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distribution assumption. This change of support model is
difficult to validate. The method is more useful if there is
production or closely spaced data that can be used to validate
the results.
Often IK is not used to provide the entire block ccdf
[
F
V
(
u
;
z
|
(
n
))]
however, the economic cutoff grades themselves need not
used; (3) each class should have a sufficient number of data
for robust interpolation. It is common to use between 8 and
15 thresholds, although there are exceptions.
∗
, but simply the e-type average of the block.
This is calculated from the point e-type averages within the
block. In this case, the process is similar to linear kriging,
and the practitioner must decide whether the extra work re-
quired in defining and solving the
k
-indicator kriging sys-
tems is worthwhile.
The results from an e-type ordinary IK can be quite dif-
ferent from a linear ordinary kriging. The geology of the
deposit and the characteristics of the spatial distribution of
the z-variable typically explain the differences. The criteria
used to decide whether an e-type IK estimate is worthwhile
are: (1) there is sufficient number of samples available for all
grade ranges defined by the k-thresholds; (2) the
z
-variable
shows a highly variable distribution, typically characterized
by a coefficient of variation above 1.0 and often above 2.0;
and (3) the indicator variogram models and other statistics
suggest overprinting of mineralization styles. A good refer-
ence on the application of Indicator Kriging is Zhang (
1994
).
Step 3: Modeling the indicator variograms
There are
K
indicator variograms to be modeled, one for each threshold
used. These variograms tend to have reasonably well-defined
structures and are easy to interpret, since there are no outli-
ers. There should not be any proportionality between them;
if there is, the proportional variogram models will result in
the same kriging weights, and thus one of the thresholds can
be dropped.
Indicator variograms standardize all points and models to
a unit variance of
p
p
)
,
where
p
is the proportion or
average value at that threshold. It is common practice to
work on standardized indicator variograms, where the sill of
each variogram is divided by their respective variance
p
k
·
(1
−
p
k
). In this case, the model is interpreted to the
theoretical sill of 1.0, making the joint modeling of the indi-
cator variograms easier and more consistent.
It is advisable also to use the same type of structure for
the models, since they all relate to the same RF. They can be
spherical, exponential, or any other licit model; however, the
contribution of each structure will vary from one threshold to
the next, usually modeled as a smooth transition.
The changes in terms of variance contribution and anisot-
ropy ranges and directions should be smooth from one indica-
tor to the next. It is good practice to plot the relative nugget ef-
fects, short scale ranges, and the anisotropy parameters for all
thresholds to check the transitions from one model to the next.
For positively skewed distributions, higher thresholds
will show less continuity, so a de-structuring effect is ex-
pected and can be quantified.
While smooth transitions from one model to the next are
desirable because they minimize order relation problems,
different patterns of anisotropy can and should be expected.
This is particularly true if multiple populations co-exist, or
an explainable change in geologic control is evident. For ex-
ample, low values in many mineral deposits tend to be more
disseminated into the host rock; while higher grade miner-
alization tends to be more spatially restricted and located in
favorable structures, veins, or higher porosity areas of the
host rock. In such a case, the thresholds for lower grades will
be more isotropic, while the anisotropy ratios will increase
for higher grade thresholds.
·
(1
−
9.5
The Practice of Indicator Kriging
There are several steps required to complete an indicator
kriging estimate as described in the literature (Journel
1983
;
Deustch and Lewis
1992
), whether the final objective is a point
ccdf, an e-type point or block estimate, or a composite ccdf.
Step 1: Obtaining an unbiased (declustered) global histo-
gram
The available samples over the domain
A
may not be
representative of the domain due to spatial clustering. The idea
of IK can be seen as a modification of the prior cdf derived
from the sample for the entire domain to a posterior ccdf spe-
cific to each location. The more representative prior cdf used
should result in a better, more representative local ccdf.
Step 2: Choosing the
K
threshold values
Indicator kriging
begins with establishing a series of thresholds. The number
of thresholds chosen is a trade-off between the amount of
data available, and the resolution sought in the ccdf model
F
(
u
;
z
k
). Also, the indicator variograms should be suffi-
ciently different in spatial structure.
While there should be a sufficient number of thresholds
to obtain good resolution, too many cutoffs induce more
order relation problems. Criteria used to define the thresh-
olds include: (1) the classes should define approximately
equal-quantity of metal, and not equal amplitude; (2) addi-
tional thresholds are usually placed around z-values that are
consequential to the project, such as economic cutoff grades;
Step 4: Kriging plans
The strategy for implementing the
estimation follows the same general criteria discussed in
Chap. 8. However, there are specific recommendations for
implementing IK.
The same search neighborhood and the same number of
data should be used for all
K
thresholds. Even if the indicator
