Geoscience Reference
In-Depth Information
metallurgical processing should provide the maximum
possible economic benefit given all operational constraints.
Other possible optimization criteria, such as maximizing
resource utilization, is not applicable in the case of grade
control, since the decision is short-term in nature, and aims
at making the most out of the current operation on a daily
basis.
Loss Functions can be used to optimize based on
pre-determined functions that assign value to estimates,
or equivalently, costs to mistakes. They were described in
Chap. 12, and further reading can be found in Journel (
1988
),
Isaaks (
1990
), and Goovaerts (
1997
). Conditional simulation
is used to provide a model of uncertainty that can be used
to optimize grade control. One alternative is the Minimum
Loss/Maximum Profit method as presented below, which
has been implemented with success in several open pit
operations. The expected profit calculation is
of low and high grade zones. These are critical variables for
the optimization of ore/waste selection because it depends
on accurately predicting the variability of high to medium to
waste grade transitions.
Typical grade control simulation grids can be 1 m by 1 m
by bench height (corresponding to the sampled blast hole
column). These are used directly in obtaining the uncertainty
model for ore/waste selection panels. Larger grid sizes may
be used and sometimes required because of time or general
computer hardware limitations, still providing reasonable es-
timates when enough simulated points are included within
the selection panels.
Given that conditional simulation models are sensitive to
departures from its stationarity assumption, it is critical that
they be controlled by geologic models. The use of geologic
boundaries may introduce issues of ergodicity, which should
be carefully handled. A constantly updated geologic model,
in addition to constant geologic control at the pit is required
to ensure that the uncertainty models derived from the con-
ditional simulations are realistic and also representative of
local geology.
Other important aspects include the behavior of the
high-grade population, which is required to control the sim-
ulated high grades, see Parker (
1991
) and Rossi and Parker
(
1993
). Issues such as limiting the maximum simulated
grade should be carefully considered, since it may signifi-
cantly impact the selection panels. The issue should be re-
solved through calibration with existing production data.
A small number of realizations, perhaps 20 or 30, are
typically used. This reflects practical limitations, since grade
control is a process that has to be completed in a short period
of time; but it may also be a sufficient number of simulations
to adequately describe the model of uncertainty, given the
data density available.
Recall that the model of uncertainty provides the prob-
ability of that node in the grid of being above (or below) any
grade
z
:
∑
( )
( )
l
P
=
−
c
−
c
+
prz
u
ore
o
t
ore mining cost
milling cost
all realizations
revenue
∑
P
=
−
c
−
c
waste
w
lo
waste mining cost
lost opportunity
all realizations
establish
c
by calculating revenue if it were milled
lo
l
( )
( )
rev
=
prz
u
0
if
rev
<
0
=
�
c
lo
rev
if
rev
>
0
�
13.4.1
Maximum Revenue Grade Control
Method
The Maximum Revenue grade control method is a two-step
procedure, first outlined by Isaaks (
1990
), and applied with
success at some mine operations, for example Aguilar and
Rossi (
1996
). Initially, a set of conditional simulations is ob-
tained from the blast hole data available. These conditional
simulations provide an uncertainty model for grades at any
specific point within the blast. Second, an economic optimi-
zation process is implemented using loss functions to obtain
the optimal ore/waste selection. The Loss Function quanti-
fies the economic consequences of each possible decision.
The simulations are used to build models that reproduce
the histogram and spatial continuity of the conditioning
data. By honoring the histogram, the model correctly rep-
resents the proportion of high and low values, the mean,
the variance, and other statistical characteristics of the data.
By honoring the variogram, it correctly portrays the spatial
complexity of the orebody, and the two-point connectivity
{
}
F z x
( ;
| ( ))
n
=
Pr
ob
Z x
( )
≤
z
| ( ),
n
α
=
1,...,
n
(13.3)
where
F
(
z;
x
|
(
n
)) is the cumulative frequency distribution
curve for each point
x
of the simulated grid and obtained
using the (
n
)
,
∀
= 1,…, n
conditioning blast holes.
In grade control, the selection decision (which material is
ore and which is waste) has to be based on grade estimates,
z
*
(
x
), while still attempting to minimize miss-classification.
Since the true grade value at each location is not known, an
error can and will likely occur. The loss function attaches an
economical value (impact or loss) to each possible error, as
described in Chap. 12.
The minimum expected loss can be found by calcu-
lating the conditional expected loss for all possible val-
ues for the grade estimates, and retaining the estimate
