Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 192

proposed to minimize the mean of the squared distances of the grid nodes to their
nearest sampling location (Mean Squared Shortest Distance)

D ij ,

N

1
N

J MSSD = min

j

(4.36)

i = 1

where n is the total number of nodes of the interpolation grid, and D ij is the distance
between the i th grid node and the j th sampling location. This distance measure can
be minimized by the simple and fast k-means algorithm. The same algorithm was
used in Section 4.2.1.2 to construct compact geographical strata for stratified simple
random sampling. In STSI, one or more sampling locations are selected randomly
from each cluster of nodes (stratum), whereas here for each cluster the mean x -
and mean y coordinate is calculated, and these centroids are used as sampling loca-
tions. Whereas in random sampling we may want to have strata of equal surface area
(clusters with equal numbers of nodes), so that the sampling design becomes self-
weighting, here this constraint should not be used, as it may lead to samples with
suboptimal spatial coverage. Existing sampling locations can easily be accommo-
dated in the k-means algorithm, by using them as fixed centroids. Figure 4.8 shows
an example.
Alternatively, spatial coverage or spatial infill samples can be designed by the
spatial simulated annealing algorithm, as proposed by van Groenigen and Stein
( 1998 ). Optimization with spatial simulated annealing requires more skills com-
pared to k-means. There are several parameters in the annealing algorithm that must
be chosen by the user and that affect the final sampling pattern. A big advantage
of spatial simulated annealing is that it is very flexible with regard to the optimiza-
tion criterion. For instance, the minimum squared distances in Eq. ( 4.36 ) can be
weighted to prioritize certain sub-areas, so that the sampling density in these sub-
areas is relatively high. This leads to the Weighted Mean Squared Shortest Distance
criterion

D ij ,

N

1
N

J WMSSD = w i min

j

(4.37)

i = 1

with w i the weight attached to node i . van Groenigen et al. ( 2000 ) used a similar cri-
terion, Weighted Mean Shortest Distance (note that the distances are not squared), to
design a spatial infill sample in an area of ca 30 ha in the old harbour of Rotterdam.
Harbour activities are increasingly shifted towards other locations, giving space for
house and office building. Existing measurements at 201 locations showed that the
concentrations of three heavy metals (Pb, Cu and Zn) and of two carbohydrates
(mineral oil and PAH's) frequently exceeded the legal threshold concentrations. To
map these contaminants, van Groenigen et al. ( 2000 ) designed a first additional
sample of 80 locations. A priority map for sampling was made, based on expected
contamination and urgency of remediation (Fig. 4.9 ). The weights attached to the
four priority classes were 1.0, 1.5, 2.0 and 3.0. Figure 4.10 shows the pattern of the
spatial infill sample. 46 of the 80 locations are located in the most urgent priority