Environmental Engineering Reference
In-Depth Information
probability that the hot spot exists to obtain the probability that the hot spot exists
and is detected by the sample.
In some situations, information on land use, or a walkover survey of visual or
organoleptic indicators of high concentrations can be used to subdivide the site into
sub-areas with different probabilities of containing the hot spot. The grid spacing
can then be adapted to these a priori probabilities as follows. For all sub-areas, there
must be an equal probability that a hot spot exists when none is detected by the
sample.
T
his probability is referred to as the a posteriori probability and denoted
by
P
(
A
B
). Bayes' formula can now be used to calculate from the a priori and a
posteriori probabilities for each sub-area the probability of not hitting the hot spot
if it exists, the consumer's risk
|
β
is
P
(
A
)
P
(
A
)
1
P
(
A
1
−
β
=
1
.
(4.42)
B
)
−
|
Hence, when
P(A)
differs between subregions, and given a constant
P
(
A
|
B
) for all
sub-areas, for instance 0.05,
differs between sub-areas, and this leads to different
grid spacings. Sub-areas with large a priori probabilities will be given smaller grid
spacings than sub-areas with small a priori probabilities. An alternative to purposive
grid sampling is to optimize the sampling pattern by minimizing the sum of the a
priori probabilities outside the zones of coverage of the sampling locations (Tucker
et al.
1996
).
β
4.5.1.1 Adding Sampling Locations to the Grid
If none of the grid data exceed the threshold concentration, one may want to add
new sampling locations to become more certain about the presence or absence of
hot spots. A practical method for selecting new sampling locations is to transform
the data so that the univariate distribution is approximately Gaussian, then krige the
transformed data, and select the
n
locations with the smallest values for
y
t
−
Y
0
V
OK
{
Y
0
−
ζ
0
=
,
(4.43)
Y
0
}
where
Y
0
is the estimated value at the new sampling location obtained by ordinary
kriging, and
V
OK
{
Z
0
−
is the ordinary kriging variance (Watson and Barnes
1995
). Equation (
4.43
) shows that, depending on the threshold concentration, loca-
tions are selected either near sampling locations with a large estimated concentration
(
Y
0
is large), or in the empty space, where
V
OK
{
Y
0
−
Z
0
}
Y
0
}
is large.
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