Geoscience Reference
In-Depth Information
lognormal distributions; (3) data are second order stationary.
The second condition is important when the data distribution
is not unconstrained in
If
X
follows an additive logistic normal distribution, then
Y
= alr(
X
) follows a multivariate normal distribution, then
the properties of
Y
being minimum estimation variance and
unbiased are achieved. However, there is no interpretation
for the expected value and variance of estimates of the origi-
nal compositions. No analytical back-transformation process
exists to calculate E{
x
(
u
0
)} from E{
y
(
u
0
)} and Var{
x
(
u
0
)}
from Var{
y
(
u
)} (Pawlowsky-Glahn and Olea
2004
). Ap-
proximations are available, but Pawlowsky-Glahn and Olea
(
2004
) identify that it is unverified if optimality conditions
hold for back-transformed values in
℘
since the Euclidean
metric does not apply in simplex space. The alr cokriging
estimator is
ℜ
and a transformation is required
to do so. For Lognormally distributed data for example are
restricted to positive real space and are transformed to fol-
low a normal distribution via natural logarithms. Valid in-
terpretation of results requires a back-transformation pro-
cess for the expected value and variance of the estimates in
transformed space. It must also be true that estimates in both
spaces are exact, minimum estimation variance best and un-
biased.
For compositional data, cokriging is not as well devel-
oped. Most available work applies cokriging to alr trans-
formed compositions (Pawlowsky
1989
; Pawlowsky et al.
1995
; Martin-Fernandez et al.
2001
). Direct cokriging of
compositions has also been formulated (Walvoort and de
Gruijter
2001
), although direct statistical analysis of compo-
sitions is not advocated by Aitchison (
1986
) or Pawlowsky-
Glahn and Olea (
2004
).
The kriging method explained by Walvoort and de Gru-
ijter (
2001
) is a reformulation of cokriging as a constrained
optimization problem. The objective is to minimize the
estimation variance while adhering to the following con-
straints:
• Components of the compositions are positive:
†
N
‡
(
)
(
)
∑
∗
−
1
x u
=
alr
c
+
alr
xu
ˆ
‰
0
k
k
Š
‹
k
=
1
N
(
)
∑
(
)
∗
y u c
=+
yu
0
k
k
k
=
1
Application of the previous equation will result in an exact
interpolation of compositions. The smoothing effect occurs
as with kriging applied to unconstrained data. No other as-
sertions can safely be made about results in the space of in-
terpolated compositions.
xu 0
∗
()
≥
T
∗
• The constant sum property of compositions:
1xu
()
=
c
∑
• Ordinary kriging formulation:
It is difficult to state if this method provides the best or cor-
rect solution; however it is an alternative and has some ad-
vantages such as the ability to handle zeros in the data.
k
=
I
8.3.6
Grade-Thickness Interpolation
When using service variables, the interpolation of the two
variables involved, grade*thickness and thickness, are es-
timated independently, as long as the thickness and grade
variables are not correlated. If they are, then some form of
co-estimation (cokriging) is required. These variables can be
estimated with any technique, geostatistical or not, but the
most commonly used is Ordinary Kriging.
The final estimated grade is then obtained by dividing the
two estimated variables:
8.3.5.1 Additive Logratio Cokriging
This method was derived for alr transformed compositions.
It can be applied to compositions following normal, lognor-
mal or additive logistic distributions. The disadvantage of
the technique is a lack of an analytical back-transformation
of cokriging results, limiting its use to basic interpolation
and mapping. Recall the alr transform and its inverse (alr
-1
)
where
c
is the constant sum constraint of the compositions,
()
*
Acc
()
x
D
d
∈ℜ
xu
∈℘
and
yu
()
:
*
Gx
()
=
Tx
*
()
(
)
r()
xu
=
yu
()
k
k
where
Acc
*
(
x
) is the estimate accumulation variable
(grade*thickness),
T
*
(
x
) is the estimated thickness, and
G
*
(
x
)
is the final estimated grade. This is uncomplicated.
However, if we are interested in obtaining the estimation
variance of
G
*
(
x
), then we need to calculate the estimation
variance of a quotient. This is less trivial, and the detailed
development can be found in Journel and Huijbregts (
1978
,
pp. 424-428). The final calculation will depend on whether
grade and thickness are intrinsically coregionalized, and also
whether we are calculating the estimation variance block by
block, or globally for entire domains.
†
‡
x
()
u
u
i
k
=
log
,
i
=
1,...,
dk
,
=
1,...,
N
ˆ
‰
x
()
Š
‹
Dk
(
)
r
−
1
yu
()
=
xu
()
k
k
=
(
)
exp
y
(
u
u
)
1
k
,...,
d
(
)
∑
exp
y
(
)
+
1
i
k
i
=
1
(
)
exp
y
(
u
)
d
∑
dk
,
c
−
x
()
u
i
k
d
(
)
∑
exp
y
(
u
)
+
1
i
=
1
i
k
i
=
1
