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N
()
h
The estimation implies therefore the determi-
nation of the weights in such a way as to satisfy
the following two conditions:
1
2
( )
=
()
( )
-+
é
ë
ê
(
)
ù
û
ú
å
g
h;z
iu z
;
i
u
hz
;
1
k
a
k
a
k
2
N
h
a
=
1
(3)
1.
the condition of unbiasedness of the
estimator:
The indicator variogram value 2γ
1
(h; z
k
) mea-
sures how often two z-values separated by a vector
h are on opposite sides of the threshold value z
k
.
The greater is γ
1
(h; z
k
) the less connected in space
are the small or large values.
*
Ez
[()()]
u
-
z
(6)
Where z*(u) and z(u) are, respectively, the
estimated and the true value in the interpolated
point u; the (6) lead to the following relation:
Kriging
The variogram, expressing the statistical depen-
dency of two points as function of their distance is
used by kriging, a sort of improved form of inverse
distance weighting, in order to optimize prediction.
Kriging consists essentially in a weighted moving
average for the estimation of a not sampled point
z(u) from the nearby points z(u
α
). In the case of
strictly stationary random function, with global
mean
m
, the linear estimation kriging, referred to
as Simple Kriging, assumes the form:
å
l
i
=
1
(7)
That ensures that kriging is an exact interpo-
lator, in the sense that in the sampled points it
returns the measured value.
2.
the condition of minimum variance, that
implies the minimization of the estimation
variance.
An important property of kriging is that the
estimation variance depends only on the semivar-
iogram model and on the configuration of the data
locations in relation to the interpolated point and
not on the observed values themselves.
Another property of kriging is represented by
the fact that the interpolated value can be used with
a degree of confidence, because an error term is
calculated together with the estimation.
Anyway it needs to be pointed out that kriging
is optimal and unbiased only on the condition that
the model is correct. This can represent both one
of the strengths of the procedure of kriging but
also a weakness, because error variances can be
seriously affected by the choice of the model.
In the case of randomly sparse sampling it is
quite probable that estimation variances will be
large and additional sampling will be necessary.
Nugget or random variance, though it does not
influence estimation, sets a lower limit to estima-
é
ù
n
n
=
()
+-
å
å
ê
ê
ú
ú
*
z( )
u
» zu
1
»
m
a
a
a
ë
û
(4)
a
=
1
a
=
1
Where the symbol * indicates the calculated
value; n is the number of measured values z(u
α
)
that take part in the estimation of the interpolated
value, λ
α
are the weights and m the global mean,
supposed known and constant in all the examined
area.
The most commonly used form of kriging is
Ordinary Kriging, that takes into account the local
fluctuations of the mean, limiting therefore the
condition of stationariety to a neighbourhood of
a point u. Differently from Simple Kriging, the
mean is not supposed known and the algorithm
of interpolation assumes the form:
n
()
=
()
=
å
a
*
z
u
» z
u
a
(5)
a
1
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