Geoscience Reference
In-Depth Information
Fig. 9.5
Data types used in indicator coding
expression of simple kriging. For example, if
I
(
u
;
s
k
) repre-
sents the indicator of presence/absence of category
k
at loca-
tion
u
, and
p
(
u
;
s
k
) is a prior probability of presence of cat-
egory
k
at
u
, the updated probability is given by simple IK:
variances
C
I
(
h
;
z
k
) could be significantly different from one an-
other. Occasionally, however, sample indicator variograms ap-
pear proportional or are similar to each other. The correspond-
ing continuous RF model
Z
(
u
) is the so-called mosaic model:
ρ
()
h
=
ρ
(; )
h
z
=
ρ
(; ;
h
zz
),
∀
zz
,
∗
IK
[
Prob
{
u
∈
k
|
(
n
)
}
]
=
p
(
u
;
s
k
)
·
[
i
(
u
α
;
s
k
)
−
p
(
u
;
s
k
)]
Z
I
k
I
kk
'
kk
'
where
p
(
u
;
s
k
) is the prior indicator mean at location
u
. The
SK system remains as above assuming a stationary residual
indicator covariance model (Goovaerts
1997
; Deutsch
2002
).
For ordinary indicator kriging, the expected value of the
indicator transform for each category is assumed unknown
but constant within a local neighborhood.
where
ρ
Z
(
h
) and
ρ
I
(
h
;
z
k
;
z
k
) are the correlograms and in-
dicator cross correlograms of the continuous RF
Z
(
u
).
The single correlogram function is better estimated either
directly from the sample
z
correlogram or from the sample indi-
cator correlogram at the median cutoff
z
k
= M
, i.e.,
F
(
M
)
= 0.5
.
Indeed, at the median cutoff, the indicator data are evenly dis-
tributed as 0 and 1 values with, by definition, no outlier values.
Indicator kriging under the mosaic model is called me-
dian indicator kriging. It is a particularly simple and fast
procedure since it calls for a single easy-to-infer median in-
dicator variogram that is used for all
K
cutoffs. Moreover, if
the indicator data configuration is the same for all cutoffs,
one single IK system needs to be solved with the resulting
weights being used for all cutoffs. For example, in the case
of simple IK,
(
u
;
z
k
)
=
n
1
−
n
∗
i
λ
α
·
i
(
u
α
;
z
k
)
+
F
(
z
k
)
λ
α
α
=
1
α
=
1
n
subject to
λ
α
=
1. The resulting Ordinary IK system of
α
=
1
equations is
�
n
∑
λ
C
(
uu
− += −
;)
z
µ
C
(
uu
…
; ;
z
α
=
,
,
n
š
�
š
β αβ
I
k
I
α
k
β
=
1
n
n
n
∑
∑
*
∑
[( ;
i
u
z
)]
=
λ
( )(
uu
i
;
z
)
+−
1
λ
( )
u
Fz
(
)
λ
1
=
k
SK
α
α
k
α
k
š
�
β
α
=
1
α
=
1
β
=
1
Ordinary indicator kriging is common because it is more ro-
bust with respect to departures from stationarity and, unlike
multi-Gaussian kriging, there is no theoretical requirement
that simple kriging be used.
where the
λ
α
(
u
)'s are the SK weights common to all cutoffs
z
k
and are given by the single SK system:
n
∑
u u u uu
λ
( )
C
(
−=− =
)
C
(
),
α
1,...,
n
β
β
α
α
β
=
1
9.4.3
Median Indicator Kriging
The covariance
C
(
h
) is modeled from either the
z
sample
covariance or, better, the sample median indicator covari-
ance. Note that the weights
λ
α
(
u
) are also the SK weights of
the simple kriging estimate of
z
(
u
) using the
z
(
u
α
) data.
One of the considerations used in indicator kriging to choose
the
K
cutoff values
z
k
is that the corresponding indicator co-
