Geoscience Reference
In-Depth Information
relation violations. Although the order relation problems can
be minimized, they cannot be completely avoided.
Correcting for order relations in the case of categori-
cal variables is simpler compared to continuous variables.
If the estimated probability of a category
s
k
is outside the
licit bounds, then the solution is to reset the estimated value
F*
(
u
;
s
k
|
(
n
)) to the nearest bound, 0 or 1. This resetting cor-
responds exactly to the solution provided by quadratic pro-
gramming.
The other constraint is more difficult to resolve because it
involves
K
separate krigings. One solution consists of krig-
ing only (
K − 1
) probabilities leaving aside one category
s
ko
,
chosen with a large enough prior probability
p
ko
, so that:
Fig. 9.6
Order relation problems and their correction. The dots are the
ccdf values returned by IK. The corrected ccdf is obtained by averaging
the forward and downward corrections
∑
Fs
*(;
u
|()) 1
n
=−
Fs
*(; |()) [0,1]
u
n
∈
k
k
variograms show different anisotropies due to the mixing of
geologic controls, the search neighborhood should be the
same. If quadrant or octant searches are used, the condi-
tions applied to those also must be the same for all thresh-
olds. Changing the search neighborhood between thresholds
changes the data used and causes the estimated probabili-
ties to vary significantly in a non-physical manner; there are
more order relation deviations, see below.
It is reasonable to orientate the samples search according
to the interpreted anisotropy model. However, this should
only be done if no significant changes in anisotropy direc-
tion is observed for all
K
variogram models; and in any case,
the search neighborhood should be made more isotropic than
what the variogram models suggest. This is to ensure that all
directions away from the point being estimated are repre-
sented in the sample pool.
0
kk
≠
o
Another solution, which should be applied after the estimat-
ed distribution is corrected (if necessary) to the interval [0,1],
is to re-standardize each estimated probability
F*
(
u
;
s
k
|
(
n
))
by the sum
∑
u
.
Correcting for order relations of continuous variable
ccdfs is more delicate because of the ordering of the cumu-
lative indicators. Figure
9.6
shows an example with order
relation problems.
The following correction algorithm implemented in
GSLib (Deutsch and Journel
1997
) considers the average of
an upward and downward correction:
1. Upward correction resulting in the upper line in Fig.
9.6
showing order relations problems:
• Start with the lowest cutoff
z
1
.
• If the IK-returned ccdf value
F
(
u
;
z
1
|(
n
)) is not within
[0,1], reset it to the closest bound.
• Proceed to the next cutoff z
2
. If the IK-returned ccdf
value
F
(
u
;
z
2
|(
n
)) is not within [
F
(
u
;
z
1
|(
n
)),1], reset it
to the closest bound.
• Loop through all remaining cutoffs
z
k
,
k
= 3, …,
K
.
2. Downward correction resulting in the lower line in
Fig.
9.6
showing order relations problems:
• Start with the largest cutoff
z
K
.
• If the IK-returned ccdf value
F
(
u
α
;
z
K
|(
n
)) is not within
[0,1], reset it to the closest bound.
• Proceed to the next lower cutoff
z
k −1
. If the IK-
returned ccdf value
F
(
u
α
;
z
k −1
|(
n
)) is not within [
F
(
u
α
;
z
K
|(
n
)),1], reset it to the closest bound.
• Loop downward through all remaining cutoffs z
k
,
k
=
K
− 2, …, 1.
3. Average the two sets of corrected ccdfs resulting in the
thick middle line in Fig.
9.6
.
Practice has shown that the majority of order relation prob-
lems are due to a lack of data; more precisely, to cases when
IK is attempted at a cutoff
z
k
which is the upper bound of
k
F
*(; |()) 1
k
sn
<
Step 5: Correcting for order relation deviations
It is nec-
essary to ensure that the IK-estimated ccdf at each location
u
respects the axioms of a cdf:
Fzn Fzn
(; |())
u
≥
(; |()),
u
'
∀
z > z
'
and
[
]
F zn
∈
(; |())
u
0,1
Since the
K
thresholds are estimated independently of each
other, the estimated ccdf values may not satisfy these order
relations. While the number of deviations may be large, per-
haps ½ of the total ccdfs, the absolute value of those devia-
tions should not be large. Journel (
1987
) recommends check-
ing the implementation of IK if the deviations are greater
than 0.01; in practice a limit of 0.05 is more reasonable.
There are several sources of order relation problems. Most
commonly, they are due to inconsistent variogram models and
the kriging implementation strategy. Also, negative indicator
kriging weights and lack of data in some classes increase order
