CHAPTER 4: Estimation
TWO-DIMENSIONAL EXAMPLES
In two dimensions we have again a set of auxiliary functions which are mostly
generalisations of the one-dimensional ones. These are shown in Fig. 4.9.
Fig. 4.9. The two-dimensional auxiliary
functions.
g(l;b) is the two-dimensional
analogue of g(l) --- sometimes read as ‘g(l) stretched through length b’. It is the average semi-variogram value between all
points on one length, b, and all points on another
parallel to it and of the same length. This function is useful for parallel
boreholes, channel samples, drives, raises and so on. The generalisation of c(l) becomes c(l;b) and is the average semi-variogram
between a length b (drive, raise, core etc.) and an
adjacent panel l by b. The function F(l,b) has been introduced in Chapter 3
for such terms as 
(A,A) for rectangular panels. We also
introduce a ‘new’ function H(l,b) which does duty for two
rather different situations. It represents the average semi-variogram value
between a panel and a point on one corner of it; it also
represents the average semi-variogram value between two lengths l and b at right angles to one another.
This is simply a mathematical accident.
A few examples will be given here to show how to
‘manoeuvre’ situations into the required form. Figure 4.10 shows a stoping
panel in a cassiterite vein, 100 ft by 125 ft, with a development drive passing
through it parallel to the
sides. The drive is 25 ft from the ‘bottom’ of the panel.
Fig. 4.10. Estimation of the panel
average from the drive average.
The drive
is so closely sampled that we can assume we know its average grade. We wish to
use that average grade as an estimate of the average grade of the panel, so
that:
(A,A) is the average semi-variogram value
between every point in A and every point in A, which is the definition of the two-dimensional
function F(l,b). Thus if we have a model for the
semi-variogram we can evaluate this. Let us suppose that in the above example
the semi-variogram is a spherical model with a range of influence of 80 ft and
a sill of 450 (lb/ton)². The table given in Chapter 3 for the F(l,b) function (Table 3.4) is for the ‘standardised’
spherical model with a=1 and C=1. We require F(125,100) for a
model in which a=80 and C=450. To convert to a=1 we divide the measurements l and b by the range a (80 ft). Thus we require F(125/80,100/80) =F(1.54,1.20) for a=1 and C=450. From the table (interpolating
linearly) we find that F(1.54,1.20)=0.7807 when C=1. Therefore, if C=450, our F value must be 0.7807\450=351.3 (lb/ton)². This finally is (A,A).
Now let
us consider the term (S,S). Our sample is a ‘line’ of length 125 ft,
and the (S,S) is the average semi-variogram value
between every point in the line and every point in the line, i.e. the
one-dimensional function F(l). For a spherical semi-variogram
model, where the length is greater than the range of influence, we know that:
Substituting the values l=125 ft, a=80 ft and C=450 (lb/ton)², gives F(125) =270.9 (lb/ton)², and this is the value used for (S,S).
Finally,
we must turn to (S,A) which presents a problem, since the
auxiliary function c(l;b) is only for situations where the
‘line’ is on one side of the panel. We must employ the same sort of manoeuvre
as before:
|
|
= average semi-variogram value between
the line and a panel 125 ft × 100 ft. = (sum of the semi-variogram
values between the line and the panel) ÷ (100
× 125) =(sum of the semi-variogram
values between the line and the points in the ‘upper’ panel + sum of the
semi-variogram values between the line and the points in the ‘lower’ panel) ÷ (100 × 125) |
However,
the average semi-variogram value between the line and the ‘upper’ panel (125 ×
75 ft) is given by c(75;125), so that the sum of the semi-variogram values
between the line and every point in the upper panel will be 75×125×c(75;125. Similarly, the sum of the
semi-variogram values between the line and every point in the ‘lower’ panel
(125 ft × 25 ft) will be given by 25×125×c(25;125). This gives:
Table 4.2 shows the values of the c(l;b)
function for the ‘standardised’ spherical model with a=1 and C=1. This table is used in the same
way as the F(l,b) table. It should be noted that the order
of the arguments is important. The value of c(25;125) is obviously different from the value of c(125;25). Standardising the
measurements of the panel in the same way as before, we require the values of c(0.94;1.54) and c(0.31;1.54) from the table. Interpolating linearly gives
the values 0.8196 and 0.6538 respectively. Multiplying by the sill of 450
(lb/ton)² gives 368.8 (lb/ton)² and 294.2 (lb/ton)² respectively. Putting these
together in the expression for (S,A) gives a value of
350.2(lb/ton)².
Having
evaluated all of the individual terms, we can now calculate the extension
variance for the setup in Fig. 4.10, so that:
This gives an estimation standard deviation or ‘standard error’ of 8.8 lb/ton
for the estimation of the panel grade. If we are willing to assume Normality
for the errors (sic) we
could be 95% certain that the ‘true’ grade of the panel lay between T* ± 2se equal to T*± 16.6 lb/ton. This standard
error would be correct for any panel having the same sampling setup, anywhere
within this deposit, since the estimation variance depends not on
the actual grades involved but on the geometric positioning of the sample and
the panel.
Fig. 4.11. Estimation of the panel
average from two drive averages.
As a second example,
consider a disseminated nickel deposit in the late stages of development. On a
particular underground level, we have a stoping block 40m by 30m to be
estimated. If we consider only the one level we can treat the problem as a
two-dimensional one. Suppose this ‘panel’ has been developed along two sides,
and the information available consists of (i) the average grade along the 40m
drive, g1 and (ii) the average grade along
the 30m drive, g2. Suppose we use the average of
these two grades to estimate the value inside the panel. Then:
For this particular deposit we have a spherical
semi-variogram with a range of influence of 60m, a sill of 0.75(%)² and a nugget effect of 0.10(%)². This implies a standard deviation for the ‘point’
sample values of 0.92%. The extension variance, as always, is given by:
(A,A)is, as before, the two-dimensional function F(l,b), but now we have two components to the
evaluation of this term. We can calculate the F(40,30) for the spherical part of the model, with a=60m and C=0.75(%)². This turns out to be 0.4336 × 0.75=0.325(%)². To this we must add the nugget effect of 0.10(%)², so that (A,A) =0.425(%)².
The average semi-variogram value between each sample and
each sample now contains four terms which have to be averaged, i.e.:
It is generally easier to look at each term separately and then
to combine them to produce the final answer. The first term (drive1,drive1) is the average semi-variogram between
all points within a length of 40m. That is, F(40).
For a
spherical model, when the length of the ‘line’ is shorter than the range of influence,
F(l) is given by:
Substituting l=40 and a=60, C=0.75(%)² gives a value of 0.239 (%)². Adding on the nugget effect produces (drive1,drive1) =0.339(%)². Similarly, (drive2,drive2) =0.283(%)². The two terms (drive1,drive2)
and ( drive2,drive1)
are identical and have been defined as the auxiliary function H(l,b). Using Table 4.3 for the H(l,b) function in the same way as the other tables
and adding in the nugget effect gives:
Putting all the individual terms together, we find that (S,S) =0.428(%)².
Finally, to calculate the estimation variance, we also
require the term (S,A). This is the average semi-variogram value between each
sample and the panel, so that:
The extension variance for the problem illustrated in Fig. 4.11
is thus:
giving a standard error for the prediction of the panel
average of 0.376% nickel.
A third
example is shown in Fig. 4.12.
Fig. 4.12. Estimation of the panel
average from two raise averages.
This time
the metal is zinc, the semi-variogram is spherical with a range of 20m and a
sill of 49(%)². The value to be estimated is the average
over the 30m by 15m panel, and the information available is the average grade
of each of the two raises through the ‘stope panel’. In the idealised situation
chosen, the raises are both 7.5m from the edges of the panel. Very briefly, the
extension variance is produced as follows:
The term (A,A) =F(30,15) when a=20 and C=49(%)², which is 0.7021×49 = 34.4(%)².
The g(l;b) function is given for the ‘standardised’
spherical model in Table 4.4. The last term to be evaluated is:
Since the setup is symmetrical, (raise1,A)= (raise2,A), so that
This gives an extension variance of:
This gives a standard error for the estimation of the panel
average of 1.14% Zn. Thus although the original sample standard deviation for
‘point’ samples was 7% Zn, two ‘samples’ within the panel produce a standard
error one-sixth of this Figure.
One last
brief example: a porphyry copper deposit has been explored by means of vertical
boreholes. To simplify the problem, each ‘bench’ in the deposit is considered
separately as a plane, and the borehole intersections as points within the
plane. Take a typical small block, 25m by 25m, with a borehole passing through it
as shown in Fig. 4.13.
Fig. 4.13. Estimation of panel average from one ‘point’ sample.
Suppose
we ‘extend’ the value of the core intersecting the block to the whole block.
Then:
Then
From previous investigation we know that the semi-variogram
has a spherical form with a range of influence of 90m and a sill of 0.6(%)² Cu. (A,A) is yet another F(25,25) function, whose value can be calculated to be
0.2145 × 0.6 = 0.129(%)² Cu. (S,S) =0 since the sample is supposed to be a
point. (S,A) must be calculated with the now familiar
(?) manoeuvring, as follows:
So, finally, the extension variance becomes:
so that the standard error for the estimation is 0.288 %Cu.
The same exercise can be repeated for a sample outside the panel, using the same logic as that applied to the
one-dimensional problem in Fig. 4.7.