Exploring the Fractal Parameters of Urban Growth and Form with Wave-Spectrum Analysis

Table 1

The numerical relationships between different fractal dimensions, scaling exponents, and autocorrelation coefficients.

Radial dimension (𝐷𝑓)	Profile dimension (𝐷𝑠)	Spectral exponent (β)	Hurst exponent (H)	Autocorrelation coefficient(𝐶Δ)	Correlation function [C(r)]
1.00	(2.50)	0.0	(−0.50)	−0.750	𝑟^−1.00
1.05	(2.45)	0.1	(−0.45)	−0.732	𝑟^−0.95
1.25	(2.25)	0.5	(−0.25)	−0.646	𝑟^−0.75
1.50	2.00	1.0	0.00	−0.500	𝑟^−0.50
1.70	1.80	1.4	0.20	−0.340	𝑟^−0.30
1.75	1.75	1.5	0.25	−0.293	𝑟^−0.25
1.95	1.55	1.9	0.45	−0.067	𝑟^−0.05
2.00	1.50	2.0	0.50	0.000	1.00
(2.25)	1.25	2.5	0.75	0.414	𝑟^0.25
(2.50)	1.00	3.0	1.00	1.000	𝑟^0.50

(1) The autocorrelation coefficient (𝐶_Δ) is defined at the micro level and associated with 𝐷_𝑠 while the correlation function C(r) is defined at the macro level and associated with 𝐷_𝑓.(2) The values in the parentheses are meaningless because they go beyond the valid range.