The angle measure technique: A new method for characterizing the complexity of geomorphic lines

A new method for characterizing the complexity of geomorphic phenomena is presented. This method, termed the angle measure technique, involves measuring the angularity of a digitized line for a wide range of scales. In this manner, the technique is capable of delineating changes in the complexity of geomorphic lines with scale, from which the characteristic scale(s) of the lines can be identified. Unlike fractal analysis, values produced by the angle measure technique correspond to single scales. Therefore, no assumptions are made concerning the relationship between complexity and scale, and the technique can resolve variations in complexity over small ranges of scale. The technique is illustrated using both computer-generated curves and natural lines, including the trace of a river channel, and is compared to fractal analysis on a contour line crossing two lava flows.     

On Fractal Dimensions of China's Coastlines

The fractal dimensions of China's coastlines are preliminarily discussed on the basis of GIS in this paper. Some significant conclusions are drawn. The fractal relationship between the length of China's continental coastline L and the yardstick r is log L = 3.99 – 0.16 log r on the scale 1:2,500,000 map. The fractal relationship between the length of Jiangsu province's coastline and the yardstick may be established as log L = 2.82 – 0.10 log r on the scale 1:50,000 map. Using the divider method, the fractal dimension of China's continental coastline is 1.16, Taiwan Island is 1.04, etc. The fractal dimensions of coastlines of the Bohai sea, the Yellow sea, the East China sea, and the South China sea tend to increase from north to south, indicating that the complexity of China's coastlines increases toward low latitudes. The substantial components of coast, biological function, and climate from north to south result in a change in fractal dimensions along the coasts of China. The fractal dimension of a coastline is different from the average fractal dimension of all its parts. The more parts of a coastline, the larger the difference between the fractal dimension of the original coastline and the average fractal dimension of all its parts. Faults control the basic trends and fractal dimensions of coastlines as a whole in the studied areas of Taiwan Island and Changle-Lufeng. The more the controlling effect of the faults, the smaller the fractal dimension of the coastline. The less the controlling effect of the faults, the larger the fractal dimension of the coastline. The results indicate that the faults control not only the basic trend of a coastline but also the complexity in the studied areas of Taiwan Island and Changle-Lufeng.     

新疆东准噶尔地区断裂的多重分形研究

结合东准噶尔地区强应变构造带的空间展布情况和断裂的空间分布特征 ,用多重分形模型测算了各带的多重分维值。研究结果表明 :强应变构造带在不同尺度上具有自相似性 ;断裂的空间分布为多重分形结构 ,断裂的演化有从复杂几何结构的次级断裂组合向单一连续的大断裂过渡而分维值逐渐减小的趋势 ;不同的大地构造单元有不同的分维值 ,而且有靠近板块缝合带 (或海沟 )分维值减小、远离板块缝合带 (或海沟 )分维值增大的趋势     

Weathering: Toward a Fractal Quantifying

Weathering occurs over a wide range of scales. To link features through these scales is a major challenge for interdisciplinary weathering studies. Fractal approach seems to be specially useful for this purpose. We introduce a multistep fractal weathering assessment scheme devoted to extract fractal weathering classifiers from texture analysis of the mineral's image. Our scheme enables to quantitatively estimate the global and local information about the geometry of the weathering pattern. This information is basic to develop geometrical indices of weathering, which can significantly enrich the common qualitative and semiquantitative weathering assessment schemes. To justify the fractal approach, a strong statistical self-similarity has been documented for both the weathering and fresh features of two common silica minerals: quartz and biogenic A-opal (phytolith) over four orders of length scales. The procedure is fast, drastically reduces thresholding bias, promises to be universal, it is valid for genetically different minerals and rock types, scale independent, and specially useful for monitoring the changes in the mineral's roughness during the alteration. Two of the proposed classifiers seem to be potentially useful for direct application in the field and be used by nonspecialist.     

岩石节理粗糙度系数的分形特征

岩石节理粗糙度系数JRC是估算节理抗剪强度和变形指标最重要的参数。通过对简易纵剖面仪获取的节理表面轮廓曲线的分形研究,讨论了节理表面轮廓曲线的自相似性和JRC的自相似性,并根据实测统计资料的分析,指出了分形理论研究JRC的适用条件和有效的使用方法。由实测统计资料的JRC尺寸效应自相似性分析,认为JRC尺寸效应具分形结构。本文介绍了一种确定JRC尺寸效应分维数D的方法,由此确定的分维数D具有明确的物理意义。     

A Grey Fuzzy Comprehensive Model for Evaluation of Geological Structure Complexity

Several structure sets (faults and folds) are characterized by their self-similarity properties. Herein, we discuss the degrees of complexity of fractures by introducing the box-counting fractal dimension of faults as a key criterion to be used in comprehensive fuzzy analysis model for evaluation of the complexity of structures. Totally, eight criteria including density, intensity, length of faults, types and box-counting fractal dimension of faults, the intersection angle between faults and coal beds, gradient coefficients, dip angles of the coal beds, and variation coefficients of dip angles of the coal seams, were used for the evaluation purpose. The grey fuzzy comprehensive assessment model was used to rank the relative importance of these criteria. Scores indicating the complexity of structure were calculated on the base of criteria values and their weights for each sub-area of the study area in the Pansan (潘三) coal mine district in the southern Anhui (安徽) Province, China. The result on the calculated complexity of structure is useful for mining planning in the study area.     

Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping

Fractional Brownian surfaces have been widely discussed as an appropriate model for the statistical behavior of topographic surfaces. The fractals model proposes that topographic surfaces are statistically self-similar, and that a single parameter, the fractal dimension, applies at all scales. This paper presents the results of empirical examinations of 17 topographic samples. Only one of these samples shows the statistical behavior predicted by the fractals model; however, in 15 of the 17 samples, the surfaces' variograms could be adequately described by ranges of scales having constant fractal dimension, separated by distinct scale breaks. For scale ranges between adjacent breaks, surface behavior should be that predicted by the fractals model; the breaks represent characteristic horizontal scales, at which surface behavior changes substantially. These scale breaks are especially important for cartographic representations and digital elevation models, since they represent scales at which there is a distinct change in the relation between sampling interval and the associated error.     

First and Second Orders of Suture Complexity in Ammonites: A New Methodological Approach Using Fractal Analysis

The study of septal patterns in ammonoids has been centered on functional and/or constructional issues. Complexly fluted septa have been considered as complementary structures that reinforce the ammonite shell, their frilled sutures possibly manifesting the demand for strength. Ammonitic sutures display features that denote typical fractal behavior, since they can present very long perimeters relative to the contiguous shell areas, and most provide evidence of statistical self-similarity when observed at varying scales of magnification. However, there is a lower limit of scale measurements below which the fractal behavior of the curve no longer holds, and the perimeter length/step size relationship approaches an Euclidean geometry. This paper describes a new methodology that allows the accurate characterization of suture complexity in ammonoids using the technique of fractal analysis (step-line procedure). The proposed methodology helps to fix the position of this cut-off point, allowing for independent estimates of the fractal dimensions of the curve for both large and small measurement scales (i.e., first and second orders of suture complexity). This approach improves the resolution of fractals in the analysis of suture complexity, thus facilitating the potential interpretation of suture patterns in functional/constructional, evolutionary and paleoecological terms.     

Some Distinctions Between Self-Similar and Self-Affine Estimates of Fractal Dimension with Case History

Compass, power-spectral, and roughness-length estimates of fractal dimension are widely used to evaluate the fractal characteristics of geological and geophysical variables. These techniques reveal self-similar or self-affine fractal characteristics and are uniquely suited for certain analysis. Compass measurements establish the self-similarity of profile and can be used to classify profiles based on variations of profile length with scale. Power spectral and roughness-length methods provide scale-invariant self-affine measures of relief variation and are useful in the classification of profiles based on relative variation of profile relief with scale. Profile magnification can be employed to reduce differences between the compass and power-spectral dimensions; however, the process of magnification invalidates estimates of profile length or shortening made from the results. The power-spectral estimate of fractal dimension is invariant to magnification, but is generally subject to significant error from edge effects and nonstationarity. The roughness-length estimate is also invariant to magnification and in addition is less sensitive to edge effects and nonstationarity. Analysis of structural cross sections using these methods highlight differences between self-similar and self-affine evaluations. Shortening estimates can be made from the compass walk analysis that includes shortening contributions from predicted small-scale structure. Roughness-length analysis reveals systematic structural changes that, however, cannot be easily related to strain. Power-spectral analysis failed to extract useful structural information from the sections.     

矿床储量规模分布的分形模型及其分维的数理分析

矿床储量规模预测在矿产勘查和评价中占有十分重要的位置,而其预测的最主要手段是建立矿床储量规模分布的数学模型.根据矿床储量的统计自相似性,应用概率统计方法建立了矿床储量规模的分形模型.同时,通过模型及其分维值的数理分析,得出不同分维值的地学解释,讨论了分维的线性与非线性最小二乘回归估计法.     

JRC分形估测方法的实用性

基于分形几何的码尺法分维数与岩石节理粗糙度系数的物理意义剖析，认为Ｄ－ＪＲＣ之间不存在必然的相关性．分析标准轮廓曲线的分维数，发现其分维数差级微小，难以实行粗糙度系数分级．根据实测资料阐述了岩石节理表面轮廓曲线的“自相似”是统计意义而不是绝对的，它要求ＪＲＣ分形估测应统计求取，而过繁的分维数测量步骤削弱了ＪＲＣ的分形统计估测的可行性．建立在实测资料统计分析基础上的ＪＲＣ尺寸效应分形模型ＪＲＣｎ＝ＪＲＣ０（Ｌｎ／Ｌ０）－Ｄ客观而真实地刻画了粗糙度系数随取样长度增大而降低的规律，其中，ＪＲＣ尺寸效应分维数（Ｄ）具明确的物理意义，它描述了ＪＲＣ随结构面规模增大而降低的衰减速率．最后，运用ＪＲＣ尺寸效应分维数（Ｄ）探讨了岩石节理粗糙度系数尺寸效应的各向异性规律．     

排土场散体岩石粒度分布与剪切强度的分形特征

应用分形理论研究了矿山排土场散体岩石粒度分布的分维规律,建立了分维数与排土场散体物料剪切强度参数的定量关系式。研究表明,排土场岩石块度分布具有良好的分形结构,分维数值大小随着排土场高度的增加而增加,但不超过3。当采样尺度范围一定时,分维数越大,散体中细颗粒含量越多,平均粒径也越小。分维数与散体岩石的剪切强度参数摩擦角?呈负指数关系。分维数值可用于排土场粒度资料的统计分析与剪切力学强度参数的预测。     

分形统计模型的理论研究及其在地质学中的应用

本文提出了一般分形模型和一般分维数的概念,认为许多地质模型是一般分形模型的特例,指出幂函数分布和帕累托分布是分形统计模型的数学基础,论证了幂函数分布在高端截尾条件下具有尺度不变的分形性质,根据非线性回归模型参数估计的方法,提出了求分维数的新方法,该方法具有许多优点。通过在计算机上产生随机数对分形统计模型进行模拟研究,以及通过实例说明分形统计模型应用的方法及步骤,并解释了分维数的实际意义。     

多维自仿射分布及其在地球化学中的应用

提出了多维自仿射分布的概念,指出多维自仿射分布是自仿射分形的数学基础,论证了多维自仿射分布在截尾条件下具有尺度不变的分形性质,交菜理论研究推广到多维情况。通过实例,说明多维自仿射分布在实际问题中的方法和步骤。并解释了分维数的实际意义。分维数是反映区域化变量在某方向变化程度的定量指标。该方法不仅适用于地球化学金元素和银元素数据,而且还适用于其它元素和地质数据,具有普遍的意义。     

遥感线性构造分形统计和蚀变信息提取在桂东地区金铅锌锡多金属成矿预测中的应用

在桂东地区ETM+遥感影像742波段融合的基础上,进行了线性构造和环形构造解译,运用分形几何学的原理和方法对研究区的遥感线性构造进行定量分析,利用计盒维数法求得研究区的线性构造分维值,得出该区线性构造具有良好的统计自相似性和分形特征,利用Surfer软件求得线性构造分维等值线图,采用主成分分析和比值法组合提取遥感蚀变异...     

Structural and lithological controls on coastline profiles in Fife, Eastern Britain

The coastline of southern Fife on the east coast of the British Isles is presented here as an illustration of geological influences on spatial variability at different lengthscales. The coastline profile of Britain is widely referred to as an example of fractal geometry, as presented in Mandelbrot's (1967) original paper on statistical self-similarity. I show here that at different lengthscales, different geological processes dominate the character of the coastline profile, and that estimates of the fractal dimension are strongly dependant on the scale and choice of sample. Significant periodic components in the tortuosity of the coastline are evident, and can be related to the causative, scale-specific, geological processes.     

一种新的粒度指标:沉积物粒度分维值及其环境意义

根据分形理论，对1000个不同类型沉积物样品的粒度分形结构及其分维值作了定量计算。结果表明：各类型沉积物都具有良好的统计自相似性，并且各自有着不同的分维值。粒度分维值的大小与分选好坏密切相关。没有经过外力搬运与分选的风化破碎物质(碎屑物)——断层泥、冻土的粒度分维值接近于碎形体的分维值2．60左右；经过外力搬运但分选作用较弱的冰碛物、水石流、泥石流堆积物的粒度分维值也接近2．60；而经过强烈外力分选作用的河床、湖泊沉积物、潮滩及海底沉积物的分维值明显偏小。鉴于此，本认为分维值可以作为沉积物类型判别的重要参数之一。     

内蒙古卫境地区遥感线性构造分形特征

在综合分析多源地学信息的基础上,利用遥感影像对内蒙古卫境地区进行线性构造目视解译,结合Arc GIS与MATLAB软件对已解译的线性构造进行分形研究,快速获取研究区线性构造的分形维数、不同方向线性构造分形维数及不同区域线性构造分维值,并对计算结果进行了分析。结果表明:研究区内线性构造、不同走向线性构造及不同区域线性构造都具有自相似性;南北向、北西向线性构造分维值较大,复杂性较高,为区域主要构造方向;铀矿化主要分布在分维值0.82的区域内,分维值越大,越有利于热液活动和成矿。     

Fractally Invariant Distributions and an Application in Geochemical Exploration

Fractal modelling has been applied extensively as a means of characterizing the spatial distribution of geological phenomena that display self-similarity at differing scales of measurement. A fractal distribution exists where the number of objects exhibiting values larger than a specified magnitude displays a power-law dependence on that magnitude, and where this relationship is scale-invariant. This paper shows that a number of distributions, including power-function, Pareto, log-normal and Zipf, display fractal properties under certain conditions and that this may be used as the mathematical basis for developing fractal models for data exhibiting such distributions. Population limits, derived from fractal modelling using a summation method, are compared with those derived from more conventional probability plot modelling of stream sediment geochemical data from north-eastern New South Wales. Despite some degree of subjectivity in determining the number of populations to use in the models, both the fractal and probability plot modelling have assisted in isolating anomalous observations in the geochemical data related to the occurrence of mineralisation or lithological differences between sub-catchments. Thresholds for the main background populations determined by the fractal model are similar to those established using probability plot modelling, however the summation method displays less capacity to separate out anomalous populations, especially where such populations display extensive overlap. This suggests, in the geochemical data example provided, that subtle differences in the population parameters may not significantly alter the fractal dimension.     

新亚欧大陆桥新疆段环境灾害时空分维特征研究

基于区域自然灾害发生的随机性和不确定性,运用分形、混沌理论,对新亚欧大陆桥新疆段近38年的环境灾害受损次数进行了时空分维特征研究。结果表明陆桥新疆段受损状况不仅在时间序列上具有自相似性,其容量维Df在0.8302～1.437之间,而且在空间序列上也具有自相似性,存在明显分维结构,其信息维D1在0.0997～0.1272之间。灾害严重区段的容量维值较大,并且容量维值越大,灾害事件中等级的复杂程度就越高,这一特征将有助于认识灾害发生规律和动力学特征。