Geoscience Reference
In-Depth Information
Because the ccdf returned by IK honors both hard
z
data and
constraint intervals, the corresponding expected value-type
(E-type) estimate also honors that information. More pre-
cisely, at a datum location
u
α
, [
z
(
u
α
)]
*
=
z
(
u
α
), if the
z
datum
is hard; and
9.4.4
Using Inequality Data
Normally the indicator data
i
(
u
α
;
z
) originate from data
z
(
u
α
)
that are deemed perfectly known; thus the indicator data
i
(
u
α
;
z
) are hard in the sense that they are valued either 0 or 1
and are available at any cutoff value
z
.
There are applications where some of the
z
information
takes the form of inequalities such as:
)
*
if the information avail-
able at
u
α
is the constraint interval
z
k
(
u
α
)
∈
(
a
α
,
b
α
]
.
In
practice, the exactitude of the E-type estimate is limited by
the finite discretization into
K
cutoff values
z
k
. For example,
in the case of a hard z datum, the estimate is:
(
(
z
u
ab
α αα
∈
(
,
]
k
E
)
*
z
( )[ , ]
u
ab
α αα
∈
∈
u
with
z
k
being the upper bound of the
interval containing the datum value
z
(
u
α
).
z
(
z
, ]
z
α
k
−
1
k
E
or
z
(
u
α
) ≤
b
α
equivalent to
z
(
u
α
)
∈
(
−∞
,
b
α
), or
z
(
u
α
) >
a
α
equivalent to
z
(
u
α
)
∈
(
a
α
,
+∞
). The indicator data corre-
sponding to the constraint interval are available only outside
that interval:
9.4.7
Change of Support with IK
0 for
za
≤
�
š
The indicator variable
I
(
u
;
z
) results from a non-linear trans-
form of the original
z
(
u
) samples; therefore, the block in-
dicator variable
I
V
(
u
;
z
) is not a linear average of point in-
dicators
I
(
u
;z
)
.
Thus, the averaging of discretization points
within a block, as done for linear variables, does not result in
the estimated block indicator:
α
i
(
u
; )
z
=
undefined for
z ab
∈
(
,
]
�
š
α
αα
1 for
zb
>
�
α
The use of inequality data does not pose any complication. The
undefined or missing indicator data in the interval (
a
α
, b
α
] are
ignored, and the IK algorithm applies identically. The constraint
interval information is honored by the resulting ccdf.
The IK solution is particularly fast if median IK is used.
However, the data configuration may change if constraint in-
tervals of type are considered; in such case, one may have to
solve a different IK system for each cutoff.
1
(
)
∫
(
)
*
i
u
;
z
≠
i
u
′
;
z
⋅
du
′
V
|
V
|
Vu
()
Or, equivalently:
*
N
1
*
∑
(
)
(
)
≠
V
F
u
; |( )
zn
F
u
′
; |( )
zn
α
N
α=
1
9.4.5
Using Soft Data
The ccdf that results from averaging the proportions of point
values within the block
V
(
u
) is called a “composite” ccdf,
[
F
N
(
u
;
z
|
(
n
))]
Hard indicator data and inequality data are treated similarly—
except for missing values at some thresholds. Soft indicator
data, however, are truly different data types and a form of
cokriging should be adopted to combine them in the esti-
mate. The Markov-Bayes formalism is commonly consid-
ered (Zhu
1991
; Zhu and Journel
1992
).
∗
(Goovaerts
1997
), and is an estimate of the
proportion of point values within
V
(
u
) that do not exceed the
threshold
z
. This is quite different than what a true block ccdf
would give, which is by definition the probability that the
average value is no greater than the threshold
z.
Practitioners have attempted to correct the IK-derived
point ccdf within the block using a change of support meth-
od, the most common being the affine correction discussed
in Chap. 7. The process is simply to correct the IK-estimated
point ccdf to represent a “block” ccdf. All
z
k
thresholds are
corrected without changing any of the estimated indicator
values
i
(
u
;
z
k
):
9.4.6
Exactitude Property of IK
Assume the location
u
to be estimated coincides with a
datum location
u
α
, whether a hard datum or a constraint in-
terval of type. Then, the exactitude of kriging (simple, ordi-
nary, or median IK) entails that the ccdf returned is either a
zero variance cdf identifying the class which the datum value
z
(
u
α
) belongs to; or a ccdf honoring the constraint interval up
to the cutoff interval amplitude:
*
*
*
z
()
u
=
fz m
−
()
u
+
m
()
u
Vk
,
k
where
f
is the classical variance reduction factor (Isaaks and
Srivastava
1989
). The dispersion variance is derived from
the
z
value variogram models.
In one published case (Hoerger
1992
), the affine correc-
tion is applied in the log-space under the permanence of
[
]
*
i
(
u
;
z
)
=
Prob
*{
Z
(
u
)
≤
z
( )}
n
α
k
α
k
0, if
za
≤
�
k
α
=
, if (
z
u
)
∈
(
ab
,
]
�
α αα
1, i f
zb
>
�
k
α
