Geoscience Reference
In-Depth Information
9.2
Gaussian-Based Kriging Methods
forward. If the multivariate Gaussian assumption is made,
the simple kriging estimate
y
SK
(
u
)
and simple kriging vari-
ance
σ
SK
(
u
) of the normal score data are the parameters of
the conditional Gaussian distribution frequency (Journel and
Huijbregts
1978
, p. 566).
The covariance modeled from the sample normal score
covariance must be used in the MG kriging system, but oth-
erwise is the same system of equations described in Chap. 8.
With both parameters estimated, the full ccdf is then mod-
eled using:
∗
The popularity of Gaussian methods comes from their sim-
plicity, the properties of the multivariate Gaussian distribu-
tion, and also from the fact that they can produce acceptable
estimates. While some characteristics of the Gaussian distri-
bution imply significant drawbacks, its simplicity and ease
of use make Gaussian-based probabilistic estimation and
simulation highly popular. Some of the convenient properties
of the multivariate Gaussian RF were discussed in Chap. 2.
Gaussian methods are maximum entropy methods in the
sense that, for a given mean and variance, the distribution
will tend to produce estimates that are as disorganized as pos-
sible. The implication is that in cases where the connectivity
of extreme values is high (low entropy), Gaussian methods
will not reproduce that connectivity. The geology of most
deposits is characterized by structuring, and this is reflected
in typical geologist's interpretations, geologic models, and
the theories about the genesis of deposits.
The maximum entropy or destructuring effect of the multi-
Gaussian RF can be better understood analyzing their indicator
variograms (Journel and Posa
1990
). It can be shown that for
y
p
∗
y
−
SK
(
u
)
σ
SK
(
u
)
y
∗
SK
[
G
(
u
;
y
|
(
n
))]
=
G
The MG SK estimate requires strict stationarity, that is, a
known mean at each location over the entire domain and all
possible sub-domains.
MG can also be implemented using OK and KT as op-
tions for quasi or non-stationary domains. This is done by
simply using the OK or KT algorithms (in any of its variants)
on the normal score data. However, the kriging variances of
the OK or KT systems are no longer the correct variances for
the estimated Gaussian distribution. Only the SK variance
is theoretically correct (Journel
1980
). All other variances,
because they result from constrained systems, will tend to
inflate the variance of the estimated Gaussian distribution.
This inflated variance could induce a bias in the back trans-
formed estimates. Therefore, the OK and KT variances must
be corrected to account for this difference.
An alternative for KT is to de-trend the original
z
data
first, and then normal score transform the residuals. The re-
siduals are assumed strictly stationary, with a mean of 0, and
therefore MG with SK can be more robustly applied. How-
ever, the MG estimated distribution is still dependent on the
effectiveness of the de-trending process.
→
1,
y
p
→∞
and the two-point cdf tends to
be independent. Therefore, the indicator variogram tends to its
theoretical sill,
σ
2
=p
·
(1
−
p) where p is the CDF value for
the indicator transform.
Another important property of Gaussian methods is ho-
moscedasticity, which means that the conditional variance
σ
K
(
u
)
does not depend on the actual data values. This is an
unusual property in the statistical world, since no other RF
has it. In practice, it is known that many variables are het-
eroscedastic, evidencing the proportional effect discussed
before. The dependency of the variance to the data values is
removed when transforming the data to the Gaussian distri-
bution.
If developing conditional simulations (Chap. 10), the
back-transformation is delayed until the simulation is com-
plete. However, the back-transformation is necessary when
estimating a conditional distribution for
Z
(
u
) (Verly
1984
;
Journel
1980
); the proportional effect and other characteristics
may not be reproduced adequately in the estimates. The back-
transform is quite sensitive to the variance of the conditional
distribution in Gaussian units,
σ
y
(
u
) , making the
Z-
estimate
potentially unstable. There are procedures that can be applied
to dampen this effect (Journel
1980
; Parker et al.
1979
), but
they have limitations.
→
0 or
y
p
9.2.2
Uniform Conditioning
Uniform Conditioning (UC) is a Gaussian method used to
estimate the recoverable resources of a mining (SMU) panel.
The method requires a panel estimate and a change of sup-
port model. The panel grade is assumed to be known, then
the distribution of SMUs within that panel can be established
from a bivariate Gaussian model. The name uniform condi-
tioning arises because of the assumption that the estimates of
the recoverable resources are conditioned to the same data
configuration for every panel grade.
The idea is to estimate a panel much larger than the SMUs.
The size of the panel is based on fitting a reasonable num-
ber of SMUs within it, and is generally irrespective of drill
hole spacing; however, often the actual drill hole spacing or
9.2.1
Multi-Gaussian Kriging
The multivariate Gaussian RF model is the most widely used
of the parametric models. Multi-Gaussian kriging is straight-
