Geoscience Reference
In-Depth Information
or (2) by using the fitted Hermite polynomials, see Remacre
1987.
The final grade above cutoff is simply
M
(
z
c
)
=
Q
(
z
c
)
timal solution is found with a simple kriging (SK) of each
component.
It is a mathematically involved method that will not
be described here in detail. Some additional references
are Armstrong and Matheron (
1986
), Chilès and Delfiner
(
2011
), and Rivoirard (
1994
). Also, a readable summary is
presented in Kumar (
2010
). There have been only a few
applications in the mining industry, the most significant
drawback being its theoretical complexity. Also, since DK
depends heavily on stationarity because it requires the the-
oretically correct SK, the estimates are very sensitive to
the stationarity assumption. Experience has shown that the
estimated values tend to be extremely smooth, with little
variability around the mean.
P
(
z
c
)
. In
practice, and to avoid spurious results,
P
(
z
c
) is only consid-
ered when it is greater than 1 %.
UC is based on a couple of important assumptions: (1)
that the Gaussian transformed point data, SMU and panels
are all bivariate normal; and (2) that the change of support
model for the SMUs can be extended to the panels, an as-
sumption common to other change of support models.
UC has at least two important limitations. First, it does
not provide any information regarding where the high or
low grade SMUs are within the panel, which is convenient
for most mine planning. This is not a theoretical limita-
tion, since it is part of the underlying premise of UC: panel
grades can be predicted reliably, but SMU grades cannot.
But it is a major practical limitation, since mine planners
require the location of the SMUs in order to calculate re-
coverable reserves. This is the most significant unresolved
practical issue.
The second significant limitation is that panels with the
same estimate will have the same grade and proportion curves
irrespective of the surrounding data. Surrounding drill hole
samples are used to estimate the panel grade, but they are not
used for determining the SMU distribution. For example, con-
sider two panel estimates: one panel is in a homogenous zone
where the surrounding samples have all the same values, and
the other where the surrounding samples result in the same es-
timated panel average but are highly variable. When the panel
estimates are the same the estimated recoverable reserves are
the same. This limitation implies that UC is more sensitive to
departures from stationarity compared to other methods, de-
spite using OK to estimate panel grades.
Therefore, UC is only recommended at an early stage of
project development. UC is useful when data are uniformly
sparse and the SMU blocks cannot be reliably estimated.
Other estimation methods that directly estimate smaller
blocks would not be considered reliable with widely spaced
exploration drilling. When there is sufficient infill drilling or
for an operational mine, it is likely that other methods will
result in better local estimates.
9.2.4
Checking the Multivariate Gaussian
Assumption
The normal score transform (or its equivalent anamorphosis)
ensures that the one-point distribution is Gaussian. This is
a necessary but not sufficient condition to prove that a RF
model is multivariate Gaussian. Theoretically, the normality
of the two-point, three-point, and in general
n
-point distribu-
tions should be checked (Verly
1984
).
While two-point statistics can be inferred from the data,
three-point and higher order statistics are much more difficult
to obtain. The corresponding analytical expressions exist,
but sparse data and irregular grids do not allow inference
from sampled values. Also, the practical significance of the
multivariate Gaussian assumption is limited, since the vast
majority of the MG applications are bivariate. In practice
only the two-point distribution is checked for bi-normality:
if the bi-Gaussian assumption holds, then the multi-Gaussian
formalism is adopted.
Checking the Gaussianity of two-point distributions im-
plies checking that the experimental cdf values of any set of
sample pairs separated by any vector
h
match the theoretical
Gaussian distribution. In practice, the comparison is made
for corresponding
p-
quantile values,
y
p
=
y
p
', such that the
two-point Gaussian distribution becomes:
{
}
G
(;
h
y
)
=
Prob Y
()
u
≤
y
, (
Y
uh
+
)
≤
y
p
p
p
9.2.3
Disjunctive Kriging
arcsin
C
(
h
)
2
Y
−
y
1
(9.1)
p
∫
2
=+
p
exp
d
θ
Disjunctive kriging (DK) was introduced by Matheron (
1974
,
1976
). It is a method that also relies on a bi-Gaussian assump-
tion and uses Hermite polynomials to transform the original
data into additive functions. The purpose is to estimate recov-
erable grades and tonnages above cutoffs for any size block.
The method is based in decomposing the variable into a
sum of uncorrelated orthogonal factors, for which the op-
2
π
1
+
sin
θ
0
The steps required to complete the actual checking process
are:
1. The variogram
γ
Y
(
h
) of the normal score data is computed
and modeled, and from this the covariance model
C
Y
(
h
) is
obtained.
