Environmental Engineering Reference
In-Depth Information
The concentrations at the nodes of the interpolation grid can be estimated by point
kriging. There are several versions of kriging, building on slightly different models
of the spatial variation, see Goovaerts (
1997
). The most common version is “ordi-
nary kriging” (OK). In OK it is assumed that the expectation of the concentration
is the same everywhere (“stationarity in the mean”). In practice not all sample data
are used to estimate the concentration at a given node, but only the sample data
in some predefined neighbourhood. This implies that the stationarity assumption is
relaxed to the often more realistic assumption of a constant mean within neighbour-
hoods. In kriging, the concentration at an interpolation node,
Y
0
, is estimated as a
weighted average of the concentrations measured at the sampling locations within
the neighbourhood
n
Y
0
=
1
λ
i
Y
i
,
(4.39)
i
=
where
Y
i
is the measured concentration at the
i
th sampling location, and
λ
i
is the
weight attached to this location. The weights should be related to the strength of
correlation of the concentrations at the sampling location and the interpolation node.
The stronger this autocorrelation (“auto” refers to the fact that we consider the same
variable at both locations), the larger the weight must be. So if we have a model
for this autocorrelation, then we can use this model to find the optimal weights.
Usually, not an autocorrelation model is used, but a variogram, which is a model of
the dissimilarity of the concentrations at two locations as a function of the distance
between the two locations, see Fig.
4.6
for an example. The smaller the semivari-
ance of the concentrations at the interpolation node and the sampling location, the
larger the weight must be. Further, if two sampling locations are very close, the
weight attached to these two locations should not be twice the weight attached to
a single, isolated sampling location at the same distance of the interpolation node.
This explains that in computing the kriging weights, besides the semivariances of
the
n
pairs of interpolation node and sampling location, also the semivariances of the
n
2 pairs that can be formed with the
n
sampling points are used. For OK,
the optimal weights, i.e., the weights that lead to the estimator with minimum error
variance (Best Linear Unbiased Estimator), can be found by solving the following
(
n
·
(
n
−
1)
/
+
1) equations
j
=
1
λ
j
γ
n
(
h
ij
)
+
ν
=
γ
(
h
i
0
),
i
=
1,
...
,
n
,
(4.40)
i
=
1
λ
i
=
n
1,
where
(
h
ij
) is the semivariance of the
i
th and
j
th sampling location separated by
distance
h
ij
,
γ
(
h
i
0
) the semivariance of the
i
th sampling location and the interpo-
lation node separated by distance
h
i
0
, and
γ
an extra parameter to be estimated,
referred to as the Lagrange multiplier. This Lagrange multiplier is needed as the
error variance is minimized under the constraint that the kriging weights sum to 1,
ν
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