城镇体系等级结构的分形维数及其测算方法*

文章探讨了城镇体系等级结构的分形研究方法。首先,讨论了区域城镇规模分布的Zipf模型,并通过分形退化分析将其应用范围加以拓广,从而与非分形研究接口;第二,引进了分形结构因子,以此开创了城镇体系等级结构的FSF分析;第三,提出了表征城镇体系等级差异的差异度概念和度量方法。文章给出实例说明了各种方法的应用,并比较了三种方法的异同。

FRACTAL DIMENSIONS OF HIERARCHICAL STRUCTURE OF URBAN SYSTEMS AND THE METHODS OF THEIR DETERMINATION

Three parameters were presented in the paper to characterize the hierarchical structure, especially size distribution of cities, of an urban system, including Zipf's dimension, fractal structure factor (FSF), and difference degree.1.Zipf's dimension.Zipf's law is very familiar to urban geographers, it is equivalent to Pareto's distribution and can be expressed mathematically as P_(k)=P₁K^-q, where q is sometimes called Zipf's dimension, which is actually the reciprocal of fractal dimension, namely, q=1/D.2.FSF.The parameter is put forwards by two Chinese geo scientists and has been introduced to the studies of urban geography by the authors of the paper.FSF can be defined as I_(r)=I₀+D_I lnr, where I_(r) is information capacity of size distribution of cities corresponding to a certain scale (r, when r=1, I_(r)=I₀), and D_I is what is called FSF.3.Difference degree.The parameter is given by the authors of the paper, it can be defined as follows: C=1-I/I_max, where I=- P_ilnP_i, I_max=lnN, and P_iP_(k)/ P_(k),P_(k) is the population of the kth city of an urban system (k=1, 2, …, N).Difference degree C can be linked with fractal dimension D by means of Zipf's formula under some conditions.The geographical meanings of the three parameters were illuminated, the methods of using them were illustrated, in particular, the degenerational forms of Zipf's model were discussed so as to connect fractal studies with non-fractal studies of urban systems, and finally, a preliminary comparison was made between them.