【作者】 周侗；

【导师】 龙毅；

【作者基本信息】 南京师范大学 ， 地图学与地理信息系统， 2007， 硕士

【摘要】 美籍法国数学家B．B．Mandelbrot创立的分形几何学(Fractal geometry)，为在形态、分布、结构上具有非规则的自然对象定量化分析提供了一种新的思路和有力工具，这一理论起源于对海岸线与河流长度量测的研究，因此，可以说从它诞生之时起就和地学，特别是地貌学研究关系紧密，同时推动了分形地貌学的发展。目前分形几何理论在地貌学中的研究主要集中在地貌边缘线(海岸线、湖泊轮廓)的分维规律变化、流域地貌中河流平面形态及河网的分维值变化规律研究、基于盒维数与分数布朗模型等的自然地貌起伏度与粗糙度的分形描述、一些特殊地貌类型区域(喀斯特地貌、沙漠地貌、黄土地貌等)的分形特征与地貌成因的关系以及分形地貌生成技术等方面。在这些研究中，往往将研究对象视为一个整体，进而在无标度区间内进行分维估值计算，探讨对象自身或对象之间分维值所反映出的规律，很少涉及对象内部分形特征差异的研究，而这种内部特征差异往往能够更好地反映地貌形态的复杂性与地貌成因及发育过程的关系。要实现对局部分维的空间差异分析，依靠传统的分形方法存在着一定的困难，需要通过对分数维的扩展以获取更多的细节信息。2002年首次提出的元分维概念就是在此基础上，针对局部分维变化所开展的研究内容之一，经过在一系列等高线簇和单线河流中的初步试验，其结果表明该方法具有较好的分析特性。本文以分形几何理论和元分维模型作为理论基础，将元分维分析方法引入到地貌特征的数字地形分析之中，提出了基于DEM的元分维概念模型和基本理论框架，并采用VC++6.0程序设计语言，设计开发了DEM元分维模型的构建与分析模块，并利用该模块探讨了元分维用于地形复杂性变化的描述机理。本文有针对性地选取了陕北黄土高原若干典型地貌区域作为实验样区，将计算得到的元分维指数作为一种新的地形因子，将其应用于黄土高原地区地貌特征研究中。实验结论如下：首先，元分维指数可以用于地形复杂性变化情况的定量描述，地形变化越剧烈的地区，元分维值越大，但小于3；反之当地形平坦时，则元分维值越小，接近于常数2；其次，通过实验反映出元分维指数在陕北黄土高原地区的宏观分布规律和黄土发育阶段有较好的对应关系；最后，将元分维指数作为分析指标，初步进行了黄土地貌类型单元的自动分区实验，效果良好。本文的研究说明DEM元分维模型是对分形分析方法及其应用的一种扩展，同时也为数字地形分析提供了非线性分析方法的尝试。更多还原

【Abstract】 Mandelbrot was the first person who discovered the self-similarity in complex phenomena by analyzing the length of Britain coastline, which led to the establishment of fractal geometry, so there was a close relationship between the fractal geometry and cartography. Fractal Dimension can be used to describe the degree of spatial complexity of the map objects as a quantitative index that reflects the ability of occupying the space. Fractal methods have had some applications in the field of cartography and GIS, such as the calculation of fractal dimension value of river and coastline, the analysis of urban structure, map generalization etc. However, all of these have been limited to the hypothesis that geography objects have relatively strict fractal character, namely self-similarity. But now some researches have shown that not all natural geographical phenomena possess simple linear self-similarity; rather, many depend on spatial or scale features. Further studies showed that when single geographical object or objects set was cross-affected by multiple factors, they would tend to show non-uniformity on spatial shape and distribution. In fact, different character inside of the object reflects the relation between the landform causes, the growth courses and the terrain complexity. Therefore a special attention must be paid to such a variable character when we try to apply the fractal theory to solve problems in the field of geography, and it is difficult to research this inside difference through the conversional fractal dimension methods. Then in 2002 a new method named Meta fractal dimension model is put forward by professor Long, and it is extensional of the fractal geometry methods.In this paper, based on the fractal analysis method, Meta fractal dimension method was applied into the digital terrain analysis through the DEM data. The concept model and the theory framework of the Meta Fractal Model were designed in the paper and the analysis software was also developed on the platform visual studio 6.0. It was proved that this method can describe the change rule of terrain complexity through the experiments. And the Meta fractal dimension index obtained by the calculation based on the Meta fractal model was applied into the landform research in northern Shaixi province. After many experiments, several conclusions were drawn in this paper. Firstly, the Meta fractal dimension index can quantificationally describe the terrain complexity, the higher of the value, more obvious of the terrain change. But the value will be less than 3 and when the terrain surface is flat, the value will be 2. It has a well corresponding relation between the growth courses of Loess landform and the distribution rule reflected by the Meta fractal dimension index through the multiple samples experiments. The research in the paper is the extension of the fractal analysis theory and its application, and also provide and new landform factor for digital terrain analysis.更多还原