Geoscience Reference
In-Depth Information
where
ar
.A/ denotes the area of the region A
I
assuming a uniform density of
individuals along the full region A under study seems to be rather unrealistic;
instead, one may better split the region A into smaller and more homogeneous
subregions A
j
(e.g. polygons), give a weight !
j
to each A
j
; and assume a uniform
distribution f
j
for each A
j
W
X
r
f.a/
D
!
j
f
j
.a/;
(6.14)
jD1
where each f
j
is uniform on A
j
; and thus its expression is given in (
6.13
).
Let us particularize (
6.14
) for the all-or-nothing case in which the covering
function is given by (
6.4
), and each c
i
is given by (
6.2
), i.e., c.a;
x
/ takes the value
1 if at least one facility i is at a distance from a below the threshold R
i
; and takes
the value 0 otherwise. Then, for any
x
, C.
x
/ takes the form
C.
x
/
D
Z
c.a;
x
/f.a/
da
Z
X
r
1
ar
.A
j
/
D
!
j
c.a;
x
/
da
(6.15)
A
j
jD1
X
r
1
ar
.A
j
/
ar
.A
j
\[
iD1
B
i
.x
i
//;
D
!
j
jD1
where, for each i
D
1;:::;p, B
i
.x
i
/ gives the set of points covered by facility i;i.e.,
the disc centered at x
i
and radius R
i
: Hence, the problem is reduced to calculating
areas of intersections of discs B
i
.x
i
/ with the subregions A
j
: Such calculation,
although cumbersome in general, are supported in GIS, see Kim and Murray (
2008
),
Murray et al. (
2009
), Tong and Murray (
2009
).
Needless to say, the density f does not need to be piecewise constant, and one
can take, for instance, a mixture of bivariate gaussians, f.a/
D
P
jD1
!
j
f
j
.a/;
where each f
j
is a bivariate gaussian density centered at some
u
j
and with
covariance matrix S
j
;
1
2
p
j
S
j
j
e
1
2
.a
u
j
/
>
S
j
.a
u
j
/
;
f
j
.a/
D
(6.16)
or, more generally, a radial basis function (RBF) density,
f
j
.a/
D
g
j
.
k
a
u
j
k
/
(6.17)
for some decreasing function g
j
; so that the density is the highest at some knot point
u
j
and decreasing in all directions.
