Geoscience Reference
In-Depth Information
8.3 Cokriging
Let y(
u
) be the secondary variable; the trend model is
then:
The term kriging is reserved for estimation using data from
the same attribute as that being estimated. For example, an
unsampled gold grade value
z
(
u
) is estimated from neighbor-
ing gold grade values.
Cokriging is a similar estimate that uses data defined on
different attributes. For example, the gold grade
z
(
u
) may be
estimated from a combination of gold and copper samples
values. There must be a spatial correlation between the pri-
mary and secondary variables that can be inferred from avail-
able information. As is the case when considering a single
variable, there are three basic variants of cokriging: simple
cokriging (SCK), ordinary cokriging (OCK), and cokriging
with a trend model (CKT). Conceptually these cokriging
methods are the same as the ones explained above; however,
there is the additional complication of dealing with at least
two variables, which is reflected in the heavier notation.
EZ
{ ()}
u
= =+
m
()
u
a
a y
1
()
u
0
1
y
(
u
) is assumed to reflect the spatial trends of the
z
variabil-
ity up to a linear rescaling of units, corresponding to the two
parameters
a
0
and
a
1
.
The estimate of the z variable and the corresponding sys-
tem of equations are identical to the KT estimate and system
with K = 1, and f
1
(
u
) = y(
u
), i.e.:
n
∑
*
(
KT
)
Z
( )
u uu
=
λ
( )
Z
(
), and
KT
i
i
i
=
1
�
š
n
∑
(
KT
)
λ
u uu u uu
uu
() (
C
−+ +
)
µ
()
µ
()( )
y
j
Rj
i
0
1
i
š
=
j
=
1
C
(
−
), i
=
1,...,
n
š
�
š
R
α
∑
∑
n
(
KT
)
λ
() 1
u
=
j
j
n
=
1
š
š
(
KT
)
λ
()( )
uu u
y
=
y
()
j
j
�
j
=
1
8.3.1
Simple Cokriging
KT
λ
's are the kriging (KT) weights and the
µ
's
are Lagrange parameters.
Kriging with an external drift is an efficient algorithm
to incorporate a secondary variable in the estimation of the
primary variable
z
(
u
), and is appropriate for linearly related
secondary data. The fundamental relation between the two
variables must make physical sense.
In mining, there are few cases where this technique has
been applied. There are two reasons for this: (1) primary
z
(
u
)
data sets in mining tend to be large, and the addition of a
linearly-related trend from a secondary variable is difficult
to justify; and (2) there are few variables where this linear
relationship can be safely assumed. Kriging with an external
drift is more common in other applications. For example, if
the secondary variable
y
(
u
) represents the travel time to a
seismic reflective horizon, assuming a constant velocity, the
depth
z
(
u
) of that horizon should be proportional to the travel
time
y
(
u
). Hence a relation of this type makes sense.
Two conditions must be met before applying the external
drift algorithm: (1) The external variable must vary smoothly
in space, otherwise the resulting KT system may be unstable;
and (2) the external variable must be known at all locations
u
0
of the primary data values and at all locations
u
to be estimated.
Note that the residual covariance rather than the covari-
ance of the original variable
Z
(
u
) must be used in the KT
system. Both covariances are equal in areas or along direc-
tions where the trend
m
(
u
) is deemed non-existent. Note also
that the cross covariance between variables
Z
(
u
) and
Y
(
u
)
plays no role in this system; this is different from cokriging.
In a sense, the
Z
(
u
) variable borrows the trend from the
Y
(
u
)
variable. Therefore, the
Z
*
(
u
) estimates reflect the trends of
the
Y
(
u
) variability, not necessarily the
z
variability.
(
)
where the
Consider a linear combination of primary and secondary data
values:
n
n
0
l
∑
∑
*
0
0
l
l
y
()
u
=
λ
⋅
y
( )
u
+
λ
⋅
y
( )
u
0
i
0
i
j
l
j
i
=
1
j
=
1
The estimation variance may be written as
n
{
}
0
∑
*
0
0
Var Y
−= − ⋅ −
Y
ρ
(0)
2
λ ρ
(
uu
)
0
0
0,0
i
0,0
i
i
=
1
n
nn
j
0
0
∑
∑∑
l
l
00
0 0
− ⋅ −
2
λρ
(
uu
) +
λλρ
⋅
⋅ −
(
uu
)
j
1,0
j
i
j
0,0
i
j
j
=
1
i
==
11
j
nn
nn
l
l
0
l
∑∑
∑∑
l
l
l
l
l
l
0
0
+ ⋅
λλρ
⋅ −
(
uu
)+2
λλρ
⋅
⋅ −
(
uu
)
i
j
1,1
i
j
i
j
0 ,1
i
j
i
==
11
j
i
==
11
j
Minimizing this estimation variance results in the simple
cokriging system of equations:
�
nu
()
1
()
∑
∑
∑
1
λ
()
uC
(
u
−
u
)
š
š
š
+
β αβ
ZZ
1
1
1
β
=
1
nu
2
'
'
λ
()
uCuu Cuu
(
−= − =
)
(
),
α
1, , ()
…
nu
�
š
β αβ α
ZY
ZZ
1
1
β
1
()
=
2
1
2
1
2
1
nu
n u
()
∑
'
2
'
'
'
λ
()
uC
(
u
−+
u
)
λ
()
uC
(
u
−
u
)
š
š
š
=
β αβ
YZ
β αβ
YY
β
=
1
1
2
1
β
=
1
2
2
2
1
2
'
C u
(
−
u
) ,
α
=
1,
…
,
nu
(
)
�
YZ
α
2
2
2
The cokriging estimator and the resulting estimation vari-
ance are
