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In-Depth Information
and
makes no such assumption. The discrete Gaussian model
(DGM) has been proposed as a more robust method to obtain
the volume-variance correction.
The key idea of the DGM is that the distributions for
different supports will be Gaussian after transformation
to Gaussian units. The transformation to Gaussian units is
achieved in two steps: (1) a normal scores transformation
like that described in Chap. 2, then (2) fitting the relationship
between the original grades and the normal scores transform
with a series of Hermite polynomials. These polynomials are
orthogonal, which is important because the variance of the
original grades is then a simple summation of the squares of
the coefficients. A change to the variance is achieved by scal-
ing the coefficients of the Hermite polynomials by a change
of support coefficient related to the factor
f
. As expected, the
corrected distribution gradually becomes more Gaussian in
shape as the scale increases.
The fitting of Hermite polynomials and the details of the
mathematics are embedded in widely used computer pro-
grams and documented in references such as Armstrong
and Matheron (
1986
); Rivoirard (
1994
) or Machuca-Mory
et al. (
2007
). An overview will be presented here. An ana-
morphosis function needs to be fit to the sample data. The
anamorphosis function is defined by a Hermite polynomial
expansion fit to the data. Hermite polynomials are related
to the Gaussian distribution and are defined by Rodrigues'
formula (Abramovitz and Stegun
1964
, p. 773). The anamor-
phosis function is equivalent to the normal score transforma-
tion in that it provides a mapping of the point variable
Z
to
the Gaussian variable
Y
and vice-versa:
ln(
f
CV
2
+
1)
ln(
CV
2
+
1)
·
b
=
where
m
is the mean,
CV
is the coefficient of variation of
the point distribution, and
f
is the variance correction factor
(VCF) previously defined.
However, since the distributions will not in general be
exactly lognormal, then the transformation of Eq. 7.9 will
not result in the same mean for the transformed and untrans-
formed distributions. So, a final step is required to ensure
that the original mean is obtained:
=
m
m
(7.10)
q
·
q
After applying Eq. 7.10, the quantiles of the SMU distribu-
tion have been rescaled to the correct mean. Interestingly,
the differences between the first transformed mean and the
rescaled mean can be used as a measure of the dissimilar-
ity between the original distribution and a Log-normal dis-
tribution. The final correction may cause the variance to be
slightly different than the target variance.
7.3.3
Other Permanence of Distribution Models
As a generalization of the previous methods, the same prin-
ciple can be applied to other distributions, most practically
to those that are characterized by two parameters, such as the
Gaussian, Lognormal, and even Gamma distributions.
Under the assumption that a sample distribution can be ap-
proximated by a multivariate Gaussian distribution, then the
resulting block distribution will also be multi-Gaussian, with
the same mean and corrected variance, as described before.
Similarly, the sample distribution can be assumed to be
multi-Lognormal, in which case the resulting SMU distribu-
tion is also assumed to be multi-Lognormal (although, as in
the case of the affine correction, this is an assumption known
to be incorrect), with the same mean and corrected variance.
As these methods have had little use in practice, the read-
er is referred to Journel and Huijbregts (
1978
, pp. 468-469)
for the specific formulae and further details on the limita-
tions of these methods.
∞
z
(
u
)
=
(
y
(
u
))
≈
p
H
p
(
y
(
u
))
p
=
0
where
p
is the coefficient of each polynomial term,
and
H
p
(
y
(
u
)) is the Hermite polynomial value. This fitting
can be thought of as a polynomial fit to the Q-Q plot between
the original grades and the normal scores.
The anamorphosis function is fit by calculating the value
of the
Φ
coefficients of the Hermite polynomials. The first
coefficient is simply the mean of the
Z
samples:
Φ= Φ =
E
{
(
Y
( ))}
u
EZ
{
( )}
u
0
Higher order coefficients are found with the following ap-
proximation:
7.3.4
Discrete Gaussian Method
{
}
Φ=
EZ
()
u
⋅
H Y
(())
u
p
p
The permanence of distribution assumption is a limitation
because most real-life mining distributions cannot be easily
fitted with a two-parameter distribution (Gaussian or Log-
normal). They have multiple modes and mixtures of popu-
lations that can only be overcome by using a method that
∫
=Φ ⋅
(())
y
u
H
(()) (())
y
u
⋅
g y
u
⋅
dy
()
u
p
n
1
∑
≈
( (
zu
)
−
zu
(
))
⋅
H
(
y
(
u
))
⋅
g y
(
(
u
))
α
−
1
α
p
−
1
α
α
p
α
=
2
