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and also the cross-variograms that describes the correlation
of one variable to the other. The schematic illustration in
Fig. 6.10 below shows the K by K matrix of bivariate rela-
tions that must be inferred. There are K direct relationships—
one for each variable—and K(K − 1) cross relationships. The 
cross relationships are almost always taken to be symmetric,
that is, the relationship between i and j is the same as j and i.
There are some interesting circumstances (referred to as the
lag effect in some literature) when this is not true.
While calculating the number of variograms is not a major
computational effort, the problem lies in the fact that the var-
iograms cannot be modeled independently from each other.
Similar to the case of a single variable, there are a num-
ber of permissible models that can be used to model cross
correlations (Journel and Huijbregts 1978 ; Goovaerts 1997 ).
The requirement that must be met is that the variance of
each variable is non-negative, and that the matrix of var-
iogram models must be mathematically valid. The linear
model of co-regionalization (LMC) is the result of a spe-
cific set of correlated variables, and was first proposed by
Journel and Huijbregts ( 1978 , p. 172). Other types of co-
regionalizations can be assumed, see for example Zhu and
Journel ( 1993 ), or the Markov models mentioned elsewhere
in this topic.
The direct and cross variograms between variables k and
k' where k, k' = 1,…,K are defined as follows:
Fig. 6.10 Schematic illustration of a Multivariate Covariance Matrix.
The diagonal terms are direct covariances, and the cross-covariances
populating the rest of the matrix
performs poorly in the re-estimation, it will not necessar-
ily perform poorly in the final estimation run.
• The sill of the variogram cannot be cross-validated from
the re-estimation.
• Variogram values smaller than the minimum distance
between samples cannot be validated, such as the nugget
effect and the behavior of the model near the origin.
The second option also shares some of the drawbacks men-
tioned, including the fact that we are re-estimating samples;
in addition, there has to be a good number of samples within
the domain to be able to split the data in test and a “ground-
truth” subsets and carry out the exercise.
It is difficult to define a useful goodness of fit test for a
variogram model. The user's experience, subjective geologic
information, and consideration of the objectives of the study
must be considered to result in a robust variogram model.
All subjective decisions must be clearly documented, and is
generally preferable to skirting responsibility by relying on
blind tests or automatic variogram model fitting software.
2
γ
h u uh
u uh
()
=
EZ
{[ ()
Z
(
+
)]
kk
,'
k
k
[
Z
( )
Z
(
+
)]},
kk
,
'
=
1,...,
K
k
'
k
'
The calculation principles explained above can be applied to
the full set of K(K + 1)/2 direct and cross variograms; how-
ever, note that the data must be equally sampled, that is, data
for variable k and k' must be available at the same data loca-
tions. In presence of unequally sampled data, it is necessary
to directly compute cross covariances and convert them to
variograms for fitting. The covariance would be calculated
directly as
6.4
Multivariate Case
Most deposits have more than one variable of interest, and
the inference and prediction of one variable can be improved
using information from a second, or secondary, variable.
Moreover, it is important to respect the relationships be-
tween the variables when creating models of more than one
variable. The different variables may be of economic value,
contaminants, density, or processing characteristics such
as hardness. Consider an expanded notation to deal with
k = 1,…,K variables.
C
( )
h
=
E Z
{
( )
u uh
Z
(
+− =
)}
mm
,
k k
,
'
1,...,
K
kk
,'
k
k
'
k
k
'
The relationship between direct variograms and direct co-
variances was given above. In case of cross variograms and
covariances, the collocated covariance is required to convert
the two:
γ
( )
h
=
C
(0)
C
( ),
h
kk
,
'
=
1,...,
K
kk
,'
kk
,'
kk
,'
The collocated covariance between a variable and itself is
the variance of that variable:
= The collocated
cross covariances can be calculated directly when the data
are equally sampled. They must be estimated when the data
are not equally sampled.
C
(0)
2
.
{(
Z
uu
)
,
in A,
k
=…
1,
,
K
}
k
k, k
k
Quantifying the spatial structure of all K variables requires
developing the direct variogram models as discussed above,
 
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