Geoscience Reference
In-Depth Information
nu
()
Another reason that cokriging is not used extensively in
practice is the screen effect of the better correlated data (usu-
ally the z samples) over the data less correlated with the z
unknown (the y samples). Unless the primary variable, that
which is being estimated, is under-sampled with respect to
the secondary variable, the weights given to the secondary
data tend to be small, and the reduction in estimation vari-
ance brought by cokriging is not worth the additional infer-
ence and modeling effort.
Other than tedious inference and matrix notations, cokrig-
ing is the same as kriging. Cokriging with trend models and a
cokriging that filters specific components of the spatial vari-
ability of either
Z
or
Y
could be developed. These notation-
heavy developments will not be given here, but can be found
in Journel and Huijbregts (
1978
) and Goovaerts (
1997
).
The three most commonly applied types of cokriging are
as follows:
1.
Traditional ordinary cokriging:
the sum of the weights
applied to the primary variable is set to one, and the sum
of the weights applied to any other variable is set to zero.
In the case of two variables, these two conditions are
1
∑
*
z u
()
−=
m
λ
() (
u
zu
)
−
m
z
α
α
z
1
1
α
=
1
()
1
nu
2
∑
'
'
+
λ
() (
u
yu
)
−
m
α
α
y
2
2
α
=
1
2
nu
()
1
∑
2
σ
()
u
=
C
(0)
−
λ
()
uC
(
u
−
u
)
ZZ
α
ZZ
α
1
1
α
=
1
()
1
2
nu
∑
'
'
−
λ
()
uC
(
u
−
u
)
α
ZY
α
2
2
α
=
1
2
The equations for simple cokriging are essentially the same
as for simple kriging, but taking into account the direct and
cross covariances. As before, the system of equations must
lead to a valid result and the cokriging variance has to be
positive, which means that the covariance matrix is positive
definite. The condition is satisfied when using permissible
coregionalization model and no two data values (of the same
variable) are collocated.
We often avoid full cokriging because it is tedious to cal-
culate, interpret, and fit the necessary variograms. The linear
model of coregionalization (Chap. 6) is restrictive, and there
is no real benefit in cases where the same amount of data
is present for both primary and secondary variables. We are
motivated to consider cokriging when there are many more
secondary data than primary data and when a simple trend/
local mean model is considered inadequate.
∑
∑
λ
( )
u
=
1 and
λ
'
( )
u
=
0
α
α
1
2
α
α
1
2
The problem with this traditional formalism is that the sec-
ond condition tends to limit severely the influence of the sec-
ondary variable(s).
2.
Standardized ordinary cokriging:
often, a better
approach consists of creating new secondary variables
with the same mean as the primary variable. Then all the
weights are constrained to sum to one.
In this case the expression could be rewritten as
8.3.2
Ordinary Cokriging
n
In the case of a single secondary variable (Y), the ordinary
cokriging estimator of Z(
u
) is written as
1
∑
*
Z
()
u uu
=
λ
() (
Z
)
COK
α
α
1
1
α
=
1
1
n
2
n
n
∑
1
2
+
λ
'
( [('
uu
Y
)
+−
mm
]
∑
∑
*
Z
()
u uu
=
λ
() (
Z
)
+
λ
' ()( ' )
uu
Y
α
α
Z
Y
2
2
COK
α
α
α
α
1
1
2
2
α
=
1
2
α
=
1
α
=
1
1
2
∑ ∑
u u
,
where
m
Z
= E{Z(
u
)}
and
m
Y
= E{Y(
u
)}
are the stationary means
of
Z
and
Y
respectively.
3.
Simple cokriging:
there is no constraint on the weights.
Just like simple kriging, this version of cokriging requires
working on data residuals or, equivalently, on variables
whose means have all been standardized to zero. This is
the case, for example, when applying simple cokriging in
a Gaussian method, such as MG or UC, because the nor-
mal score transforms of each variable have a stationary
mean of zero.
Except when using traditional ordinary cokriging, covari-
ance measures should be inferred, modeled, and used in the
cokriging system rather than variograms or cross variograms.
n
n
λ
()
+
λ
' () 1
=
where the
λ
's are the weights applied to the
n
1
z
samples
with the single condition
1
2
α
α
α
=
1
α
=
1
1
2
1
2
and the
'λ
's are the weights applied to the
n
2
y
samples.
Cokriging requires a joint model for the matrix
of covariance functions including the
Z
covariance
C
Z
(
h
)
, the
Y
covariance
C
Y
(
h
)
, the cross
Z-Y
covariance
C
ZY
(
h
) = Cov{Z(
u
),Y(
u
+
h
)}
, and the cross
Y - Z
cova-
riance
C
YZ
(
h
)
.
More generally, the covariance matrix requires
K
2
covari-
ance functions when
K
different variables are considered in
a cokriging exercise. The inference becomes extremely de-
manding in terms of data and the subsequent joint modeling
is particularly tedious. Algorithms such as kriging with an
external drift and collocated cokriging have been developed
to shortcut the tedious inference and modeling process re-
quired by cokriging.
2
