Geoscience Reference
In-Depth Information
The surfaces were interpreted individually from the top
down. In the case of the units above TDS, they were interpo-
lated probabilistically using indicator kriging (see details in
Sect. 14.3). Care must be taken to ensure that the surfaces do
not cross each other as the interpretation progresses from one
section to the next. The proto-ore limit TDCpy shown above
the topographic surface (Fig. 3.4c ) is only an interpretational
technique to produce a fully enclosed volume.
There are different manual and semi-automatic tech-
niques that can be used to produce an interpretation such as
this. The specific procedure depends in part on the software
available to do the work. Regardless of the details, thorough
checking and validation is necessary to ensure that the mod-
eling process occurred as intended.
produces a single boundary. This is much like doing a tradi-
tional interpretation and wireframing. However, a modified
distance function, DF mod can consider the uncertainty using
parameters C and β , creating a range of probable boundaries.
The vein geometry and corresponding tonnage uncertainty
can be calculated using these different vein boundaries.
To calculate the distance function, assume that a first
sample is non-vein and has an indicator of 0, VI = 0. The
distance function is the distance to the nearest sample with
indicator of 1, VI = 1. This sample could exist next to the
original sample if located at the contact between vein and
non-vein or in a nearby drillhole if located at some distance
from the vein, see Fig. 3.5 . The actual distance is then modi-
fied depending on the value of the indicator VI. Consider
the DF:
3.2.1
Distance Functions and Tonnage
Uncertainty
(
)
dx
2
+++ ∀
dy
2
dz
2
C
VI = 0
=
DF
(
)
2
2
2
dx
+++⋅ ∀
dy
dz
C
-1
VI = 1
Geologic modeling with extensive interpretation and digiti-
zation is recommended; trained professionals can understand
a great deal about the geometry of the deposit. Stochastic geo-
statistical techniques such as indicator simulation, truncated
(pluri)Gaussian or other techniques often create models that
are very random. A relatively recent approach to geologic
modeling is to use a signed distance function (DF) that maps
the location of boundaries and at the same time allows for an
assessment of the uncertainty. This uncertainty is represent-
ed spatially by a zone (or bandwidth) that is quantifiable and
needs to be calibrated. The DF is calculated directly from in-
dividual drillhole samples coded with a distance, rather than
a wireframe model. This approach is currently applicable to
binary geologic systems, with only two geologic domains, as
in for example vein-type deposits, although further develop-
ment into multivariate systems is ongoing.
Changing the DF impacts the size and shape of the zone
of uncertainty. Two parameters, the distance function uncer-
tainty component, C, and the distance function fairness com-
ponent beta, β , are used to modify the DF. The C parameter
controls the bandwidth and therefore the uncertainty. The β
parameter controls the position of bandwidth. With proper
calibration, values of C and β can result in models that are
both accurate and precise.
The DF is the Euclidean distance between different types
of samples. The distance is the shortest distance to a sample
with a different rock type (vein or non-vein). The distance is
given a positive sign in one rock type and a negative sign in
another. The contact between samples has a distance func-
tion of zero. An isoline connecting successive 'zero' points
defines the iso-zero surface or shell.
The vein geometry (and tonnage) uncertainty cannot be
calculated directly using an Euclidean distance because it
2
2
2
where,
++ is the Euclidean distance between
the current point and the closest point with a different VI, C
is the uncertainty parameter. When the indicator VI is 0, or
non-vein, the DF returns a positive value equal to the dis-
tance plus the uncertainty parameter C . If the indicator I V is
1 signalling the presence of vein, the DF returns a negative
value equal to the distance plus the uncertainty parameter C.
The distance from −  C to + C is defined as the uncertainty
bandwidth.
dx
dy
dz
3.2.1.1 Uncertainty Parameter C
The parameter C must be calibrated so that the width of un-
certainty is neither too large nor too small. Consider two drill
holes (Fig. 3.6 ), one with a vein intercept, the other without,
separated by a distance ds that represents the typical drill
hole spacing. The true vein boundary, or iso-zero bound-
ary of the vein must exist at some location between the two
drillholes. The drill hole distance, ds , is the maximum geo-
logically reasonable distance that can be assigned to C and
is equal to the drill hole spacing. For example, in Fig. 3.6 ,
the mid-point between the holes could be a likely position of
the contact, the iso-zero boundaries. However, the vein could
extend to almost any point in between, with higher or lower
probability depending on the local geology.
The uncertainty parameter C is not designed to define
the location of the iso-zero boundaries but rather to define a
reasonable bandwidth of uncertainty associated with it. The
uncertainty bandwidth cannot be greater than the drill hole
spacing.
Widely spaced drill holes would suggest a large band-
width, which would produce large vein boundary and ton-
 
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