Geoscience Reference
In-Depth Information
Since the focus is on the mean density throughout each ring, there is a suppression
or smoothing out of such peaks.
2
Obviously, in a regression involving a
two-parameter function such as Eq. (
6.2
) any secondary peak is necessarily absent.
6.4
The Regional Boundary as a Point
In this section and the next, attention is focused on the possibility of using regional
density functions to establish the boundary between two adjoining regions. We
consider two regions
A
and
B
, with
A
0
and
B
0
representing their respective metro-
politan areas,
A
0
having the larger population. It is our contention that a point on the
boundary between the two regions will be at the location where their respective
regional density functions intersect. In other words, at the boundary of two regions
the density is the same for both regions, reflecting comparable levels of interaction
with
A
0
and
B
0
.
Attention in this section is focused on the relatively simple case, where the
boundary is represented by a single point. In the following example
T
, the distance
between
A
0
and
B
0
, is 300 miles, and the single-point boundary occurs at location
x*
along the line
A
0
B
0
. We retain the above-mentioned Bogue generalization that the
slope
n
across regions increases with increasing regional populations and therefore
with higher values of
N
. This is reflected in the parameter values for the density
functions of regions
A
and
B
, which are as follows: central densities (
N
A
¼
13,200;
N
B
¼
1,900) and slopes (
a
¼
1.15;
b
¼
0.9). The distance variables for regions
A
and
B
are expressed as
x
A
and
x
B
.
The boundary as a point is derived with the aid of Fig.
6.2
. Here density is
measured on the two logarithmically-scaled vertical axes, the left axis referring to
region
A
and the right axis to region
B
. Each of the two horizontal axes measures the
distance from metropolitan areas
A
0
or
B
0
. In order to obtain a solution, neither
distance axis is scaled logarithmically, as was the case in Fig.
6.1
for the single
region. The curves thus both decrease with distance at a decreasing rate, a feature of
the inverse power function discussed in the preceding section. The two curves
intersect at location
x*
, which is 200 miles from metropolitan area
A
0
and 100 miles
from metropolitan area
B
0
. The “reach” of metropolitan area
A
0
is thus greater than
that of metropolitan area
B
0
, so that
x
*
x
*
where
x
h
is the halfway point
between
A
0
and
B
0
.
Further discussion of the point boundary, as defined by the
equality of densities, is found in Appendix 6.1.
>
x
h
>
2
Such a tendency was also reflected in a study of major metropolitan areas by Clark (
1951
,
pp. 491-493). This used concentric rings “generally drawn at each mile radius.” Out of the
35 cases examined, only five exhibited secondary peaks (due to the presence of sub-centers),
and in each case this was relatively slight.
