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for each class (right). Weights applied to conditional Pb distribution all
shown in light blue, gray, and orange shaded regions of the Zn histo-
grams (Leuangthong et al.
2006
)
Fig. 14.41
Schematic illustration showing multivariate calibration
data, declustered Zn and Pb histograms to be determined (left). The
division of multivariate calibration data into multiple classes, with dis-
tributions on the right representing the conditional distribution of Pb
The transformation order reflects the significance of each
variable. All variables other than Zn (main variable) and Pb
(main secondary variable) were transformed conditional to
either Zn and Pb or Zn and Fe. The choice of the second-
ary variable in each transform order reflects the relationship
between the secondary and tertiary variable. This is not mea-
sured by the correlation coefficient alone; non-linearities and
constraint features (if present) need to be examined as well in
crossplots between the different elements. The determination
of the secondary and tertiary variable was based on careful as-
sessment of the relevant bivariate and trivariate distributions.
Declustering was performed to compensate for prefer-
ential drilling. Given the multivariate nature of this dataset
and the intended application of a multivariate transforma-
tion technique, declustering must be consistent between all
variables. Declustering was performed within domains, and
the declustered Zn distribution obtained using the accumu-
lated weights from kriging within a rock type. This approach
considers drill hole data redundancy through the spatial vari-
ability model and by domain.
Secondary variables were declustered through a bivari-
ate calibration of the Pb distribution using both the repre-
sentative distribution of Zn and the crossplot of Zn and Pb.
Specifically, the declustered Zn distribution is divided into
a series of classes and the corresponding conditional distri-
butions of Pb are determined. The declustered Pb distribu-
tion is then constructed by accumulating all the conditional
distributions weighted by the declustered Zn probability for
the corresponding class (see Fig.
14.41
). For all tertiary vari-
ables, the same rationale was applied, and the declustered
distributions for Fe through TOC were determined using the
declustered distributions of the two dependent variables plus
the trivariate calibration data.
The Stepwise Conditional Transformation was then per-
formed on the declustered distributions. Figure
14.42
shows
the scatterplots of the variables resulting from the first trans-
form sequence of Zn, Pb and Fe (see Table
14.17
). The trans-
formed variables are independent and multiGaussian, which
translates to a circular shape in the crossplot. From Fig.
14.42
,
crossplots with the third variable (Fe, in this case) show some
banding; however this is simply a numerical artefact of hav-
ing many classes and consequently fewer data within each
class (Leuangthong
2003
). This banding does not impact
data reproduction. Independence of the transformed variables
means that each variable can be simulated independently.
Variograms were then calculated and modeled for each of
the transformed variables. Figure
14.43
shows an example
of the horizontal and vertical variogram models for the step-
wise conditionally transformed Zn, Pb, Fe and Ba for one
rock type. Note that secondary and tertiary variables exhibit
relatively high nugget effect; this is explained by the inde-
pendence imposed by transforming each class separately
(Leuangthong and Deutsch
2003
).
Sequential Gaussian simulation was then independently
performed for the seven transformed variables. A total of 40
realizations were generated for each variable within each do-
main. Only those blocks belonging to the specific rock type
were simulated. Each realization was then
back transformed
to the original units of the data. The back transformation for
each simulated realization is also conditional. For example,
the back transform of Fe is conditional to the simulated val-
ues for Zn and Pb.
The simulations were thoroughly checked to ensure re-
production of (1) the composite values at their respective lo-
cations, (2) the histogram and associated summary statistics,
and (3) the variograms in Gaussian space of the stepwise
transform scores. For this multivariate simulation, the mul-
tivariate relations were also checked. The simulated models
were then upscaled to 25 × 25 × 25 ft blocks to compare to the
existing long term model.
