Geoscience Reference
In-Depth Information
where
gy
u
is the probability value
y
corresponding to a
standard Gaussian distribution. Since the polynomials are
orthogonal and thus there is no correlation between them, the
variance of the
Z
samples can be identified to:
(
(
))
kriging plan used to estimate the blocks to get the predicted
dispersion variance, to the use of probabilistic estimation
techniques (Chap. 9), to the application of conditional simu-
lations (Chap. 10).
{
}
{
}
Var
Φ
(())
Y
u
=
Var
Z
()
u
7.3.6
Restricting the Kriging Plan
nn
n
{
}
∑∑
∑
2
≈
φφ
cov
H Y
(
( )),
u
HY
(
( ))
u
=
φ
pq
p
q
p
The concept is based on tuning the kriging plans to control
smoothing to match the resulting block distribution to the
expected SMU distribution as closely as possible.
This method was proposed originally by Parker and is
discussed in Rossi and Parker (
1993
) and Rossi et al. (
1993
).
It utilizes the notion that the smoothing property of kriging
(see Chap. 8, and Journel and Huijbregts
1978
, pp. 450-452)
can be controlled to obtain an estimated block distribution
that closely matches the predicted SMU distribution. Certain
parameters of the kriging plan, such as search neighbor-
hoods, minimum and maximum number of samples and drill
holes, the use or not of octant searches, etc. can impact the
degree of smoothing of the resulting block distribution.
Restricting the kriging plan has the advantage of being
simple, although rarely the kriged block distribution
will match exactly the desired SMU distribution. More
commonly, the matching is achieved for certain cutoffs of
interest along the grade-tonnage curve. It is local in the sense
that the method is estimating individual block grades, which
combine to form a distribution similar to the desired SMU
distribution.
One of the disadvantages of the method, as pointed out
by Journel and Kyriakidis (
2004
), is that it is specific to each
mineral deposit, and cannot be formulated in general terms.
Also, the increased restrictions on the Kriging plans result in
higher variance of the resulting block distribution, typically
at the expense of higher conditional bias. The spatial distri-
bution of estimates is still smooth, that is, the variogram of
the estimates will show a significantly lower nugget effect
and continuous behavior at the nugget effect.
It is important to note that the requirement of conditional
unbiasedness of the kriged block model is incompatible with
the requirement of predicting tons and grade received at a fu-
ture date by the processing plant, see for example Isaaks and
Davis (
1999
) and Isaaks (
2004
). This has been empirically
verified in practice. Still, too much conditional bias in the
output kriged model can lead to significant prediction biases
that should be avoided.
The SMU estimates at this time are interim estimates
awaiting much more data from blast hole sampling or in-
fill drilling. At the time of final estimation for grade control,
care should be taken to avoid conditional bias. It is often
more important at the prefeasibility and feasibility stage of
resource estimation to get predictions that reasonably reflect
the recoverable resource that will ultimately be obtained.
pq
==
11
p
=
1
The modeled anamorphosis function can be checked against
the original data by comparing the distributions resulting
from the samples to the distribution from the anamorphosis.
The distributions should be identical, although in practice
extreme values can be difficult to model.
Then, the sample histogram at the SMU block support is
obtained using the bi-Gaussian assumption. To correct the
sample distribution to a predicted-SMU distribution the ana-
morphosis function is modified by adding a change of sup-
port coefficient
r
:
∞
∑
p
Zv
( )
=Φ
(
y v
( ))
≈
r
⋅Φ
H Y
(
( ))
v
v
pp
p
=
0
The calculation of
r
requires the dispersion variance of the
SMU-sized blocks, in obtained from the variogram model de-
rived from samples values (Chap. 7). The anamorphosis func-
tion corresponding to the SMU support
v
assumes that the
distribution of
[
YY
( ),
u
( )]
v
is bi-Gaussian, and is found with:
nn
pp
r
2
2
σσγ φφ
=−≈∑∑
r
cov
vu
pq
vv
,
pq
==
11
n
{
}
p
2
2
H Y
(
( )),
u
HY
(
( ))
u
=
∑
r
φ
p
q
p
p
=
1
from which the
r
coefficient can be obtained. The distribu-
tion of grades representing SMU volumes is easily deter-
mined with the obtained
r
coefficient, the fitted coefficients
and the Hermite polynomials. Although apparently complex,
the procedure is automated and widely available in different
programs.
The DGM is deemed to be more robust than the affine
or indirect lognormal correction because the normal scores
transform is general, and no additional assumptions are nec-
essary for the original or the SMU distributions.
7.3.5
Non-Traditional Volume-Variance
Correction Methods
There are other methods used for volume-variance correc-
tion, some of them empirical. These range from adjusting the
