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and many samples is that block estimates are very smooth;
the farther apart the drill holes, the greater the smoothing.
In some special circumstances, the price to pay for smooth-
ing may be unavoidable if making
final
estimates, but it will
generally be unacceptable for
interim
estimates.
One approach for interim estimates is to modify the es-
timation procedure to match the smoothing to the ultimate
block distribution. The block distribution is predicted ana-
lytically, see Chap. 7. The search in estimation is reduced
in size to increase the variability of the estimates, using an
iterative approach until the adequate amount of smoothing
is achieved. These estimates will have conditional bias, but
that is not a concern since decisions are being made on the
accuracy of the overall grade distribution, and not on block
by block estimates.
Although there are many papers on conditional bias, the
subject continues to be poorly understood and controversial
in the geostatistical community. Authors like Krige (
1994
,
1996
,
1999
) claim that resource estimates should have no
conditional bias at all. Sinclair and Blackwell (
2002
) argue
that conditional bias contributes to the discrepancies be-
tween resource models and production. Guertin (
1984
) and
Pan (
1998
) proposed different types of corrections for con-
ditional bias. Isaaks (
2004
) argues that a conditionally unbi-
ased estimate that is also an accurate recoverable resource
predictor, except for one theoretical case that in practice is
never found, is an oxymoron.
These authors have found that smoothing in mineral re-
source (interim) estimates in long term open pit mine plan-
ning is less acceptable than conditional bias, and in fact that
some conditional bias is necessary for the resource estimates
to better predict the mined tonnages and grades. These au-
thors have also found that for final estimates using tightly
spaced data, large search radii and many samples do not im-
prove the estimates. The practitioner should understand the
purpose of the estimates and strive to manage and understand
the consequences of smoothing and/or conditional bias.
There are alternate methods to avoid smoothing. Chap-
ter 9 presents methods based on probabilistic estimation
that avoids considering a single kriged estimate as the block
grade that will be encountered. Chapter 10 presents methods
based on simulation, which by construction do not smooth.
In Chap. 13 it will be argued that, for ore/waste selection
and grade control, simulation-based methods are preferable
to any form of kriging.
of the data. However, in mining, the overriding interest is in
estimating a certain selective mining unit (SMU), a volume
of material of a specific size that characterizes mining se-
lectivity. The definition of the selective mining unit (SMU)
volume size is
the minimum volume of material on which ore
and waste can be separated, which is a function of mining
method and selectivity
. This size is related to the ability of
the equipment to select material; but it is also based on the
data available for ore/waste classification (blast holes and/
or dedicated grade control drilling), the procedures used to
translate that data to mineable dig limits, and the efficiency
with which the mining equipment excavates those dig limits.
Several sources of dilution must be accounted for, includ-
ing internal dilution due to grade variability within the SMU,
external dilution resulting from geological and geometric
contacts, and planned and unplanned operational dilution.
Dilution and estimation domains definition (Chap. 4) are the
two most important factors for accurately estimating recov-
erable resources.
Recoverable resources imply that we are interested in
evaluating a truncated statistic of the overall grade distri-
bution. The classical formulae are found after defining an
economic cutoff for any set of SMU estimates (Journel and
Huijbregts
1978
, p. 480). The tonnage is simply the sum of
all unit tonnages (or area of the histogram) that are above
that threshold:
Tz T Fz
() [
=⋅ −
1
(
)]
0
0
Z
0
N
+∞
1
A
∑
∫
t
u
z
=⋅
T
f zdzT
N
()
=⋅
(;)
0
Z
0
i
i
c
z
c
A
i
=
1
where
T
0
is the total in-situ tonnage at cutoff 0 and
z
c
is the
grade cutoff applied.
The quantity of metal is calculated as the summation of
the quantity of metal of each individual unit:
N
+∞
1
A
∑
∫
Q z
( )
=⋅
T
z
⋅
f
()
z dz
=⋅
T
z t
⋅
( ; )
u
z
0
0
Z
0
i
i
c
N
z
c
A
i
=
1
where z is the grade of the unit. Finally, the average grade of
the recovered material is:
m
(
z
0
)
=
Q
(
z
0
)
T
(
z
0
)
8.1.2
Volume Support of Estimation
8.1.3
Global and Local Estimation
In some cases we are interested in point estimation, that is,
estimation at the scale of the data. The smoothing of most
estimation from widely spaced drill holes implies that the
variability of the estimates is not the same as the variability
The estimation methods mentioned here yield local esti-
mates, in the sense that the estimated values are specific to a
location within the deposit, and are derived from nearby sam-
ples. A global estimate is an estimate for an entire domain or
