Statistical Sampling Strategies for Survey of Soil Contamination - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 188

with n h and n hu the numbers of sampling locations in stratum h and in the inter-
section of stratum h and block u , respectively. For estimators of the sampling
variance of the synthetic and regression estimators, I refer to de Gruijter et al. ( 2006 ,
pp. 145, 146).
4.3.2 Model-Based Approach
If the number of sampling locations with a laboratory analysis of the contaminant
concentration is large enough to estimate a reliable variogram, a model-based sam-
pling strategy for estimating the block-means is recommendable. In this case there
is no need for random sampling, and in general it will be suboptimal. Purposive
sampling, such that the sampling locations cover the study area optimally, is a good
option in this case. This can be achieved by purposive grid sampling or by spatial
coverage sampling, see Section 4.4.1 . In purposive grid sampling the grid is not
placed randomly on the study area, but located such that the grid nodes optimally
cover the study area. For square grids in a square area this is achieved by center-
ing the grid. Regular grids can be suboptimal for irregular shaped areas, and when
existing sample data at point locations are available. In these situations a spatial
coverage, or in case we have prior measurements at point locations, a spatial infill
sample can be a good alternative. They can be designed by forming compact geo-
graphical strata by the k-means algorithm, as described in Section 4.2.1. However,
in this case the sampling locations are not selected randomly from the strata, but the
centroids of the strata are used as sampling locations. For more details on sampling
patterns for spatial interpolation, see Section 4.4.1.
In a model-based sampling strategy, the spatial means of the blocks are esti-
mated by block kriging, see Section 4.4.2 for a short introduction to kriging and for
references to literature on this geostatistical estimation technique.
4.3.3 Required Number of Sampling Locations
For a design-based approach, constraints can be imposed on the sampling vari-
ances of the estimated block-means, on the probability of the error in the estimated
block-means, or on the error rates in testing hypothesis on the block-means. If prior
estimates of the spatial variances within the blocks, or within the intersections of the
blocks and the strata, are available, we can estimate the required number of sampling
locations for each block with the procedures described in Section 4.2.5.
To decide on the required number of sampling locations of a model-based sam-
pling strategy, we must choose a variogram before sampling starts. Let us consider
first estimating the means of square blocks from a square grid sample. There is no
simple equation that relates the grid spacing to the variance of the estimated block
means (block kriging variance). What can be done, however, is to calculate the block
kriging variance (of the block centred on the midpoints of the grid cells) for a range
of grid spacings, plot the block kriging variances against the grid spacing, and use
this plot inversely to determine, given a constraint on the variance, the maximum