Geoscience Reference
In-Depth Information
conditions there is usually a variation of density within each ring, a feature that will
be considered in a later section. The value of
N
M
(1) is the central density or the
density at one unit of distance from the center of the region, while
n
is the value of
the slope of the density function.
The inverse power function may be contrasted with the negative exponential
function, the form of the density function approximating the pattern of distance-
density decay within the metropolitan area (Clark
1951
). This may be expressed as
¼
x
0
Mx
ðÞ
¼
C
exp
ð
bx
Þ
ð
C
>
0
b
>
0
0
x
Þ
ð
6
:
3
Þ
;
;
or
ln
Mx
ðÞ
¼
ln
C
bx
ð
6
:
4
Þ
where
Ci
s the density at the center of the metropolitan area and
b
is the slope, with
x
0
representing the limit of the metropolitan area, as defined by some minimum
density.
6.3.1 Parameters of the Regional Density Function
Returning to the inverse power function of Eq. (
6.2
), the graph of this function is
shown as curve
M
in Fig.
6.1
, where
R
refers to an unspecified distance from the
center, and
M
(
R
) is the density at this distance. Within the metropolitan part of the
region the inverse power function tends to exaggerate density levels, so that the
central-density parameter,
N
in Eq. (
6.2
), should be seen as an extrapolated value.
However,
N
may be taken as an indication of the population of the metropolitan
area which, in turn, may be regarded as an indication of the population of the
region, though not of its territorial extent. In other words, across regions the value
of
N
increases with regional population. This regularity is similar to the situation
across metropolitan areas, where the value of the central density,
C
in Eq. (
6.4
),
increases with metropolitan-area population (Clark
1951
).
In the case of the slope parameter,
n
in Eq. (
6.2
), Bogue indicated at several
points that across regions this would increase with the population of the metro-
politan area, and thus with the population of the region. The following general
explanation for this variation in slope across regions was offered: “the more rapid
decline of settlement [density] with distance in the case of the larger units [regions
with greater populations and thus with higher values of
N
] results from a much
higher level of land occupancy in the inner zones and only a moderately higher
level of land occupancy in the outer zones” (Bogue
1950
, p. 33). We consider this
apparent regularity in Sects.
6.4
and
6.5
. It will be noted that across regions the
increase in the slope of the density function with population stands in marked
contrast to the situation across metropolitan areas, where the slope,
b
in Eq. (
6.4
),
declines with population (Clark
1951
; Weiss
1961
).
