Geoscience Reference
In-Depth Information
ods. Bulk density is sometimes forgotten and few good sam-
ples are available. Parrish (
1993
) has a good discussion on
bulk densities and their importance to resource estimation.
Another group of raw variables that are commonly mod-
eled are auxiliary variables. Some examples include thick-
nesses of formations, typical of sedimentary deposits; eleva-
tions of the top or bottom of particular surfaces of interest,
such as bedrock contact; the geometry of diamondiferous
pipes; top of sulfides enrichment; footwall and/or hanging
wall positions in tabular deposits. Sometimes, variables
such as grade multiplied by thickness (proportional to metal
content) is used in tabular deposits. This grade times thick-
ness variable transforms a 3-D modeling problem into a 2-D
exercise, because the third spatial dimension, usually much
smaller than the other two, is incorporated into the variable
being estimated.
Another important group of variables are metallurgical
performance variables. Resource and reserve models should
include predictions of rock characteristics, crushing/grind-
ing throughputs, final product recoveries, and other variables
as a more realistic basis for cash flow predictions. There is
a trend in the industry to model geometallurgical variables
and include them in resource models. They can be ore and
gangue mineralogical variables, useful for better predicting
plant performance, concentrate grade, and heap or vat leach
performances. Some of these variables do not average lin-
early, and thus require special consideration.
Resource models must consider a breadth of issues that
were simplified in the past including all types of dilution
(Chap. 7) and geologic variables that affect mine and plant
performance including geo-mechanical and geo-metallurgi-
cal variables. A resource model is much more than a geologi-
cal in-situ model.
close-spaced while the hard data will tend to be located at
a much larger spacing. This is characteristic of geophysical
data, where a dense grid of data is available.
Second, the soft variable may be a simple condition, such
as “within this rock type, the Au grade will be no larger the
1.0 g/t”. The soft information is qualitative in nature and
somehow must be expressed in a numeric format before it
can be used explicitly in the modeling process to any ad-
vantage.
A common procedure is to turn the soft information into
an indicator or a series of indicators that allows merging it
with the hard data. The details of the modeling techniques
are discussed in Chap. 9.
5.5.3
Compositional Data
The following discussion is a summary of CCG (Centre for
Computational Geostatistics) Guidebook 7 (Manchuk
2008
).
Compositional data are multivariate data where the variables
or components represent some part of a whole (Pawlowsky,
1989
; Pawlowsky et.al.,
1995
). All variables from a compo-
sition are measured on the same scale and unit system and
are constrained by a constant sum property. The sum de-
pends on the measurement scale. Some common ones are 1
for fractional data, 100 for percentages, and 10
6
for parts per
million or ppm. A set of variables summing to a constant is
also referred to as a closed array (Chayes
1962
). The concept
of a compositional data set
X
with
D
components and
N
ob-
servations can be written as:
X
D
≡
{
x
,...,
x
:
x
≥
0
,...,
x
≥
0 X1
;
D
=
1
N
}
1
D
1
D
Note that
1
D
is a column vector of ones of size
D
and
1
N
of
size
N
.
Two issues for statistical analysis are raised by this equa-
tion: (1) variables are not free to range in (− ∞, + ∞), thus the
relationships are not free to vary independently, and (2) the
constant sum constraint must force at least one covariance or
correlation to be negative; when one component gets large,
the others must necessarily decrease. Correlations are not
free to range in [−1,1] causing spurious correlations (Aitchi-
son
1986
).
5.5.2
Soft Data
Soft data is a term used in reference to information that pro-
vides imprecise or indirect measurements of the variables of
interest. Some specific examples include geophysical read-
ings such as magnetic anomalies associated with an iron
(magnetite) deposit. Another example is the use of radiomet-
ric readings (a gamma probe) to obtain the ratio of parent-to-
daughter products of Uranium
238
(U
238
) decay chain, from
which, and after proper calibration, the U
238
grade may be
estimated.
Soft data does not have the same quality of information as
compared to hard data, which is typically the assayed miner-
al grade. Depending on the specifics of the indirect measure-
ments, there are two general characteristics that determine
the methodology used to analyze and apply the soft data.
First, the quantity of soft data may be significantly larger
than the hard data, but of poorer quality. The data may be
5.5.3.1 Compositions in Natural Resources
In a natural resources context, compositions are geochemi-
cal, geophysical, or lithological. Whole rock geochemistry is
an example, and may come in various forms depending on
the depositional environment and target resource. Consider
mining to extract metal products, such as copper suplhides.
Several mineralogical species may be responsible for all the
Cu in a sample, such as chalcopyrite, chacolsite, covelite,
