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.

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.

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.

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.

THE MULTIFRACTAL HAUSDORFF AND PACKING MEASURE OF GENERAL SIERPINSKI CARPETS

In this paper, authors study the properties of multifractal Hausdorff and packing measures for a class of self-affine sets and use them to study the multifractal properties of general Sierpinski carpet E, and they get that the multifractal Hausdorff and packing measure are mutual singular, when they are restricted on some subsets of E.

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.

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.

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.

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.

Morphology and multifractal features of a guyot in specific topographic vicinity in the Caroline Ridge,West Pacific

Massive seamounts have been surveyed and documented in the last decades.However,the morphologies of seamounts are usually described in qualitative manners,yet few quantitative detections have been carried out.Here,based on the high-re solution multi-beam bathymetric data,we report a recentlysurveyed guy ot on the Caroline Ridge in the West Pacific,and the large-scale volcanic structures and smallscale erosive-depositional landforms in the guyot area have been identified.The multifractal features of the guyot are characterized for the first time by applying multifractal detrended fluctuation analysis on the surveyed bathymetric data.The results indicate that the multifractal spectrum parameters of the seafloor have strong spatial dependency on the fluctuations of local landforms.Both small-and large-scale components contribute to the degree of asymmetry of the multifractal spectrum(B),while the fluctuations of B are mostly attributed to the changes in small-scale roughness.The maximum singularity strength(α0)correlates well with the roughness of large-scale landforms and likely reflects the large-scale topographic irregularity.Comparing to traditional roughness parameters or monofractal exponents,multifractal spectra are able to depict not only the multiscale characteristics of submarine landforms,but also the spatial variations of scaling behaviors.Although more comparative works are required for various seamounts,we hope this study,as a case of quantifying geomorphological characters and multiscale behaviors of seamounts,can encourage further studies on seamounts concerning geomorphological processes,ocean bottom circulations,and seamount ecosystems.

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

Fractal and Multifractal Propertiesof the Spatial Distribution of Natural Fractures——Analyses and Applications

The Cantor's dust theory is applied to investigate the scaling properites of the spatial distribution of natural fractures obtained from detailed scanline surveys of 27 field sites in the Appalachian Plateau of western New York, USA. The results obtained in this study indicate: 1) fracture spacing is characterized by fractal and multifractal properties. On small scales analyses yield an average fractal dimension of 0.15, which suggests a very high degree of clustering. In contrast, on large scales, fractures tend to be more regular and evenly distributed with an average fracture dimension of 0.52; 2) fractal dimension varies with different sets in different orientations, which can be attributed to interactions between pre-existing fractures and younger ones, as well as variations of the intensity of the stresses under which the fractures were formed; 3) a time sequence of fracture set formation can be proposed based on fractal and multifractal analyses, which consists of (from old to young): N-S, NW, ENE, and NE-striking sets. This time sequence is confirmed by the study of the abutting relationships of different fracture sets observed in the field.

Multifractal analysis of surface EMG signals for assessing muscle fatigue during static contractions

This study is aimed at assessing muscle fatigue during a static contraction using multifractal analysis and found that the surface electromyographic （SEMG） signals characterized multiffactality during a static contraction. By applying the method of direct determination ofthef（a） singularity spectrum, the area of the multifractal spectrum of the SEMG signals was computed. The results showed that the spectrum area significantly increased during muscle fatigue. Therefore the area could be used as an assessor of muscle fatigue. Compared with the median frequency （MDF）--the most popular indicator of muscle fatigue, the spectrum area presented here showed higher sensitivity during a static contraction. So the singularity spectrum area is considered to be a more effective indicator than the MDF for estimating muscle fatigue.

Mechanical Fault Diagnosis Based on Band-phase-randomized Surrogate Data and Multifractal

The vibration signals of machinery with various faults often show clear nonlinear characteristics.Currently,fractal dimension analysis as the common useful method for nonlinear signal analysis,is a kind of single fractal form,which only reflects the overall irregularity of signals,but cannot describe its local scaling properties.For comprehensive revealing of internal properties,a combinatorial method based on band-phase-randomized（BPR） surrogate data and multifractal is introduced.BPR surrogate data method is effective to eliminate nonlinearity in specified frequency band for a fault signal,which can be utilized to detect nonlinear degree in whole fault signal by nonlinear titration method,and the overall nonlinear distribution of fault signal is displayed in nonlinear characteristic curve that can be used to analyze the fault signal qualitatively.Then multifractal theory as a quantitative analysis method is used to describe geometrical characteristics and local scaling properties,and asymmetry coefficient of multifractal spectrum and multifractal entropy for fault signals are extracted as new criterions to diagnose machinery faults.Several typical faults include rotor misalignment,transversal crack,and static-dynamic rubbing fault are analyzed,and the results indicate that those faults can be distinguished by the proposed method effectively,which provides a qualitative and quantitative analysis way in the field of machinery fault diagnosis.

Multifractal analysis of the Yellow River flows

This paper deals with time series of the Yellow River daily flows at Tongguan hydrological station, from the year 2000 to 2005. Power spectrum analysis and statistical moment scaling function on a range of scales revealed scaling qualities of the data. The partition function, which displayed a convex curvature, and the generalized dimension function showed that multifractality is presented. The singularity spectrum, which is single-humped, has shown strong multifractality degree.

Multifractal characteristics and spatial variability of soil particle-size distribution in different land use patterns in a small catchment of the Three Gorges Reservoir Region,China

Characterizing soil particle-size distribution is a key measure towards soil property.The purpose of this study was to evaluate the multifractal characteristics of soil particle-size distribution among different land-use from a purple soil catchment and to generalize the spatial variation trend of multifractal parameters across the catchment.A total of 84 soil samples were collected from four kinds of land use patterns(dry land,orchard,paddy,and forest)in an agricultural catchment in the Three Gorges Reservoir Region,China.The multifractal analysis method was applied to quantitatively characterize the soil particle size distribution.Six soil particle size distribution(PSD)multifractal parameters(D(0),D(1),D(2),(35)a(q),(35)f[a(q)],α(0))were computed.Additionally,a geostatistical analysis was employed to reveal the spatial differentiation and map the spatial distribution of these parameters.Evident multifractal characteristics were found.The trend of generalized dimension spectrum of four land use patterns was basically consistent with the range of 0.8 to 2.0.However,orchard showed the largest monotonic decline,while the forest demonstrated the smallest decrease.D(0)of the four land use patterns were ranked as:dry land<orchard<forest<paddy,the order of D(1)was:dry land<paddy<orchard<forest,D(2)presented a rand-size relationship as dry land<forest<paddy<orchard.Furthermore,all land-use patterns presented asΔf[α(q)]<0.The rand-size relationship ofα(0)was same as D(0).The best-fitting model for D(0),D(1),D(2)andΔf[α(q)]was spherical model,forΔα(q)was gaussian model,and forα(0)was exponential model with structure variance ratio was 1.03%,49.83%,0.84%,1.48%,22.20%and 10.60%,respectively.The results showed that soil particles of each land use pattern were distributed unevenly.The multifractal parameters under different land use have significant differences,except forΔα(q).Differences in the composition of soil particles lead to differences in the multifractal properties even though they belong to the same soil texture.Farming behavior may refine particles and enhance the heterogeneity of soil particle distribution.Our results provide an effective reference for quantifying the impact of human activities on soil system in the Three Gorges Reservoir region.

Research on the multifractal characteristics of the temporal-spatial distribution of earthquakes over New Zealand area

Using the Hill estimator, general multifractal characteristics of events in the New Zealand area have been dis-cussed. Results show that the spatial distribution of shallow events has apparent clustering characteristics, inde-pendent of the threshold magnitude; but for deep events these characteristics are not clear. While the time interval distribution has obvious clustering characteristics both for deep and shallow events, although with a different scal-ing range, the Hill estimates tend to indicate that the time interval distribution has a unifractal rather than a multifractal nature. All above reveal that the seismicity nature for shallow and deep events is apparently different.

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.