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parameters in order to gain more knowledge and
reduce the uncertainty by means of a more ac-
curate sampling.
•
Concentrations:
Z=values of the continu-
ous variable (pollutant concentration in
groundwater);
•
Locations:
u
α
is one of n locations, with
α=1,2,…,n;
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ASSeSSMent: geoStAtIStIcAl
APProAch
•
Concentrations at locations:
z(u
α
) is the
value of Z at location uα (pollutant concen-
tration in groundwater at location u
α
).
Geostatistical approaches have proved to be very
useful in the case of environmental variables where
the prediction uncertainty is required to support
decision-making regarding remediation and the
eventuality to proceed with further sampling in
some areas.
In a risk assessment procedure a fundamental
step is that of the characterization of the source of
contamination both as far as its spatial definition
and its Representative Concentration (CRS) are
concerned. Many environmental investigations
are aimed at making important decisions, as for
example declaring an area potentially contami-
nated. A commonly used procedure consists in
circumscribing all the locations in which the
contamination exceeds a given value retained
tolerable. The application of Indicator kriging
permits to determine and to map the value of the
probability of exceeding a given threshold value.
This helps the responsible of the decisional process
to delimit the vulnerable areas on the basis of the
knowledge of the uncertainty associated to the
examined phenomenon. The Indicator Kriging
lends itself well also to the analysis of qualita-
tive variables, allowing integration with data of
quantitative nature.
Spatial patterns are usually described using the
experimental semivariogram γ(h) which measures
the average dissimilarity between data separated
by a vector h. It is computed as half the average
squared difference between the components of
data pairs (Goovaerts, 1999):
h
1
N
()
2
()
=
é
ë
ê
()
-
( )
ù
û
ú
å
g
h
z
u
z
u
h
()
a
a
2
N
h
(1)
a
=
1
where N(h) is the number of data pairs within
a given class of distance and direction. The semi-
variogram is theoretically a function that has to
begin from 0 because coincident samples have
the maximum similarity; anyway it shows a dis-
continuity at origin due to the nugget variance,
that is to say the random component not spatially
correlated, connected to the error measurement.
In environmental decision making processes
there is often the necessity to know the locations
where a variable exceeds a certain limit. The
characterization of the spatial distribution of z-
values above or below a given threshold value
z
k
requires a prior coding of each observation
z(u
α
) as an indicator datum i(u
α
; z
k
) defined as
(Goovaerts, 1999):
Modeling the Spatial Pattern
of a continuous Attribute
ì
í
ï
ï
î
ï
ï
1 if z(u)
³
z
I(u;z)
=
a
k
a
k
0 otherwise
(2)
Consider the problem of describing the spatial
pattern of a continuous attribute z, say a pollutant
concentration in groundwater. The information
available consists of a set of n observations {z(u
α
),
α=1,2,….n, that can be explained in such a way:
Indicator semivariograms can then be com-
puted by substituting indicator data i(u
α
; z
k
) for
z-data z(u
α
) in Eq. (1) (Goovaerts, 1999):
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