Advances in Earth Science ›› 2006, Vol. 21 ›› Issue (1): 1-9.

• Articles •

### Power Spectra Analyses of Spatial Auto-Correlations of Urban Density: An Application to the Hangzhou Metropolis

Chen Yanguang 1, Liu Jisheng 2

1. 1.Department of Geography, Peking University, Beijing  100871,China;2.College of Urban and Environment Science, Northeast Normal University, Changchun  130024,China
• Received:2004-10-18 Revised:2005-09-28 Online:2006-01-15 Published:2006-01-15

Chen Yanguang, Liu Jisheng. Power Spectra Analyses of Spatial Auto-Correlations of Urban Density: An Application to the Hangzhou Metropolis[J]. Advances in Earth Science, 2006, 21(1): 1-9.

The standard negative exponential distance-decay relationship of urban population density, namely the Clark empirical law, ρ(r)=ρ0exp(-br), is proved to be a correlation function reflecting the relation between the center of the city and the point at a given distance. Therefore the Fourier transformation of the urban density should give an index/exponent of power spectra such as β≈2 theoretically. However, when the fast Fourier transformation (FFT) is applied to the urban densities of the Hangzhou metropolis in 1964, 1982, 1990 and 2000 (according to Census time), it turns out to be that the β values vary from 1.44 to 1.80, not often approximating to 2. If the exponential function with power,ρ(r)=ρ0exp(-brσ), is employed to fit the urban density data of Hangzhou instead of Clark’s model, the results are the restraint parameter σ values vary from 0.45 to 0.78, not often approximating to 1. A semi-log relation between σ and β values can be expressed as β≈2+0.7 lnσ . Going a further step, the fractal dimension of self-similar curves, D, and Hurst exponent, H, can be given by means of the formulae β=5-2D and D=2-H. The conclusions can be drawn from the mathematical transformations and computations as follows: First, Urban density has a long-distance dependence, but it is passive/negative and tends to become weak with the lapse of time; Second, the action of urban population in space is localized as a city grows. Third, since the Clark model can be derived using entropy-maximizing method, the calculated results imply that the so-called entropy-maximization is only a developing tendency instead of a realistic state. Last but not least, the idea of action-at-a-distance has been introduced into the various urban simulations based on cellular automata (CA) and all that, but the localization of urban population activity means that the long-range effect should be reflected and revised in the light of amplitude-spectra analyses of urban density equation as an auto-correlation function.

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