how human consumption of recharge runoff affects the Aral Sea and provides scientific perspective on its ecological conservation and sustainable development.

Contact Angle Prediction Model for Underwater Oleophobic Surfaces Based on Multifractal Theory

Traditional microstructure scale parameters have difficulty describing the structure and distribution of a roughmaterial’s surface morphology comprehensively and quantitatively. This study constructs hydrophilic and underwateroleophobic surfaces based on polyvinylidene fluoride (PVDF) using a chemical modification method, and the fractaldimension and multifractal spectrum are used to quantitatively characterize the microscopic morphology. A new contactangle prediction model for underwater oleophobic surfaces is established. The results show that the fractal dimension ofthe PVDF surface first increases and then decreases with the reaction time. The uniformity characterized by the multifractalspectrum was generally consistent with scanning electron microscope observations. The contact angle of water droplets onthe PVDF surface is negatively correlated with the fractal dimension, and oil droplets in water are positively correlated.When the fractal dimension is 2.0975, the new contact angle prediction model has higher prediction accuracy. Themaximum and minimum relative deviations of the contact angle between the theoretical and measured data are 18.20%and 0.72%, respectively. For water ring transportation, the larger the fractal dimension and spectral width of the materialsurface, the smaller the absolute value of the spectral difference, the stronger the hydrophilic and oleophobic properties, andthe better the water ring transportation stability.

Multifractal estimation of NMR T_(2) cut-off value in low-permeability rocks considering spectrum kurtosis: SMOTE-based oversampling integrated with machine learning

The transverse relaxation time (T_(2)) cut-off value plays a crucial role in nuclear magnetic resonance for identifying movable and immovable boundaries, evaluating permeability, and determining fluid saturation in petrophysical characterization of petroleum reservoirs. This study focuses on the systematic analysis of T_(2) spectra and T_(2) cut-off values in low-permeability reservoir rocks. Analysis of 36 low-permeability cores revealed a wide distribution of T_(2) cut-off values, ranging from 7 to 50 ms. Additionally, the T_(2) spectra exhibited multimodal characteristics, predominantly displaying unimodal and bimodal morphologies, with a few trimodal morphologies, which are inherently influenced by different pore types. Fractal characteristics of pore structure in fully water-saturated cores were captured through the T_(2) spectra, which were calculated using generalized fractal and multifractal theories. To augment the limited dataset of 36 cores, the synthetic minority oversampling technique was employed. Models for evaluating the T_(2) cut-off value were separately developed based on the classified T_(2) spectra, considering the number of peaks, and utilizing generalized fractal dimensions at the weight <0 and the singular intensity range. The underlying mechanism is that the singular intensity and generalized fractal dimensions at the weight <0 can detect the T_(2) spectral shift. However, the T_(2) spectral shift has negligible effects on multifractal spectrum function difference and generalized fractal dimensions at the weight >0. The primary objective of this work is to gain insights into the relationship between the kurtosis of the T_(2) spectrum and pore types, as well as to predict the T_(2) cut-off value of low-permeability rocks using machine learning and data augmentation techniques.

Analysis of the USD/JPY and EUR/JPY Exchange Rates Using Multifractal Analysis and Extreme Value Theory

We performed a multifractal analysis using wavelet transform to detect the changes in the fractality of the USD/JPY and EUR/JPY exchange rates, and predicted their extreme values using extreme value theory. After the 1997 Asian financial crisis, the USD/JPY and EUR/JPY became multifractal, then the USD/JPY became monofractal and stable, and yen depreciation was observed. However, the EUR/JPY became multifractal and unstable, and a strong yen depreciation was observed. The coherence between the USD/JPY and EUR/JPY was strong between 1995 and 2000. After the 2007-2008 financial crisis, the USD/JPY became monofractal and stable, and yen appreciation was observed. However, the EUR/JPY became multifractal and unstable, and strong yen appreciation was observed. Various diagnostic plots for assessing the accuracy of the GP model fitted to USD/JPY and EUR/JPY are shown, and all the diagnostic plots support the fitted GP model. The shape parameters of USD/JPY and EUR/JPY were close to zero, therefore the USD/JPY and EUR/JPY did not have finite upper limits. We predicted the maximum return level for the return periods of 10, 20, 50, 100, 350, and 500 years and their respective 95% confidence intervals (CI). As a result, the 10-year and 100-year return levels for USD/JPY were estimated to be 149.6 and 164.8, with 95% CI [143.2, 156.0] and [149.4, 180.1], respectively.

Long memory of price-volume correlation in metal futures market based on fractal features

An empirical test on long memory between price and trading volume of China metals futures market was given with MF-DCCA method. The empirical results show that long memory feature with a certain period exists in price-volume correlation and a fittther proof was given by analyzing the source of multifractal feature. The empirical results suggest that it is of important practical significance to bring the fractal market theory and other nonlinear theory into the analysis and explanation of the behavior in metal futures market.

On approximating multifractal traffic burstiness with Markov modulated Poisson processes

We investigate the approximating capability of Markov modulated Poisson processes (MMPP) for modeling multifractal Internet traffic. The choice of MMPP is motivated by its ability to capture the variability and correlation in moderate time scales while being analytically tractable. Important statistics of traffic burstiness are described and a customized moment-based fitting procedure of MMPP to traffic traces is presented. Our methodology of doing this is to examine whether the MMPP can be used to predict the performance of a queue to which MMPP sample paths and measured traffic traces are fed for comparison respectively, in addition to the goodness-of-fit test of MMPP. Numerical results and simulations show that the fitted MMPP can approximate multifractal traffic quite well, i.e. accurately predict the queueing performance.

Nuclear magnetic resonance T_2 spectrum:multifractal characteristics and pore structure evaluation

Pore structure characteristics are important to oil and gas exploration in complex low-permeability reservoirs. Using multifractal theory and nuclear magnetic resonance （NMR）, we studied the pore structure of low-permeability sandstone rocks from the 4th Member （Es4） of the Shahejie Formation in the south slope of the Dongying Sag. We used the existing pore structure data from petrophysics, core slices, and mercury injection tests to classify the pore structure into three categories and five subcategories. Then, the T2 spectra of samples with different pore structures were interpolated, and the one- and three-dimensional fractal dimensions and the multifractal spectrum were obtained. Parameters a （intensity of singularity） andf（a） （density of distribution） were extracted from the multifractal spectra. The differences in the three fractal dimensions suggest that the pore structure types correlate with a andf（a）. The results calculated based on the multifractal spectrum is consistent with that of the core slices and mercury injection. Finally, the proposed method was applied to an actual logging profile to evaluate the pore structure of low-permeability sandstone reservoirs.

Multifractal Analysis of Element Distribution in Skarn-type Deposits in the Shizishan Orefield,Tongling Area,Anhui Province,China

A series of element concentrations sampled from four drill cores with a length about 1000 m into different skarn-type deposits were selected from the Shizishan orefield, central Tongling, southeastern part of Anhui Province. Using the multifractal method, the distribution and migration characteristics of the major and trace elements are analyzed. The multifractal spectrum of the major elements is left-skewed, whereas the spectrum of the trace elements is right-skewed, which shows that in the process of skarn formation, the trace elements were enriched only locally, and major elements transported within a much larger range. The correlation coefficients of the multifractal parameters Aa （width of the multifractal spectrum） of the four drill cores are relatively low, but the correlation coefficients of the multifractal parameters R （spectrum symmetry parameter） and Af are relatively higher, indicating that although the non-homogeneous intensity of the distribution of elements is inconsistent, their spatial accumulation patterns are almost the same during the ore-forming process. The statistics of the mnltifractal parameters of various elements in the different locations show that the ore-forming processes and element migration pattern in the Shizishan orefield are consistent, and that the migrations of trace elements and major elements exhibit some differences.

Nanopore structure comparison between shale oil and shale gas:examples from the Bakken and Longmaxi Formations

In order to analyze and compare the differences in pore structures between shale gas and shale oil formations, a few samples from the Longmaxi and Bakken Formations were collected and studied using X-ray diffraction, LECO TOC measurement, gas adsorption and field-emission scanning electron microscope. The results show that samples from the Bakken Formation have a higher TOC than those from the Longmaxi Formation. The Longmaxi Formation has higher micropore volume and larger micropore surface area and exhibited a smaller average distribution of microsize pores compared to the Bakken Formation. Both formations have similar meso-macropore volume. The Longmaxi Formation has a much larger meso-macropore surface area, which is corresponding to a smaller average meso-macropore size. CO_2 adsorption data processing shows that the pore size of the majority of the micropores in the samples from the Longmaxi Formation is less than 1 nm, while the pore size of the most of the micropores in the samples from the Bakken Formation is larger than 1 nm. Both formations have the same number of pore clusters in the 2–20 nm range, but the Bakken Formation has two additional pore size groups with mean pore size diameters larger than 20 nm. Multifractal analysis of pore size distribution curves that was derived from gas adsorption indicates that the samples from the Longmaxi Formation have more significant micropore heterogeneity and less meso-macropore heterogeneity. Abundant micropores as well as mesomacropores exist in the organic matter in the Longmaxi Formation, while the organic matter of the Bakken Formation hosts mainly micropores.

Multifractal Simulation of GeochemicalMap Patterns

Using a simple multifractal model based on the model De Wijs, various geochemical map patterns for element concentration values are being simulated. Each pattern is self-similar on the average in that a similar pattern can be derived by application of the multiplicative cascade model used to any small subarea on the pattern. In other experiments, the original, self-similar pattern is distorted by superimposing a 2-dimensional trend pattern and by mixing it with a constant concentration value model. It is investigated how such distortions change the multifractal spectrum estimated by means of the 3-step method of moments. Discrete and continuous frequency distribution models are derived for patterns that satisfy the model of De Wijs. These simulated patterns satisfy a discrete frequency distribution model that as upper bound has a continuous frequency distribution to which it approaches in form when the subdivisions of the multiplicative cascade model are repeated indefinitely. This limiting distribution is lognormal in the center and has Pareto tails. Potentially, this approach has important implications in mineral and oil resource evaluation.

Wave height statistical characteristic analysis

When exploring the temporal and spatial change law of ocean environment, the most common method used is using smaller-scale observed data to derive the change law for a larger-scale system. For instance, using 30-year observation data to derive 100-year return period design wave height. Therefore, the study of inherent self-similarity in ocean hydrological elements becomes increasingly important to the study of multi-year return period design wave height derivation. In this paper, we introduced multifractal to analyze the statistical characteristics of wave height series data observed from oceanic hydrological station. An improvement is made to address the existing problems of the multifractal detrended fluctuation analysis (MF-DFA) method, where trend function showed a discontinuity between intervals. The improved MFDFA method is based on signal mode decomposition, replacing piecewise polynomial fitting used in the original method. We applied the proposed method to the wave height data collected at Chaolian Island, Shandong, China, from 1963 to 1989 and was able to conclude the wave height sequence presented weak multi-fractality. This result provided strong support to the past research on the derivation of multi-year return period design wave height with observed data. Moreover, the new method proposed in this paper also provides a new perspective to explore the intrinsic characteristic of data.

Fractal and Multifractal Modeling of Hydrothermal Mineral Deposit Spectrum: Application to Gold Deposits in Abitibi Area, Ontario, Canada

A number of fractal/multifractal methods are introduced for quantifying the mineral deposit spectrum which include a number-size model, grade-tonnage model, power spectrum model, multifractal model and an eigenvalue spectrum model. The first two models characterize mineral deposits spectra based on relationships among the measures of mineral deposits. These include the number of deposits, size of deposits, concentration and volume of mineral deposits. The last three methods that deal with the spatial-temporal spectra of mineral deposit studies are all expected to be popularized in near future. A case study of hydrothermal gold deposits from the Abitibi area, a world-class mineral district, is used to demonstrate the principle as well as the applications of methods proposed in this paper. It has been shown that fractal and multifractal models are generally applicable to modeling of mineral deposits and occurrences. Clusters of mineral deposits were identified by several methods including the power spectral analysis, singularity analysis and the eigenvalue analysis. These clusters contain most of the known mineral deposits in the Timmins and Kirkland Lake camps.

Incipient mechanical fault detection based on multifractal and MTS methods

An incipient mechanical fault detection method, combining multifractal theory and Mahalanobis-Taguchi system （MTS）, which is based on statistical technology, is proposed in this paper. Multifractal features of vibration signals obtained from machine state monitoring are extracted by multifractal spectrum analysis and generalized fractal dimensions. Considering the situation of mass samples of normal mechanical running state and few fault states, the feature parameters corresponding to different mechanical running states are further optimized by a statistical method, based on which incipient faults are subsequently identified and diagnosed accurately. Experimental results proved that the method combining multifractal theory and MTS can be used for incipient fault state recognition effectively during the mechanical running process, and the accuracy of fault state identification is improved.

Random Difference of the Trace Element Distribution in Skarn and Marbles from Shizishan Orefield,Anhui Province,China

Spatial distribution patterns of element concentrations can reflect the information of the mineralization processes. Both the Hurst exponent calculated by R/S analysis and the generalized fractal dimension calculated by using the multifractal model are important parameters for describing the spatial distribution of elements. Five long drill holes, named as M1, S1, S2, S3, and S4, have been selected in the Shizishan （狮子山） skarn orefield in Tongling （铜陵）, Anhui （安徽） Province, China. Marbles are well developed around M1 and skarn rocks are largely distributed along S1, S2, S3, and S4 drill holes. The drill holes were sampled evenly with an interval of 10 m and 16 trace elements have been measured. The mean of the △D（q） （the height of the generalized dimension spectrum） in the M1 drill hole is the lowest. In addition, the mean of the Hurst exponents of the 16 elements in the M1 drill hole is also much smaller than that of S1, S2, S3, S4 drill holes, which is in accordance with the analysis of the generalized dimension. It is indicated by the generalized dimension and Hurst exponent that the distribution of trace elements in the marbles is more random than that in the skarn. The result suggests that the mineralization process can change the randomness and persistence features of the element distribution.

Multifractal characteristics of shale and tight sandstone pore structures with nitrogen adsorption and nuclear magnetic resonance

Based on the experiments of nitrogen gas adsorption(N_2 GA) and nuclear magnetic resonance(NMR),the multifractal characteristics of pore structures in shale and tight s andstone from the Chang 7 member of Trias sic Yanchang Formation in Ordos Basin,NW China,are investigated.The multifractal spectra obtained from N2 GA and NMR are analyzed with pore throat structure parameters.The results show that the pore size distributions obtained from N2 GA and NMR are different,and the obtained multifractal characteristics vary from each other.The specific surface and total pore volume obtained by N2 GA experiment have correlations with multifractal characteristics.For the core samples with the similar specific surface,the value of the deviation of multifractal spectra Rd increases with the increase in the proportion of large pores.When the proportion of macropores is small,the Rd value will increase with the increase in specific surface.The multifractal characteristics of pore structures are influenced by specific surface area,average pore size and adsorption volume measured from N2 GA experiment.The multifractal characteristic parameters of tight sandstone measured from NMR spectra are larger than those of shale,which may be caused by the differences in pore size distribution and porosity of shale and tight sandstone.

Selection of Multifractal Scaling Breaks andSeparation of Geochemical andGeophysical Anomaly

Spatially superimposed multiple processes such as multiplicative cascade processes often generate multifractal measures possessing so-called self-similarity or self-affinity that can be described by power- law type of functions within certain scale ranges The multifractalities can be estimated by applying multifractal modeling to the measures reflecting the characteristics of the physical processes such as the element concentration values analyzed in rock and soil samples and caused by the underlying mineralization processes and the other geological processes. The local and regional geological processes may result in geochemical patterns with distinct multifractalities as wall as variable scaling ranges. Separation of these multifractal measures on the basis of both the distinct multifractalities and the scaling ranges will be significant for both theoretical studies of multifractal modeling and its applications. Multifractal scaling breaks have been observed from various multifractal patterns. This paper introduces a technique for separating multifractal measures on the basis of scaling breaks. It has been demonstrated that the method is effective for decomposing geochemical and geophysical anomalies required for mineral exploration. A dataset containing the element concentration values of potassium and phosphorus in soil samples was employed for demonstrating the application of the method for studying the fertilizer and yield optimization in agriculture