Geoscience Reference
In-Depth Information
(
)
exp
y
s
k
(
u
)
ficient if one data is being used. Recall
that
σ ρρ
2
=− ⋅
1
(
)
M
+
1
k
i
,0
i
,0
x
s
(
u
)
=
alr
−
1
y
s
(
u
)
=
,
k
=
1,...,
d
2
(
)
in presence of one data, thus
=−
given that the
estimation variance is known, as is the case here.
The single super secondary variable is used with the pri-
mary data in the well known collocated cokriging equations:
ρ
1
σ
k
M
+
1
k
M
+
1
∑
d
exp
y
s
i
(
u
)
+
1
i
,0
k
M
+
1
i
=
1
d
(0.1)
∑
s
s
x
(
u
)
=−
c
x
(
u
)
DM
+
1
i
M
+
1
i
=
1
n
∑
y
*
= ⋅ +⋅
a y
cy
i
i
while a simulated model can be obtained, there are a number
of unresolved and unknown issues of this method. To begin
with, kriging of alr transformed data does not necessarily
result in an optimal solution of the original compositional
data in the simplex (Chap. 8). Also, the covariance between
known compositions and simulated compositions is not
necessarily correct, and the covariance between two simu-
lated compositions is not necessarily correct. And it is not
known whether the global statistics such as the variogram,
mean, variance and other moments are reproduced in the
simplex.
super
secondary
i
=
1
n
∑
2
σ
=− ⋅ −⋅
1
a
ρ
c
ρ
K
i
i
,0
super
secondary
i
=
1
10.6.7
Simulation Using Compositional Kriging
Compositional kriging is done without any transform of
compositions from the simplex to real space. Solving the
compositional kriging system of equations gives a vector
that adheres to the constraints of compositions. These do not
necessarily form the vector of expected values
m
*
and we
are not working with multivariate normal data or distribu-
tions. A covariance matrix can be calculated. Unfortunately,
the space of compositions is the simplex and there is no
definition of covariance and cross-covariance in that space
(Pawlowsky-Glahn and Olea
2004
).
Defining the multivariate distribution from which a
composition is simulated using this method is an outstand-
ing problem. We know two constraints on the distribu-
tion:
10.7
Post Processing Simulated Realizations
Conditional simulation generates many possible outcomes,
all with the correct variability. Dealing with the multiple
realizations has proven a difficult practical issue, typically
underestimated by geostatistical researchers.
Simulation corrects for the smoothness of kriging and is
theoretically free of conditional bias. Multiple realizations
allow for uncertainty assessment, so conditional simulation
techniques are required to assess joint uncertainty. Uncer-
tainty visualization is one aspect that needs to be considered
(Caers
2011
).
The following discusses in general terms some of the
most common CS applications in mining, including assess-
ment of point scale uncertainty, change of support, uncer-
tainty around existing mine plans and schedules, optimiza-
tion studies of various types, and recoverable reserves cal-
culation (Dimitrakopoulos and Ramazan
2008
; Dowd
1994
).
The multiple realizations represent at each location the
most information that can be gathered. The full range of pos-
sible values, described as a conditional cumulative distribu-
tion frequency (ccdf) is a model of local uncertainty.
Simulation allows for a better understanding of the vol-
ume variance relations. The basic idea is to simply simulate
values at a tight grid spacing, and then average up to the
relevant block size. This “brute force” approach is best to
observe volume-variance relations, since all other change
of support methods require assumptions that are difficult to
verify, or may be theoretical approximations.
More importantly, CS allows mimicking the mining pro-
cess itself, either through blast hole drilling and sampling, or
through production drilling in underground mines. Mining
∑
x
, but the shape of multivariate condi-
tional distributions is also needed. It is also not clear if
m
*
(the estimated averages) and
S
*
(the estimation variance) are
correctly parameterizing these distributions.
≥
0;
i
xc
=
i
Simulation using alr Cokriging
Under certain assump-
tions, simulation using alr cokriging can be accomplished. If
the distribution of compositions is assumed additive logistic
normal, then the alr transformed data is multivariate normal
distributed. At location
u
M
+1, kriging gives a multivariate
conditional distribution with mean and covariance param-
eters given by:
N
N
(
)
(
)
(
)
∑
∑
µ
∗
=
y u
∗
=+
c
Φ
alr
xu c
=+
Φ
yu
y
M
+
1
k
k
k
k
k
=
1
k
=
1
MM
{
}
(
)
∑∑
∗
∗
T
S
=
Cov
yu
(
)
=
Φ
Cu u
−
Φ
y
M
+
1
k
l
k
l
kl
==
11
Despite there being no analytical back transform of
m
*
and
S
*
, a vector,
y
s
(
u
M
+1), can still be simulated from the mul-
tivariate conditional distribution. The inverse alr transform
can be applied to recover a simulated composition
x
s
(
u
M
+1).
