  Urban population density always follows the exponential distribution and can
be described with Clark's model. Because of this, the spatial distribution of
urban population used to be regarded as non-fractal pattern. However, Clark's
model differs from the exponential function in mathematics because that urban
population is distributed on the fractal support of landform and land-use form.
By using mathematical transform and empirical evidence, we argue that there are
self-affine scaling relations and local power laws behind the exponential
distribution of urban density. The scale parameter of Clark's model indicating
the characteristic radius of cities is not a real constant, but depends on the
urban field we defined. So the exponential model suggests local fractal
structure with two kinds of fractal parameters. The parameters can be used to
characterize urban space filling, spatial correlation, self-affine properties,
and self-organized evolution. The case study of the city of Hangzhou, China, is
employed to verify the theoretical inference. Based on the empirical analysis,
a three-ring model of cities is presented and a city is conceptually divided
into three layers from core to periphery. The scaling region and non-scaling
region appear alternately in the city. This model may be helpful for future
urban studies and city planning.

Physics and Society