CHAPTER 2: The Semi-Variogram
OTHER VARIABLES
It has been said time and again that geostatistics -- Kriging and so on -- can
be applied equally well to other variables which are spatially or temporally
distributed. This book has been more or less devoted to mining applications,
because this is still the major field. However, many other variables can be
handled, and I should like to give one or two examples here. Even in mining
applications, grade or economic value of the mineral is not always the sole
variable of interest. In many deposits the ‘thickness’ of the deposit is as
important as the grade, and in many sedimentary deposits this factor is far
more important. In the Cornish tin example described above, the width of the
vein is fully as important a variable as the cassiterite content. Both variables
are required to assess the economic viability of the lode or portions of it.
Figure 2.19 shows the overall experimental semi-variogram calculated for the
nine levels, 6-14. To this semi-variogram I fitted a model which consisted of:
a small nugget effect, which was slightly surprising; one spherical component
with a range of influence of about 30ft and another with a range of 150ft.
Fig 2.19. Experimental
semi-variogram constructed on the lode widths in the cassiterite vein.
As an example of other types of spatially
distributed data which might be considered, Fig 2.20 shows an experimental
semi-variogram which was produced during a study of the rainfall
characteristics and runoff in a catchment area in the Pennines in England.
Fig 2.20.
Experimental semi-variogram constructed on the measured rainfall at rain gauge
sites.
The data observations are the recorded
monthly rainfall figures at rain gauges scattered over the catchment area. The
semi-variogram has been constructed without regard to the direction
of the
pair of samples. That is, the author has assumed that his variable shows the
same continuity down the long axis (direction of flow of the major river) as it
does across the valley. The erroneous nature of this assumption is immediately
apparent when given the information that the catchment area measures about 30km
across the valley (short axis). The marked discontinuity in the experimental
curve suggests that there is a definite difference between the two major
directions. Semi-variograms ought to be constructed for at least two different
directions to check for this. The second conclusion which can be drawn from
this experimental semi-variogram is that if the same shape is shown by the new
individual strong trend is in evidence which must be taken into consideration.
When considering the nature of rainfall it does seem sensible to expect
different amounts of rain to fall on the tops of mountains than lower down in
the valleys. This is a good example of when the ‘trend’ cannot be ignored in
the geostatistical estimation procedure.
And now for a completely different type of
application we can take a time series example, rather than one which is
spatially distributed. A series of readings have been taken at the same site in
a large river, of various different variables of interest. This is a
‘one-dimensional’ situation, in which the dimension is time rather than space.
Instead of distance between samples, we now have time between samples, so that
the horizontal axis of the experimental semi-variogram will now read ‘time
between observations’. The estimation procedure will be used to predict values
of these variables forward into the future, or to fill in gaps in the records
caused by machine failure. Figure 2.21 shows two experimental semi-variograms
calculated in one case for the temperature of the water, and in the other for
the amount of suspended solids contained in the water. The latter looks like an
ideal case for a spherical type of model, with a suggestion of a ‘trend’ at the
weekly scale i.e., fairly homogeneous within any specified week, but varying in
level from week to week. The experimental semi-variogram for the temperature
shows a perfect daily cycle in temperature, with a little drift coming in after
3 or 4 days.
Fig 2.21. Experimental semi-variograms
calculated on water quality variables measured over time.