Geoscience Reference
In-Depth Information
12
Uncertainty and Risk
Abstract
This chapter shows how multiple realizations can be used to support the assessment of
uncertainty and risk.
12.1
Models of Uncertainty
The minimized estimation variance or kriging variance
can only be equated to a local estimation error if the error dis-
tribution is Gaussian and the estimation error does not depend
on the actual sample values, a property called homoscedastic-
ity, discussed in Chap. 8. In this case, the estimation variance
could be associated to the variance of the error distribution.
This is seldom found in practice because most grade distri-
butions are positively skewed and the local uncertainty will
depend on the local grades; more uncertainty will be expected
in high grade areas. The estimation variance does not provide
a reliable uncertainty model for small blocks.
The kriging variance may be used in instances where the
distribution is likely to be Gaussian. This may apply if very
large volumes of material are considered, since most spatial
distributions will tend to become more symmetric, and there-
fore become more Gaussian-like as more small scale values
are averaged together. The reasonable limits of application
are not known ahead of time, see Davis (
1997
) among others.
Other, more recently developed techniques, have attempt-
ed to introduce local measures of uncertainty by making the
kriging estimation variance data dependent. Most of these
techniques have been applied in the context of resource clas-
sification (for example, Arik
1999
).
Non-linear geoestatistical techniques rely on data trans-
formation to obtain a probabilistic estimate that carries
uncertainty (Chap. 9). Except for the case of the indicator
transform, the uncertainty model is developed in the trans-
formed space, most commonly Gaussian.
Conditional simulation provides a model of uncertainty at
each location by a set of simulated realizations. The uncer-
tainty is better described when a large number of realizations
are available, but a relatively small number (say 100) is suf-
ficient to provide a reasonable approximation.
All estimates have some error or uncertainty. Predictions are
always inaccurate, with errors stemming from widely spaced
data, geological variability, lack of knowledge to determine
the best parameters for estimation, approximations made in
the estimation procedure, and limitations of the models used.
Although the error will never be known except at
locations where data are collected in the future, traditional
statistics and geostatistics provide models of uncertainty.
Chapters 8-10 discussed estimation, estimation variances,
and methods to obtain a conditional distribution of uncer-
tainty for a random variable:
F
(; |( )
z
u
n
=
Pr
ob Z
{ ()
u
≤
z
|()}
n
(12.1)
Equation 12.1 is a complete description of uncertainty in the
variable
z
based on our random function model. Obtaining
reliable models for the conditional distributions denoted in
Eq. 12.1 has proven difficult, particularly for small volumes
(one block at a time), as opposed to large deposit-scale vol-
umes.
Early attempts in geostatistics to characterize uncer-
tainty relied on the kriging variance, typically in the form
of confidence intervals attached to each estimated block
grade:
(
µ
x
|
(
n
)
−
d
)
≤
µ
x
|
(
n
)
≤
(
µ
x
|
(
n
)
+
d
)
(12.2)
where
d
is the difference from the average value that defines
the confidence level. For example,
d
= 2*
σ
(twice the standard
deviation of the random variable) represents the 95 % confi-
dence level if the distribution has a Gaussian shape (Chap. 2).
