豫北地区城镇体系空间结构的多分形研究

以河南省北部为研究区，对城镇体系空间结构的多分形性质进行了实证研究，证明城镇体系是地球表面分形支体上发育的多标度分形系统。计算结果表明，豫北地区城镇体系的多分维D_q和奇异谱f(a)在参数q≈-1处发生了标度间断，多分形测度只在q∈0, +∞)范围内发育，当q≤1时，标度紊乱，f(a)函数和D_q谱均不收敛，从而证明：多分形是由单分形演化而来，城镇体系的多分形结构是由测度集中区向测度疏散区不断发展、逐渐形成的。另外，从人地关系、系统演化以及测算方法等角度对有关问题进行了初步探讨。

A Study of Multifractal Measures of the Spatial Structureof the Urban System in Central Plains

The study is made of the spatial structure of the systems of cities and towns in Central Plains, mainly including North Henan, China, using the theory of multifractals. By means of the box counting method and μ-weight formulae, we calculate the values of the Lipschitz-Hölder 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 D_q of the urban systems in the studied area. The resultant values show that there is a scaling breakdown in the f(α) curve as well as the spectrum of fractal dimensions D_q when the moment order q≈-1, where q_c=-1 perhaps is a critical value for q, i.e., the multifractals come on well when q∈0, ∞], as for q≤-1, the multifractal measures are abnormal (dysplasia or hypolasia): the f(α) curve and the D_q function cannot converge, which maybe implies a sort of phase transition from a rural to urban settlement system during the course of regional urbanization. This research demonstrates that the spatial structure of urban systems can be characterized with multifractal geometry, and moreover, wavelet transformation can be used to analyse the multifractal structure of urban systems.