Geoscience Reference
In-Depth Information
ficients for the SMU and panel sized blocks; (4) transform
the
Z
(
V
)
using the panel anamorphosis function; (5) transform the
Z
c
(
v
) cutoff grades to
Y
c
(
v
) using the SMU anamorphosis
function; and (6) calculate the proportion and quantity of
metal above each cutoff.
The first step in UC is to estimate the panel grades, most
commonly using ordinary kriging. Uniform conditioning re-
lies on a robust estimate of the panel grade. Also, the panel
grade estimates have to be uniformly robust, that is, with
nearly the same amount of data. Large panels are used be-
cause the block kriged panel grades are more robust for larg-
er panels. Also, in UC the SMUs are seen as discretizing the
larger panel, and there should be a sufficient number of them.
Ordinary kriging in the original grades units is the most pop-
ular option used, although the panel grade can also be esti-
mated directly in Gaussian units, see, for example, Guibal
(
1987
). The size of the panel and the SMUs must first be
chosen to complete the OK of the panel. Also, the grade di-
rectional variograms and corresponding models
∗
(
V
)
panel estimates to the Gaussian variable
Y
∗
Fig. 9.1
Schematic showing the setting for UC. Notice that there are
three support sizes of interest: the data, the SMUs and the panels
γ
h
must
be obtained. Although not required, it is common practice to
set the discretization of the block (panel) kriging to the SMU
resolution within the panel.
The second step is to fit a DGM model to the point scale
data, which is used to perform the change of support from
sample support to the larger panels and also to the SMU-size
blocks. The DGM model was described in Chap. 7, and can
also be found in several references, for example Journel and
Huijbregts (
1978
, p 472). Figure
9.2
shows a graphical rep-
resentation of Gaussian anamorphosis.
The DGM model is applied to get the anamorphosis func-
tion for the SMU sized blocks. The same procedure is re-
peated to calculate the change of support for the panel sized
blocks, using the
r
' coefficient corresponding to the larger
panel size
V
. The dispersion variance calculated from the
variogram models for the panels provides the theoretical
variance to be used to obtain
r
'.
The panel variance is found as before solving for
r
' in the
following equality:
()
Fig. 9.2
Graphical representation of anamorphosis and the transforma-
tion of grades
slightly larger is used. The panel grades should be estimated
with little uncertainty. Figure
9.1
shows a schematic of the
scales that uniform conditioning considers.
The earliest reference to UC is Matheron (
1974
). Some
further references for UC are Remacre (
1987
), Guibal and
Remacre (
1984
), Rivoirard (
1994
), Vann and Guibal (
1998
),
Chilès and Delfiner (
2011
), and Roth and Deraisme (
2000
).
In general, there are few published detailed references on
the theory and practice of UC. The method has gained some
popularity due to its simplicity and availability in commer-
cial software. The following description is based on the more
recent compilation by Neufeld (
2005
).
There are 6 basic steps required to complete UC: (1) esti-
mate the panel grades; (2) fit the Discrete Gaussian Model
(DGM) to the data; (3) determine the change of support coef-
n
=
∑
2
22
p
σ
()
r
φ
'
V
p
p
=
1
It can be shown that the correlation coefficient of the bi-
Gaussian distribution
[
]
Y
vV
is:
( ),
(
)
r
'
ρ
(, )
vV
=
r
Figure
9.3
shows the anamorphosis transformation for both
the panel and the SMU grades.
