Geoscience Reference
In-Depth Information
than two categories are defined, because the additional in-
dicator covariances would not reproduce the correct spatial
continuity.
As with the truncated Gaussian method, the order and
spatial sequence of the categories is fixed. While this may be
reasonable in sedimentary stratigraphic environments, may
be unreasonable in a more general setting.
�
š
0, if
za
≤
α
j
(
u
; )
z
=
undefined (missing), if
z ab
∈
(
,
]
�
š
α
αα
1, i f
zb
>
�
α
• Local soft indicator data y(
u
α
; z) originating from ancil-
lary information providing prior probabilities about the
value z(
u
α
):
10.6
Co-Simulation: Using Secondary
Information and Joint Conditional
Simulations
y
(
u
; )
z
=
Prob Z
{
(
u
)
≤
z
local information}
∈
[0,1]
α
α
• Global prior information common to all locations
u
within
the stationary area A:
There are two distinct ways in which multiple variables can
be accounted for. The first is considering secondary informa-
tion by conditioning the primary variable to both primary
and secondary information. Secondary information can refer
to the same primary variable, but presented in a different
format, or more generally any other variable to which the
primary variable is correlated. This is different than jointly
simulating the primary and secondary information, which
will be discussed later.
Gaussian techniques (Wackernagel
2003
) are commonly
used due to their simplicity, but often indicator-based meth-
ods may be preferable because of how easily the secondary
information can be incorporated into the process.
F z
( )
=
Prob Z
{
( )
u
≤ ∀∈
z
},
u
A
At any location
u
in A, prior information about the value
z(
u
) is characterized by any one of the four previous types
of prior information. The process of building the ccdf with
indicator kriging consists of a Bayesian updating of the local
prior into a posterior cdf:
¢
*
[Prob{
Z
( )
u
≤+ =
zn n
(
)}]
IK
¢
n
n
∑
∑
¢
λ
() ()
u u uu
Fz
+
λ
(;)
z
−
ν
(;)(
zy
;)
z
0
α
¢
α
α
α
=
1
¢
α
=
1
10.6.1
Indicator-Based Approach
The λ
α
(
u
; z)
s are the weights attached to the n neighboring
hard indicator data, the ν
α
(
u
; z)
s are the weights attached
to the n neighboring soft indicator data, and λ
0
is the weight
attributed to the global prior cdf. To ensure unbiasedness, λ
0
is usually set to:
A major advantage of the indicator kriging approach to gen-
erating posterior conditional distributions (ccdf's) is its abil-
ity to account for secondary or soft data. As long as the soft
data can be coded into prior local probability values, indica-
tor kriging can be used to integrate that information into a
posterior probability value.
The prior secondary information can take one of the fol-
lowing forms:
n
n
¢
∑
∑
λ
() 1
u
=−
λ
(;)
u
z
−
ν
(;)
u
z
0
α
¢
α
α
=
1
¢
α
=
1
The ccdf model is thus an indicator co-kriging that pools in-
formation of different types: the hard i and j indicator data
and the soft y prior probabilities. When the soft information
is not present or is ignored (n
= 0), the expression reverts to
the known IK expression.
If the spatial distribution of the soft y data is modeled by
the covariance C
I
(
h
; z) of the hard indicator data, then there
is no updating of the prior probability values y(
u
α
; z) at their
locations
u
α
, i.e.,
• Local hard indicator data i(
u
α
; z) originating from local
hard data z(
u
α
):
i
(
u
; )
z
=
1, if
z
(
u
)
≤=
z
,
0
if not
α
α
or (
i
u
;
s
)
=
1, if
u
∈
category
s
, = 0
if not
α
k
α
k
• Local hard indicator data j(
u
α
; z) originating from ancil-
lary information that provides hard inequality constraints
on the local value z(
u
α
). For example, if z(
u
α
) can only
take values within the interval (a
α
, b
α
], then:
¢
¢
*
¢
[Pr
ob Z
{
(
u
)
≤+ ≡ ∀
z
(
n
n
)}]
y
(
u
;
z
),
z
α
IK
α
Most often, the soft z data originate from information related
to, but different from, the hard data z(
u
α
). In this case, the
