Geoscience Reference
In-Depth Information
Fig. 10.12
Truncated Gaussian simulation
Fig. 10.13
Consequence of ordering, implicit in the truncated Gauss-
ian technique
The sequential indicator simulation (SIS) algorithm is iden-
tical for continuous or categorical variables. The key differ-
ence is, in the latter case, that the
K
categories can be defined
in any order; the local ccdf obtained from indicator kriging
provides a cdf-type ordered set of the probability interval [0,1]
discretized in
K
intervals. The simulated category at location
u
is defined by the interval in which the random number
p
falls,
which is drawn from the uniform distribution [0,1].
Note that the ordering of the K categories is arbitrary and
does not affect the simulated model. The ordering does not
affect which category is drawn at location
u
or the spatial
distribution of categories (Alabert
1987a
) because the ran-
dom number
p
has a uniform distribution.
10.5.3
Truncated PluriGaussian
A variant of truncated Gaussian is the truncated pluri-Gauss-
ian method. This method uses multiple Gaussian variables,
which allow for using different variograms, each with its
own spatial variability model, including different relative
nugget effects, anisotropies, ranges, and other variogram pa-
rameters.
For practical reasons, the number of Gaussian functions
is usually kept to 2, one category being the complement of
the other. A non-conditional simulation of the indicator vari-
able
i(
u
)
can be obtained by truncating a simulation of the
standard Gaussian RF
Y(
u
):
10.5.2
Truncated Gaussian
()
l
()
l
i
()
u
==
1, ify ()
0,
u y
≤
The truncated Gaussian technique simulates a continu-
ous standard Gaussian field and truncates it at a series of
thresholds to get a categorical variable realization. Only one
(Gaussian) variogram model can be specified in this tech-
nique. Continuous Gaussian realizations are generated and
truncated with the proportions of the different rock types.
The conditioning data are coded as the normal score value
at the centroids of each category (or rock type) (Fig.
10.12
).
An important feature of truncated Gaussian simulation is
the ordering of the resultant probability density function (pdf)
models. The codes used to characterize the individual classes
are generated from an underlying continuous variable. There-
fore, normally class 2 will occur between classes 1 and 3.
Only rarely would code 1 be next to code 3 (Fig.
10.13
). There
are specific applications where this could be an advantage,
such as in simulating a sedimentary sequence. However, the
most common applications in mining are the simulation of
lithology; mineralization type; and alteration. In these cases,
rarely ordering is part of the natural process being simulated.
p
=
otherwise
−
=
being the standard normal p quantile,
and at the same time the desired proportion of indicators,
{
p
y Gp
1
()
with
E Iu
=
.
The Gaussian RF model is fully characterized by its cova-
riance
C
Y
(h)
, and there is direct relationship between
C
Y
(h)
and the indicator covariance after truncation at the
p
quantile.
Thus, by inversion of the indicator variograms, the Gaussian
variograms are obtained (Fig.
10.14
).
Multiple truncations of the same Gaussian realization
at different thresholds would result in multiple categorical
indicators each with the right proportions (marginal proba-
bilities). But the indicator covariances of the additional cat-
egories would be controlled by the Gaussian RF, because
the covariance
C
Y
(h)
is the single parameter that defines the
Gaussian RF, and thus can only be used to define one in-
dicator covariance. There is a significant drawback if more
()
}
