Geoscience Reference
In-Depth Information
obtained from a calibration scattergram of z values versus
collocated v values (Fig.
10.15
):
The range of v values is discretized into L classes (v
l−1
,
v
l
], l = 1,…,L. For class (v
l−1
, v
l
], the y prior probability
cdf can be modeled from the cumulative histogram of
primary data values z(
u
α
) such that the collocated sec-
ondary data values v(
u
α
) fall into class (v
l−1
, v
l
]:
One is to decompose the original variables Z
i
(
u
) into orthog-
onal factors, that is, obtain the principal components of the
original Z-variable correlation matrix at |h| = 0 (Luster
1985
).
The significant assumption here is that the orthogonality of
the principal components at lag 0 extends to all possible lags.
Alternative options is to use either the super-secondary
variable (Babak and Deutsch
2009
), or a stepwise condi-
tional transformation (Leuangthong and Deutsch
2003
). The
choice between these two transformations depends on the
shape of the cross plot between the two variables. If there is a
non-linear or constrained relationship between the variables,
stepwise conditioning provides a more flexible, albeit data-
hungry, method; if the cross plots show mostly a relationship
correctly and fully characterized by its linear correlation co-
efficient, then a simpler super-secondary variable approach
may be preferred (Fig.
10.17
).
¢
¢
¢
y
(; ) r{()
u
z
=
ob Z
u
≤
z v
()(,
u
∈
v
v
}
10.15
α
α
l
−
1
l
Note that the secondary variable v(
u
) need not be con-
tinuous. The classes can be in fact categories of v values; for
example, if the information v relates to different lithologies
or mineralization types.
The calibration scattergram that provides the prior y prob-
ability values may be borrowed from a different and better
sampled field. That calibration scattergram may be based
on data other than those used to calibrate the covariance
parameters B(z).
10.6.5
Stepwise Conditional Transform
The stepwise conditional transform (SCT) was introduced by
M. Rosenblatt in 1952 and was re-introduced by Leuangth-
ong and Deutsch in 2003 for use in geostatistics. The moti-
vation for it is that it produced independent model variables,
thereby avoiding cosimulation.
In a bivariate case, the normal transform of the second
variable is conditional to the probability class of the first
or primary variable. Extending to a k-variate case, the kth
variable is conditionally transformed based on (k − 1) first
variables.
The following is an example of a bivariate case where
Zn is the primary variable and Pb is the secondary. Zn is
transformed to a Gaussian distribution using the Normal
Scores procedure as before. Pb is transformed as shown in
Fig.
10.18
. The stepwise conditional transform results in a
non-correlated variable at
h
= 0, which indicates that Cosim-
ulation is not required.
The major incentive to use SCT in practice is that it is ro-
bust when dealing with complex multivariate distributions.
This means that SCT is able to solve issues of non-conformity
to multi-Gaussian assumptions, as well as significantly sim-
plify the joint simulation of multiple variables. The major
disadvantage is that it is data-intensive, while care must
be used not to define classes with less than 50 data values
(Leuangthong and Deutsch
2003
), which could make the re-
sulting conditional distributions not representative. Another
possible limitation is that SCT may produce artifacts in the
transformation of secondary variables. This should thor-
oughly checked for before proceeding with any subsequent
simulation.
The more important implementation aspects of SCT are:
1. The number of classes; it is recommended than no less
than 10 are used. More classes result in a correlation that
10.6.4
Gaussian Cosimulation
If a Gaussian method is used, the data must be transformed
to a standard normal distribution. If two variables are con-
sidered, the cross correlation between
Y
i
and
Y
j
should show
a bivariate normal distribution, i.e., an elliptical probability
contours along a line through the origin. For third and higher
orders, a distribution of k Gaussian variables should show
probability contours following a hyper-ellipsoid in k-dimen-
sions.
The first possible approach is a direct co-simulation, as
proposed by Verly (
1993
), and is similar to that for conven-
tional simulation, see Fig.
10.16
. It begins with establishing a
random path through all the grid nodes. At each grid node the
nearby data and previously simulated grid nodes are found,
a conditional distribution is constructed by cokriging, and a
simulated value is drawn from this distribution. The simulat-
ed value is added to the conditioning data, and the process is
repeated until all nodes are simulated. The final steps are to
back-transform the simulated values and to check the results.
The next approach for a joint simulation is to define a hi-
erarchy of variables. In this case, the variables are not simu-
lated simultaneously, but in order according to a pre-defined
hierarchy and conditionally to the previously simulated
variables (Almeida and Journel
1994
). This idea allows for
the implementation of a full cokriging, a collocated cokrig-
ing approximation, or a further approximation based on the
Markov-Bayes model.
The third approach is to apply a transformation that would
make the correlated variables independent (Luster
1985
). To
obtain the non-correlated variables, there are several options.
