Geoscience Reference
In-Depth Information
There are several variants of kriging with a trend model,
including random trend model, kriging the actual trend,
using a secondary variable to impose a trend on the primary
variable, and others. Most of these are found in non-mining
applications since they are more difficult to justify given the
amount of data normally encountered in mining.
The random trend model is equivalent to the Bayesian
kriging model, but is simpler to implement. The weakness
of both models lies in the inference of the statistics of
M
(
u
),
whether interpreted as a random trend or as prior guess on
z
data, and on the key hypothesis of independence of the
M
(
u
)
and
R
(
u
) values. As usual, the only physical reality is
z
, not
M
or
R
.
8.2.4
Local Varying Mean
8.2.6
Kriging the Trend and Filtering
Kriging with a locally varying mean (LVM) is a variant of
SK that works with the residuals, but is different in that the
mean is not constant everywhere. In this sense, it is similar
to OK, particularly if the mean is constant within certain re-
gions.
The general approach is to model the trend so that at every
location the value
m
(
u
) is known and possibly different. As
before, a covariance model for the residuals is needed. The
influence of the local mean depends on the amount of pri-
mary data available in the neighborhood. If there are a large
number of nearby samples, then the influence of the mean
is mitigated; in areas where the primary data is scarce, the
mean has a large influence. LVM is appropriate for modeling
geological trends and smooth secondary data, and is more
commonly applied in simulations (Chap. 10).
Rather than estimating the sum
Z
(
u
)
= m
(
u
)
+ R
(
u
) one could
estimate only the trend component m(
u
). Starting directly
from the original
z
data the KT system shown above is easily
modified to yield a KT estimate for
m
(
u
):
n
=
∑
u uu
*
()
m
m
()
λ
() ( )
Z
KT
i
i
i
=
1
and the KT system:
�
∑
n
∑
K
λ
()
m
( )
uuu
C
(
−+
)
µ
m
( )
uu
f
(
)
==
0, i
1,...,
n
�
j
R j
i
k
k
i
j
=
1
k
=
0
n
∑
()
m
�
λ
( )
uu u
f
(
)
=
f
( ),
k
=
0,...,
K
j
k
j
k
j
=
1
λ
's are the KT weights and the
m
()
m
where the
µ
's are La-
grange parameters. Note that this system differs from the KT
system of the variable
Z
(
u
).
This algorithm identifies the least-squares fit of the trend
model when the residual model
R
(
u
) is assumed to have no
correlation:
C
R
(
h
)
= 0
for all
h
≠ 0
.
The direct KT estimation of the trend component can
also be interpreted as a low-pass filter that removes the ran-
dom (high-frequency) component
R
(
u
). The same principle
underlies the algorithm of factorial kriging and that of the
Wiener-Kalman filter (Kalman
1960
).
Factorial kriging is a technique that aims to either extract
features for separate analysis or filter features from spatial
data. The technique was originally proposed by Matheron
in the early days of geostatistics (1971), and takes its name
in relation to factor analysis (Journel and Huijbregts
1978
;
Goovaerts
1997
). Factorial kriging is of greatest interest to
geophysicists and those concerned with image analysis.
8.2.5
Random Trend Model
A similar model to KT results from interpreting the trend as a
random component. The random trend is denoted
M
(
u
), and
is added to a residual
R
(
u
) independent from it:
{
}
{
}
Z()
u
=
M()
u
+
R(), w th EZ()
u
u
=
EM()
u
Prior data that allows describing the trend are assumed avail-
able. For example, they can be prior information
m
(
u
) about
the local
z
data. These trend data allows inference of the M-
covariance
C
M
(
h
), and the corresponding residual data can
be used to infer the covariance of the residuals
R
(
u
). Based
on the independence assumption, the
z
data covariance is
then
C
()
h
=
C
()
h
+
C
()
h
Z
M
R
Kriging is then performed using the
z
data and the covari-
ance model
C
Z
(
h
). The resulting kriging estimates and vari-
ances depending on the
E
{
M
(
u
)} and
C
M
(
h
)
.
The variance
of
M
(
u
) can be made non-stationary and used to measure
the reliability of the prior guess
m
(
u
). However, in this case
the M-covariance is not anymore stationary and its inference
may become problematic.
8.2.7
Kriging with an External Drift
Kriging with an external drift is a particular case of KT
above. It considers a single trend function
f
l
(
u
) defined at
each location from some external, secondary variable.
The trend model is limited to two terms
m
(
u
)
= a
0
+ a
1
f
1
(
u
), with the term
f
1
(
u
) set equal to the secondary variable.
