Statistics - Field Sampling: Principles and Practices in Environmental Analysis - page 166

the line becomes flat with a slope of 0 or the y axis has a maximum value. The variance at
0 distance is called the nugget. The distance from 0 to where the slope becomes 0 is
called the range. In this case the variance at the point at which the line has a slope of 0 is
made up of the nugget and an additional variance, which is still called the sill. It is
important to remember that many of the components being sampled for in the
environment occur naturally at a low level. This natural low level and its variance will
lead to the occurrence of a nugget in the variogram of this component.
The third model is the linear model (Figure 6.8), which is simply a straight line with no
sill. A fourth model (Figure 6.9) is one that is curved between the 0 variance and 0
distance to a distance at which the line again has a slope of 0. The range is from 0 to the
place at which the slope becomes 0, and this is the sill. Typically one calculates the
variance and plots it versus the distance, and the curve is fitted to one of the appropriate
models.
When using data from field sampling the component variogram may not look like any
of those shown in Figures 6.6 through 6.9. There are a number of reasons for this. If the
concentration of the component goes to 0, there will be no variation, and so the variance
will also go to 0. This would be the case where there is a spill of a component that is not
commonly found in the environment. At some distance the concentration will be 0 and so
will the variance. In this case the variance will increase for some distance and at some
further distance fall back to 0. There are other situations that produce variograms
significantly different from those shown. In all these cases the graphed variogram is
compared to these idealized variograms. The one it fits best is the one used in carrying
out calculations.
To carry out a calculation using kriging the distance from the unknown point to all of
the known points must be known. The distances between all of the known points are also
needed, and thus must be calculated. Two tables are produced, one of which contains the
coordinates of each point and a table of distances between all known points. These
distances are used in the calculation of both a covariance function and a variogram, and
are the basis of the final weighting factors. The variogram is then fitted to a model
variogram.
Commercially available software packages are available that fit variograms to a model
variogram semiautomatically. However it is important for the researcher to know what is
being done, because a number of interpolation methods will be available to choose from.
Kriging can be chosen as one of the interpolation methods. (See Chapter 7 on modeling
and Appendix A for software sources.)
Kriging methods are used extensively in producing surface and subsurface maps of
contamination, thus this methodology will be associated with a wide variety of
commercial modeling programs. For these programs actual calculation is not necessary. It
will be beneficial to know something about the methods in interpreting the results of this
modeling, however. Some understanding of kriging will also help in understanding the
steps a commercial program is asking you to carry out.
6.12.3. Kriging Map
The values for a characteristic can be plotted on a map of the area sampled. The maps