Geoscience Reference
In-Depth Information
distribution (Eq. 12.1) derived from the realizations, the ex-
pected loss can be found by:
monthly basis. Resource classification, as discussed below,
is generally meant to be a global guideline of confidence,
meant mostly for the benefit of shareholders and investors,
and should not be used as an uncertainty model to provide a
detailed risk assessment of the mine schedule.
Figures
12.4
and
12.5
illustrate how risk may change as
a function of the volumes considered. Figure
12.4
shows the
monthly probability intervals of Cu grades for an operating
copper mine. The graph shows the two values that corre-
spond to the P
90
(90th percentile) and P
10
(10th percentile)
of the conditional distribution derived from the conditional
simulations. It also shows the resource model grade for the
same period, as well as the Mine Plan grade, which is gen-
erally a lower value than the resource model grade. This
is because the mine planner sometimes adds dilution and a
safety factor to the grade predicted by the resource model,
typically on a monthly basis, not block by block. Mine plan-
ners may consider the monthly average grade provided by
the resource model as risky, thus penalizing in some fashion
the estimate. But the practice is variable and no standard
methodology exists. It is dependent on the experience and
prejudices of the engineer that defines the budgeted grade.
Figure
12.5
shows a similar graph for yearly periods of a
5-Year Mine Plan. Note that Year 1 in Fig.
12.5
is obtained
by simply averaging the grades of the 12 months shown in
Fig.
12.4
.
Note how Fig.
12.4
shows much more variability than
Fig.
12.5
. As expected, the smaller volumes represented by
the 12 months in Fig.
12.4
are more variable than grades
averaged over a yearly volume (Year 1, Fig.
12.5
). Also, it is
interesting to note that the grades predicted by the resource
model and the mine plan do no necessarily fall within the in-
terval defined by the P
90
and P
10
limits. This occurs both for
monthly and yearly volumes, and more so when considering
periods further away in time. This is to be expected, since
periods further away in time are likely to have less drilling
and thus be more uncertain.
The risk of not achieving the predicted production for
each period can be mitigated through further infill drilling.
The infill drilling can be directed to those areas with higher
uncertainty. A global confidence measure as used on most
resource classification schemes would not allow optimiza-
tion of the infill drilling to that level of detail.
{
}
+∞
−∞
∫
E Lz
(
*
−
Z
)()
n
=
Lz
(
*
−
z dF zx n
)
·
(; |())
N
1
real
∑
*
≅
Lz
(
−
z
)
(12.2)
l
N
l
=
1
real
where
N
real
is the number of realizations and
z
* is the re-
tained estimate.
The minimum expected loss can then be found by sim-
ply calculating the conditional expected loss for all possible
values of the estimates
z
*, and retaining the estimate that
minimizes the expected loss. As explained in Isaaks (
1990
),
the expected conditional loss is a step function whose value
depends on the assumed costs of each bad decision, and the
relative costs of misclassification. This implies that the ex-
pected conditional loss depends only on the classification of
the estimate
z*
(
u
), not on the estimated value itself, as long
as all benefits and costs are constant with respect to grade.
The Loss Function thus quantifies the consequences of
false positives and false negatives, weighs the probability
and relative impact of each, and then provides the minimum
cost solution under the loss model used. For example, the
loss incurred when an ore grade panel is sent to the waste
dump is a type of lost opportunity cost, measured by the
profit that should have been realized. If the same panel is
waste, but is sent to the mill, the loss is a combination of the
loss incurred in processing material that does not produce the
metal to pay for itself, plus the loss derived from the oppor-
tunity lost in processing payable material, if any.
Loss functions are in general asymmetrical, since the con-
sequences of under- or overestimation have different costs.
In metal mining, where small volumes of ore may have high
value, it is typically costlier to send ore to the waste dump
than to process waste. Precious and most base metals mines
have this characteristic, which is more notable if high eco-
nomic cutoffs are used. There are other cases where the op-
posite is true, such as high volume, direct-shipping iron ore
mines, who prefer to avoid dilution in the shipment.
Optimal estimates can be derived for a Loss Function if
the conditional distribution of the random variable is avail-
able. The uncertainty model as described by the realizations
provides all the information required to optimize decision-
making under uncertainty.
When assessing uncertainty and risk it is also important
to consider the scale of interest, i.e., the volume of mate-
rial being assessed. There are differences between a global,
deposit-wide geologic confidence assessment and a more
local, mine production-oriented risk assessment. A global
confidence measure cannot be used for local, block-by-
block risk assessments. A typical example is the resource
classification scheme, often used by mining engineers as a
measure of confidence on mine schedules, for example on a
12.3
Resource Classification and Reporting
Standards
Public disclosure of estimated resources requires that re-
source estimates be classified according to degrees of con-
fidence and allocated as measured, indicated and inferred.
Reserves must be classified as either proven or probable re-
serves, derived under certain rules from resource categories.
Different resource classification standards are used in differ-
