Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 178

As mentioned above, estimation of the sampling variance of ¯ y SY is cumbersome. A

simple, often applied procedure is to calculate the sampling variance as if the sample
is a simple random sample, i.e. the variance is estimated by Eq. ( 4.4 ). In general this
procedure over-estimates the sampling variance, so that we are on the safe side.
Alternatively, if a variogram can be estimated from the random grid sample, then
this variogram can be used to estimate the sampling variance, see de Gruijter et al.
( 2006 , p. 131) for details.
For SY, confidence intervals can be approximated by Eq. ( 4.6 ).

4.2.2.2 Areal Fraction with Concentrations Exceeding Threshold
Concentration

The above formulas for estimating means can also be used to estimate the fraction
(proportion) of the area with concentrations exceeding a given threshold concen-
tration. The measured concentrations at the sampling locations j n are

transformed to a 0/1 indicator variable, with value 1 if the concentration exceeds
the threshold and 0 otherwise

= 1 ···

1if y j >

y t

i j , y t = (4.14)

0if y j ≤ y t

where y t is the threshold concentration. The above formulas are then simply applied
to this indicator variable. For SI and SY this boils down to computing the proportion
of sampling locations with concentrations exceeding the threshold

n

1
n

p SI =ˆ ˆ p SY = i j , y t . (4.15)

j = 1

For STSI the areal fraction is estimated as a weighted mean of the sample
proportions per stratum, using the relative areas of the strata as weights.
For SI the sampling variance of the estimated areal fraction is estimated by

S 2 ( i y t )
n

V ˆ

p SI =

, (4.16)

− 1

where S 2 ( i y t ) is the estimated spatial variance of the indicator

S 2 ( i y t )

=ˆ p SI (1 −ˆ p SI ). (4.17)

For STSI the sampling variance of the estimated fraction is estimated by

L

a h S 2 h ( i y t )

V (

p STSI ) ˆ = , (4.18)

n h − 1

h = 1

where S 2 h ( i y t ) is the estimated spatial variance of the indicator within a stratum