Geoscience Reference
In-Depth Information
There are a number of opportunities for potential improve-
ment on the modeling methodology presented in this case
study: (1) optimize the merging of the variables at the two
different levels. The merging of the variables was done using
logical groupings of the 112 variables. An optimization pro-
cedure could be developed to select ideal subsets of variables
to increase the predictive power of the regression model.
(2) Improve the selection of the set of variables to use for
each variable predicted. In this work, all 112 variables were
used for all 6 output variables. Eliminating some of the less
significant variables may reduce noise and increase model
accuracy.
modeled using sequential Gaussian co-simulation for the p
20
,
p
50
and p
80
values of each mineral.
14.7.5
Modeling 23 Head Grade Variables
The plant performance modeling requires 23 head grade
variables for input into the linear regression models: Cu,
U3O8, Ag, Au, Co, Mo, Pb, Zn, Ba, Fe, Al, Si, K, Ca, S,
Co2, La, Mg, Mn, Na, P, Ti, Ce. These 23 variables are simu-
lated on a grid with the following dimensions: x
min
= 56,105;
y
min
= 30,515; z
min
= -1932.5; x
siz
= 10; y
siz
= 10; z
siz
= 15;
n
x
= 360; n
y
= 624; n
z
= 119. There are a total of 111,572 head
assay samples used in the modeling. The K:AL ratio and
B
adj
S are also required, but are simply calculated from the
realizations of K, Al, Ba, and S.
The head grade variables are considered composi-
tional because all chemical and mineral rock components
must sum to 100 %. Because not all elements in a sample
are assayed, the sum of the head grades is always less
than 100 %. However, in geostatistical modeling if this
constraint is not explicitly imposed it can be violated in
some areas of the model. For this reason a logarithmic
transform of 24 head grade variables is considered, with
the 24th variable imposing the 100 % constant sum (23
variables listed above + 1 filler variable). The logarithmic
transform is:
14.7.4
Part II: Multivariate Compositional
Simulation of Non-additive
Geometallurgical Variables
As shown, recovery and plant performance outcomes are
influenced by a large number of variables, including head
assays, mineralogy and mineral associations. Models that
utilize all these variables outperform models based on head
assays alone. The compositional nature of the variables must
be accounted for, and many of the variables are correlated.
In the proposed methodology, data transformations are
used to maintain the compositional nature of the variables
and PCA analysis is used to decorrelate the relationships
between variables to make geostatistical modeling more
straightforward.
Modeling methodologies are developed for a total of 135
variables, separated into three groups: head grade assay val-
ues; grain size measurements; and mineral associations. Sig-
nificantly more samples exist for the head grade variables,
therefore they are modeled first. The grain size and associa-
tion variables are modeled using the head grade realizations
as secondary information. This ensures that the spatial dis-
tribution of these variables are consistent with the deposit
overall.
The head grade and mineral association data are consid-
ered compositional, that is, they are non-negative and sum to
100 %. A logarithmic transform is used to deal with this con-
stant sum constraint. Normally, these variables would then
be co-simulated with sequential Gaussian simulation (SGS;
Isaaks
1990
); however, the large number of variables avail-
able and the large grid size makes this procedure too compu-
tationally intensive. An alternative is to perform a principal
component (PCA) transform on the logarithmic data to gen-
erate uncorrelated variables. SGS is then preformed on the
uncorrelated PCA values. The values are back-transformed
into original units to generate the realizations. This proce-
dure is used to model the head grade and mineral association
data. The grain size data, which are not compositional, are
†
‡
x
y
=
ln
i
ˆ
‰
i
x
Š
‹
filler
where
y
i
is the new variable to be modeled and
x
i
are each of
the 23 variables to be modeled. This transformation requires
that there are no zero values for any variable as ln(0) is un-
defined. The back transformation is
y
e
i
x
=
i
24
∑
y
e
+
1
i
i
=
1
There are now 23 logarithmic transformed variables.
There are complex relationships among these 23 variables
(Fig.
14.65
). It would be difficult to reproduce all these re-
lationships with traditional SGS. The PCA transform is used
to generate 23 new
uncorrelated
variables. These variables
are linear combinations of the 23 logarithmic variables but
are uncorrelated. An assumption of independence between
the 23 variables is then made and all 23 PCA variables are
modeled independently with SGS. This ensures good repro-
duction of the correlation between the 23 variables in the
final realizations.
