Multifractal Model Research Articles

Intermittency of gravity wave turbulence on the surface of an infinitely deep fluid: directional effects

This study investigates the influence of surface wave characteristics, specifically wave steepness and directional spreading, on intermittency in deep-water gravity wave turbulence through long-term numerical simulations of three-dimensional potential fully nonlinear periodic gravity waves. We conducted this investigation by estimating the scaling exponent of the surface elevation under different sea state conditions. With our numerical methods, we were able to evaluate the scaling exponents of the structure-function up to 12th order. The observed increased intermittency in directionally narrower sea states and in higher steepness conditions aligns with known effects of quasi-resonant wave–wave interactions and wave breaking. Comparative analyses reveal that both the conventional She–Leveque model and the multifractal models, also used to represent intermittency in wave turbulence of a different nature, exhibit a strong correlation in this study. This observation underscores the universality of intermittency phenomena within wave turbulence.

Correlation analysis of multifractal stock price fluctuations based on partition function

Studying the correlation analysis of stock price fluctuations helps to understand market dynamics better and improve the scientific nature of investment decisions and risk management capabilities. Most existing methods use multifractals to explore the correlation between different economic entities. However, the study of multifractals fails to fully consider the weight of each entity’s impact on the market, resulting in an inaccurate description of the overall market dynamics. To address this problem, this paper creatively proposes a weighted multifractal analysis method (WMA). The correlation analysis of government regulation, market supply and demand, and stock price index is performed using the data of A-share listed companies in Shenzhen and Shanghai as samples. First, we consider the amplitude fluctuation information the signal carries and weigh the partition function according to the proportion of variance in the segment for different amplitude changes. Secondly, we derive the theoretical analytical form of the classical multifractal model (SMA) of the scaling indicator under WMA. Finally, through numerical simulation experiments, it is confirmed that WMA is equally effective as SMA. In addition, the re-fractal correlation analysis of real financial time series also confirms that WMA can effectively utilize the amplitude fluctuation information in the series and outperforms the classical SMA method in distinguishing different signals.

On a multi-fractional model for biogas production for a cellulose-based substrate

On a multi-fractional model for biogas production for a cellulose-based substrate

Effect of water occurrence in coal reservoirs on the production capacity of coalbed methane by using NMR simulation technology and production capacity simulation

The occurrence of gas and water is a critical factor affecting the production capacity of coalbed methane. Therefore, occurrence and dynamic interaction between gas and water in coal reservoirs need to be studied. T2 spectrum of all the samples were studied by NMR technique at 0, 5, 9, and 28 h of natural evaporation. Then, multifractal model was used to characterize water distribution heterogeneity, and a novel evaluation coefficient was proposed to characterize distribution heterogeneity of water content, the correlation among evaluation coefficient, water distribution heterogeneity and pore structure parameters was analyzed. The results are as follows. In the saturated state, the moisture distribution of three types is heterogeneous. Type A of adsorption pore-developed exhibits strong moisture variation heterogeneity of small pores, in contrast to the uniform moisture variation in large pores. Type B of fracture-developed has the highest coefficient of variation in moisture distribution. The coefficient of variation A1 for moisture content can determine the speed of moisture transport and evaporation in the pore structure. A lower A1 indicates a greater moisture transport volume and faster moisture transport rate over the same time. The coefficient of variation A2 for moisture distribution represents the heterogeneity of moisture content variation in the pore structure. That is, a smaller A2 indicates better heterogeneity in the distribution of moisture content in the pore structure. Numerical simulation indicates the average gas production and the peak gas production time increase and decrease with the initial moisture content (Sgrw) increases, respectively. These findings provide practical guidance for coalbed methane development, especially in optimizing coalbed methane production capacity prediction and improving recovery efficiency.

Identification of Au geochemical anomalies in the Wulonggou area of Qinghai Province based on the W-N multifractal model

Identification of Au geochemical anomalies in the Wulonggou area of Qinghai Province based on the W-N multifractal model

Pore-Fracture System Distribution Heterogeneity by Using the T2 Spectral Curve under a Centrifugal State

In this paper, 12 sandstone samples are collected from the Taiyuan Formation in Qinshui Basin, and sample types using the T2 spectral under LF-NMR saturation and centrifugation conditions are classified. Moreover, single and multifractal models were used to calculate fractal parameters of saturated and centrifugal T2 spectra, and the correlation between different fractal parameters, pore structure, T2cutoff value, and pore permeability parameters was studied. The results are as follows. (1) According to the T2 spectrum curves under centrifugation and saturation conditions, all the samples can be divided into three types. There are significant differences in the uniform pore size distribution. However, the non-uniformity of small pore distribution in type B samples is stronger than that of other types, while heterogeneity of large pore distribution is weaker than that of different types. The centrifugal T2 spectrum curve exhibits both single-fold and multifractal characteristics. The results of a single fractal by using a centrifugal T2 spectrum are consistent with those of a saturated T2 spectrum, indicating that single fractal features by using centrifugal and saturated T2 spectra are consistent. Unlike the single fractal parameters, the correlation between the saturation and centrifugal T2 spectrum’s multifractal parameters is weak. This suggests that the physical significance conveyed by the centrifugal T2 spectrum’s multifractal parameters differs from that of the saturated T2 spectrum.

Method for Predicting Bound Water Saturation in Tight Sandstone Reservoirs Using Morphology and Fractal Models

The nuclear magnetic resonance T2 spectrum was used to identify the T2 cut-off value, which is the key to determining the irreducible water saturation of a reservoir. In this paper, the saturation and centrifugal T2 spectra of sandstone and coal samples were used to explore the correlation between each parameter and the T2 cut-off value, using a single fractal dimension, a multifractal dimension and a spectral morphology discrimination method. The conclusions are as follows: (1) The T2 spectra of nine sandstone samples in this paper can be divided into four types. Type A is represented by sample 2, wherein the T2 spectrum shows a bimodal state and the area of the right T2 spectrum (2.5~100 ms) is larger than that of the left T2 spectrum (T2 < 2.5 ms), indicating that the sample has good pore connectivity and belongs to the macroporous development sample. The B-type T2 spectrum is unimodal, and the pore connectivity is poor, indicating that it is a large-pore development sample. The T2 spectrum of the C-type sample is unimodal, and the pore connectivity is very poor, indicating that it is a mesoporous development sample. The T2 spectrum of the D-type sample shows a single peak state, and the main T2 is distributed within 0.1~2.5 ms. The pore connectivity is very poor, which indicates that it belongs to the small pore development type sample. (2) The single fractal model shows that, compared with other single fractal parameters, D2 increases with the increase in the T2 cut-off value, but the correlation is weak. Therefore, it is not feasible to predict the T2 cut-off value using the single fractal dimension parameter. (3) The multifractal model shows that D−10–D10 increases linearly with the increase in D−10–D0, but there is no obvious linear correlation between D0–D10 and D−10–D10, indicating that the low pore volume area in this kind of sample controls the overall heterogeneity of pore distribution. (4) The related parameters affecting the T2 cut-off value include D−10–D10, D−10/D10, D−10–D0, TM and D2. Therefore, based on the above five parameters, a T2 cut-off value prediction model is constructed. The T2 cut-off value calculated by the model is highly consistent with the experimental value, which proves the reliability of the model.

Diagnostics of Ship Engines Based on Wavelet Neural Network and Image Scanning Using Programmable Logic Circuit

The article is devoted to a diagnostic system for ship engines based on a wavelet neural network and image scanning using a programmable logic circuit and considers a method for analysing multifractal wavelet models. The combination of wavelet neural networks with a programmable PLIC-based (programmable logic integrated circuit) real-time image processing platform has a significant potential for the purposes of non-destructive testing, which makes it possible to accurately diagnose faults and take effective measures for predictive maintenance, which in turn makes it possible to effectively increase safety and reliability of equipment and reduce maintenance costs. The article proposes an improved approach to the diagnosis of ship engines, which is based on a wavelet neural network and image scanning using a programmable logic circuit. Wavelet packet decomposition is a method for local time and frequency analysis. It gradually refines the signal at multiple scales through scaling and conversion operations, and it can automatically adapt to the requirements of time-frequency signal analysis to focus on any detail of the signal. It has the advantage of good diagnostic accuracy for information with different noise levels, as well as high reliability since image data from multiple engine signals is used.

Applicability of a Fractal Model for Sandstone Pore-Fracture Structure Heterogeneity by Using High-Pressure Mercury Intrusion Tests

Pore structure heterogeneity affects the porosity and permeability variation of tight sandstone, thereby restricting sandstone gas production. In total, 11 sandstone samples were taken as a target in the northwest margin of the Junggar Basin. Then, scanning electron microscope and high-pressure mercury injection tests are used to study the distribution of a pore and fracture system in the target sandstone. On this basis, single and multifractal models are used to quantitatively characterize the heterogeneity of pore structure, and the applicability of the classification model in characterizing the heterogeneity of the pore-fracture structure is explored. The results are as follows. (1) The target samples are divided into two types, with the mercury removal efficiency of type A samples ranging from 44.6 to 51.8%, pore size mainly distributed between 100 and 1000 nm, and pore volume percentage ranging from 43 to 69%. The mercury removal efficiency of type B samples ranges from 14 to 28%, and pore diameter distribution is relatively uniform. (2) Different fractal models represent different physical meanings. The calculation results of sponge and thermodynamic fractal models indicate that the heterogeneity of pore structure distribution in the type B sample is significantly stronger than that in type A, which is inconsistent with the conclusions of the Sierpinski model. This is because the aforementioned two models characterize the complexity of pore surface area, while the Sierpinski model characterizes the roughness of pore volume. The comparison shows that there is a significant correlation between the thermal dimensionality value DT and the volume percentage of macropores and mesopores. Therefore, the thermodynamic model can better quantitatively characterize the heterogeneity of macropore and mesoporous pore distribution. (3) The results indicate that higher pore volume range is mainly influenced by mesopores and macropores. From the relationship curve between mercury removal efficiency and single fractal dimension, it can be seen that mercury removal efficiency is greatly affected by distribution heterogeneity of the lower value area of pore volume, and it has no obvious relationship with distribution heterogeneity in the lower value area of the pore volume.

Analysis of financial markets by coupled criticality approach

Analysis of the criticality in finance and economics is an important topic for scholars. The crises are conditions in which unpredictable behavior have been emerged. On the other hand, the contagion from one market to another one due to the complex nature of them is a crucial concept. Therefore, we should consider economies and financial markets as entangled systems that interact with each other, and studying one’s behavior without considering its interacting structures will not provide us with adequate information. In this paper, we have considered the critical states of some emerging and mature markets’ indices. S&P500 and Nikkei 225 as representatives of mature markets, and SSE Composite and TEDPIX as representatives of emerging markets have been analyzed. Coupled criticality emerges when two interacting systems reach their critical state due to their interactions. One of the efficient methods to study the coupled criticality is the bi-variate multifractal random walk model. This method is useful to assess the correlation of rare events and coupled criticality in entangled systems such as financial markets. The observations in this paper show that TEDPIX and S&P500 have higher critical states among others in most time lags by assessing criticality parameter explained in the model, and in longer time lags criticality parameter decreases in all indices. Coupled criticality parameter explained in the model is assessed between each index mentioned and TEDPIX. Results show that TEDPIX and S&P500 have the highest level of coupled criticality. To figure out how long-range time correlation between rare events in each index affect its critical behavior, the data of each index are shuffled, and the criticality parameter is calculated for the shuffled data. From the observations, it can be concluded that this matter has more substantial effect on mature markets. As it is known, shuffling removes the effect of the long-range time correlations. Therefore, the coupled criticality parameter is calculated for the shuffled S&P500 data and TEDPIX. Results show that the long-range correlation of S&P500 data has a significant effect on the coupled criticality behavior of S&P500 and TEDPIX.

Local phase-constrained convolutional autoencoder network for identifying multivariate geochemical anomalies

Autoencoder is a powerful tool for identifying multivariate geochemical anomalies. However, existing autoencoder-based geochemical anomaly detection methods primarily rely on a global reconstruction error (e.g., mean square error) to define the lower limit of geochemical anomalies, neglecting the common, local structure information of geochemical data. This limitation inevitably results in the decreased accuracy of geochemical anomaly identification. This study proposed a local Phase-Constrained Convolutional AutoEncoder network (PC-CAE) for the identification of multivariate geochemical anomalies. Initially, we employed a local Fourier transform to extract phase information from both the original and the reconstructed data. Subsequently, a convolutional autoencoder network was utilized to learn the latent representation of geochemical background, using the local phase difference between the original and reconstructed data to preserve the local data structure related to geology setting. Additionally, an adaptive weighting strategy was employed to mitigate the overfitting issue. The training samples with high reconstruction errors were finally identified as anomalies. We tested the validity of PC-CAE using the stream sediment geochemical dataset collected in the Jiaodong gold province, Eastern China. The results demonstrated that PC-CAE outperforms existing convolutional autoencoder network and spectrum–area multifractal model in identifying multivariate geochemical anomalies associated with Au mineralization.

A multifractal model for predicting the permeability of dual-porosity media with rough-walled fractures and variable cross-sectional pore channels

The seepage of rock strata is greatly influenced by the pore network and fracture network; however, the prediction of permeability becomes challenging due to the changes in the cross section of pore channels and the morphology of fractures. In this study, a novel pore-fracture permeability model based on a fractal theory is proposed, and the analytical solutions of the model are given. In contrast to the traditional smooth parallel plate and uniform cross section straight capillary, this model not only considers the roughness of the fracture surface, but also the cross section variation and tortuosity of the pore channel. The comparisons between the calculated results and the experimental data verify the reliability of this model. The quantitative analyses of microscopic parameters indicate a positive correlation between the permeability and the fractal dimension, size, and proportion of pores and fractures. Conversely, there is a negative correlation with roughness, tortuosity, and cross-sectional changes. The range in which the seepage contribution of pores cannot be ignored is determined. Two logarithmic relationship expressions for permeability are presented. This study contributes to explore the effects of the geometry and morphology of the pore-fracture media on seepage and supplements the studies on the permeability models.

Mapping exploration targets through multifractal modelling of soil geochemical data in the Xiaohongshilazi Pb–Zn–(Ag) ore district, Jilin Province, NE China

The Xiaohongshilazi Pb–Zn–(Ag) ore district, located in the Jizhong–Yanbian Metallogenic Belt in NE China, presents challenges in identifying geochemical anomalies and prospecting minerals due to thick covers, complex geological evolution and mining activities. To overcome these obstacles, we adopted an innovative approach that combines multifractal theory with concentration-area (C-A) modelling and local singularity analysis (LSA) to characterize the geochemical signature. Initially, factor analysis of log-transformed data was used to identify ore-forming factors, which were then modelled using C-A and LSA. Subsequently, modelling results were compared to conventional maps generated using inverse distance weighted (IDW) interpolation. Additionally, the C-A model was employed to establish thresholds for singularity indices generated by LSA for target mapping. The results revealed a close relationship between Factor 1 (F1) and mineralization. The C-A model of F1 failed to detect weak anomalies compared to the IDW-interpolated map, while LSA not only identified weak anomalies but also enhanced the detection of reliable anomalies. Three prospecting targets (PT-01, PT-02 and PT-03) were identified. Validation of PT-01 using new drillholes and existing mineralization confirms the effectiveness of targets and the accuracy of the model. Our approach provides a robust alternative for prospecting at ore districts encountering similar challenges.

Effect of Pore Structure Heterogeneity of Sandstone Reservoirs on Porosity–Permeability Variation by Using Single–Multi-Fractal Models

Effect of Pore Structure Heterogeneity of Sandstone Reservoirs on Porosity–Permeability Variation by Using Single–Multi-Fractal Models

Uncovering the multifractality of Lagrangian pair dispersion in shock-dominated turbulence

Lagrangian pair dispersion provides insights into mixing in turbulent flows. By direct numerical simulations (DNSs) we show that the statistics of pair dispersion in the randomly forced two-dimensional Burgers equation, which is a typical model of shock-dominated turbulence, is very different from its incompressible counterpart because Lagrangian particles get trapped in shocks. We develop a heuristic theoretical framework that accounts for this—a generalization of the multifractal model—whose prediction of the scaling of Lagrangian exit times agrees well with our DNS. Published by the American Physical Society 2024

Quantitative Description of Pore and Fracture Distribution Heterogeneity Using Mercury Removal Curve and Applicability of Fractal Models

Many studies have used fractal theory to characterize pore structure distribution heterogeneity through mercury intake curves. However, there is relatively little research on the fractal model calculation of mercury removal curves. In this study, a high-pressure mercury intrusion test is used to describe the pore and fracture distribution heterogeneity (PFDH). The fractal physical meaning of the mercury removal curve was determined by calculating the change in the curve’s fractal dimension value. The results are as follows. (1) According to mercury removal efficiency and porosity, samples can be divided into types A (mercury removal efficiency above 35%) and B (mercury removal efficiency below 35%). In general, type A sample belongs to micro-pore-developed types, and type B samples belong to the macro-pore-developed type. (2) The Menger model (M) represents the complexity of a specific surface area, while the Sierpinski model (S) represents the roughness of the pore volume. Among all the samples, the lower-pore-volume region controls PFDH. (3) According to the calculation results of the single fractal model, it can be seen that the PFDH of type B is stronger than that of type A, which is similar to the results of mercury intrusion. According to the calculation structure of the multifractal model, it can be seen that the volume distribution heterogeneity of type B under various pores is significantly stronger than that of type A. This is opposite to the result of mercury injection. (4) DM has a relationship with the pore volume percentage at different stages, so the M model at the mercury inlet stage can better characterize PFDH at the mercury inlet stage.

Linear multifractional stable motion for modeling of fluid-filled regions in retinal optical coherence tomography images

Fluid appears in several retinal disorders can be visualized by optical coherence tomography (OCT) images. Since manual survey of OCT images is challenging, automatic fluid detection is desirable. This paper develops a generalized multifractal framework to model the local scale-invariant property of retinal OCT signals and accordingly proposes a simple methodology for automatic localization of fluid-filled regions. The proposed framework comprises both local self-similarity and heavy tailed distribution of OCT signals by generalization of fractional Levy stable motion (fLsm) to the Riemann-Liouville multifractional Levy stable motion (RL-mLsm). In order to measure the multifractality appears in RL-mLsm, the scaling properties of Levy flights and stochastic differential equations (SDEs) are used to develop a fractional momentbased algorithm namely, multi-fractional Levy fluctuation analysis (MLFA) algorithm. Here, we apply MLFA to model OCT images. Modeling results indicate the significant difference between the multifractal bandwidth in non-fluid and fluid regions. In addition, by utilizing various surrogate series of the signals, we discover the source of multifractality of OCT signals. As an application of the proposed multifractal modeling, we propose a new method, based on simple clustering of estimated local scaling exponent of modeled OCT signals, to localize fluid regions in OCT images. The localization method is validated, and the experimental results reveal the superiority of proposed method. Overall, the proposed multifractal framework has full advantages on excellent improvement of fluid localization, which promises the proposed model is suitable for effective description of self-similar nature of normal and abnormal OCT images independent of capturing device.

RespirAnalyzer: an R package for analyzing data from continuous monitoring of respiratory signals.

The analysis of data obtained from continuous monitoring of respiratory signals (CMRS) holds significant importance in improving patient care, optimizing sports performance, and advancing scientific understanding in the field of respiratory health. The R package RespirAnalyzer provides an analytic tool specifically for feature extraction, fractal and complexity analysis for CMRS data. The package covers a wide and comprehensive range of data analysis methods including obtaining inter-breath intervals (IBI) series, plotting time series, obtaining summary statistics of IBI series, conducting power spectral density, multifractal detrended fluctuation analysis (MFDFA) and multiscale sample entropy analysis, fitting the MFDFA results with the extended binomial multifractal model, displaying results using various plots, etc. This package has been developed from our work in directly analyzing CMRS data and is anticipated to assist fellow researchers in computing the related features of their CMRS data, enabling them to delve into the clinical significance inherent in these features. The package for Windows is available from both Comprehensive R Archive Network (CRAN): https://cran.r-project.org/web/packages/RespirAnalyzer/index.html and GitHub: https://github.com/dongxinzheng/RespirAnalyzer.

Pricing Options with the Multifractal Model of Asset Returns

Pricing Options with the Multifractal Model of Asset Returns

Single and multi‐fractal dimension variation of tight sandstone by using centrifuge T2 spectral curve

AbstractPore‐fracture structure distribution heterogeneity (PFSH) affects dynamic variation of porosity‐permeability of tight sandstone reservoirs, then restricting gas production performance. A fractal model by low‐field nuclear magnetic resonance technology (LF‐NMR) has been used in the quantitative characterization of PFSH. Among some literature, PFSH was studied by using a saturated T2 spectrum. However, there are few studies on fractal characteristics of T2 spectral morphology in a centrifugal state and its influence on porosity‐permeability parameters. In this paper, 30 tight sandstone samples were collected from Taiyuan Formation in Qinshui Basin. Then LF‐NMR technology was used to analyze PFSH, and sample types were divided by using T2 spectra difference under saturated and centrifugal conditions. Meanwhile, single (model 1 and 2) and multi‐fractal model are adopted to calculate fractal parameters of saturated and centrifugal T2 spectra, and then a difference in fractal parameters under different water conditions was compared. Correlation between different fractal parameters, pore structure and T2 cut‐off value are studied, and a mathematical prediction model for T2 cut‐off value by using fractal and pore structure parameters are established. The results are as follows. (1) All the samples are divided into four types A/B/C/D. For example, the type A sample is characterized by a single peak of T2 spectrum and T2 value is less than 10 ms, which indicates that this type belongs a smaller‐pore developed. Type B sample is characterized by a single peak of T2 spectrum and T2 value is10–100 ms, which indicates that this type belongs to mesopore developed. (2) In saturated state (DS), PFSH of type A sample by using model 1 and 2 is stronger than that of type B, followed by type C and D. Then the multifractal model shows that PFSH of type B sample is stronger than that of other sample types. Correlation between fractal dimension calculated by using single fractal and pore structure parameters is stronger than that of multifractal dimension. (3) T2 spectrum in centrifugal state has fractal characteristics (Di), and there are certain correlation Di with Ds. Therefore, a mathematical prediction model for T2 cut‐off value by using fractal and pore structure parameters is established. © 2023 Society of Chemical Industry and John Wiley & Sons, Ltd.