160
ϕ
β
β
β
β
1
1
β β
2
β β
2
β β
2
1
1
12
n
OMCK
ϕ
n 2
C 1
=
( )
x
=
( )
x
∑
∑
ϕ β
ϕ ϕ
β
1
β
1
1
11
OMCK
ϕ
n 1 1
C
=
( )
x
( ) (
)
1
1
β
α
1
α
β
α
( )
x
11
1
1
1
(
)
α
x
1
2
1 1
α α α α −
+
( ) (
)
1
−
+
( )
(
)
1
= 1
) +
( C (
C
)
n 1
OMCK
OMCK
2
1 2
)
2 2
( ) ( 1
2
) +
( C 12 (
1
ϕ
O
2
μ
α
11
1
( ) 11 (
)
1
C 1
) = 1
−
OMCK
1
( )
u
( (
( ) (
)
( )
=
( )
=
( )
∑
)
( ) (
∑
∑
( ) (
) +
=
( )
ϕ
∑
( ) (
) +
β
β
β
β
1
1
β β
2
β β
2
β β
2
1
1
22
n ( (
OC
ϕ
K
C C
n (
2
) )
−
C
) ) 2
ϕ
μ
α
2
2 2
21
2
2
α α 1
(
)
x α
β
α α 2
2
α
β
α
β
α α −
+
( ) (
)
α 1
α α −
+
( )
(
)
α α 2
α α α
= 1
ϕ 2
ϕ
)
β
+
( C 22 (
C
)
α
n 2
OMCK
ϕ
OC
μ
K
C
( )
u
( (
( ) +
( ) =
⎧
⎨
⎪
⎧ ⎧
⎪
⎪ ⎪
=
( )
( )
∑
∑
( ) +
=
( )
ϕ
∑
( ) +
ϕ
β
β
β
1
β β
2
β β
1
β
1
1
=
1
( ) ∑
( ) ∑
( )
β
β
1
1
β β
2
ϕ
1
n ( (
OC
ϕ
K
C C
n (
1 1
OMCK
ϕ
(
ϕ
∑
) +
β
OCK C C
⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎪
⎨ ⎨
⎪ ⎪ ⎪ ⎪
⎩
⎪
⎨ ⎨
⎪
⎪ ⎪
⎪
⎪ ⎪
⎪
⎩ ⎩
⎪ ⎪
(2)
with the same notations as in the previous subsection.
2.3 Coregionalization modeling
Solving the cokriging system in both cases, requires the knowledge of the direct and crosscovariances between the primary and secondary variables. In this respect, the linear model of
coregionalization is widely used to fit such covariances, owing to its mathematical simplicity
and tractability (Journel and Huijbregts, 1978; Goovaerts, 1997; Wackernagel, 2003). In this
model, the direct and cross-covariances C ij
C C ( )
h (
)
i j =
j
are defined as weighted sums of L
basic covariances, also called basic nested structures:
C
b c
ij
C C
ij
b b
l
l
l
L
( )
h =
( )
h
=
∑ 1
(3)
where, for each structure (
)
l
L
… , ( ) ,
ij
l
i j
, =1 2
, is a 2 × 2 real-valued, symmetric, positive semidefinite matrix (coregionalization matrix) and c h
l ( )
h is a permissible stationary covariance
model (basic nested structure). In practice, such a model can be fitted to a set of experimental
direct and cross-covariances by means of semi-automated algorithms (Goulard and Voltz,
1992; Emery, 2010).
2.4 Proposed algorithm
The multi-collocated cokriging is subjected to availability of the secondary variable at
target locations. This requirement is frequently met in oil reservoir characterization, since
the secondary variable (e.g. seismic) is exhaustively available entire the region. Taking into
account this valuable information, the primary variable such as porosity and permeability can
be modeled in 3-dimension by multi-collocated cokriging. However, in the mining industry,
the secondary information may be rarely available exhaustively at target location and instead,
one, commonly is dealing with different sampling patters such as heterotopic, in which the primary and secondary variables share partially some locations and leading to more availability
of secondary variable. In this case, an algorithm proposed in this study, namely hierarchical
searching strategy, in order of utilizing the multi-collocated cokriging system for mining applications. This theory is inspired from hierarchical joint simulation algorithm already proposed
by Almeida and Journel (1994) for simulating the variables in turn rather than simultaneously
taking into account the collocated cokriging. Interested readers are referred to Goovaerts
(1997) for more information about the latter approach. Based on this, the proposed algorithm
for cokriging of two variables in multi-collocated cokriging, proceeds as follows:
1. Define the hierarchy of variables, starting with the most important or better autocorrelated variable with more availability, Y 2
2. Estimate Y 2 at target locations x by ordinary kriging to define the parameters of conditional
cumulative distribution function of the secondary variable Y 2 at all the target locations.
The conditioning information in this step is just the values of secondary variable.
3. Use ordinary multicollocated cokriging to estimate Y 1 taking into account the information available from Y 2 , already estimated at target locations and also the original Y 2
accompanying the primary variable Y 1 at sample locations.
ϕ
β
β
β
β
1
1
β β
2
β β
2
β β
2
1
1
12
n
OMCK
ϕ
n 2
C 1
=
( )
x
=
( )
x
∑
∑
ϕ β
ϕ ϕ
β
1
β
1
1
11
OMCK
ϕ
n 1 1
C
=
( )
x
( ) (
)
1
1
β
α
1
α
β
α
( )
x
11
1
1
1
(
)
α
x
1
2
1 1
α α α α −
+
( ) (
)
1
−
+
( )
(
)
1
= 1
) +
( C (
C
)
n 1
OMCK
OMCK
2
1 2
)
2 2
( ) ( 1
2
) +
( C 12 (
1
ϕ
O
2
μ
α
11
1
( ) 11 (
)
1
C 1
) = 1
−
OMCK
1
( )
u
( (
( ) (
)
( )
=
( )
=
( )
∑
)
( ) (
∑
∑
( ) (
) +
=
( )
ϕ
∑
( ) (
) +
β
β
β
β
1
1
β β
2
β β
2
β β
2
1
1
22
n ( (
OC
ϕ
K
C C
n (
2
) )
−
C
) ) 2
ϕ
μ
α
2
2 2
21
2
2
α α 1
(
)
x α
β
α α 2
2
α
β
α
β
α α −
+
( ) (
)
α 1
α α −
+
( )
(
)
α α 2
α α α
= 1
ϕ 2
ϕ
)
β
+
( C 22 (
C
)
α
n 2
OMCK
ϕ
OC
μ
K
C
( )
u
( (
( ) +
( ) =
⎧
⎨
⎪
⎧ ⎧
⎪
⎪ ⎪
=
( )
( )
∑
∑
( ) +
=
( )
ϕ
∑
( ) +
ϕ
β
β
β
1
β β
2
β β
1
β
1
1
=
1
( ) ∑
( ) ∑
( )
β
β
1
1
β β
2
ϕ
1
n ( (
OC
ϕ
K
C C
n (
1 1
OMCK
ϕ
(
ϕ
∑
) +
β
OCK C C
⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎪
⎨ ⎨
⎪ ⎪ ⎪ ⎪
⎩
⎪
⎨ ⎨
⎪
⎪ ⎪
⎪
⎪ ⎪
⎪
⎩ ⎩
⎪ ⎪
(2)
with the same notations as in the previous subsection.
2.3 Coregionalization modeling
Solving the cokriging system in both cases, requires the knowledge of the direct and crosscovariances between the primary and secondary variables. In this respect, the linear model of
coregionalization is widely used to fit such covariances, owing to its mathematical simplicity
and tractability (Journel and Huijbregts, 1978; Goovaerts, 1997; Wackernagel, 2003). In this
model, the direct and cross-covariances C ij
C C ( )
h (
)
i j =
j
are defined as weighted sums of L
basic covariances, also called basic nested structures:
C
b c
ij
C C
ij
b b
l
l
l
L
( )
h =
( )
h
=
∑ 1
(3)
where, for each structure (
)
l
L
… , ( ) ,
ij
l
i j
, =1 2
, is a 2 × 2 real-valued, symmetric, positive semidefinite matrix (coregionalization matrix) and c h
l ( )
h is a permissible stationary covariance
model (basic nested structure). In practice, such a model can be fitted to a set of experimental
direct and cross-covariances by means of semi-automated algorithms (Goulard and Voltz,
1992; Emery, 2010).
2.4 Proposed algorithm
The multi-collocated cokriging is subjected to availability of the secondary variable at
target locations. This requirement is frequently met in oil reservoir characterization, since
the secondary variable (e.g. seismic) is exhaustively available entire the region. Taking into
account this valuable information, the primary variable such as porosity and permeability can
be modeled in 3-dimension by multi-collocated cokriging. However, in the mining industry,
the secondary information may be rarely available exhaustively at target location and instead,
one, commonly is dealing with different sampling patters such as heterotopic, in which the primary and secondary variables share partially some locations and leading to more availability
of secondary variable. In this case, an algorithm proposed in this study, namely hierarchical
searching strategy, in order of utilizing the multi-collocated cokriging system for mining applications. This theory is inspired from hierarchical joint simulation algorithm already proposed
by Almeida and Journel (1994) for simulating the variables in turn rather than simultaneously
taking into account the collocated cokriging. Interested readers are referred to Goovaerts
(1997) for more information about the latter approach. Based on this, the proposed algorithm
for cokriging of two variables in multi-collocated cokriging, proceeds as follows:
1. Define the hierarchy of variables, starting with the most important or better autocorrelated variable with more availability, Y 2
2. Estimate Y 2 at target locations x by ordinary kriging to define the parameters of conditional
cumulative distribution function of the secondary variable Y 2 at all the target locations.
The conditioning information in this step is just the values of secondary variable.
3. Use ordinary multicollocated cokriging to estimate Y 1 taking into account the information available from Y 2 , already estimated at target locations and also the original Y 2
accompanying the primary variable Y 1 at sample locations.