河南省城镇体系空间结构的多分形特征及其与水系分布的关系探讨

Multifractal Measures Based on Man-Land Relationships of the Spatial Structure of the Urban System in Henan

基于河南省城镇体系空间结构的多分形性质及其与水系的分维关系的实证探讨发现如下地理规律:①在较大标度范围内,河南省城市体系的空间结构为简单分形,但在较小标度范围内,系统已经具有明确的多重分形特征;②多分维谱在参数q≈-4处出现标度间断,从而城市体系的空间结构发生对称破缺。③城市体系空间结构的分维小于水系空间结构的分维。由此证明:第一,多分形是由单分形演化而来,城镇体系的多分形结构是基于地球表面的分形支体由测度集中区向测度疏散区渐近发育的。第二,城市体系的分形发育与水系的分维结构具有一定的数理关系,城市体系的分维理当小于水系维数。

By means of the theory of multifractals, this paper is devoted to studying the spatial structure of urban systems, taking man-land relationships into consideration. Taking the system of cities and towns in Henan Province, China, as a example, and using the box counting method and μ-weight formulae, we calculate the values of the Lipschitz-Hlder exponent α(q), the fractal dimension of the support of singularities f(α), the sequence of mass exponent τ(q), and the dimensions of fractal measures Dq of the urban systems in the studied area. The data processing reveals that the scaling range reflecting log-linearity of complex fractals is narrower than that of simple fractals. This denotes that fractal systems are some kinds of evolving systems, and multifractals usually come from common fractals at least where geographical phenomena are concerned. The computation results show that the spatial structure of the Henan urban system has multifractality to some extent, with a scaling breakdown in the f(α) curve as well as the spectrum of fractal dimensions Dq when the moment order q=-4. That is to say, q_c=-3 perhaps is a critical value for q, the multifractals come on well only when q∈[-3 ,∞], as for q≤-4, the multifractal measures are abnormal: the f(α) curve and the D_q function fail to converge, which maybe implies a sort of phase transition from a rural to urban settlement system during the course of regional urbanization. Moreover, this is another evidence that multifractals generate by evolution from simple fractals. A discovery is made that the fractal dimension of spatial structure of urban systems is less than that of river systems in the same studied region, which maybe means that urban systems as fractals must be included by hydrographical nets, i.e. water systems. We can develop what is called 'inclusion principle' about man-land relations that is as important as the 'matching principle' about urban systems. Both the matching principle and inclusion principle will be supposed to be the basic principles of fractal urban geography, which will play a significant role in urban plan and geographical space optimization in the future.