• 论文 •

### 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

1. 1. 北京大学城市与环境学系,北京100871;
2. 东北师范大学地理系,长春130024
• 收稿日期:2002-07-02 修回日期:2002-09-25 出版日期:2002-12-15 发布日期:2002-12-15
• 作者简介:陈彦光(1965-),河南罗山人,副教授。从事地理分形和地理系统的空间复杂性研究。E-mail: ygchen@water.pku.edu.cn。
• 基金资助:

国家自然科学基金资助项目(40071035)

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

CHEN Yan-guang1, LIU Ji-sheng2

1. 1. Department of Geography, Peking University, Beijing 100871, China;
2. Department of Geography, Northeast Normal University, Changchun 130024, China
• Received:2002-07-02 Revised:2002-09-25 Online:2002-12-15 Published:2002-12-15

Abstract:

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.