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.

Fractal dimensions of flocs between clay particles and HAB organisms

The impact of harmful algal blooms （HABs） on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency （RE） was influenced by many factors, including clay type and concentration, algal growth stage, and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema eostatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay. For the original clay, the two-dimensional fractal dimension （D2） was 1.92 and three-dimensional ffactal dimension （D3） 2.81, while for the modified clay, D2 was 1.84 and D3 was 2.50. The addition of polyaluminum chloride （PAC1） lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D2 and D3. Due to the decrease of D3, and the increase of the effective sticking coefficient, the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D3 and D2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D3 and D2 of flocs were the lowest.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

Entropy Production and Fractal Dimensions in Heavy Ion Nuclear Reaction at Intermediate Energies

The characteristics of the nonlinear dynamics in the Heavy Ion Collision (HIC) at intermediate energies have been studied by evaluating the productions of the Generalized Entropy (GE) and the Multifragmentation Entropy (ME) as well as the features of the information and fractal dimensions within the Isospin Quantum Molecular Dynamical Model compensated by the lattice methods. Results demonstrate from various views that the existence of deterministic chaos in the dynamical process of reaction.

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.

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.

A Method for Quantifying the Emotional Intensity and Duration of a Startle Reaction with Customized Fractal Dimensions of EEG Signals

The assessment of emotions with fractal dimensions of EEG signals has been attempted before, but the quantification of the intensity and duration of sudden and short emotions remains a challenge. This paper suggests a method for this purpose, by using a new fractal dimension algorithm and by adjusting the amplitude of the EEG signal in order to obtain maximal separation of high and low fractal dimensions. The emotion was induced by embedding a scary image at 20 seconds in landscape videos of 60 seconds length. The new method did not only detect the onset of the emotion correctly, but also revealed its duration and intensity. The intensity is based on the magnitude and impulse of the fractal dimension signal. It is also shown that Higuchi’s method does not always detect emotion spikes correctly;on the contrary, the region of the expected emotional response can be represented by fractal dimensions smaller than the rest of the signal, whereas the new method directly reveals distinct spikes. The duration of these spikes was 10 - 11 seconds. The magnitude of these spikes varied across the EEG channels. The build-up and cool-down of the emotions can occur with steep and flat gradients.

Responses of fractal dimensions of Picea koraiensis seedlings to different light environments

The changes of fractal dimension ofPicea koraiensis seedlings under different light intensities in natural secondary forests was studied. The results showed that with the change of light environment, crown characters ofPicea koraiensis seedlings exhibited a greater plastic in lateral number, lateral increment, lateral dry weight, and specific leaf area. The range of calculated fractal dimensions of seedling crowns was confined between 2.5728 and 2.1036, but maximum of fractal dimension achieved in term moderate shading and in extreme low light conditions fractal dimension was least.

Speaker Identification Based on Fractal Dimensions

This paper discusses application of fractal dimensions to speech processing. Generalized dimensions of arbitrary orders and associated fractal parameters are used in speaker identification. A characteristic vactor based on these parameters is formed, and a recognition criterion definded in order to identify individual speakers. Experimental results show the usefulness of fractal dimensions in characterizing speaker identity.

Multi-Scale Pore Structure Heterogeneity in Tuff Reservoirs Investigated with Multi-Experimental Method and Fractal Dimensions in Chang 7 Formation,Southern Ordos Basin

The tight tuff reservoir is an unusual type of unconventional reservoir with strong heterogeneity.However,there is a lack of research on the microscopic pore structure that causes the heterogeneity of tuff reservoirs.Using the Chang 7 Formation in Ordos Basin,China as a case study,carbon-dioxide gas adsorption,nitrogen gas adsorption and high-pressure mercury injection are integrated to investigate the multi-scale pore structure characteristics of tuff reservoirs.Meanwhile,the fractal dimension is introduced to characterize the complexity of pore structure in tuff reservoirs.By this multi-experimental method,the quantitative characterizations of the full-range pore size distribution of four tuff types were obtained and compared in the size ranges of micropores,mesopores and macropores.Fractal dimension curves derived from full-range pores are divided into six segments as D1,D2,D3,D4,D5 and D6 corresponding to fractal characteristics of micropores,smaller mesopores,larger mesopores,smaller macropores,medium macropores and larger macropores,respectively.The macropore volume,average macropore radius and fractal dimension D5 significantly control petrophysical properties.The larger macropore volume,average macropore radius and D5 correspond to favorable pore structure and good reservoir quality,which provides new indexes for the tuff reservoir evaluation.This study enriches the understanding of the heterogeneity of pore structures and contributes to unconventional oil and gas exploration and development.

Fractal Study on the Evolution of Micro-Pores in Concrete Under Freeze-Thaw

After exposure to freeze-thaw cycles, scanning electron microscopy(SEM) and nuclear magnetic resonance(NMR) were used to test the four mixtures. The microstructure is qualitatively analyzed from the 2D SEM image and the 3D pore distribution curve before and after freezing and thawing. The fractal dimension is utilized to characterize the two-dimensional topography image and the three-dimensional pore distribution, quantitatively. The results reveal that the surface porosity and volume porosity increase as the freeze-thaw action increases. Self-similarity characteristics exist in micro-damage inside the concrete. In the fractal dimension, it is possible to characterize pore evolution quantitatively. The fractal dimension correlates with pore damage evolution. The fractal dimension effectively quantitatively characterizes micro-damage features at various scales from the local to the global level.

A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

Damage Characteristics Analysis and Fractal Study of Shale With Prefabricated Fractures under Thermal-mechanical Coupling

To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.

Research status and prospects of the fractal analysis of metal material surfaces and interfaces

As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions.Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces.However,fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials.From the perspective of two-dimensional planes and three-dimensional curved surfaces,this paper reviews the current research status of the fractal analysis of metal precipitate interfaces,metal grain boundary interfaces,metal-deposited film surfaces,metal fracture surfaces,metal machined surfaces,and metal wear surfaces.The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized.Starting from three perspectives of fractal analysis,namely,research scope,image acquisition methods,and calculation methods,this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed.It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.

Microstructure Characterization of Bubbles in Gassy Soil Based on the Fractal Theory

The microscopic characterization of isolated bubbles in gassy soil plays an important role in the macroscopic physical properties of sediments and is a key factor in the study of geological hazards in gas-bearing strata.Based on the box-counting method and the pore fractal features in porous media,a fractal model of bubble microstructure parameters in gassy soil under different gas con-tents and vertical load conditions is established by using an industrial X-ray CT scanning system.The results show that the fractal di-mension of bubbles in the sample is correlated with the volume fraction of bubbles,and it is also restricted by the vertical load.The three-dimensional fractal dimension of the sample is about 1 larger than the average two-dimensional fractal dimension of all the slices from the same sample.The uniform porous media fractal model is used to test the equivalent diameter,and the results show that the variation of the measured pore diameter ratio is jointly restricted by the volume fraction and the vertical load.In addition,the measured self-similarity interval of the bubble area distribution is tested by the porous media fractal capillary bundle model,and the fitting curve of measured pore area ratio in a small loading range is obtained in this paper.

Flow behavior of a coupled model between horizontal well and fractal reservoir

Many research findings have proven that the system of porous medium reservoirs exhibits different heterogeneous structures at various scales,demonstrating some form of self-similarity with fractal characteristics.In this paper,fractal theory is incorporated into the reservoir to investigate coupled flow between reservoir and horizontal well.By examining the pore structure of highly heterogeneous reservoirs,the fractal dimension can be determined.Analytical methods are utilized to solve the Green function of a point source in a reservoir with fractal characteristics.Employing Green's function and the principle of spatial superposition,a finite flow model for a horizontal well coupled with a fractal reservoir is developed to calculate the flow rate and flow profile of the horizontal well.The model also accounts for the impact of wellbore friction and is solved numerically.A specific example is used for calculation to analyze the influence of fractal parameters on the production and flow rate of the horizontal well.When considering the fractal characteristics of oil reservoirs,the flow rate of the horizontal well is lower than that in Euclidean space.As the fractal dimension increases,the connectivity of pores in the reservoir improves,making it easier to drive the fluid into the wellbore,and the flow distribution along the wellbore becomes more uniform.Conversely,as the anomalous diffusion index increases,the connectivity between pores deteriorates,thus the distribution of flow rate along the wellbore becomes more uneven.

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.

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.

Fractal Analysis on Methane Hydrate Formation in Porous Media

The study of the hydrate formation process in porous media is of great significance for hydrate application.In this work,the formation process of methane hydrate in porous media was monitored in situ by a low-field magnetic resonance imaging(MRI)system.The formation characteristics of methane hydrate in porous media and the change of fractal dimension of pore space were studied through the change of residual water saturation and T2 relaxation time distribution.The experimental results show that the hydrate formation process is divided into two stages:fast and slow.During the formation process,the water in the pores is continuously consumed and transformed into hydrate,and the overall T2 distribution gradually shifts to the left.In the formation process of hydrate,the pore space becomes more complex,the change of fractal dimension from top to bottom of the reactor gradually increases,and the hydrate formation rate also gradually increases.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.