基于引力模型的城市空间互相关和功率谱分析——引力模型的理论证明、函数推广及应用实例

Derivation and generalization of the urban gravitational model using fractal idea with an application to the spatial cross-correlation between Beijing and Tianjin

空间相互作用是先于城市体系而存在的重要概念 ,引力模型是描述空间相互作用的基本函数之一 ,但引力模型的理论基础不明确而且实际应用有局限。本文首先从城市地理系统的广义分形假设出发 ,推导出引力模型的幂函数形式 ,使其从一个经验模型上升为理论模型 ;进而引入时变函数和时滞参数将引力模型推广为更为一般和更加实用的形式 ,为发展城市引力过程的空间互相关分析和功率谱分析方法奠定了理论基础。借助 194 9～ 1998年 5 0年的人口演化数据 ,以北京 -天津的空间相互作用为实例 ,对基于城市引力关系的空间作用进行了相关分析和波谱分析 ,从而提供了城市网络空间相互作用广义引力分析的典型范例。

The urban gravitational model taking the form of the law of universal gravitation has been regarded as an experiential model for lack of theoretical basis so far, in contrast with Wilson's spatial interaction model based on entropy-maximization.It is of significance to make a theoretical proof as well as an advance that will lead us to its underlying rationale.In the paper contributing to spatial interaction of cities, the gravitational model, I = GM i M j r - b , is derived out of the general function, I = A * f ( x 1 , x 2 ,..., x n ), on the assumption of general fractality of urban system, and then the model is transformed into the new expression such as F ( r , τ )= GR ( τ ) r - b = Gr -b F - 1 [ S ( ω )] by introducing time-varying function, M i = f i ( t ), M j = f j ( t ), and time-lag parameter τ , while taking into account the cross-correlation function R ( t )=∫ f i ( t ) f j ( t + τ )d t , where S ( ω ) is energy-spectral density, and F ( r , τ ) the integral of the gravitation, I .Thus the analytical method of power spectrum for urban interaction is developed based on the new gravitational model.The new model was applied to characterize the process of spatial interaction between two cities,Beijing and Tianjin,from 1949 to 1998, and the experiential analysis can be divided into three steps.First, the spatial cross-correlation was described by means of the primitive time series of urban population, and some kinds of regularity were revealed through the patterns of cross-correlation coefficient and the corresponding power spectrum.Secondly, after taking the trend component parts out of the time series using fit equations, the spatial cross-correlation and power spectrum analyses were made based on the data with some periodic change to bring to light more hidden regularity on urban spatial network.Going a step further, we can make more deep analyses by taking out the periodic component parts, this step fails to be taken as space is limited.Inspite of this, the anti-intuitional rules have been discovered conforming to which the cities interact spatially.This paper will initiate the power spectrum analysis of urban interactions based on the gravitational model by giving an entire typical example, and more studies will be expected in this field in the future.