  Urban form has been empirically demonstrated to be of scaling invariance and
can be described with fractal geometry. However, the rational range of fractal
dimension value and the relationships between various fractal indicators of
cities are not yet revealed in theory. By mathematical deduction and
transformation (e.g. Fourier transform), I find that scaling analysis, spectral
analysis, and spatial correlation analysis are all associated with fractal
concepts and can be integrated into a new approach to fractal analysis of
cities. This method can be termed 3S analyses of urban form. Using the 3S
analysis, I derived a set of fractal parameter equations, by which different
fractal parameters of cities can be linked up with one another. Each fractal
parameter has its own reasonable extent of values. According to the fractal
parameter equations, the intersection of the rational ranges of different
fractal parameters suggests the proper scale of the fractal dimension of urban
patterns, which varies from 1.5 to 2. The fractal dimension equations based on
the 3S analysis and the numerical relationships between different fractal
parameters are useful for geographers to understand urban evolution and
potentially helpful for future city planning.
