Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 181

Fig. 4.5 (Särndal et al. 1992 ). The 95% confidence interval for the cumulative fre-
quency has been computed with the Student distribution, approximating the degrees
of freedom by Eq. ( 4.12 ).

4.2.3 Using Ancillary Information in Estimation

Ancillary information such as maps with covariates either can be used at the sam-
pling stage or at the estimation stage of a sampling strategy. An example of use at
the sampling stage is stratified random sampling. This section describes how ancil-
lary information can be used at the estimation stage. The formulas (estimators) in
Section 4.2.2 do not make use of such ancillary information.

4.2.3.1 Post-Stratification Estimator

If the study area can be split up into sub-areas that have less variation within them
than in the area as a whole, the efficiency can be increased by using these sub-
areas either at the sampling stage (stratified sampling) or at the estimation stage.
In the latter case, for each sampling location the sub-area (in sampling terminology
referred to as a group) must be determined. For SI, the mean (areal fraction) can
then be estimated by the post-stratification estimator

G

¯

y pos = a g ¯ y s g , (4.22)

g = 1

where a g is the relative area of sub-area g , and y s g is the sample mean of sub-area g .

In the case of a stratified simple random sample, for which one wants to use a
second grouping at the estimation stage, the mean can be estimated by

¯

L g

G G

A gh
A g ¯

¯

a g ¯ y pos = y g = a g y s gh , (4.23)

g = 1 g = 1 h = 1

where L g is the number of strata in group g , A g is the estimated area of group g ,
and A gh and y s gh are the estimated surface area and the sample mean of group g in

stratum h , respectively. Note that the relative sizes a g must be known. Also note that
Eq. ( 4.23 ) uses the ratio of the estimated areas A gh and A g . This is because this gives
more precise estimates than the ratio of the true areas.

¯

4.2.3.2 Regression-Estimator

If quantitative ancillary information is available and is known everywhere in the
study-area, for instance from remote sensing or from a digital terrain model, then
the spatial mean (areal fraction) can be estimated by the regression estimator