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the use of factors to penalize block values along contacts.
A more direct approach is preferred, estimating the grade
of each geologic unit within the block and then obtaining
the average block grade using Eq. 7.1.
c. The information effect is usually handled with factors,
sometimes calibrated to production figures, and often ap-
plied by mining engineers to the ore resource model at the
time of developing the mine plan. In any case, the block
model documentation should clearly state its limitations
in terms of dilution, and to what extent it can be consid-
ered “recoverable”.
d. If an indirect or empirical method has been used to incor-
porate additional dilution into the model to compensate
for planned and unplanned operational dilution, such as
using a larger SMU size, this should be clearly stated in
the documentation.
In addition to the above, good practice requires:
a. A more specific method to include internal dilution into
the resource model. This can be done through any of the
methods mentioned in Sect. 7.3, and in all cases should
include a fair assessment of the uncertainties and trad-
eoffs involved.
b. Geologic contact dilution should be explicitly incorporat-
ed into the block model, and a statement about the uncer-
tainty of the position of the contacts should be included.
The information effect should be dealt with using at least
a reasonable empirical approximation, or a modification
of the estimation method.
c. All the work should be well documented and clearly pre-
sented, detailing the checks performed and the quality
control procedures in place.
Best practice consists of using uncertainty models to deal
with all three types of dilution described: block averaging,
geologic model uncertainty, and operational dilution. The
full conditional simulation study would:
1. Incorporate the uncertainty of the geologic model, thus
implicitly considering geologic dilution.
2. The internal dilution is more accurately incorporated by
direct block simulation or simply by averaging the simu-
lated values into the SMU size.
3. The simulation model should also incorporate operational
dilution and the information effect by simulating the com-
plete mining process.
Thus, most of the possible sources of dilution and ore loss
are modeled simultaneously. In such case, it is not necessary
to apply any of the volume-variance correction methods, un-
less it is done as checks on simulation models, for example.
The work is only completed when, as always, a very thor-
ough validation and checking of the models is completed and
documented. Preferably, the simulations models should be
validated against production, or at least alternative models,
and through thorough statistical and graphical checking, see
Chap 11.
7.6
Exercises
The objective of this exercise is to review change of support
calculations. Some specific (geo)statistical software may
be required. The functionality may be available in different
public domain or commercial software. Please acquire the
required software before beginning the exercise. The data
files are available for download from the author's website—
a search engine will reveal the location.
7.6.1
Part One: Assemble Variograms
and Review Theory
You will use the Cu variable from the largedata.dat
dataset. The key parameter in all scaling is the variogram;
however, the normal scores transforms of grades do not av-
erage linearly and we cannot use the normal scores vario-
grams for scaling. The variograms of the Cu grades directly
are required. Of course, the direct grade variogram should be
similar to the normal scores variogram.
Question 1:   Compute  and  it  a  3-D  Cu  variogram  (like 
that modeled in Chap. 6). Comment on the
“stationarity” of the variogram model, that is,
does it latten off at the variance of Cu grades?
Question 2: Write a short review of the key theoretical
results needed for variogram scaling: (1) the
deinition of the average variogram/ average 
covariance, (2) the deinition of the disper-
sion variance and the link to the average var-
iogram, (3) krige's relation or the additivity of
variance, and (4) the scaling of variogram sill
parameters.
Question 3:
Derive the volume scaling law of the nugget
effect, that is, demonstrate that the following
relation is exact: CV = |v|/|V| Cv . Where CV
and Cv are the nugget effects at scales V and
v , respectively.
7.6.2
Part Two: Average Variogram Calculation
Average variogram or “gammabar” values tell us the vari-
ance at any scale. The discretization required for stable nu-
merical integration is a consideration. Average variogram
values can be calculated between two disjoint volumes V
and v′ ; however, classic histogram and variogram scaling re-
quires the average variogram to be calculated for V = v′ , that
is, for the same volume and itself. This brings up the zero
effect as another complicating factor.
Question 1:
Consider your reference Cu variogram model
and a 10 m cubed block size for a num-
ber of sensitivity studies. Create a plot with
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