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.     

Scaling Behavior of Gravel Surfaces

Roughness of successively developed gravel surfaces in flume experiments is investigated using laser scanned elevation fields. Scaling behavior of these surfaces is studied using structure functions. The results show that all surfaces exhibit good scaling behavior for two scaling regions. Multifractal analysis based on singular measures is then conducted for the two scaling regions, respectively. The relatively smaller scaling region characterizing subgrain scale roughness exhibits evident nonstationarity and intermittency. In contrary, the measures for the larger scaling region characterizing the spatial distribution of grains are more stationary and less intermittent than the measures for the subgrain scaling region.     

Morpho‐sedimentary characteristics of proximal gravel braided river deposits in a Froude‐scaled physical model

A Froude‐scaled physical model of a proximal gravel‐bed braided river was used to connect the river morphological characteristics, and sedimentary processes and forms, to deposit geometry. High resolution continuous three‐dimensional topographic data were acquired from sequential photogrammetric digital elevation models paired with grain‐size surface maps derived from image analysis of textural properties of the surface. From these data, the full three‐dimensional development of the braided river deposit and grain‐size sorting patterns was compiled over an experimental time period of 41 h during which the model river reworked a large portion of the braided channel. The minimum surface of the deposit is developed progressively over time by erosion, migration and avulsion of channels, and by local scour at channel confluences. The maximum surface of the deposit is formed by amalgamation of braid bar surfaces and has less overall relief than the minimum surface. Confluence scour constitutes about 5% of the area of the minimum surface. Migration of individual confluences is limited to distances of the order of the width and length of the confluence, so that confluences do not form laterally extensive deposits and basal surfaces. Maximum and minimum surfaces have very similar grain‐size distributions, and there is no extensive basal coarse layer. Deposit maximum thickness is strongly associated with large channel confluences which occur as deeper areas along the main channel belt and make up a large proportion of the thickest portions of the deposit.     

Statistical properties of sediment bed profiles in alluvial channels

The objective of this study is to investigate in detail the statistical properties of series of bed elevations measured on gravel-bed and sand-bed alluvial channels, in order to identify means of quantifying bed roughness effects on streamflow. The semivariogram is used as the basic statistical method for investigating roughness properties of bed profiles obtained from field work and laboratory experiments. For sand bedforms, the semivariograms include exponential and periodic components from which can be obtained reliable measures of bedform spacing and height, as well as information on the degree of regularity of bedform arrangement. Because of the irregular nature of gravel-bed profiles, the approach in this case uses the semivariogram to investigate fractal properties of series of bed elevations to determine scales of bed roughness associated with grain sizes and small-scale bedforms and to estimate the Hausdorff dimension corresponding to each scale. These superimposed scales of roughness may be responsible for the greater flow resistance generally observed in gravel-bed rivers rather than predicted from the theoretical friction equation.     

Natural rock joint roughness quantification through fractal techniques

Accurate quantification of roughness is important in modeling hydro-mechanical behavior of rock joints. A highly refined variogram technique was used to investigate possible existence of anisotropy in natural rock joint roughness. Investigated natural rock joints showed randomly varying roughness anisotropy with the direction. A scale dependant fractal parameter, K _v, seems to play a prominent role than the fractal dimension, D _r1d, with respect to quantification of roughness of natural rock joints. Because the roughness varies randomly, it is impossible to predict the roughness variation of rock joint surfaces from measurements made in only two perpendicular directions on a particular sample. The parameter D _r1d × K _v seems to capture the overall roughness characteristics of natural rock joints well. The one-dimensional modified divider technique was extended to two dimensions to quantify the two-dimensional roughness of rock joints. The developed technique was validated by applying to a generated fractional Brownian surface with fractal dimension equal to 2.5. It was found that the calculated fractal parameters quantify the rock joint roughness well. A new technique is introduced to study the effect of scale on two-dimensional roughness variability and anisotropy. The roughness anisotropy and variability reduced with increasing scale.     

GIS支持下基于DEM的中等流域的划分——以富县为例

以陕西富县为例,探讨在GIS环境下从数字高程模型(DEM)中提取流域特征并进行中等流域的划分的详细过程,包括DEM的预处理、流水方向的确定、汇流累积量分析、河网的提取和子流域的划分,通过设定不同的汇流累积量的阈值,通过实验,富县地区的阈值设为20 000,得到中等流域的划分结果较为合理。     

Analysing laser‐scanned digital terrain models of gravel bed surfaces: linking morphology to sediment transport processes and hydraulics

The grain‐scale topography of a sediment surface is a key component of a fluvial system, affecting aspects including sediment transport, flow resistance and ecology. However, its effect is hard to quantify because of the need for grain‐scale elevation data from in situ fluvial gravel surfaces which are difficult to collect. The sediment surface properties are, therefore, commonly estimated as a function of the sediment grain‐size distribution; however, because of additional factors, such as grain packing and shape, there is not necessarily a unique relationship between the two. A new methodology has been developed that uses terrestrial laser scanning to collect grain‐scale topographic data from in situ fluvial gravel surfaces, from which digital terrain models are created. This paper investigates methods of analysing such digital terrain models, and possible sedimentological interpretations that can be drawn from the analysis. Eleven digital terrain models from exposed gravel surfaces in two contrasting rivers (the River Feshie and Bury Green Brook) were analysed by calculating: the distribution of surface elevations, semivariograms, surface inclinations, surface slopes and aspects and grain orientation. The distribution of surface elevations and surface slope and aspect analysis were found to be most informative. In the River Feshie, grain‐size was interpreted as being a dominant control on sediment surface structure and gravel imbrication was identified. In Bury Green Brook, the location of the digital terrain models within the riffle–pool sequence was the dominant control on surface structure and grain orientation. Such digital terrain models therefore provide a new approach to measuring and quantifying the topography of fluvial sediment surfaces.     

Quantification of Natural Fracture Surfaces Using Fractal Geometry

The purpose of this paper is to present an extensive evaluation of the methods to calculate the fractal dimension of natural fracture surfaces. Three methods; variogram analysis (VA), power spectral density (PSD), and roughness-length method (RMS) are applied to 2-D surface data (PSD) and 1-D profiles (VA and RMS) extracted from the surface data of 54 mm diameter crystallized limestone samples. Surface topography of the samples is quantified through a newly designed fully automated device. Before the application, self-affinity of the surface roughness and the applicability of these methods are validated using synthetically generated fractal surfaces. Fractal dimension values of the profiles are obtained as between 1 and 1.5 with a few exceptions. VA and RMS methods yield consistent fractal dimensions while the PSD values are lower than those of the other two methods. In terms of practical applicability, the VA is found more convenient than other two methods because there still exists shortcomings with the PSD and RMS methods due to difficulties in the mathematical analysis of the plots whose slopes are used in the computation of fractal dimension. However, it is observed that the data of limited size fracture surfaces are convenient for fractal analysis and the results are promising for further applications if the fracture surface size is restricted like cores recovered from deep boreholes.     

分形理论在水文水资源中的应用

较系统地论述了分形理论在水文水资源中的应用.主要包括:水系河网结构和流域地形地貌及其演变,河床表面形态,降水时空分布,洪水时空变化,径流过程和土壤水下渗等.概括了当前在应用中的特点:活跃思想和方法创新,同时指出值得注意的问题:在大量资料基础上进行深入分析和充分论证.     

Dislocation substructure of mantle-derived olivine as revealed by selective chemical etching and transmission electron microscopy

Cleaved and mechanically polished surfaces of olivine from peridotite xenoliths from San Carlos, Arizona, were chemically etched using the techniques of Wegner and Christie (1974). Dislocation etch pits are produced on all surface orientations and they tend to be preferentially aligned along the traces of subgrain boundaries, which are approximately parallel to (100), (010), and (001). Shallow channels were also produced on (010) surfaces and represent dislocations near the surface that are etched out along their lengths. The dislocation etch channel loops are often concentric, and emanate from (100) subgrain boundaries, which suggests that dislocation sources are in the boundaries. Data on subgrain misorientation and dislocation line orientation and arguments based on subgrain boundary energy minimization are used to characterize the dislocation structures of the subgrain boundaries. (010) subgrain boundaries are of the twist type, composed of networks of [100] and [001] screw dislocations. Both (100) and (001) subgrain boundaries are tilt walls composed of arrays of edge dislocation with Burgers vectors b=[100] and [001], respectively. The inferred slip systems are {001} 〈100〉, {100} 〈001〉, and {010} 〈100〉 in order of diminishing importance. Exploratory transmission electron microscopy is in accord with these identifications. The flow stresses associated with the development of the subgrain structure are estimated from the densities of free dislocations and from the subgrain dimensions. Inferred stresses range from 35 to 75 bars using the free dislocation densities and 20 to 100 bars using the subgrain sizes.     

Fractal properties of multi-order folding as a tool for exploration of low-grade banded iron ores in the Krivoy Rog basin (Ukraine)

The low-grade Palaeoproterozoic stratabound banded iron ores of the Krivoy Rog basin (Ukraine) underwent strong tectonometamorphic deformation into superimposed folds of several orders, with amplitudes from centimetres to hundreds of metres. The across-strike sections of bed surfaces defining the low-grade ore bodies resemble self-similar fractal curves; hence, a fractal geometrical model was developed in order to quantify the complexity and sinuosity of bed contours. Two different methods of measurement (polygonal approximation and two-dimensional grid cell counting) were used for 5–8 different scales. Factual similarity dimension D and other model parameters have been estimated by means of linear regression and compared for both measurement methods. From the fractal model a sinuosity coefficient of contours of the folded bed surfaces K _s and a coefficient of degree of exploration of iron ore bodies K _e were constructed. It is pointed out that parameters of the model can be used for determination of the optimal exploration length scales.     

江西德兴斑岩铜矿田断裂分形研究

本文将分形几何学的原理和方法应用于江西德兴斑岩铜矿田三组断裂系统的二维平面分布特征研究,发现在研究的标度范围不人有统计自相似性,而且不同矿区了区域具有孙现的分维数Ｄ值,矿化岩体产出部位分维明显高于整个矿山分维值,ＮＥ向断裂体系分维值略高于ＷＮＷ向体系分维值,结合断裂产状,时代,穿切关系与岩体侵入部位对比可知,分维值越高,越有利于矿床形成,矿床规模也越大,其中ＮＥ向和ＷＮＷ向断裂体系起主控矿作用。     

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。     

DEM在构造地貌定量分析中的应用与展望

作为一种描述地形起伏特征的数据模型,数字高程模型（Digital Elevation Model,DEM）通过有限的地形高程数据实现对地形曲面的数字化模拟（即地形表面形态的数字化表示）,为研究地表的演化过程提供了数据基础。文章归纳梳理了DEM在基本地形因子、流域地貌特征、古地貌面的重塑、构造地貌发育模式、地貌分类与制图以及地形特征提取算法等领域的应用现状。总的来看,DEM研究对象以陆地为重点,以河流地貌和山地地貌为主要内容;研究过程从早期对地貌形态的定性描述向多种地貌参数的半定量、定量分析转变;研究尺度空间上从某个小流域到整个造山带,时间上从数小时向百万年扩展。构造地貌演化时间序列的不确定性、地貌参数获取的复杂性、地形模型算法的多样化以及DEM生成过程中的误差因素,这些均影响构造地貌定量分析结果的准确度,因此在归纳梳理已有研究成果的同时,对DEM在构造地貌研究领域的应用进行了一些思考。      

丹江口水库库区及周边地区河网形态、地貌特征及构造活动性意义

河流作为构造-气候相互作用最为敏感的地貌单元,记录了丰富的水系演化、构造变形以及气候变化等信息。通过研究河流的形成与演化过程来阐述区域地貌和构造活动特征是构造地貌研究的一个突破点,河流形态的空间变化是阐述河流形成与演化特征最为直观有效的方法。丹江口水库库区地貌特征复杂、地质灾害频发、差异性构造活动较为强烈,是开展构造地貌研究的理想场所。通过对丹江口水库库区及周边地区河流形态特征、地貌特征及构造活动特征等进行综合分析发现,河网分维值空间特征与区域内构造活动性较强的断裂的空间分布高度吻合,构造活动性较强的断裂带及周边地区,河网受到构造活动的影响,发育不成熟,河网分维值出现低值,分维值均小于1.115;构造活动性较弱的断裂带及周边地区,河网发育过程中未受到构造活动的影响,发育较成熟,河网分维值高,分维值均大于1.25;而河网分维值空间变化与地形坡度、平均高程等地貌参数相关性不显著。因此,区域构造活动性特征是河网形态空间变化控制的关键因素。利用河网形态的分维特征量化区域构造活动的强弱及各区域构造活动的差异,对于河流的形成与演化、构造活动性及预测地质灾害的发生等方面的研究都有一定的参考价值。      

区域滑坡空间分布的变维分形特征研究

定量研究滑坡的空间分布特征对研究滑坡灾害的孕育发展具有重要意义。以陕西省宁强县滑坡为例,采用变维分形方法,探讨滑坡在空间分布上的变维分形特征。结果表明：区域滑坡与不同影响因子,即高程、起伏度、坡度、距河流距离、距断裂带距离和地层岩性之间全部呈一阶累积和变维分形分布特征,且分形维数分别是0.511 1、1.135 7、1.290 6、0.903 8、1.189 5、0.979 9。由此可见滑坡对于不同影响因子的敏感度由高到低分别是坡度＞距断裂带距离＞起伏度＞地层岩性＞距河流距离＞高程。上述研究表明用变维分形理论来研究滑坡的空间分布特征是完全可行的,它能够定量地反映滑坡的空间分布与各影响因素之间的敏感性关系,特别是对于GIS空间叠加分析时权重的确定有着重要的意义。     

基于改进投影覆盖法结构面粗糙度尺寸效应分析

研究结构面粗糙度的尺寸效应特征对全面理解结构面的力学性质有重要的意义。基于此,将随机数应用于投影覆盖法中,改进了投影覆盖法计算结构面分维数时三角形单元划分方法;在分辨率相同情况下,应用改进投影覆盖法计算了两个形貌特征不同的结构面（2 m×2 m）尺寸大小不同时的分维数;探讨了用取自不同区域位置相同尺寸结构面的最大分维数差值来判定结构面合理尺寸的方法。研究结果表明:（1）形貌特征不同的结构面尺寸效应特征不尽相同,一般情况下,随尺寸增大,结构面粗糙度先增大而后减小,即在小尺寸时表现为正尺寸效应,而后为负尺寸效应;（2）对于某一特定结构面,不同尺寸范围选取位置不同,其尺寸效应特征也不尽相同;（3）不同区域位置相同尺寸结构面最大分维数差值随尺寸的增大,其总体趋势为先减小而后逐渐趋于稳定。该研究结果是全面认识结构面粗糙度尺寸效应的有益补充,为确定结构面合理尺寸提供一条新途径。     

Fractal analysis of the Oyo River, cave systems, and topography of the Gunungsewu karst area, central Java, Indonesia

Gunungsewu, the southernmost subzone of the Southern Mountains, central Java, is a classic karst terrain, bounded by the Oyo River on the north and the Indian Ocean on the south. In this study, the Oyo River, the main drainage in the area, is divided into 14 segments, and the fractal dimension of each segment is determined by the box-counting method. The 14 segments show various values of fractal dimensions, which are controlled by lithology and geologic structures of the area that is being dissected by the river. Easily eroded lithologies have larger fractal dimensions, and the value changes abruptly when the river crosses faults. Fractal dimensions of six underground rivers in the Bribin, Sodong, Semuluh, Jomblang, Soga, and Sumurup caves, and the surface topography above the caves were also determined. The caves have fractal dimensions that range from 1.043 ±0.01 to 1.08±0.01; the surface topography has fractal dimensions that range from 1.49±0.01 to 1.732±0.01. The fractal dimension of an underground river is proportional to the fractal dimension of the surface topography over the passage. Larger fractal dimensions of underground rivers are associated with smaller flow rates of the rivers. Received, June 1998Revised, March 1999, August 1999, February 2000Accepted, February 2000