Environmental Engineering Reference
In-Depth Information
standard deviation of the estimated average log(Pb) concentrations for blocks of
50
50 m as a function of the grid-spacing. The plot shows that if, for instance,
the standard deviation must be smaller than 0.06 mg kg 1 , then the maximum grid
spacing is 158 m.
This procedure can also be used to obtain a rough estimate of the required num-
ber of sampling locations for spatial coverage sampling, by computing n
×
A
d 2 , with
A the surface area of the study region, and d the calculated maximum spacing of
a square grid. The sampling pattern, i.e. the x and y coordinates of the n sampling
locations, is then optimized by k-means clustering or spatial simulated annealing
(Section 4.4.1.2 ). Finally, we may also search for the sampling pattern of n loca-
tions with minimum value for the block kriging variance, averaged over all blocks
(Section 4.4.1.3 ).
=
4.3.3.1 Bayesian Data-Worth Analysis
In Sections 4.2.5 and 4.3.3 the required number of sampling locations is based on
a constraint on the uncertainty about the global or local mean. The uncertainty is
expressed in terms of variance (sampling variance or block kriging variance), proba-
bility of errors in the estimated mean, or probability of decision errors (error rates in
statistical testing of hypotheses). The alternative, especially appropriate for sequen-
tial sampling, is to base the total number of sampling locations on a cost-benefit
analysis. In this approach, the number of new sampling locations, i.e. the number of
locations in the next batch, is calculated with the help of a decision model consist-
ing of an objective function for each decision alternative, for instance remediation
or no remediation. The objective function includes a risk term, i.e., the extra costs
that would be incurred in the event of failure (no remediation of contaminated soil,
remediation of clean soil). This risk term is the product of a probability of failure and
the costs of failure. By collecting additional sample data, the probability of failure
can be reduced. As long as the reduction in the risk outweighs the costs of sam-
pling, sampling is continued. I refer to Freeze et al. ( 1992 ); Ramsey et al. ( 2002 );
Back ( 2007 ); Norberg and Rosén ( 2006 ) for more details and applications to survey
of contaminated soils.
4.4 Mapping Concentrations at Point Locations
This section is about mapping soil contamination at high spatial resolutions. In the
previous section we considered the situation where we want to decide on soil reme-
diation of delineated blocks. In that situation it is natural to take these blocks as
estimation units. If the remediation units are not delineated before sampling, but
will be delineated after the survey on the basis of the map depicting the contaminant
concentration, then in general we would like to have maps depicting mean con-
centrations of spatial units much smaller than the remediation units. For this aim,
a model-based approach is the best (and only) option. This implies that sampling
locations need not be selected randomly but typically are selected purposively, and
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