Geoscience Reference
In-Depth Information
The corresponding co-kriging system would call for a matrix
of
K
2
direct and cross indicator covariances of the type:
The so-called probability kriging (PK) estimate (Sullivan
1984
), actually a ccdf estimate for
Z
(
u
), is written in its sim-
ple kriging form:
C
( ;
h
z
,
z
)
=
Cov I
{ ( ;
u uh
z
),
I
(
+
;
z
)}
I
kk
'
k
k
'
n
[
]
[
]
*
∑
i
(;
u
z
−
Fz
( )
=
λ
(; ) ( ; )
u
z
⋅
i
u
z
−
Fz
( )
k
k
α
k
α
k
k
PK
The direct inference and joint modeling of the
K
2
covari-
ances is not practical for large
K
. The shape of the cross
variograms is quite smooth relative to the direct indicator
variograms making it impossible to fit the set of covari-
ances/variograms with the linear model of coregionaliza-
tion. Also, the kriging matrices that need to be inverted
would be significantly larger. There has been however
some solutions proposed. Suro-Pérez and Journel (
1991
)
proposed to reduce the co-IK system by working on lin-
ear transforms of the indicator variables, which are less
cross-correlated, such as indicator principal components.
Another solution calls for a prior bivariate distribution
model.
A prior bivariate distribution model amounts to forfeit-
ing actual data-based inference of some or all of the in-
dicator (cross-) covariances. Most commonly, the bivari-
ate Gaussian model after normal scores transform of the
original variable is adopted:
Z
(
u
)→
Y
(
u
)=
(
Z
(
u
)). A slight
generalization of the bivariate Gaussian model is offered
by the (bivariate) isofactorial models used in disjunctive
kriging (DK). The generalization is obtained by a nonlinear
rescaling of the original variable
Z
(
u
) by transforms
(·)
different from the normal score transform
(·). In either
case, all indicator (cross-) covariances are then determined
from the Gaussian
Y
covariance. At this point, it makes
more practical sense to simply rely on the complete multi-
variate Gaussian model.
More importantly, general experience indicates that
co-IK improves little from IK (Goovaerts
1994
). Such is
the case if primary and secondary variables are equally
sampled, as happens with indicator data defined at various
cutoff values. In addition, when working with continuous
variables, the corresponding cumulative indicator data do
carry substantial information from one cutoff to the next
one; in which case, the loss of information associated with
using IK instead of co-IK is not as large as it appears. Fi-
nally, co-IK will generally create more order relation prob-
lems, which require additional manipulation of the esti-
mated cdf.
α
=
1
n
[
]
∑
+
v z
( ;
u
)
⋅
p
(
u
)
−
0.5
α
k
α
α
=
1
(
)
(
(
)
)
[
]
α
= ∈
is the uniform (cdf)
transform of the datum value
z
(
u
α
), the expected value of which
is 0.5;
pu
F zu
0,1
where
( )
{
( )
}
F
z
=
Prob
Z u
≤
z
is the stationary cdf of
Z
(
u
).
correspond to
The co-kriging weights
and
v
α
u
(; )
k
z
λ
u
(; )
k
z
the indicator and the uniform data, respectively.
The corresponding simple PK system requires the infer-
ence and modeling of (2K + 1) (cross-) covariances:
K
in-
dicator covariances,
K
indicator-uniform cross covariances,
and the single uniform transform covariance. That modeling
is still demanding and represents a drawback in practice. For
this reason, there have been few practical applications of PK.
9.8
Summary of Minimum, Good and Best
Practices
This section presents the details of what is considered mini-
mum, good, and best practices in probabilistic estimation, as
well as discussing specific presentation and reporting issues.
The block model obtained should be thoroughly checked and
validated, as detailed in Chap. 11, according to the criteria
suggested for minimum, good, and best practices. As before,
these validations should result in, among other comments, a
statement about whether the model can be considered “re-
coverable” or fully diluted.
In the case of having an estimated distribution of possible
values for each block, it is even more important to consider
the reporting and transmittal of the resource model to down-
stream clients. One possibility is to obtain the e-type average
for the block, and present it; however, having developed a
range of possible values, other information, such as probabil-
ities of exceeding important cutoffs, should be provided. In
this sense, it is important to consider that probabilities can be
seen as proportions of blocks, a concept that may make the
understanding and manipulation of this information easier
for downstream clients.
The minimum practice in estimating distributions should
include making the estimation implementation specific to
each estimation domain. All other criteria given in Sect. 8.4
should apply here, including documentation and justification
of the Kriging method chosen, the reporting of the ore re-
sources, which should be based on E-type estimates, the use
of grade-tonnage curves, and the full documentation of the
9.7
Probability Kriging
An alternative to indicator co-kriging is to use not only the
transformed indicators, but also their standardized rank
transforms, which are distributed in [0,1]. The idea is that
greater resolution could be achieved near data locations.
