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     

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。     

A probabilistic procedure to assess the uncertainty of fractal dimensions from measurements

In order to assess the degree of uncertainty of fractal dimensionality, a procedure is given which provides a cumulative probability measure of uncertainty of fractal dimensionality in terms of assessments of range and most likely value. The procedure allows for a variation of the model proposed (i.e., variability of the parameters involved in the fractal relation) and leads to a relevant estimate of the uncertainty. The procedure is simple to apply and useful in providing a statistically valid comparison of results.As an illustration, the procedure was used on the fractal dimensions of faults measured from shear zones of different scale magnitudes. Results suggest that these shear zones are indeed quantitatively similar with a fractal dimension around 1.5. It is also shown that the range of uncertainty of the fractal dimensionality (90% likelihood range) is sensitive to the amount of data used to calculate the fractal dimension of the faults.     

Monte Carlo Studies of Relations between Fractal Dimensions in Monofractal Data Sets

—Within the fractal approach to studying the distribution of seismic event locations, different fractal dimension definitions and estimation algorithms are in use. Although one expects that for the same data set, values of different dimensions will be different, it is usually anticipated that the direction of fractal dimension changes among different data sets will be the same for every fractal dimension.¶Mutual relations between the three most popular fractal dimensions, namely the capacity, cluster and correlation dimensions, have been investigated in the present work. The studies were performed on the Monte Carlo generated data sets. The analysis has shown that dependence of the fractal dimensions on epicenter distribution, and relations among the fractal dimensions, are complex and variable. Neither values nor even inequalities among dimension estimates are preserved when different fractal dimensions are used. The correlation and the capacity dimensions seem to be good tools to trace collinear tendencies of eipicenters while the cluster dimension is more appropriate to studying uniform clustering of points.     

Fractal dimensions of individual flocs and floc populations in streams

The fractal dimension of an individual floc is a measure of the complexity of its external shape. Fractal dimensions can also be used to characterize floc populations, in which case the fractal dimension indicates how the shape of the smaller flocs relates to that of the larger flocs. The objective of this study is to compare the fractal dimensions of floc populations with those of individual flocs, and to evaluate how well both indicate contributions of sediment sources and reflect the nature and extent of flocculation in streams. Suspended solids were collected prior to and during snowmelt at upstream and downstream sites in two southern Ontario streams with contrasting riparian zones. An image analysis system was used to determine area, longest axis and perimeter of flocs. The area–perimeter relationship was used to calculate the fractal dimension, D, that characterizes the floc population. For each sample, the fractal dimension, D_i , of the 28 to 30 largest individual flocs was determined from the perimeter–step‐length relationship. Prior to snowmelt, the mean value of D_i ranged from 1·19 (Cedar Creek, downstream) to 1·22 (Strawberry Creek, upstream and downstream). A comparison of the means using t‐tests indicates that most samples on this day had comparable mean values of D_i . During snowmelt, there was no significant change in the mean value of D_i at the Cedar Creek sites. In contrast, for Strawberry Creek the mean value of D_i at both sites increased significantly, from 1·22 prior to snowmelt to 1·34 during snowmelt. This increase reflects the contribution of sediment‐laden overland flow to the sediment load. At three of the sampling sites, the increase in fractal dimensions was accompanied by a decreases in effective particle size, which can be explained by an increase in bed shear stress. A comparison of fractal dimensions of individual flocs in a sample with the fractal dimensions of the floc populations indicates that both fractal dimensions provide similar information about the temporal changes in sediment source contributions, about the contrasting effectiveness of the riparian buffer zones in the two basins, and about the hydraulic conditions in the streams. Nevertheless, determining the individual fractal dimensions of a set of large flocs in a sample is very time consuming. Using fractal dimensions of floc populations is therefore the preferred method to characterize suspended matter. Copyright © 2000 John Wiley & Sons, Ltd.     

黄庄-高丽营断层的分段性

应用分形几何理论讨论了黄庄－高丽营断层的分段性。用分数维作为定量标志把断层分成了三段（ＳＷ－ＮＥ）;第Ⅰ段和第Ⅱ段以良乡凸起为界,第Ⅱ段和第Ⅲ段以ＮＷ向的南口－孙河断层为界。用地球物理资料、大地测量和断层位移测量资料、地震活动资料等对分段的合理性进行了验证。认为分数维较大、断层活动和地震活动性均较强的第Ⅱ段应为地震危险重点监视区。     

Characterization of Soil Particle Size Distribution with a Fractal Model in the Desertified Regions of Northern China

We constructed an aeolian soil database across arid, semi-arid, and dry sub-humid regions, China. Soil particle size distribution was measured with a laser diffraction technique, and fractal dimensions were calculated. The results showed that: (i) the predominant soil particle size distributed in fine and medium sand classifications, and fractal dimensions covered a wide range from 2.0810 to 2.6351; (ii) through logarithmic transformations, fractal dimensions were significantly positive correlated with clay and silt contents (R² = 0.81 and 0.59, P < 0.01), and significantly negative correlated with sand content (R² = 0.50, P < 0.01); (3) hierarchical cluster analysis divided the plots into three types which were similar to sand dune types indicating desertification degree. In a large spatial scale, fractal dimensions are still sensitive to wind-induced desertification. Therefore, we highly recommend that fractal dimension be used as a reliable and quantitative parameter to monitor soil environment changes in desertified regions. This improved information provides a firm basis for better understanding of desertification processes.     

Fractal analysis of long-range paleoclimatic data: Oxygen isotope record of pacific core V28-239

R/S analysis of the oxygen isotope curve of Pacific core V28-239 yields a fractal dimension of 1.22. This value is considered to characterize global climatic change over the last 2 million years as expressed by changing O¹⁸ ratios and confirms that climatic variations are characterized by long-term persistence. The fractal dimension of 1.22 compares favorably with the approximate fractal dimension of 1.26 for annual precipitation records for nine major cities in the United States. Although the precipitation and oxygen isotope data are measured in different physical units and recorded at different time scales, fractal analysis allows for a mathematical comparison of the two phenomena. Additionally, since the fractal dimensions of the oxygen isotope and precipitation records are similar, it is implied that such fractal dimensions are characteristic of climate change over the spectral range of 10 to 10⁶ years. Given this temperature curves based on fractal parameters of long-term O¹⁸ data could be constructed which would allow examination of characteristics of temperature variation over tens and hundreds of years. Such studies may allow the establishment of limits on natural temperature variation and document the persistence of temperature trends through time. If these trends and limits can be resolved, long-range climatic prediction is feasible.     

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.     

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.     

分形维数计算结果的可靠性

本文以大量分形维数计算结果的实例为基础，着重讨论了影响分维值可靠性的因素，特别指出对于未知动力学系统的自然界中的分形，很难获得真正的分形维数，若研究统计分形维数的变化特征，在采取了某些措施之后，其分维值可比，可用。     

甘肃北山花岗岩裂隙几何学特征研究及岩石质量初探

文中以甘肃北山花岗岩中发育的构造裂隙(主要指节理)为研究对象,通过野外裂隙调查,应用传统的概率统计方法与分形几何学理论,利用Mapinfo,ArcGIS平台进行裂隙几何学参数(方位、长度、密度等)的统计、计算和裂隙网络的空间结构分析,研究花岗岩岩体中裂隙的几何学特征。并以此为基础,对甘肃北山花岗岩岩体质量优劣进行初步评价。结果表明:在10～200cm范围内,裂隙网络是分形的;5个测点裂隙网络的分维值分别是1.636,1.548,1.596,1.724,1.604。分维数D不仅能刻画岩体中结构面发育的数量,而且能反映结构面在岩体中分布的均匀程度和交切方式。因此,可以表征岩体的质量优劣,对岩体质量进行分级。按照分维所划分的岩体质量分级,北山花岗岩属于裂隙较发育、岩体质量等级一般的岩体     

Fractal analysis of 2-D fracture networks of geothermal reservoirs in south-western Turkey

Natural fracture patterns of producing geothermal formations in south-western Turkey are mapped at different scales. The fractal dimensions of different fracture network properties, such as spatial distribution, density, connectivity, orientation, and length are measured by different methods. Analysis of the natural fracture patterns from giga to microscales identifies the descending behavior of box-counting fractal dimension with respect to the scale. It is observed that the fracture networks represent scale-invariant properties, but fractal dimensions might notably differ when the mass dimension is measured applying different methods. Anisotropic nature of fracture networks is also included in the fractal analysis.     

祁连山与龙首山断层的分维特征

本文利用分维几何学方法对祁连山和龙首山断层进行了定量性研究。结果表明,分维数可以表征断层的不均匀性、不规则性和复杂性等,但是,与地震活动无明显的相关关系。分维数的大小与断层分枝数、断层迹线的展布面积有一定关系。     

Characterizing aquatic sediment-oil aggregates using in situ instruments

The effects of emulsified crude oil and salinity (15, 30 per thousand ) on the steady state aggregate volume distributions and fractal dimensions were determined for a range of mean velocity gradients, (G(m) =5-50 s(-1)). Aggregation was performed in a 40-L cylindrical tank with a 4-blade paddle mixer. Three-dimensional fractal dimensions (D3) and volume distributions were determined using a procedure integrating data from an electrozone and an in situ light scattering instrument. Two-dimensional fractal dimensions (D2) and derived volume distributions were determined using a recently developed submersible flow cytometer equipped with a digital camera and image analysis software. For latex beads or emulsified crude oil systems, the above listed instruments yielded consistent size distributions and fractal dimensions (D2=1.92 +/- 0.16, D3=2.94 +/- 0.12). Mean volume aggregate diameters determined using the FlowCAM were consistently larger that those determined using the LISST-100 or Coulter Multisizer due to aggregate orientations during measurements. With increasing G(m) values, all colloidal aggregates showed increasing D3 values due to reduced aggregate length. Because of the compactness of all the aggregates (D3 >2), D2 values remained constant at 2. Neither salinity nor sediment type significantly affected D2 values calculated for sediment-crude oil aggregates. However, clay-oil aggregates showed higher D3 values than clay aggregates. This suggests that colloidal oil and mixing shear are the more dominant factors influencing aggregate morphology in nearshore waters. Overall, the data suggests that the analysis methods provide consistent size distribution results. However, because of the shear and salinity of coastal waters, resulting aggregates are too compact to estimate their D3 values using image analysis alone.     

地形等高线的分形特征及其动力学含义

以天山地区不同地貌类型的地形等高线为研究对象 ,用尺度法探讨了它们的分形特征 .结果表明 :在 0 .1至 2 0 0km的标度范围内 ,不同地貌类型等高线具有不同的分形特点 ,而且与标度有关 .陆地表面地形等高线分维值与地貌侵蚀作用和不均匀堆积作用的强度密切相关 ,可作为地貌学研究中一个重要参数     

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。     

分形几何的数学基础及其在地震预报中的应用

本文就地震预报中常用的分形几何中的几个数学概念做了简要的介绍，包括测度，豪斯道夫维数、计盒维数等，并介绍了有关维数的性质和估计方法。作者试图通过了这些数学概念的简要介绍，使读者对分维有更准确的理解。     

海原断层系的分形研究

本文根据Okubo等人测量圣安德烈斯断层系所用的复盖维数法,对海原断层系进行了分形测量,求出海原断层系的整体维数D_0=1.137,其景泰段D_0=1.109,海原段D_0=1.182。计算中未得到邵家水段和李使堡段的分维数。此外,文中还着重探讨了断层几何与地震活动性及其力学环境的关系,进一步证明了自相似断层的几何复杂性与地震活动密切相关,剪切断裂带具较低分维数(1.1—1.3),而在张性环境中形成的断裂体系具较高的分维数(1.5—1.6)。最后本文讨论了断层迹线图等因素对分形测量精度的影响。     

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.