天山地区地貌系统的自仿射分形与多重分形特征研究

利用标准偏差法和固定质量法，研究了新疆天山地区跨越多个不同构造地貌单元的两条地形剖线的自仿射分形和多重分形特征。结果表明：在所研究的标度范围内，两条剖线均表现出了不同特征的多度域分形性质，多重分形谱Dq的形态和值域范围也呈现出不同特征研究认为，地貌形态并不是完全随机的，而是一种确定性随机，不同标度区间的分维值表征了内外营力作用的方式，强度和空间尺度，同时提出地貌宏观与微观作用尺度的分界点在5km左右。这些结果对地貌动力学定量研究具有重要意义。     

新疆天山地区地貌分形与多重分形特征研究

利用投影覆盖法和投影覆盖概率对新疆天山地区不同构造地貌类型进行了分形与多重分形特征的研究。结果表明：在所研究的标度范围内，不同地貌区均表现出明显的多度域分形，分维值总体上呈现出高山区＞中低山区＞盆地区特点，多重分形谱Dq的形态和值域范围也表现出不同特征。研究认为，地貌表面的分维值与地貌形成的内外力地质作用方式和强度密切相关，并提出5-6km的尺度可作为地貌学研究中宏观与微观作用的分界点。     

Is a chaotic multi‐fractal approach for rainfall possible?

An Erratum has been published for this article in Hydrological Processes 15 (12) 2001, 2381–2382. Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono‐fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi‐fractal approaches. The assumption behind multi‐fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi‐fractals. However, it has been observed recently that multi‐fractal approaches might provide positive evidence of a multi‐fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi‐fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi‐fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi‐fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi‐fractal approach for rainfall characterization. Copyright © 2001 John Wiley & Sons, Ltd.     

Self-Affine Fractals and the Fractal Dimension of Fractured Rock Profiles

The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D values of eight profiles are calculated by the ruler method and by the standard deviation method respectively.It is shown that if tc is far greater than the sampling step tc two methods yield the same results,although if it is far smaller than r,the D by the standard method will be about 1.20,while D by the ruler method will very close to 1.0,because two fractal dimensions,local and global,exist on two sides of tc In order to obtain the local fractal dimension which may be close to that of the standard deviation method,the ruler method must be modified.We propose a way to estimate the tc and to modify the ruler method.Finally,a profile having given D is generated in terms of the principle of non-integer order differential,through which the above two methods are verified and lead to the same res     

太湖水体遥感反演参数的空间异质性

空间异质性的存在,会导致水质参数遥感反演中的尺度效应,影响反演精度,因此通过分析水质参数空间异质性,对于选择适当分辨率的遥感影像,提高反演精度具有重要意义.通过2008年10月在太湖布置的3个样方,利用GIS地统计学原理和分形维数的方法,对水质遥感反演中的三要素浓度,包括叶绿素a(Chl.a)、总悬浮物(TSM)和溶解有机碳(DOC)的空间异质性及其可能产生的尺度效应进行了研究.结果表明:太湖水体的三要素浓度在不同样方单元中变异系数相差较大,存在着明显的尺度效应;三个样方内Chl.a变异函数曲线斜率在变程范围内变化都较为剧烈,分形维数较高,说明太湖水体Chl.a受到某种起主导作用的生态过程的影响和控制;Chl.a和TSM的空间结构比例都在90%左右,有较强的空间相关性,表明其空间异质性的产生主要是由于结构性因素引起的,随机性因素作用微弱;DOC空间结构比例较小,说明随机性因素对其空间异质性的产生起了主导作用.三个样方中Chl.a的变程分别为147.3m、129.3m和115.0m,TSM的变程分别为1131.7m、130.6m、149.1m,因此在遥感反演中可选择TM影像,选择5×5窗口,以150m×150m作为基本单元;而DOC的变程分别为34.3m、38.5m、26.4m,表明其自相关距离较小,建议直接选择分辨率为30m的TM影像,使实际测量值与遥感影像最小单元相对应,消除反演过程中的尺度效应带来的误差.该研究也表明,MODIS的像元尺寸(250、500、1000m)明显偏大,在太湖水体三要素反演过程中,由于空间异质性引起的尺度效应,会造成一定的误差.     

中国湖泊分形特征初探

本文分形概念入手,介绍自然界（水系,云,雪,树木）普遍存在分形特征,继而用偏差法求解中国湖泊分布的分形特征,并对其进行地理意义的解释,为湖泊信息分类和湖泊数据库建设服务。     

岩石断面的分形测量及其分维的计算

在实验室内使岩石单轴压缩直至破裂,用位移传感器测量其断面高度,用显微镜观测断面切片的剖线,所得数据用尺度法计算其分维,各种岩石的不同曲线的分维在1.002-1.028之间,与野外断层主迹线分维值相同.用切岛法计算整个断面的分维约为2.05左右.有关分析和计算认为,分维的大小与岩石破裂时所受力的方向有关,与采样步长有关,而与岩石样品的大小无关.     

台湾地区地震的空间关联维特征与构造环境研究

采用关联维方法对台湾地区地震活动的空间特征进行了研究。先利用 10 0a来台湾的地震目录计算各个地震区、带的关联维数 ,将地震空间分布的分形特征定量表达出来 ,然后综合分析地震空间分布的关联维数和孕震构造环境之间的关系 ,得出了以下结论 :1)台湾东、西部地震区由于地震属于不同的大地构造单元 ,因此关联维数有较大的差异 ;2 )在各地震区内部的各个地震带由于板块构造、地壳结构、活断层分布上的差异 ,而具有与其构造特征相对应的关联维数 ;3)各地震带内部的各个不同的部位又由于不同的构造应力场 ,而导致地震分布上出现不同的丛集性 ,表现为不同的关联维数。这些结论充分说明通过关联维分析所得到的地震活动的空间图像与地震活动所代表的不同地质构造背景有着良好的对应关系     

吉林龙岗火山碎屑分形研究

用分形理论分析了吉林龙岗火山碎屑物粒度的分形结构特征。结果显示,射气喷发碎屑物分维值>射气岩浆喷发碎屑物分维值>岩浆喷发碎屑物分维值,分维值可作为区分火山不同喷发类型的定量参数。而对于龙岗岩浆喷发碎屑物,不同火山喷发的碎屑物其分维值也有差别,晚期喷发的金龙顶子火山碎屑分维值>2,早期喷发的小金龙顶子碎屑分维值>2,火山碎屑物分维值可作为区分不同喷发源和划分火山喷发地层序列的一种指标。研究表明,分维值<2的火山碎屑中有不同含量的非等轴颗粒,且分维值与非等轴颗粒的含量呈负相关     

地震活动前兆的三种广义分形维

地震研究及其预报是多学科的交叉,而非线性科学即分形学和混浊理论给人以新启示。本文叙述了地震的广义能量分维,广义时间分维和广义空间分维,探讨了三种广义分维在地震的前兆和余震中的变化规律,进而指出三种广义分维由主震前的低层异常变为震后再度回升现象,作者对第５个地震活跃期的特点提出了一些看法。     

地震记录的广义分维及其应用

根据分形理论,对不同信噪比地震记录的分维特征进行了分析,指出地震记录中噪声背景与信号部分具有不同的分维尺度,地震道时间序列的分维数值与计算时所用的测量尺度有关,因此,可利用广义分维的概念计算地震记录的分数维.地震记录广义分维大大提高了分形算法在计算机自动识别地震波震相时的抗噪声能力.最后用本文方法对实际地震记录进行了有效的初至波自动拾取.     

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.     

Suspended sediment load estimation and the problem of inadequate data sampling: a fractal view

Suspended sediment load estimation at high resolutions is an extremely difficult task, because: (1) it depends on the availability of high‐resolution water discharge and suspended sediment concentration measurements, which are often not available; (2) any errors in the measurements of these two components could significantly influence the accuracy of suspended sediment load estimation; and (3) direct measurements are very expensive. The purpose of this study is to approach this sampling problem from a new perspective of fractals (or scaling), which could provide important information on the transformation of suspended sediment load data from one scale to another. This is done by investigating the possible presence of fractal behaviour in the daily suspended sediment load data for the Mississippi River basin (at St. Louis, Missouri). The presence of fractal behaviour is investigated using five different methods, ranging from general to specific and from mono‐fractal to multi‐fractal: (1) autocorrelation function; (2) power spectrum; (3) probability distribution function; (4) box dimension; and (5) statistical moment scaling function. The results indicate the presence of multi‐fractal behaviour in the suspended sediment load data, suggesting the possibility of transformation of data from one scale to another using a multi‐dimensional model. Copyright © 2005 John Wiley & Sons, Ltd.     

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.     

基于显式垂直比例因子的显式分形插值地震数据重建

在前人工作的基础上对分形插值方法作了详细的探讨,给出了分形插值函数的显式表达方式.在量纲分析的基础上给出了垂直比例因子的局部显式表达式,旨在提高地震道插值重建的精度及突出局部信息,并从单道地震图的角度分析其在地震道插值重建中的应用效果.研究了垂直比例因子的变化对分形插值精度的影响.数值实验表明,随着垂直比例因子的增大,分形垂直的误差逐渐增大,二者之间呈显出指数增长的趋势.该法克服了随机分形插值方法必须进行多步迭代的弱点,提高了计算效率.通过对理论地震道插值重建的分析,说明了本文分形插值方法的高精度和高效率.本文提出的显式分形插值方法既能够突出地震道数据的局部信息,又较好地保持了地震道数据的总体变化趋势.     

结构非均匀性理论模型的关联维描述

利用二维contor三分"非空"集合生成方法构建了非均匀结构理论模型。通过不断改变分形元三段访问概率及其空间构型,得到了样本值强度及其空间分布各不相同的24种样本,并采用推广G-P法分别求解了关联维。研究结果表明,利用关联维不仅可以表征结构的匀质程度,同时也可以描述其几何分布的非均匀程度,分维值越小反映了结构强度及其空间分布的不均匀性增大的物理本质。作为一种简化、有效刻画几何结构非均匀性的指标,关联维是一种较为合适的选择。     

Fractal interrelationships between topography and structure

Factal interrelationships between topography and structure are investigated in two areas of the North American central Appalachian Mountains: one in the intensely deformed Valley and Ridge province and the other in the relatively undeformed foreland area of the Appalachian Plateau province. In the Valley and Ridge province the fractal dimensions of topographic and structural relief vary systematically along the strike of major folds following a second-order polynominal trend. Cross-correlation of the fractal dimensions of topography to structure indicates that there is a significant positive correlation between the two. Fractal analysis of topography in the relatively undeformed foreland area of the Appalachian Plateau revealed no significant variation in the fractal characteristics of topography across the study area, consistent with the lack of near-surface structure. However, fractal analysis of deeper structures beneath the Plateau area undertaken using reflection seismic data revealed step-wise increases in fractal dimension from the deeply buried Precambrian basement to the near-surface. These vertical changes in fractal dimension can be related to the tectonic history of the area. Taken together, these studies indicate that fractal analysis provides a means to quantify and compare the influence of near-surface structure on topographic development and lateral and vertical structural variability. Fractal analysis provides a means to characterize the systematic changes in the complex patterns formed by topography and structure and the interrelationships between them. Similarity in their fractal characteristics implies similarity in the relative amplitude and abundance of different wavelength features in the topographic or structural profile. © 1998 John Wiley & Sons, Ltd.     

Self－affine fractal features of earthquake time series before and after moderate earthquakes

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The fractal properties of topography as controlled by the interactions of tectonic,lithological, and geomorphological processes

Landscape topography widely exhibits fractal structure. Because of the complexity of relief geometry this structure is not homogeneous in space, and the study of its spatial characteristics represents a powerful method for investigating the interrelationships between landforms and underlying processes. We explore these interrelationships using the digital elevation model (DEM) of an area located in central Italy, where landscape topography is strongly linked to its geological evolution, being characterized by alternating intermountain basins and mountain ranges trending NW–SE. A modified version of the method based on the standard deviation of relief elevations is used to evaluate the fractal parameters of relief after tiling the DEM in spatial units characterized by homogeneous fractal geometry, and statistical methods in conjunction with spatial analysis techniques are applied to the resulting terrain datasets. Both the lowest and (to a lesser extent) the highest values of fractal dimension are found to follow the ridge‐and‐valley trend. Low fractal dimension is observed in the mountain ranges characterized by massive strata of limestone, and along the fault scarps defining the contact between the intermountain basins and the surrounding slopes, where sediment deposition prevails. High fractal dimension is observed in regions characterized by highly erodible terrigenous lithology, and in areas where tectonic activity favors erosional processes mainly by rivers. The analysis of the (fractal) power law parameters also suggests that each major lithological complex has its own characteristic fractal signature. These results provide new insights into the link between the fractal properties of topography and the tectonic, lithological, and geomorphological features of the area, and show that the analysis approach proposed is useful to depict key aspects about the geomorphological and geological setting of an area, using only a DEM. Copyright © 2017 John Wiley & Sons, Ltd.     

Nonplanar Faults: Mechanics of Slip and Off-fault Damage

Stress interactions and sliding characteristics of faults with random fractal waviness in a purely elastic medium differ both qualitatively and quantitatively from those of faults with planar surfaces. With nonplanar fault models, solutions for slip diverge as resolution of the fractal features increases, and the scaling of fault slip with fault rupture dimension becomes nonlinear. We show that the nonlinear scaling of slip and divergence of solutions arise because stresses from geometric interactions at irregularities along nonplanar faults grow with increasing slip and produce backstresses that progressively impede slip. However, in real materials with finite strength, yielding will halt the growth of the interaction stresses, which will profoundly affect slip of nonplanar faults. We infer that in the brittle seismogenic portion of the Earth’s crust, off-fault yielding occurs on pervasive secondary faults. Predicted rates of stress relaxation with distance from major faults with random fractal roughness follow a power-law relationship that is consistent with reported clustering of background seismicity up to 15 kilometers from faults.