Geoscience Reference
In-Depth Information
8.3.3
Collocated Cokriging
1. Calculate a prior distribution of uncertainty based on spa-
tial information of the same type.
2. Calculate a likelihood distribution of uncertainty based
on multivariate information at the location we are pre-
dicting.
3. Merge the prior and likelihood distributions into an up-
dated (posterior) distribution.
4. Perform post processing with the updated distribution.
The prior distribution in (1) is calculated as the conditional
distribution of each variable at each unsampled location
con-
ditional to the surrounding data of the same type
. This is
kriging from the surrounding data.
The likelihood distribution in (2) is calculated as the con-
ditional distribution of each variable at each unsampled lo-
cation
conditional to other data types at the same location
.
This is accomplished using cokriging or perhaps some form
multivariate linear regression.
The updated distribution (3) is created by merging the
prior with the likelihood distribution. The arithmetic is ex-
actly the same as collocated cokriging. The separate con-
tributions of the secondary data and the data or the same
variable are more easily understood compared to collocated
cokriging.
Bayesian updating is appealing because it is simple. In
cases where multiple secondary variables are available, there
are few approaches that can be used with comparable ease.
Still, the major steps in Bayesian updating are involved, and
include: (1) data assembly and calculation of correlation;
(2) calculation of likelihoods using secondary data; (3) cal-
culation of prior probabilities of all variables, combining
likelihoods and prior distributions into posterior distribution;
(4) cross validation and checking; and (5) summarizing un-
certainty and displaying results. The resulting distributions
of uncertainty can be used for a qualitative assessment of
local uncertainty.
The results of Bayesian updating should be used to com-
plement conventional analysis. They provide a quantification
of how the secondary data merge together to predict the
primary variable(s). Bayesian updating, as do all other
estimation techniques, is as good as and does not go beyond
the data that have been input to the algorithm.
Collocated cokriging makes two simplifications (Zhu
1991
).
The first is that only one secondary variable is considered;
the second is that the cross covariance is assumed to be a lin-
ear scaling of the variance. The reasoning behind this is that
the collocated
y
value is surely more important than the other
y values available in the neighborhood and likely screens the
influence of multiple secondary data. Under this assumption,
the cross variogram is no longer needed, and the ordinary
cokriging estimator is re-written as
n
1
∑
u uu uu
*
Z
()
=
λ
() (
Z
)
+
λ
()()
Y
COK
α
α
1
1
α
1
The corresponding cokriging system requires knowledge of
only the Z covariance
C
Z
(
h
) and the
Z-Y
cross covariance
C
ZY
(
h
). A further approximation through the Markov model
allows simplifying the latter:
C
()
h
=⋅
B
C
(
h
)
, for all
h
ZY
Z
C
(0)
Z
where
are the vari-
B
=
·
ρ
(0) ;
C
(0)
and C
(0)
ZY
Z
Y
C
(0)
Y
ances of
Z
and
Y
, and
ρ
ZY
(
0
) is the linear coefficient of cor-
relation of collocated
z-y
data.
If the secondary variable
y
(
u
) is densely sampled but not
available at all locations being estimated, it may be estimat-
ed at those missing locations conditional to the
y
data. Under
the collocated model and since the
y
values are only second-
ary data, the estimation of the missing
y
values should not
impact the final estimate of the
Z
variable.
The Markov model is becoming widely used due to its
simplicity. It can only be used when collocated secondary data
will be used. If the secondary data are smooth then consider-
ing
y
values beyond the collocated values should not help.
Retaining only the collocated secondary datum does
not affect the estimate (close-by secondary data are typi-
cally very similar in values), but it may affect the resulting
cokriging estimation variance: that variance is overestimat-
ed, sometimes significantly. In an estimation context this is
generally not a problem, because kriging variances are rarely
used. In a simulation context where the kriging variance de-
fines the spread of the conditional distribution from which
simulated values are drawn, this may be a problem.
8.3.5
Compositional Data Interpolation
Interpolation of compositional data is a multivariate prob-
lem. From a geostatistical perspective, cokriging is the typi-
cal method. It is an exact linear estimator and under certain
conditions is unbiased and provides a minimum estimation
variance estimate.
Conditions that ensure these properties are true include:
(1) the domain of interpolation,
8.3.4
Collocated Cokriging Using Bayesian
Updating
Bayesian updating is a technique closely related to collocat-
ed cokriging, but is designed for many secondary variables
that are available to predict the primary data. The method
can be subdivided into the following steps:
ℜ
is unconstrained; (2)
the variables are distributed according to a model that per-
mits valid interpretation of results, for example normal or
