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Based on the actual monitoring data of the acoustic emission (AE) ground pressure monitoring and positioning system, this paper introduces fractal theory and the multifractal detrended fluctuation analysis (MF-DFA) method to estimate the waveform multifractal spectrum of goaf rock acoustic emission signals and investigate multifractal time-varying response characteristics. The research results show that the wavelet hard thresholding method has the best noise reduction effect on the original signal, and the box counting dimension has a strong waveform identification effect. Before deformation damage occurs, fractal spectral width Δα shows an increase and then decrease and the fluctuation scale factor Δf(α) decreases and then increases. When damage occurs, fractal spectral width Δα decreases and then stabilizes and concentrates. Simultaneously, the fluctuation scale factor Δf(α) keeps decreasing until the lowest point, and then shows an increasing trend until it reaches a stable state. This study is of great significance to the stability evaluation and disaster early warning of mine goaf.

The heterogeneity structural characteristics of the pores in the coal matrix largely determine the storage and transport capacity of gas. In order to quantitatively characterize the pore structure in coal matrix and analyze the influencing factors, the filled pores (0.38–1.5 nm) and diffusion pores (1.5–100 nm) of seven coal samples with different metamorphic degrees were measured by low-temperature liquid nitrogen adsorption and carbon dioxide adsorption experiments, and combined with the theory of multifractality, the filled pores and adsorbed pores of coal samples with different degrees of metamorphism were characterized and discussed. The multiple fractal characteristics and influencing factors of the filled and adsorbed pores of coal samples with different metamorphic degrees were characterized and discussed. The results show that both diffusion pores and filled pores have multiple fractal characteristics; the more metamorphism, the more developed the filled pores and the stronger the heterogeneity. High-order coals had a strong heterogeneity of diffusion pores, while low-order coals exhibited a strong heterogeneity of filled pores. The heterogeneity of filled pores was negatively correlated with specific surface area and pore volume, while the opposite was true for diffusion pores. The relationship between pore heterogeneity and fixed carbon content showed a “U” distribution.

Dilemmas in financing, risk control, and liquidity have arisen in the face of the current land system, and land securitization can provide a way of thinking to solve these problems. This paper proposes four financial instruments for land asset securitization. It calculates the transaction pricing of that land asset security by splitting it into a composite of independent financial derivative values based on the risk-free arbitrage theory in financial engineering. Finally, based on the multiple fractals and volatility spillover index method, the impact of digital currency on the trading dependence and risk spillover of land asset securities when land is securitized as an asset is studied in depth. The results show that the fractal characteristics of land stocks are 0.756, and land bonds are 0.737. The multiple fractal characteristics of the cross-correlation of different land security assets are differentiated, which indicates that the impact of digital currency on the land security assets among them is stronger in the long memory of small fluctuations. The inverse persistence of large fluctuations is stronger. The risk spillover results show that there is a small risk correlation between digital currencies and land securities transactions.

To investigate the flow characteristics of movable water in coal under the influence of micro-nano pore fractures with multiple fractal structures, this study employed nuclear magnetic resonance (NMR) and multifractal theory to analyze gas–water seepage under different injection pressures. Then, the scale threshold for mobile water entering coal pores and fractures was determined by clarifying the relationship among “injection pressure-T2 dynamic multiple fractal parameter seepage resistance-critical pore scale”. The results indicate that coal samples from Yiwu (YW) and Wuxiang (WX) enter the nanoscale pore size range at an injection pressure of 8 MPa, while the coal sample from Malan (ML) enters the nanoscale pore size range at an injection pressure of 9 MPa. During the water injection process, there is a significant linear relationship between the multiple fractal parameters log X(q, ε) and log(ε) of the sample. The generalized fractal dimension D(q) decreases monotonically with increasing q in an inverse S-shape. This decrease occurs in two distinct stages: D(q) decreases rapidly in the low probability interval q < 0; D(q) decreases slowly in the high probability interval q > 0. The multiple fractal singularity spectrum function f(α) has an asymmetric upward parabolic convex function relationship with α, which is divided into a rapidly increasing left branch curve and a slowly decreasing right branch curve with α0 as the boundary. Supporting evidence indicates the feasibility of a methodology for identifying the variation in multiple fractal parameters of gas–water NMR seepage and the critical scale transition conditions. This investigation establishes a methodological foundation for analyzing gas–water transport pathways within porous media materials.

In this paper, the fractal mechanism of soil improvement by vegetation was revealed by analyzing the soil characteristics under four typical vegetation types: Salix cheilophila , Caragana korshinskii , Hippophae rhamnoides , and Corethrodendron fruticosum in Mu Us Sandy Land. The results showed that (1) the soil of each vegetation type was mainly composed of sand (> 90%), and the content of clay and silt was less than 10%. CF had the least heterogeneity of soil particles in the 0–120 cm soil layer, HR was uniformly distributed in the 120–200 cm soil layer, and the change trend of particle composition in the 140–200 cm soil layer was consistent. (2) Soil fractal dimension (D) was positively correlated with soil clay and silt content, and negatively correlated with sand content. The generalized dimension spectrum D(q) decreases in the “S” type, and the sensitivity was higher in the region of q < 0, indicating that the particle distribution in the sparse area is more susceptible to disturbance. (3) HR had the best water holding capacity in the surface layer (0–40 cm) and deeper layer (120–200 cm), and the average soil water storage in 0–200 cm reached 791.61 mm (13.76% higher than CF). All vegetation had soil water deficit in specific soil layers, among which SC had water deficit in 20–30 cm, CK in 0–10 cm, HR in 60–80 cm, and CF in 180–200 cm soil layers (0.91 mm, 0.90 mm, 0.92 mm, and 0.93 mm). (4) HR significantly affected soil bulk density and porosity by increased silt and sand ( P  < 0.05). The fractal parameters of SC were significantly correlated with soil water content ( P  < 0.05). The pH of CK was a significant correlation with soil water storage ( P  < 0.01). CF soil particle composition and fractal parameters were significantly correlated with soil moisture content, bulk weight, and capillary porosity ( P  < 0.05). The study showed that HR achieves the best water retention and soil modification effect by optimizing soil structure and is the preferred vegetation for ecological restoration of the Mu Us Sandy Land.

Using the multifractal detrended cross-correlation analysis (MF-DCCA) method and the Empirical Mode Decomposition (EMD)-MF-DCCA method, this study quantifies the dynamic interrelation between carbon emission allowance returns in the Chinese and EU markets. The cross-correlation statistics indicate a moderate acceptance of the cross-correlation between the two carbon markets. Applying the MF-DCCA and EMD-MF-DCCA methods to the two markets reveals that their cross-correlation exhibits a power-law nature. Moreover, the apparent persistence of the cross-correlation and notable Hurst index show that the cross-correlation between long-term trends of the returns of the Guangdong and EU carbon emission markets exhibits stronger fractality over the long term, whereas the cross-correlation between the short-term fluctuations of the Hubei and EU carbon emission markets demonstrates stronger fractality. Subsequent investigations show that both fat tails and long memory contribute to the various fractals of the cross-correlation between the returns of the Chinese and EU carbon emission markets, especially for the fractals between the Hubei and EU carbon emission markets. Ultimately, the sliding window analysis demonstrates that national policy, trading activity, and other factors can make the observed multiple fractals more sensitive. The aforementioned findings facilitate an understanding of the current state of the Chinese carbon emission market and inform strategies for its future development.

We proposes a multiple fractal dimensions (MFD) method for robust object description. MFD is an effective feature extraction approach, which is first calculated based on a phase angle quantization method to categorize the points of the input image. And then fractal dimensions are calculated to describe the distribution of feature pattern characterized as the intrinsic property of the general objects, i.e., land scene, face and pedestrian. We theoretically proven that our MFD is shown to be invariant to local variations, i.e., Bi-Lipschitz, which is a desirable characteristic for objects, such as land-scene images, face and pedestrian due to the existence of scale variations, local variations and illumination variations in those images. The proposed method is extensively evaluated on land-use scene recognition, face recognition, expression recognition, and pedestrian detection. The experimental results on UC Merced 21-class scene dataset, AR, JAFFE and INRIA pedestrian databases show that our method achieves superior performances over several state-of-the-art methods in terms of recognition rates.

The study of the spatial distribution pattern of soil fractal properties and its relationship with water-salt factors contributes to an in-depth understanding of the mechanisms by which the ecohydrological processes of inland salt marsh wetlands affect soil texture. In the wetland of Sugan Lake at the northern margin of the Qaidam Basin on the Tibetan Plateau, the single and multiple fractal dimensions and their influencing factors of four sample sites with different hydrological gradients from the lake shore to the peripheral uplands of Xiaosugan Lake were analyzed by combining field investigation and quantitative analysis. The results showed that from the lake shore to the peripheral uplands of Xiaosugan Lake, the soil texture demonstrate clear multifractal behaviour, with generalized dimension and multifractal singularity spectra showing inverse S-type curves and left-deviating. The degree of heterogeneity of soil textural homogeneity and particle size distribution frequency curves decreased and then increased, and the soil singularity fractal dimensions (Dv), capacity dimensions (D0), information dimensions (D1), correlation dimensions (D2), singular spectrum width Δα, and singular spectrum shape feature Δf showed U-shaped trends. Soil Dv, D0, D1, D2, Δα, and Δf were positively correlated with clay content, silt content, vegetative belowground biomass (BGB), and Ca2+, negatively correlated with the soil sodium adsorption ratio (SAR). Under the combined influence of wetland hydrological characteristics, ionic composition, and plant community characteristics, the soil fractal dimension of the Sugan Lake wetland showed a complex distributional evolution pattern, reflecting the interactive influence mechanism between soil structure and wetland biotic and abiotic factors in inland salt marsh wetlands.

The occurrence of landslide hazards significantly induces changes in slope surface displacement. This study conducts an in-depth analysis of the multifractal characteristics and displacement prediction of highway slope surface displacement sequences. Utilizing automated monitoring devices, data are collected to analyze the deformation patterns of the slope surface layer. Specifically, the multifractal detrended fluctuation analysis (MF-DFA) method is employed to examine the multifractal features of the monitoring data for slope surface displacement. Additionally, the Mann–Kendall (M-K) method is combined to construct the α indicator and f(α) indicator criteria, which provide early warnings for slope stability. Furthermore, the long short-term memory (LSTM) model is optimized using the particle swarm optimization (PSO) algorithm to enhance the prediction of slope surface displacement. The results indicate that the slope displacement monitoring data exhibit a distinct fractal sequence characterized by h(q), with values decreasing as the fluctuation function q decreases. Through this study, the slope landslide warning classification has been determined to be Level III. Moreover, the PSO-LSTM model demonstrates superior prediction accuracy and stability in slope displacement forecasting, achieving a root mean square error (RMSE) of 0.72 and a coefficient of determination (R2) of 91%. Finally, a joint response synthesis of the slope landslide warning levels and slope displacement predictions resulted in conclusions. Subsequent surface displacements of the slope are likely to stabilize, indicating the need for routine monitoring and inspection of the site.

As a harbinger of anomalous weather regimes in the troposphere, the Northern Annular Mode (NAM) propagates from the stratosphere to the troposphere. This fact makes understanding and predicting NAM variability of great importance. In this study, the multi-fractal behaviors of NAM variability are investigated using extended self-similarity based, multi-fractal detrended fluctuation analysis (ESS-MF-DFA) and the NAM indices from 1000 to 10 hPa. The results show that there are contrasting multi-fractal behaviors between the stratosphere and the troposphere that have a transition band near 200 hPa. The stratospheric NAM variability is more complicated and has multiple multi-fractal regimes over different scales with marked contrasting warm–cold season features. To understand these contrasting stratospheric–tropospheric multi-fractal behaviors, three surrogate methods are adopted to show how temporal ordinal patterns over an annual scale contribute to these behaviors, whereas the distribution of NAM variability only plays a minor role. Further studies show that contrasting warm–cold variability may lead to these contrasting behaviors. Among them, warm–cold seasonal variations, power spectral density (PSD), and autocorrelation provide a similar conclusion. Results indicate that although predictions of the NAM index over the stratosphere are required and necessary, the complicated multi-fractal behaviors make linear prediction strategy difficult to obtain high realizable predictability of NAM variations over the stratosphere.