Geoscience Reference
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Fig. 1.19 Ripley's (
1976
) estimator illustrated for region A with events at points
x
and
y
. Radius
of both circles is equal to
kx yk
;
w
xy
¼
(
α
1
+
α
2
)/2
ˀ
;
w
yx
¼ʲ
/2
ˀ
(Source: Agterberg
1994
, Fig. 1)
originally pointed out by Ripley (
1976
),
w
ij
represents the conditional probability
that an event is observed when it is distance
r
ij
away from the
i
-th event. In general,
w
ij
is not equal to
w
ji
as illustrated in Fig.
1.19
.If
is replaced by the observed
intensity
n
/|
A
| where
n
is total number of events in
A
, then Ripley's estimator
for
K
(
r
) is obtained with
ʻ
X
ij
I
i
r
ij
^
Kr
n
2
A
w
1
ðÞ¼
jj
i6¼j
This expression is only valid if
r
is sufficiently small because for large
r
the
weights may become unbounded. The condition that
r
should not exceed the radius
of the smallest circle about an arbitrary point
x
that does not intersect the circum-
ference of
A
. Stoyan et al. (
1987
, p. 125) discuss the problem of bias for large
r
and
give a method by which it can be avoided. Diggle (
1983
) argues that the restriction
on Ripley's original estimator does not present a serious problem in practice
because the dimensions of the region
A
are generally larger than the distances for
which
K
(
r
) is of interest. For example, when
A
is the unit square,
r
should not
exceed 2
0.5
. For larger distances between events, the sampling fluctuations will
increase significantly.
As pointed out before, in many types of geological applications, the region
A
does not have a simple shape. Events (e.g., oil wells or mineral deposits) may
only occur within an environment type bounded in 2-D by discontinuities such as
intrusive contacts, faults, unconformities or facies changes. An algorithm for
polygonal
A
was programmed originally in the SPLANCS package of Rowlinson
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