Geoscience Reference
In-Depth Information
2. Recall that the two-point Gaussian cdf
G
(
h
;
y
p
) is the
non-centered indicator covariance for the threshold
y
p
(Chap. 6). Then, if expression 9.1 is evaluated for a series
of
p
-quantile values
y
p
, the corresponding indicator vario-
gram is derived with:
differs from the expected value
m
. This is due to the expo-
nentiation involved in that back-transform. It is particularly
problematic since it significantly amplifies any error in the
estimation of the lognormal estimate
y*
(
u
) or its SK variance
σ
SK
2
(
u
).
Another problem commonly encountered is that the es-
timated values increase as the kriging variance increases.
This is a serious issue because higher kriging variances cor-
respond to sparsely populated areas, and thus it can result
in overestimation of the estimated values. Among others,
this problem was critical for several gold mines in northern
Nevada (USA) in the 1980s, because the predicted grades
increased away from the main mineralized zones. A simi-
lar artificial grade trend can be produced by this method in
many other types of deposits, such as porphyry copper de-
posits.
This extreme sensitivity to the back-transform explains
why lognormal kriging is not used any more. The method
has been largely replaced by other Gaussian approaches or
the indicator kriging approach. While there are exceptions,
the use of lognormal kriging is mostly confined to South
Africa, where the method was originally developed (Sichel
1952
,
1966
; Krige
1951
).
γ =−
(;
h
y
)
pG y
(;
h
)
I
p
p
In this case, the indicator function is defined as
I
(
u
;
y
p
)
=
1
if
Y
(
u
)
≤
y
p
, 0 otherwise.
3. The experimental indicator variograms of the normal score
data
γ
I
(
h
;
y
p
) are obtained for the same
p-
quantiles.
4. The indicator variograms obtained experimentally
ˆ
(;
ˆ
γ
h
and the theoretical Gaussian-deduced
γ
I
(
h
;
y
p
)
are compared graphically. Based on the quality of the
comparison, the bi-Gaussian assumption may be rejected
or accepted.
y
)
I
p
An additional check is to verify that the pattern of indicator
spatial correlation is symmetric with respect to the
p-
quantiles,
that is
=¢
. The experi-
mental indicator variograms of the normal score data for p and
p' should match well. This checking is not common because it
is difficult to assess if the differences are significant.
p
1
p
γ
(;
h
y
)
=
γ
(;
h
y
)
, where
I
p
I
¢
p
9.3
Lognormal Kriging
9.4
Indicator Kriging
A particular case is found if the stationary multivariate RF is
assumed lognormal. In this case, the RF
The indicator kriging-based estimation methods are non-
parametric in the sense that they do not make any prior as-
sumption about the distribution being estimated. The ob-
jective of the method is not to estimate parameters of an
assumed distribution, but directly estimate the distribution
itself (Journel
1983
).
Consider the binary transform of the original
Z
(
u
) vari-
able defined as:
=
u u
has
a multivariate Gaussian distribution with mean
m
′, covari-
ance
C
′
(
h
), and variance
σ
′
2
= C
′
( 0)
.
The relationship be-
tween the arithmetic and logarithmic moments is (Journel
and Huijbregts
1978
, p. 570):
Y
() ln ()
Z
¢ ¢
2
/2
me
+
m
=
and
=
�
1, if Z(u)
≤
z
(9.2)
k
'2
I
(; )
u
z
Ch
Ch me
2
()
2
2
σ
()
= −⇒= −
1
σ
m
1
k
0, otherwise
In the method implementation the original data is log trans-
formed:
y
(
u
)
= ln z
(
u
)
.
The
z
(
u
) variable has to be strictly
positive. Simple or ordinary kriging of the log data yields an
estimate
y*
(
u
) for
ln z
(
u
). Unfortunately, a good estimate of
ln z
(
u
) is not necessarily a good estimate of
z
(
u
); in particular
the antilog back-transform
e
y
The indicator formalism consists of discretizing the con-
tinuous variable
z
with a series of
K
threshold values
z
k
,
k = 1,…, K.
The experimental cdf of the
n
samples within
the stationary domain are considered a prior distribution,
which can be obtained through an equal-weighted aver-
age:
(
u
)
is a biased estimator of
Z
(
u
)
as can be derived from the relationships above. The unbiased
back-transform of the simple lognormal kriging estimate
y*
(
u
) is actually:
∗
n
1
= ≤=
∑
{
}
F
(; )
u
z
Prob Z
()
u
z
i
( ; )
u
á
z
k
k
k
n
α=
1
∗
σ
SK
(
u
)
/
2]
∗
e
[
y
(
u
)
+
z
(
u
)
=
This is the proportion of the samples
z
(
u
α
) below the cutoff
z
k
. In this prior cumulative distribution frequency, the sam-
ples could be weighted to account for spatial clustering. The
where
σ
SK
2
(
u
) is the simple lognormal kriging variance. In
practice, the theoretically non-biased estimate
z
*
(
u
) often
