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soft data the approach is aimed at assessing the
probability that the value of z at any unsampled
area u is greater than a given critical threshold
z k . Indicating with F the conditional cumulative
distribution function (ccdf) of the variable Z, ac-
cording with the (9) it results:
local soft indicator data y
l (u 0 ; l k ) assuming
values within the interval [0,1].
The next step has been the codification in
indicator variable of the different types of infor-
mation available. For z(u α ), being a continuous
variable, the critical legislative threshold value of
10 μg/l has been used. The soft variable has been
calibrated by estimating the marginal distribution
of the samples with concentrations higher than 10
μg/l relatively to each type of area, as proposed
by Goovaerts & Journel (1995). The marginal
proportions of exceeding the critical value of 10
μg/l, relatively to each homogeneous area are
reported in Tab 1.
As illustrated before, the random function
approach amounts to modeling the two unknown
values as realizations of two spatially dependent
random variables. According to this approach,
the local hard and soft indicator data i(u 0 ;z k ) and
y l (u 0 ; l k ) are interpreted as realizations of two
correlated random functions, I(u 0 ;z k ) and Y l (u 0 ;
l k ) that can be processed together by cokriging,
where I(u 0 ;z k ) is the primary variable and Y l (u 0 ;
l k ) the secondary variable.
F (u; z k (n 1 +n 2 ))=Prob{Z(u) ≥ z k (n 1 +n 2 ) } (25)
where the notation (n 1 +n 2 ) expresses the con-
ditioning to n 1 hard data {z(uα); α=1,2,…, n 1 } and
n 2 soft data { l(u α ); α=1,2,…, n 2 } retained in the
neighbourhood of u.
In this approach the probability distribution is
regarded, according with the (10), as the condi-
tional expectation of the indicator random variable
I(u;z k ), given the information set (n 1 +n 2 ):
F(u; z k (n 1 +n 2 )) = E{ I(u; z k ) (n 1 +n 2 ) }
(26)
ì
í
ï ï
î ï ï
1 if Z(u)
³
z
0
k
with I(u;z)
=
(27)
0 otherwise
0k
The ccdf F(u; z k n 1 +n 2 )) can be estimated by
(co)kriging the indicator I(u;z k ) using indicator
transform of hard and soft data.
The indicator approach requires a preliminary
coding of the hard and soft data into local hard
and soft “prior” probabilities:
Cokriging of Hard and Soft Data
Figure 3a shows the spatial distribution of arsenic
concentration in groundwater obtained by means
of Ordinary Kriging.
It is possible to observe that the concentra-
tion of arsenic is higher in the middle and in the
southern zone, whereas the values decrease below
the critical value at north and north-east.
The arsenic distribution seems to be concen-
trated in specific areas and for this reason it has
been decided to investigate further introducing
the soft information, represented by land use in-
formation and evaluating the probability of risk.
On the basis of the available information about
the activities that characterized each section of
Prob {Z(u) ≥ z k | local information at u}
(28)
The term “local prior” means that the prob-
ability in Eq. 28 originates from hard and soft
information at location u, prior to any updating
based on neighbouring data. The final target of this
approach is updating this local prior probability
in the posterior probability (Eq. 26). Thus, the
prior information can take one of the following
forms:
local hard information data i(u
0 ;z k ), with
binary indicators defined by Eq. 27;
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