Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 179

S 2 h ( i y t )

=ˆ p h (1 −ˆ p h ). (4.19)

For large numbers of sampling locations, confidence intervals for areal frac-
tions can be estimated by assuming a normal or Student distribution, see Eqs. ( 4.6 )
and ( 4.11 ). Problems arise for small samples (say n 20), and when estimating

extreme (very small or very large) areal fractions. In these cases exact confidence
bounds based on the binomial distribution are required. For SI such exact bounds
can be easily calculated. However, for STSI this is not straightforward (Wendell and
Schmee 1996 ). For STSI confidence bounds can be approximated by assuming the
standardized estimated areal fraction has a Student distribution

<

p STSI −

ˆ p

t ( df ) ,

V (

p STSI ) ∼

(4.20)

ˆ

with the degrees of freedom df equal to Eq. ( 4.12 ). The 100(1 − α/ 2)% interval for

the areal fraction is

t ( df )
1

V (

p STSI ± ˆ p STSI ). ˆ (4.21)

− α/ 2

For SY, confidence bounds can be approximated as for STSI.

4.2.2.3 Spatial Cumulative Distribution Function

The SCDF of y can be estimated through repeated application of the indicator tech-
nique described in Section 4.2.2.2. The measured variable y is first transformed to a
series of indicator variables corresponding to a number of increasing threshold con-
centrations. Note that, contrary to Eq. ( 4.14 ) in this case the indicator has a value
of 1 if the concentration is smaller than or equal to the threshold, and a value of 0
else. The areal fractions estimated from these indicators, together with the threshold
concentrations, form an estimate of the SCFD. It is common to use all different con-
centrations in the sample as thresholds. If there are measurements below a detection
limit, then this detection limit is the smallest threshold. If there are several detection
limits, the largest detection limit is used as the first threshold.
Figure 4.4 shows the SCDF's of the Cu, Pb, Co and Ni concentrations in the
topsoils (0-10 cm, brown diamonds) and subsoils (50-100 cm, yellow squares)
in the Netherlands, estimated from the stratified simple random sample depicted
in Fig. 4.4 . The Cu and Pb concentrations in topsoils are higher than in subsoils
due to anthropogenic contamination, whereas Co and Ni have geogenic origin, and
consequently concentrations in topsoils and subsoils are comparable.
Figure 4.5 shows the estimated SCFD of Pb for the peat soils near Mijdrecht.
Six of the 30 sampling locations from three strata (Fig. 4.2 ) could not be sampled
(no permission). The Pb concentrations are much higher than in the Netherlands as
a whole, compare with Fig. 4.4 .