Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

量化三维网格模型复杂度的Fractal Dimension^(+)方法

对三维网格模型复杂度的评估及应用进行研究,提出基于Fractal Dimension^(+)方法量化三维网格模型复杂度。该方法利用分形几何学的分形维度来确定三维网格模型的不规则性;为了全面量化复杂度并且提高量化的准确性和可信度,在分形维度算法中添加孔隙比和亏格数的概念,全面考虑介质内部空间分布和几何结构的连接性以及孔洞分布的信息。对大量三维网格模型进行实验,结果表明Fractal Dimension^(+)方法可以有效计算出准确的复杂度。

Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

A Successive Shift Box-Counting Method for Calculating Fractal Dimensions and Its Application in Identification of Faults

Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.

Fractal dimension for porous metal materials of FeCrAl fiber

The FeCrA1 fiber was used to prepare porous metal materials with air-laid technology, and then followed by sintering at 1300 ℃ for a holding period of 2 h in the vacuum. In addition, a novel fractal soft, which was developed based on the fractal theory and the computer image processing technology, was explored to describe the pore structure of porous metal materials. Furthermore, the fractal dimension of pore structure was calculated by the soft and the effects of magnification and porosity on ffactal dimension were also discussed. The results show that the fractal dimension decreases with increase in the magnification, while it increases continuously with the porosity enhancing. The interrelationship between the fractal dimension and the magnification or porosity can be presented by the equation of D=α_0exp（-x/α_1）＋α_2和D=k_2-（k_1-k_2）/[1＋exp（（θ-k_0）/k_3）], respectively.

Classification of forearm action surface EMG signals based on fractal dimension

Surface electromyogram （EMG） signals were identified by fractal dimension.Two patterns of surface EMG signals were acquired from 30 healthy volunteers＇ right forearm flexor respectively in the process of forearm supination （FS） and forearm pronation （FP）.After the raw action surface EMG （ASEMG） signal was decomposed into several sub-signals with wavelet packet transform （WPT）,five fractal dimensions were respectively calculated from the raw signal and four sub-signals by the method based on fuzzy self-similarity.The results show that calculated from the sub-signal in the band 0 to 125 Hz,the fractal dimensions of FS ASEMG signals and FP ASEMG signals distributed in two different regions,and its error rate based on Bayes decision was no more than 2.26%.Therefore,the fractal dimension is an appropriate feature by which an FS ASEMG signal is distinguished from an FP ASEMG signal.

Study on the Spatial Pattern Changes of Land Use Based on Fractal Dimensions in Tianjin New Coastal Area

[Objective] The aim was to discuss the spatial pattern changes of land use in Tianjin new coastal area based on fractal dimensions.[Method] By dint of remote and geographic information system technology to obtain the data of urban land use in new coastal area from 1993 to 2008,the boundary dimension,radius dimension and information dimension of each land use type were calculated based on fractal dimension.In addition,the revealed land use spatial dimension changes characteristics were analyzed.[Result] The spatial distribution of each land use type in new costal area had distinct fractal characteristics.And,the amount and changes of three types of dimension values effectively revealed the changes of complicatedness,centeredness and evenness of spatial pattern of land use in the study area.The boundary dimension of unused land and salty earth increased incessantly,which suggested its increasing complicatedness.The boundary of the port and wharf and shoal land was getting simpler.The radius dimension of the cultivated land was larger than 2,which suggested that its area spread from center to the surroundings;the one in salty land and waters distributed evenly within different radius space to the center of the city;the one in other land use types reduced gradually from center to the surroundings.The information dimension value in the woodland and orchard land,unused land and shoal land was small,and was in obvious concentrated distribution;the spatial distribution of cultivated and salty land concentrated in the outside area;the construction area in the port and wharf spread gradually on the basis of original state;the spatial distribution of waters and residents and mines were even.[Conclusion] Applying fractal dimensions to the study of spatial pattern changes of urban land use can make up for some disadvantages in classical urban spatial pattern quantitative research,which has favorable practical value.

ACCURATE SPEECH SEGMENTATION VIA the IMPROVED SHORT-TIME FRACTAL DIMENSION

Objective To improve the accuracy of speech segmentation through the improved short time fractal dimension. Methods An equation was established for window size selection of speech analysis. Dynamic Window Step (DWS), a novel method to determine the sliding window steps adaptively in agreement with the local properties of signals, was proposed. Results The influence of the window step on the short time fractal dimension was discussed. Compared with fixed window steps, more accurate and efficient fractal dimension trajectories were obtained with dynamic window steps. Conclusion The proposed method was applied to a number of speech signals. It shows promise in speech segmentation, speech recognition and other transient signal analysis.

INFLUENCE OF SERIES OF SQUARE GRIDS ON FRACTAL DIMENSIONS-A Case Study of Mountains of China's Mainland

MANDELBROT enunciated the uncertainty of the length of a coastline in his paper "How long is the coastline of Britain?" published in Science in 1967. The fractal concept was presented for the first time in that paper and has been applied to many fields ever since. Although fractal dimensions of lots of phenomena were calculated by the box-counting method,the quantitative influence of series of square grids on them is ignored. The issue is systematically discussed as a case study of the mountains of China′s Mainland in this paper. And some significant conclusions are drawn as follows: 1) Although the fractal character objectively exists in the mountains of China′s Mainland,and it does not vary with the changes of series of square grids,the fractal dimensions of the mountains of China′s Mainland are different with these changes. 2) The fractal dimensions of the mountains of China′s Mainland vary with the average lengths of sides of series of square grids. The fractal dimension of the mountains of China′s Mainland is the function of the average length of side of square grid. They conform to the formula D=f(r) (where D is the fractal dimension,and r is the average length of side of square grid). 3) Different dots of data collection can affect the fractal dimension of the mountains of China′s Mainland. 4) The same range of length of side of square grid and dots of data collection can ensure the comparison of fractal dimensions of the mountains of China′s Mainland. The research is helpful to get the more understanding of fractal and fractal dimension,and ensure that the fractal studies would be scientific.

One-Dimensional Fractal Curves and Fractal Dimension Determination

Various methods for evaluating the fractal curves were reviewed and simulated on computer. It is shown box-counting and power spectrum methods generally give poor measuring results, while yard and variation methods could obtain good results. However,owing to multiple influencing factors, further study needs to be done before widespread application of variation method. In order to improve the measuring accuracy of yard method, a new method has been propoed to measure the fractal dimension by changing the instrumental resolutions.

Fractal Dimension Analysis Based Aging State Assessment of Insulating Paper with Surface Microscopic Images

The insulating paper of the transformer is affected by many factors during the operation,meanwhile,the surface texture of the paper is easy to change.To explore the relationship between the aging state and surface texture change of insulating paper,firstly,the thermal aging experiment of insulating paper is carried out,and the insulating paper samples with different aging times are obtained.After then,the images of the aged insulating paper samples are collected and pre-processed.The pre-processing effect is verified by constructing and calculating the gray surface of the sample.Secondly,the texture features of the insulating paper image are extracted by box dimension and multifractal spectrum.Based on that,the extreme learning machine(ELM)is taken as the classification tool with texture features and aging time as the input and output,to train the algorithm and construct the corresponding relationship between the texture feature and the aging time.After then,the insulating paper with unknown aging time is predicted with a trained ELMalgorithm.The numerical test results show that the texture features extracted from the fractal dimension of the micro image can effectively characterize the aging state of insulating paper,the average accuracy can reach 91.6%.It proves that the fractal dimension theory can be utilized for assessing the aging state of insulating paper for onsite applications.

The Research of ECG Signal Automatic Segmentation Algorithm Based on Fractal Dimension Trajectory

In this paper a kind of ECG signal automatic segmentation algorithm based on ECG fractal dimension trajectory is put forward.First,the ECG signal will be analyzed,then constructing the fractal dimension trajectory of ECG signal according to the fractal dimension trajectory constructing algorithm,finally,obtaining ECG signal feature points and ECG automatic segmentation will be realized by the feature of ECG signal fractal dimension trajectory and the feature of ECG frequency domain characteristics.Through Matlab simulation of the algorithm,the results showed that by constructing the ECG fractal dimension trajectory enables ECG location of each component displayed clearly and obtains high success rate of sub-ECG,providing a basis to identify the various components of ECG signal accurately.

A Method for Calculating Fractal Dimension of Amicrons and Fractal Simulation of Boundary Micro-Topography

A simple and practical method to calculate the fractal dimension （FD） of amicron＇s projective surface profile based on fractal theory is proposed. Taking AI（OH）3 material particles as an example, the scanning electron microscope （SEM） photos of particles were processed using an image.processing software （IPS） Photoshop. Taking the pixel as a fixed yardstick with the enlargement of the size of the particle image, the box-dimension and circumference-area （C-S） methods were used to calculate the FD of the surface profile of the particle. The FD of 1.2623 of the classic Koch curve is obtained, which approximates the true value of 1.2628. The complexities of the object＇s boundary and surface micro-topography are simulated successfully by a generator method.

Analysis of Multi-Scale Fractal Dimension for Image Interpolation

This article presents a novel image interpolation based on rational fractal fimction. The rational function has a simple and explicit expression. At the same time, the fi＇actal interpolation surface can be defined by proper parameters. In this paper, we used the method of ＇covering blanket＇ combined with multi-scale analysis; the threshold is selected based on the multi-scale analysis. Selecting different parameters in the rational function model, the texture regions and smooth regions are interpolated by rational fractal interpolation and rational interpolation respectively. Experimental results on benchmark test images demonstrate that the proposed method achieves very competitive performance compared with the state-of-the-art interpolation algorithms, especially in image details and texture features.

Volume fractal dimension of soil particles and relationships with soil physical-chemical properties and plant species diversity in an alpine grassland under different disturbance degrees

Fractal geometry is an important method in soil science,and many studies have used fractal theory to examine soil properties and the relationships with other eco-environmental factors.However,there have been few studies examining soil particle volume fractal dimension in alpine grasslands.To study the volume fractal dimension of soil particles （D） and its relationships with soil salt,soil nutrient and plant species diversity,we conducted an experiment on an alpine grassland under different disturbance degrees：non-disturbance （N0）,light disturbance （L）,moderate disturbance （M） and heavy disturbance （H）.The results showed that （1） Ds varied from 2.573 to 2.635 among the different disturbance degrees and increased with increasing degrees of disturbance.（2） Shannon-Wiener diversity index,Pielou＇s evenness index and Margalef richness index reached their highest values at the M degree,indicating that moderate disturbance is beneficial to the increase of plant species diversity.（3） In the L and M degrees,there was a significant positive correlation between D and clay content and a significant negative correlation between D and soil organic matter （SOM）.In the H degree,D was significantly and positively correlated with total salt （TS）.The results suggested that to a certain extent,D can be used to characterize the uniformity of soil texture in addition to soil fertility characteristics.（4） For the L degree,there was a significant negative correlation between D and the Shannon-Wiener diversity index; while for the M degree,there was a significant negative correlation between D and Pielou＇s evenness index.

Calculating changes in fractal dimension of surface cracks to quantify how the dynamic loading rate affects rock failure in deep mining

The split-Hopkinson pressure bar(SHPB)and digital image correlation(DIC)techniques are combined to analyze the dynamic compressive failure process of coal samples,and the box fractal dimension is used to quantitatively analyze the dynamic changes in the coal sample cracks under impact load conditions with different loading rates.The experimental results show that the fractal dimension can quantitatively describe the evolution process of coal fractures under dynamic load.During the dynamic compression process,the evolution of the coal sample cracks presents distinct stages.In the crack propagation stage,the fractal dimension increases rapidly with the progress of loading,and in the crack widening stage,the fractal dimension increases slowly with the progress of loading.The initiation of the crack propagation phase of the coal samples gradually occurs more quickly with increasing loading rate;the initial cracks appear earlier.At the same loading time point,when the loading rate is greater,the fractal dimension of the cracks observed in the coal sample is greater.

Relevance between abutment pressure and fractal dimension of crack network induced by mining

Based on the geological conditions of coal mining face No.15-14120 at No.8 mine of Pingdingshan coal mining group,the real-time evolution of coal-roof crack network with working face advancing was collected with the help of intrinsically safe borehole video instrument.And according to the geology of this working face,a discrete element model was calculated by UDEC.Combining in situ experimental data with numerical results,the relationship between the fractal dimension of boreholes'wall and the distribution of advanced abutment pressure was studied under the condition of mining advance.The results show that the variation tendency of fractal dimension and the abutment pressure has the same characteristic value.The distance between working face and the peak value of the abutment pressure has a slight increasing trend with the advancing of mining-face.When the working face is set as the original point,the trend of fractal dimension from the far place to the origin can be divided into three phases:constant,steady increasing and constant.And the turning points of these phases are the max-influencing distance(50 m)and peak value(15 m)of abutment pressure.

Introducing Fractal Dimension to Estimation of Soil Sensitivity to Preferential Flow

Food dye Brilliant Blue was introduced as the tracer in a dye-tracing experiment to obtain dye profile patterns of sandy loam soil, aeolian sandy soil, percolating paddy soil and permeable paddy soil. The dyed soil profiles were then photographed and the photos were scanned into a computer. Edited with certain software, only the dyed areas were left on the profile photos, which indicted the preferential flow paths for water and solute transport. Fractal dimensions of the dye patterns were calculated according to Arnold's function. Soil particle size distribution was analyzed by pipette method. The regression analysis showed that there was significant relationship between soil clay content and fractal dimension D of the dye pattern of soil profile. Based on the experiment results, the possibility of introducing fractal dimension to estimation of soil sensitivity to preferential flow is discussed.

Fractal Characteristics and Fractal Dimension Measurement on Broken Surfaces of Aluminum Electric Porcelain

The characteristics of broken surfaces were r esearched by a scanning electron microscope (SEM) and a reflection microscope, a nd the fractal dimensions of broken surfaces were measured by the Slit Island me thod. The experimental results indicate that the broken surface of aluminum elec tric porcelain is a fractal body in statistics, and the fractal dimensions of br oken surfaces are different with the different amplification multiple value.In a ll of measured fractal dimensions,both of values measured in 100× under reflect ion microscope and in 500× under SEM are maximum, whereas the values measur ed in 63× under reflection microscope and in 2000× under SEM are obviously min imum. The fractal dimensions of broken surfaces are also affected by the degrees of gray comparison and the kinds of measuring methods. The relationships betwee n the fractal dimensions of broken surfaces and porcelain bend strengths are tha t they are in positive correlation on the low multiples and in negative correlat ion on the high multiples.

Characterizing the Spheroidization Grade and Strength of 15CrMo Steel through Determining Fractal Dimension

The fractal dimension（FD） of surfaces has been widely used to characterize the properties of materials.However,most of the previous researches were concentrated on the correlation between the FD of surfaces and mechanical properties of materials,such as impact energy and fracture toughness,etc.The aim of this paper is to characterize the spheroidization grade and strength of 15CrMo steel through determination of FD of cementite phase on the basis of two-dimension microstructural image.Two methods,namely slit-island method（SIM） and box-counting method（BCM）,are used to determine the value of FD.It is found that the FD value evaluated by using BCM is generally higher than that evaluated by SIM.This phenomenon may be due to the difference in the principles used in different methods.Whether SIM or BCM is used,the spheroidization grade in 15CrMo steel linearly increases with decreasing the value of FD.The relationship between the FD value,D,and spheroidization grade,Sg,can be approximately expressed as D≌-0.11Sg＋A,where A is a constant value which is depended on the evaluation method.Both the ultimate strength and the yielding strength of 15CrMo steel increase with increasing FD of cementite phase.There may be a common relationship between the FD of cementite phase and strength of 15CrMo steel.When the FD of cementite phase in 15CrMo steel is determined,the strength of this steel can be evaluated.The present paper can provide a novel method to evaluate the strength and spheroidization grade of carbon steel through determination of fractal dimension（FD） of cementite phase.