Geoscience Reference
In-Depth Information
A special case is the probability of exceeding certain thresh-
olds, such as the economic cutoffs. Naturally, the e-type
mean and other truncated statistics can also be obtained.
In mining and other earth science applications it is criti-
cal to know the probability of the block grade exceeding a
cutoff grade:
{
}
=−
z
(;
u
z
)
= Prob
Z zn Fz
() |( )
u
≥
1
( ;
u
| ())
n
V
c
V
c
V
c
which results in the probability of the estimated block grade
being recoverable.
The uncertainty about any value
z
(
u
) can be derived from
probability intervals, such as:
Fig. 9.7
Order relation problems and their correction ignoring the class
(z
6
, z
7
] that did not contain any
z
data. The ccdf value
F
(
u
α
; z
7
|
(
n
)) is
ignored and the correction is applied to the remaining cutoff values
{
}
Prob Z
() [,]|()
u
∈
a b
n
=
F
(; |())
u
b
n
−
F
(; |())
u
a
n
V
V
V
where
a
and
b
are the thresholds that define the interval of
interest. For example, there is frequently more than one eco-
nomic cutoff. Aside from the higher grade mill cutoff, there
may be a lower grade cutoff used to determine the mate-
rial that is to be processed using a lower cost option, such a
leaching, or that may be simply stockpiled for later process-
ing. To adequately engineer the stockpile or heap leach it is
important to know the probability of the block falling within
the two economic cutoffs.
Probably the most common application of IK, however, is
an e-type estimate. The e-type estimate can be written:
a class [
z
k −1
, z
k
] that contains no
z
data. In such case the
indicator data set is the same for both cutoffs
z
k −1
and
z
k
and yet the corresponding indicator variogram models are
different; consequently, the resulting ccdf values will likely
be different with a good chance for order relation problems.
A solution would be to retain only those prior cutoff val-
ues
z
k
such that the class (
z
k
−1
,
z
k
] has at least one datum as
in Fig.
9.7
:
Order relation problems represent the most severe draw-
back of the indicator approach. They are the price to pay for
trying to reproduce (even approximately) more than a single
sample covariance.
K
+
1
∑
*
m
()
u
=
mi
(; )
u
z i
−
(;
u
z
)
k
k
k
−
1
k
=
1
Step 6: Interpolation between the
K
values of the
ccdf
Once the K ccdf values have been corrected, it is neces-
sary to interpolate values between thresholds and extrapolate
beyond the first and last threshold to obtain a complete dis-
tribution. A more complete description of the models com-
monly implemented can be found in Deustch and Journel
(
1997
).
Some implementations of IK assumes a particular distri-
bution model for the interval between two consecutive thresh-
olds such as the linear model, which assumes a uniform dis-
tribution and is generally accepted between thresholds; the
power model is commonly used for extrapolating the lower
tail, between 0 and threshold
z
1
, and sometimes the upper tail,
between threshold
z
k
and 1; and the hyperbolic model, which
is most used to control the extrapolation of the upper tail. In
practice, it is better to use the non-parametric global distribu-
tion shape adapted to the IK-estimated ccdf values.
where
i
(
u
;
z
k
) is the kriged indicator value for threshold
k
,
i
(
u
;
z
0
)
=
0,
i
(
u
;
z
K
+
1
)
=
1, and
m
k
is the (declustered)
class mean, that is, the mean of the data falling in the interval
[z
k −1
, z
k
]
. Note that if only the e-type is required, then no
correction for change of support is necessary, since
m
V
(
u
)
∗
=
∗
m
(
u
)
for variables that upscale linearly.
9.6
Indicator Cokriging
The IK estimators discussed above independently krige each
threshold and thus they do not make full use of the informa-
tion contained in the series of indicators. The full informa-
tion contained in the cross indicators can be accounted for
by using a co-kriging estimator across defined thresholds.
Note that this co-IK is not between multiple data types, but
between the same data type considering all possible thresh-
olds.
The co-indicator kriging (co-IK) estimate is then defined as:
Step 7: Using the IK model and calculating recoverable
resources
The ccdf distribution can be used to provide any
statistic of interest at location
u
. The ccdf is a measure of
uncertainty, from which probability intervals can be derived.
Kn
∑∑
*
[(;
i
u
z
)]
=
λ
==
(;
u
z i
) (; )
⋅
u
z
k
coIK
α
,
k
k
k
o
0
k
11
α
