Geoscience Reference
In-Depth Information
indicator RV
I
(
u
;
z
k
) has only two possible outcomes, 1 or 0.
Thus, by definition of the expected value:
hard indicator data
i
(
u
α
;z
) originating from local hard data
z
(
u
α
), as in Eq. 9.2.
The second type is a local hard indicator data
j
(
u
α
;
z
) origi-
nating from ancillary information that provides hard inequali-
ty constraints on the local value
z
(
u
α
). If
z
(
u
α
)
∈
[
a
α
,
b
α
] then:
{
}
{
}
{
}
EI
(; )
u
z
= ⋅
1
bI
(; ) 1
u
z
=
+ ⋅
0
bI
(; ) 0
u
z
=
k
k
k
{
}
{
}
{
}
=
E
I
(; )
u
z
=⋅
1
Prob
I
(; ) 1
u
z
= =
Prob Z
()
u
≤
z
k
k
k
0 if
za
≤
These relations still hold for conditional expectations, such
that:
š
α
[
]
j
(
u
; )
z
=
missing if
z
(
u
)
∈
ab
,
�
š
α
α αα
1 if
zb
>
E
{
I
(
u
;
z
k
)
|
(
n
)
} =
Prob
{
Z
(
u
)
≤
z
k
|
(
n
)
} =
F
(
u
;
z
k
|
(
n
))
�
α
The practical consequence is that a conditional cdf can be
built by assembling
K
indicator kriging estimates. This ccdf
represents a probabilistic model for the uncertainty about the
unsampled value
z
(
u
).
A third type is the coding of soft indicator data
y
(
u
α
;
z
) origi-
nating from ancillary information providing prior probabili-
ties about the value
z
(
u
α
):
y
(
u
α
;
z
)
=
Prob
{
Z
(
u
α
)
≤
z
|
local information
} ∈
[0, 1]
∗
=
I
(
u
;
z
k
)
|
(
n
)
}
∗
=
∗
{
[
i
(
u
;
z
k
)]
E
{
Prob
Z
(
u
)
≤
z
k
|
(
n
)
}
The fourth type of coding is the global prior information
common to all locations
u
within the stationary area:
which can be obtained with a weighted linear average. The opti-
mal weights are given by a kriging system on the indicator data:
F
(
z
)
=
Prob
{
Z
(
u
α
)
≤
z
}
,
∀
u
∈
A
n
{
}
*
∑
*
EI
(; )|()
u
z
n
=
i
(; )
u
z
=
λ
(; )(
u
z i
u
; )
z
k
k
α
k
α
k
This global prior is different than the local prior histogram
of samples used in the indicator kriging system. Figure
9.5
shows a graphical representation of all four.
α
=
1
Since several thresholds
k
are used, this is usually called
multiple indicator kriging (MIK). The weights and the ccdf
F
(
u
;
z
k
) are dependent on both the location and a number of
thresholds
z
k
,
k = 1,…, K.
Thus, there is one indicator vario-
gram
γ
I
(
u
;
z
k
) and one kriging system per threshold. While
the inference is more time-consuming, the flexibility is
greater, and no prior assumption of any type of distribution is
made.
Another advantage of indicator methods is that no back-
transformation is required, since working with indicator
variables directly yields a ccdf model for the RV
Z
(
u
). An-
other important aspect of the IK method is that it can be ap-
plied equally to continuous or categorical variables. In what
follows, references made to continuous variables also apply
to categorical ones.
There are challenges with IK including (1) inference of
the distribution details, particularly above the highest thresh-
old used in the kriging, (2) the greater effort required to infer
all of the required parameters such as the variograms, (3) the
inevitable multivariate Gaussian flavor in high order distri-
butions (> 2) because of averaging and (4) the practical use
of either probabilities or a smooth estimator.
9.4.2
Simple and Ordinary IK with Prior Means
In Simple Indicator Kriging, the expected value of the in-
dicator transform for each category is assumed known and
constant throughout the study area. The linear estimate is
then a linear combination of the
n
nearby indicator RVs and
the CDF value.
F
(
z
k
)
=
n
∗
i
(
u
;
z
k
)
−
λ
α
[
i
(
u
α
;
z
k
)
−
F
(
z
k
)]
=
α
=
1
n
1
−
n
∗
i
(
u
;
z
k
)
=
λ
α
·
i
(
u
α
;
z
k
)
+
F
(
z
k
)
λ
α
α
=
1
α
=
1
The SIK system of equations is then:
n
∑
uu uu
…
λ
C
(
−
;)
z
=
C
(
−
; ;
z
α
=
,
,
n
β αβ
I
k
I
α
k
β
=
1
where
C
I
(
h
;
z
k
) =
Cov
{
I
(
u
;
z
k
)
, I
(
u
+
h
;
z
k
)} is the indicator co-
variance at cutoff
z
.
If
K
cutoff values are retained, simple IK requires
K
indi-
cator covariances in addition to the
K
cdf values.
There are cases when the prior means can be considered
non-stationary and can be inferred from a secondary vari-
able. In such cases one can use the general non-stationary
9.4.1
Data Integration
The indicator formalism allows for a more straightforward
integration of different data types. There are four types of
data that can be used in indicator coding. The first is local
