Linear Richardson plots from non-fractal data sets

Richardson analysis, one of the principal methods of fractal analysis, is performed by measuring the perimeter of a curve with strides of varying length and constructing a log-log plot of perimeter against stride length. Certain simple geometrical forms can produce linear plots that mimic fractal behavior, and two smooth curves have been discovered that produce linear Richardson plots for strides varying by two orders of magnitude or more. The existence of such curves was not suspected before this study. Richardson analyses that suggest fractal geometry of low dimension or over a limited range of stride length should be checked against the source data for independent evidence of self-similarity.     

鄂西清江上游NE向活动断层束分段分形特征

在地质调查的基础上,初步厘定了清江上游ＮＥ向断层束７条区域性活动断层的轨迹结构,并依据断层几何轨迹的间断、连接和重叠特性,划分断层带的相对独立断裂段;进而利用分形分维分析,分别估算活动断层和相对独立断层段的轨迹结构分维值。计算结果表明：每条断层的轨迹结构具有其特征性的分维值,轨迹结构越复杂,分维值越高,咸丰断层分维值最高（Ｄ＝１．２６８）,齐岳山断层带分维值最低（Ｄ＝０．９８０）;每个相对独立的断层段也具有特征性的分维值,其中,郁江断层带的Ⅱ段的分维值最高（Ｄ＝１．３１８）,齐岳山断层带的Ⅰ段的分维值最低（Ｄ＝０．９６２）;断层轨迹结构越复杂、分段阶区分支断裂越多、分维值越高,其现今活动性越强;活动断层分段分形分析是进行断层活动性评价的有效途径之一。     

Fractal dimensions and geometries of caves

Lengths of all caves in a region have been observed previously to be distributed hyperbolically, like self-similar geomorphic phenomena identified by Mandelbrot as exhibiting fractal geometry. Proper cave lengths exhibit a fractal dimension of about 1.4. These concepts are extended to other self-similar geometric properties of caves with the following consequences.Lengths of a cave is defined as the sum of sizes of passage-filling, linked modular elements larger than the cave-defining modulus. If total length of all caves in a region is a self-similar fractal, it has a fractal dimension between 2 and 3; and the total number of linked modular elements in a region is a self-similar fractal of the same dimension. Cave volume in any modular element size range may be calculated from the distribution.The expected conditional distribution of modular element sizes in a cave, given length and modulus, also is distributed hyperbolically. Data from Little Brush Creek Cave (Utah) agree and yield a fractal dimension of about 2.8 (like the Menger Sponge). The expected number of modular elements in a cave equals approximately the 0.9 power of length of the cave divided by modulus. This result yields an intriguing parlor trick. An algorithm for estimating modular element sizes from survey data provides a means for further analysis of cave surveys.     

Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies

On the basis of results of simple box-counting analysis, it has been suggested in several recent publications that natural fracture patterns are fractal. Fractal patterns are characterized by self-similarity of structure on a range of scales and provide straight-line distributions on box-counting plots. New analysis of a fracture pattern that provided the most convincing straight-line box-counting curve previously published, shows that the curve is non-linear and, therefore, that the fracture pattern is not fractal. Non-linear box-counting curves are also characteristic of other natural fracture patterns analysed but spurious linear curves can be obtained if the area analysed extends beyond the mapped area.     

Complexity and scale in geomorphology: Statistical self-similarity vs. characteristic scales

Two models of the relationship between complexity and scale of geomorphic lines are compared, one based on statistical self-similarity (in which complexity is invariant for some range of scale), and the other on the concept of characteristic scales (in which complexity changes continuously with scale). Two corresponding techniques are used in the comparison, fractal analysis utilizing the divider method, and an angle measure technique. These techniques are applied to three types of coastlines: fiord, volcanic, and tectonic, in order to ascertain which model, statistical self-similarity or characteristic scales, is more useful in understanding variations in coastline complexity for scale. Apparently linear log-log plots of number of steps against steplength produced by fractal analysis display slight but significant curvature. Upon closer examination, it is determined that using fractal dimension to compare even the same types of features is unreliable because of the dependency of fractal dimension on scale of measurement, even if the same steplengths are used throughout the study. These results are corroborated by the use of the angle measure technique, a method based on measuring angles between points along a digitized line. It is concluded that the coastlines examined display no evidence of statistical self-similarity and that the characteristic scales model is more useful in investigating complexity and scale in geomorphology.     

基于分形理论的堆石料级配优化研究

建立颗粒粒径的质量分布分形模型,收集不同堆石坝工程32条堆石料级配曲线,统计分形维数D的分布特性。各堆石坝工程堆石料级配具有良好的分形特性,可考虑将D作为反映和描述堆石料级配特性的新指标,D在2.348～2.699之间,大多数堆石料的分形维数在2.6左右。以古水垫层料为研究对象,设计了6组不同分形维数的级配曲线,采用随机颗粒不连续变形方法（SGDD）研究不同分形维数对堆石料压实性能和宏细观力学特性的影响。分形维数从2.0到2.8,孔隙比呈先减后增趋势,在D =2.7时压实性能最优,而力链的非均匀程度随分形维数的增大而增大。综合考虑试样压实性和力链的非均匀程度,确定D =2.7时的级配为优化级配。     

The fractal shape of sutured quartz grain boundaries: application as a geothermometer

Along grain boundaries of quartz from metamorphic and igneous rocks complex interfingering (sutures) may occur. Under the light microscope the lengths of the sutures range from approximately 10–1000 m. The sutured grain boundaries are statistically self-similar over one to two orders of magnitude. They represent fractals. Their mathematical counterpart are Koch curves which are developed after two to four iterations. The fractal (Hausdorff-Besicovitch) dimensions D of sutured quartz grain boundaries from different grades of metamorphism are between ca. 1.05 and 1.30. The D-value decreases with increasing temperature during formation of the sutures. On a statistical basis, D can be used as a measure of this temperature and, therefore, as a deformation-related thermometer.     

湘中断裂体系分维与锑矿分布

笔者将分形几何学的原理和方法,应用于湘中地区断裂体系的二维平面分布特征研究,发现在研究的标度范围内（ｒ约２～８０ｋｍ）不仅具有统计自相似性,并且同一地区,不同分区具有不同的分维值Ｄ,分维值分布特征为东南区分维值高于西北区分维值,将本区断裂体系的这种分维值分布特征与锑矿床分布特征对比,可发现锑矿床（点）产出较多,矿床规模较大的地区,其分维值较大。     

断层体系分维与油田分布

本文将分形几何学的原理和方法，应用于东营凹陷Ｔ２，Ｔ６和ＴＲ３个不同构造层的断层体系的二维平面分布特征研究，发现在研究的标度范围内（ｌ＝０．５－４ｋｍ）不仅具有统计自相似性，并且同一构造层不同子区域不同的分维值Ｄ．分维值分布特征为南区分维值略低于北区分维值，东区分维值略高于西区分维值，将本区断层体系的这种分维值布特征与油田分布特征对比，可发现分维值越高，油田规模越大。     

基于分形理论的1915—2000年渭河泾河口—潼关段河道演变研究

渭河泾河口—潼关段是渭河平面形态中最活跃的部分。运用近百年来的多种地形资料,在调校民国地图测绘误差的基础上重建了1915—2000年本段渭河平面形态的空间矢量数据,使用盒维数法测算了其近百年来的分维D,发现D区间为[1.0432,1.0774]。1915—1958年是本段河型在相对天然环境下的发展过程,D呈现动态平衡态...     

Fractal Analysis of the Complexity of United States Coastlines

Coastlines have long been used as a principal example of a natural feature that exhibits fractal structure. With the advent of large digitized databases, it has become possible to examine in detail large regions of coast and to examine differences in complexity, as measured by the fractal dimension, among regions. In this study, we have determined the fractal dimension of the Atlantic and Pacific coastlines of the conterminous United States. The traditional divider method was used in obtaining the fractal dimension of each coastline arc from the NOAA Medium-Resolution Shoreline Data Set. On average, the Atlantic coast has much higher fractal dimension than the Pacific coast. The results also indicate that the complexity of the Atlantic coast increases toward low latitudes. These results have implications for the interpretation of species distributions and diversity patterns along the coast and for the understanding of the dynamics of biotic recovery from mass extinctions.     

一种测量断裂构造信息维的方法及其应用

基于分形理论,提出了一种快速测量单区和多区断裂构造信息维的方法。按直线、正方形、Koch曲线及Sierpinski垫片4种图形的测量结果,信息维测量值与其理论值之间最大的相对误差绝对值仅为0.5%,表明采用该方法测量出的信息维具有较高的可靠性和准确度。在某火山岩型铀矿田开展了该方法的应用试验。该铀矿田内有超过92%的铀矿床均位于断裂信息维大于1.24的区域内,说明断裂构造信息维越大,越有利于火山岩型铀矿成矿,从而为建立新的铀矿预测标志提供了新的思路和方法。     

On the practice of estimating fractal dimension

Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape.     

Natural fracture profiles,fractal dimension and joint roughness coefficients

Summary Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.     

基于图像分割的煤岩割理CT图像各向异性特征

利用工业CT对自然煤岩样进行断层扫描观测。针对煤岩裂隙系统的多尺度、各向异性特征,应用Canny算子图像分割与方向性边缘检测技术,提取煤岩CT图像割理的总体特征、水平方向和垂直方向特征;根据特征图像计算了煤岩样的总体分形维数、孔隙度,各向异性分形维数与孔隙度及其在三维空间中的分布;讨论了分形维数与孔隙度、渗透率之间的关系,并根据煤岩样的分形维数、孔隙度对实际工程岩体的孔隙度和渗透率进行了外推计算。研究表明,煤岩样不同扫描断面的分形维数和孔隙度不同,同一煤样同一断面不同方向的分形维数与孔隙度亦不相同。利用图像分割与边缘检测对工业CT图像进行分析,可以对煤岩的各向异性分形维数与孔隙度在2D与3D空间进行精细描述。      

堆石料缩尺方法的分形特性及缩尺效应研究

采用6种不同缩尺方法将同一条现场级配曲线缩制成不同的数值模拟级配曲线,建立以不同粒径范围内颗粒数为测量数的分形模型,研究了颗粒级配分布的分形特性;基于颗粒流方法,生成与级配曲线相对应的6组数值试样进行双轴压缩试验,研究了缩尺方法对数值试样宏观力学特性及细观力学响应的影响;并讨论了颗粒级配分布的分形特性与数值试样力学特性之间的关系。结果表明：采用不同方法缩尺后数值试样的颗粒级配分布具有分形特性,分维数D为1.463～1.783;随着缩尺方法相似比尺M的增大,数值试样中细颗粒数量增多,粗细颗粒的充填关系得到改善,力学特性逐步提高;颗粒级配分布分维数D与数值试样力学特性指标之间存在较好的线性关系。     

Random fractal characters and length uncertainty of the continental coastline of China

A coastline is a random fractal object in a geographical system whose length is uncertain. To determine the coastline length of a country or a region, the scaling region and fractal dimension of the coastline is first calculated, and then, the length of the coastline is measured using the scale at the lower limit or near the limit of the scaling region. For this study, the scaling region of the continental coastline of China is determined. The box-counting dimension was calculated with ArcGIS software using 33 scales and a map scale of 1:500,000, and the divider dimension calculated by a C language program. Moreover, the reliability of the Chinese coastline length value, which is widely used currently, is discussed in this paper. The results show that the scaling region of the continental coastline of China is from 0.1 to 400 km. In the scaling region, the box-counting dimension and the divider dimension of the coastline are 1.2004 and 1.0929, respectively. According to fractal theory, the divider dimension more accurately represents the irregularity of a coastline. The length of the continental coastline of China is approximately 21,900 km when the measurement scale is 0.1 km; however, the length is 18,214 km when the scale is 0.25 km, and this value approaches the continental length of China (18,400 km) in popular use today. Although the coastline length is shorter than 21,900 km, the length is acceptable because the measurement scale (0.25 km) is close to the lower limit of the scaling region.     

应用多重分形法确定老挝窑房地区水系沉积物异常下限

文章阐述了传统的统计学方法与分形含量-求和法求取异常下限的理论模型及步骤,并利用Q-Q图直观地判定了老挝丰沙里省窑房地区水系沉积物中Au,Cu,Pb,Zn元素在地球化学场中不满足正态分布或对数正态分布,而是满足两个层次的分形分布,从而利用最小二乘法拟合各个层次的直线方程求出分维数D,直线相交处对应的ri为各元素的异常下限。地球化学异常场分维数的大小可代表地球化学过程的强烈程度;分形方法圈定的异常范围较大、较连续,利于圈定隐伏、半隐伏矿体形成的弱异常,在红层建造的地层中效果更为显著。     

New techniques in rock mass classification: application to welded tuffs at the Nevada Yucca Mountain

Summary Many rock mass classification systems exist to assist the engineer in assessing the rock support requirements for underground design. On-going research in this area is directed at attempting to utilize the fractal dimension and the acoustic emission response of the tuffs at the Nevada Yucca Mountain to further aid in rock mass classification. Acoustic emission response is shown to be correlated with the porosity of the sample. Engineering behaviour of the rock varies dramatically with porosity; events and peak amplitude offer a means to distinguish between fracture porosity and pore porosity and consequently the engineering behaviour of the rock. Fractal dimension is used to characterize the roughness of fracture surfaces. Two fractal dimension calculation methods, one based on the semi-variogram for the surface and the other based on the use of dividers, are applied for this purpose. The divider method is shown to resolve deviation from a straight line; the semi-variogram method is shown to identify statistical similarity to various types of noise.Nomenclature D fractal dimension - AE acoustic emission - b b-value determined from log(frequency) against log(amplitude) plots - (h) semi-variogram function - h lag distance for semi-variogram function - H an exponent term related to fractal dimension asD=2 –H     

构造断裂的分形分析

本文介绍了分形理论在构造断裂问题研究中的应用及构造问题中分形维数的计算方法与测试方法,并对当前这一研究工作中的一些问题进行了评论。