CHAPTER 4: Estimation
SOME MORE COMPLEX PROBLEMS
Let us return to the original problem posed in Figs. 4.1
and 4.2. We showed that if we used the grade of sample 1 to estimate the point
A, we obtained a standard error of 25.4 p.p.m. We then introduced the problem
of estimating a 60 ft by 30 ft block centred at A. After the examples above, it
should be easy enough to calculate that the extension standard error will now
be 19.2 p.p.m. [
(S,A)=356, (A,A)=344, (S,S)=0] , when estimating the average grade of the panel from the
single sample point 1. This is over 20% lower than when trying to estimate the
central point. The real conclusion is simply that it is easier to estimate the
average grade over a block than to specify the grade at a single point. Now, if
we consider the arithmetic mean of the 5 point-samples, we obtain the
following:
If we use
this estimator to predict the central point of the block, the extension
standard deviation is 21.8 p.p.m. However, if we estimate the panel, the
extension standard deviation reduces to 12.8 p.p.m. Notice that both of these
Figures are lower than when only considering sample 1. It would appear that,
even though the other samples are a lot further away from the centre of the block, they are
contributing a fair amount of information about the block grade.
To conclude this chapter, an example on a slightly grander scale, on which the
reader can exercise his new-found knowledge. For this deposit a simulation has
been used, since in that case we know the
semi-variogram and the value at every point within the deposit. This enables us
to compare estimates with ‘actual values --- a situation which is rare to the
point of extinction in the real world. It also enables us to produce a set of
samples on any given sampling scheme proposed. The simulated deposit
is a low grade sedimentary iron ore, with an overall average of about 35% Fe, a
standard deviation of 5% Fe, a range of influence of 100m and a sill of 25(%)² Fe---obviously. The semi variogram is spherical (yet
again) with no nugget effect. An area 400 metres square has been simulated and
a set of 50 samples taken from it at random. The positions and values of these
are shown in Fig. 4.14 and Table 4.5.
Fig 4.14. A set of
random samples taken from a simulated iron ore deposit
The initial estimation, at the
pre-feasibility stage, is to be done on 50m by 50m blocks. For this first
example, each block has been allocated the average grade of all interior
samples. Where a sample falls on the edge it has been allocated to both blocks.
Figure 4.15 shows the estimated value for each block.
Fig 4.15. Estimates
of block values formed by averaging all interior samples in the iron ore
deposit, and the corresponding extension standard deviations.
Blocks without internal sampling have been
shaded in. The upper Figure shown in each block is the estimator T*, and the lower is the extension standard
deviation. Since this deposit is actually Normally
distributed, 95% confidence limits would be given (approximately) by T*± 2se. For comparison, the ‘true average grade
for each block is shown in Fig. 4.16.
Fig 4.16. ‘Actual’
average values within each block in the simulated iron ore deposit.
It can be seen that in most of the 37
estimated blocks, the true value is within the 95% confidence interval. Four or
five blocks lie just outside the interval, and three or four are considerably
outside. This is a little higher than would be expected, since we would only
expect about two blocks to lie outside a 95% interval. However, if we consider
the 99% confidence interval (3 standard deviations) only one block is
significantly outside the interval, i.e. the lower left-hand block of the area
(south-west corner).
For comparison, Fig. 4.17 shows a set of
samples taken on a regular grid from the same ‘deposit’. In this case, each 50m
block has two samples in opposing corners.
Fig 4.17. A set of
samples taken on a regular grid from the simulated iron ore deposit.
Using the average of these two samples to
estimate the block results in an extension standard deviation of 2.5% Fe, Fig.
4.18 shows the estimated values in each block.
Fig 4.18 Estimated
block values from regular samples.
It can be seen that although five or six
blocks lie outside the 95% confidence interval, not one lies more than 2.25
standard deviations from the true value.
Having
exhausted the possibilities of extension in idealised circumstances, let us
move on to some more interesting situations.