Geoscience Reference
In-Depth Information
Fig. 7.8 An example of block dis-
persion variances resulting from
different discretization grids. The
variogram model and the block
size are fixed. The discretization
in Z is always 1 because bench
height and composite length are
the same in this example. Note
that a 3 × 3 × 1 grid in this case
is sufficient to approximate the
block dispersion variance
high and low grades within an SMU. The opposite is true for
low f values.
As volume increases from the data support to an SMU
support, the mean stays the same and the variance changes
by a predictable amount (summarized in the f factor). The
shape of the distribution also changes. The influence of the
central limit theorem is felt to some extent, since the average
of identically distributed values tends to a normal distribu-
tion. The grades inside an SMU in fact are not independent;
therefore, the distribution of SMU grades does not always
approach a normal distribution.
where z is any value of the original distribution, z′ is the cor-
responding value of the SMU distribution, f is the variance
correction factor , and m is the mean of both sample and
SMU distributions.
According to Journel and Huijbregts ( 1978 , p. 471), the
affine correction can be applied up to about a correction
factor of 30 % (  f > 0.7), although in the experience of these
authors this is optimistic. Even for volume-variance correc-
tions much smaller than 30 % the affine correction seems to
provide the wrong prediction, see Rossi and Parker ( 1993 )
and the example below.
7.3.1
Affine Correction
7.3.2
Indirect Log-normal Correction
The affine correction is the simplest method for volume-
variance correction. It is based on the concept that the
distribution does not change its shape while the variance
is reduced, therefore assuming that there is no increase in
symmetry of the resulting distribution. Although there is no
additional explicit assumption about the point and SMU dis-
tributions, the permanence of shape assumption is limiting,
since it is known that the distribution shape will change as
the variable is averaged within larger volumes. Therefore, in
practice, the range of application of this method is limited to
small changes in variances, for which changes in distribution
shape are small.
The affine correction works by transforming each value
of the sample distribution into a different value of the SMU
distribution, according to the following relationship:
The indirect log-normal correction (ILC) is based on
the idea that the change of support is described by two
Log-normal distributions that have the same mean, but
different variances. This is considered true regardless of
the characteristics of the two original distributions (point
and SMU support), except that they need be positively
skewed.
The quantiles of the original distribution are transformed
into the SMU distribution following an exponential equation:
=
(7.9)
aq b
q
with the coefficients a and exponent b given by:
b
m
CV 2 + 1
m
a
=
f
·
CV 2 + 1
(7.8)
z
'
= −+
f zm m
(
)
 
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