Environmental Engineering Reference
In-Depth Information
N
V
K
Y
0
i
−
Y
0
i
,
1
N
J
MeanV
=
(4.38)
i
=
1
where
V
K
Y
0
i
−
Y
0
i
is the kriging variance at interpolation node
i
. Alternatively,
the maximum of the kriging variances at the grid nodes may be minimized. To
find the sampling pattern with minimum value for Eq. (
4.38
), van Groenigen and
Stein (
1998
) and van Groenigen et al. (
1999
) proposed the spatial simulated anneal-
ing algorithm. The kriging variance holds for the variogram used in kriging; if we
change the variogram, this will affect the kriging variance. Consequently, the opti-
mized sampling pattern also depends on the chosen variogram, although the effect
may be rather small.
The problem is that many contaminant concentrations often have distributions
that deviate from normal, for instance, the right tail of the distribution can be longer
(positive skew or right-skewed). In this case kriging the untransformed concentra-
tions is suboptimal, and more accurate predictions can be obtained by transforming
the concentrations such that the transformed concentrations are approximately
normal distributed. For this reason concentrations with positive skew are often
log-transformed before interpolation by kriging. Using a variogram for the log-
transformed concentrations, we may optimize the sampling pattern by minimizing
Eq. (
4.38
). However, it must be stressed that the optimized sampling pattern
obtained with the variogram of log-transformed concentrations generally will be
suboptimal for the mean variance of the interpolation errors on the original scale.
Brus et al. (
2007
) found that the difference in the mean kriging variances,
Eq. (
4.38
), of a spatial coverage sample designed with the k-means clustering
algorithm and of a geostatistical sample designed by directly minimizing the mean
kriging variance with spatial simulated annealing, was small. So, in practice, a use-
ful procedure might be to optimize the sampling pattern with the k-means algorithm
first. If this is unsatisfactory for one of the reasons mentioned in Section
4.4.1.2
,we
might proceed with optimizing the sampling pattern with spatial simulated anneal-
ing, either using a distance-criterion such as Eq. (
4.36
) or, if a prior variogram is
available, a variance-criterion such as Eq. (
4.38
).
A different situation is when one or more maps of covariates are available, think
for instance of a map depicting clay content or a digital elevation model. It is well
known that contaminant concentrations are often correlated with the clay content or,
in floodplains and terraces along rivers, with altitude. In this case interpolation by
universal kriging can be advantageous. The optimal sampling pattern for universal
kriging generally differs considerably from the optimal pattern for ordinary kriging.
The optimal sampling pattern for universal kriging will contain several locations
with extreme values for the covariates, enhancing the estimation of the spatial trend
parameters (Brus and Heuvelink
2007
).
4.4.1.4 Supplementary Sample for Estimating the Variogram
In practice the sample data are used both for estimating the variogram, and for spa-
tial interpolation. In the previous section on geostatistical sampling, it was assumed
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