CHAPTER 5: Kriging
SIMULATED IRON ORE
To round out this chapter, let us return
to the simulated iron ore deposit mentioned at the end of Chapter 4. Two sets
of samples had been taken. The 50 ‘random’ samples were shown in Fig. 4.14 and
the regular grid in Fig. 4.17.
The regular grid actually comprises 41
samples. As before, the 400-metre-square area was divided into 50-metre-square
blocks. However, this time the block values were estimated by the method of
kriging. The range of influence of the semi-variogram model for this deposit
was 100m, so all samples within this distance of a block were included in its
estimation. The results are shown in Fig. 5.5, and once again the upper Figure
in each block is the estimated value, whilst the lower is the kriging standard
deviation, or standard error. For comparison, Fig. 5.6 shows the kriging
solution for the case where the area is divided into 100-metre-square blocks.
Notice that the kriging standard deviations in all cases are much lower than
for the 50-metre blocks.
|
Fig. 5.5. Simulated
iron ore deposit --- block averages kriged from the set of random samples and
the corresponding kriging standard deviations --- 50 metre blocks. |
Fig. 5.6. Simulated
iron ore deposit --- block averages kriged from the set of random samples and
the corresponding kriging standard deviations --- 100 metre blocks. |
This once again bears out the principle that
it is ‘easier’ to estimate large areas rather than small ones. Figure 5.7 shows
the estimated values for the blocks when using the sample values from the
regular grid. The kriging standard deviation for all of these blocks is 2.4%Fe.
This is not markedly different from the extension standard deviation of 2.5%Fe.
Perhaps our conclusion here must be that with a regular grid of this particular
size, the arithmetic mean of the two ‘corner’ samples is as good an estimator
as a weighted average of all samples within 100m of the block. The exterior
samples would seem to be superfluous in the circumstances. This conclusion does
not hold for the irregular sampling, for which some great improvements in
accuracy result from the application of kriging.
Fig.
5.7. Simulated iron ore deposit --- block averages kriged from the set of
regular samples --- 50 metre blocks.
The usual requirement in ore reserve
estimation is the production of block values. However, in many other possible
applications of kriging --- such as geochemistry or hydrology --- the
estimation desired is in the form of ‘point’ values, or a contour map of the
variable of interest. Kriging can be used to produce the close grid of values
necessary to the plotting of contour maps. In fact, the procedure is very much
easier than ‘area’ estimation, since all the ‘average semi-variogram values’
reduce to simple values of the semi-variogram model itself. Since all of the
observations are made at specified points, the left hand side of the kriging
system is ‘point-with-point’ semi-variogram values. Since the value to be
estimated is also at a point, the right hand side is also ‘point-with-point’
semi-variogram values. Figure 5.8 shows the contour map produced using the 50
randomly chosen samples. The blacked-out area at the top of the map is outside
the range of influence of any sample, and hence cannot be estimated.
|
Fig.
5.8. Simulated iron ore deposit --- contour map kriged from random samples. |
Fig.
5.9. Simulated iron ore deposit --- kriged standard deviation map for the
kriged contour map from random samples. |
One of the advantages of kriging as an interpolation
technique is that every estimate is accompanied by a corresponding kriging
standard deviation. Thus, for any contour map of values, a companion map of
‘reliability’ can be produced. This is shown in Fig. 5.9. The location of the
sample points can easily be seen by the concentration of the low value contours
in Fig. 5.9 --- the 1%Fe and 2%Fe contours. The highest possible contour value
would be 5Ï2=7.07%Fe, which is the boundary
around the blacked out area. This corresponds to trying to estimate the value
at a point which is just on the range of influence away from the nearest
sample.
The ‘unreliable’ areas are quite clearly
outlined by the 5%Fe contour (the sample standard deviation). A standard error
which is larger than the original sample standard deviation denotes a rather
unreliable prediction.
|
Fig.
5.10. Simulated iron ore deposit --- contour map kriged from regular samples. |
Fig.
5.11. Simulated iron ore deposit --- map of kriging standard deviations for
map kriged from regular samples. |
Figure 5.10 shows the interpolated contour map using the regular
samples, and Fig. 5.11 the corresponding map of the kriged standard deviation.
Note that, even though only 41 samples are available, and even though these are
on a very large grid (71m), the highest contour value in Fig. 5.11 is only
4%Fe.
An additional
advantage of kriging as an estimation technique is that the maps and/or
calculations of the ‘standard errors’ can be produced without actually taking the samples. For
example, if infill drilling were proposed on the regular grid, it is fairly
obvious from Fig. 5.11 where the new samples should be taken. If a decision was
taken to reduce the grid to 50m, i.e. put a hole in the centre of each 4%Fe
contour, a complete new map of the resulting standard error could be drawn
before setting foot in the field.