Geoscience Reference
In-Depth Information
The price to pay for this choice is that the variance of the
Gaussian simulated values is inflated, and that the OK esti-
mator does no longer provide the exact Gaussian mean, only
an approximation. While these authors generally discour-
age simulating using OK, if there is a significant number of
original data, such as blast holes, the variance inflation issue
may be less significant.
As with all other simulation techniques, the most critical
and often time-consuming step is to check the results. Hon-
oring the declustered histograms of the Gaussian values and
of the original distribution is a first check. The reproduction
of the variogram model for both the Gaussian and the origi-
nal data should be checked as well. The simulated spatial
distribution should show the characteristics and trends of the
original data, adequately reproducing trends and local means
and variances.
If there is reliable production information, the grade-
tonnage curves derived from the simulated models should
reproduce well the actual values from production. Likewise,
if grade control data exists and is not used in the simulation,
it can be used to validate the conditional simulation models.
It is good practice to implement the simulation initially
on a small area, fine-tuning the simulation parameters that
result in a better reproduction of histograms and variogram
models, as well as production data if available. After the
implementation details are defined, then the full, multiple
simulation runs are completed.
from the existing data. Initially,
N
lines are defined in the
three-dimensional space:
D
i
, i
=
1, N
. On each line, a one-
dimensional RF
Y(
u
Di
), i
=
1, N
is defined; these
N
RFs are
independent from each other. For each line, there is also a
3-D Random Function:
Z
i
(
u
), i
=
1,… N
.
At first, the line
D
i
is simulated at each point
u
Di
on the
line. Moving averages is typically used for the initial 1-D
simulation. Second, the simulated value
y(
u
Di
)
at
u
Di
is as-
signed to all the points inside the slice or band perpendicular
to line
D
i
at
u
Di
:
Z
()
u
=
y
(
u
)
i
Di
Then, at each point
u
in the 3-D space, sum all the values from
the
N
slices or bands to obtain a realization for this point:
N
Z
s
(
x
)
=
√
Z
i
(
x
)
N
i
=
1
The unconditional simulation is generated after obtaining the
values for all the points in the 3-D model.
Data conditioning of the TB simulations is done by devel-
oping two kriging runs: first, the conditioning data is kriged
to obtain
y
kc
(
u
), which reproduces data trends. The second
step is to krige with the unconditional simulated values at
these conditioning data locations to obtain
y
ku
(
u
). Then, the
conditional simulation values
y
cs
(
u
) are calculated as the
unconditionally simulated values adjusted by the difference
between both kriged values:
10.2.2
Turning Bands
[
]
The Turning Bands (TB) method was the first 3-D geosta-
tistical simulation method, originated by Matheron (
1973
)
and developed by Journel (
1974
). Although the sequential
simulation method has been popular for many years, turn-
ing bands simulation is still used. The turning bands method
generates a 3-D simulation results from several independent
1-D simulations along lines that can be rotated in 3-D space.
This unique way of simulating provides 3-D unconditional
realizations.
After transforming the original data to Gaussian values,
TB consists of two main steps: (1) develop an unconditional
simulation in Gaussian units, with the experimental histo-
gram reproduced by transformation and the covariance or
variogram from the data being also reproduced; and (2) con-
dition the turning bands simulation through a post-processing
using kriging (Journel and Huijbregts
1978
). This method
exactly honors the conditioning data and also preserves the
variability of the unconditional simulation realizations. This
method has been adapted to conditioning multiple-points
structures (Ren et al.
2004
; Ren
2005
).
The initial step in Turning Bands is to obtain an uncon-
ditional simulation based on the covariance models derived
y
(u)
=
y
(u)
+
y
(u)
−
y
(u)
cs
uc
kc
ku
The simulation is performed in the Gaussian space to ensure
histogram reproduction, while covariance reproduction is
ensured by the use of the unconditional simulation.
The conditional simulation procedure can be understood
using a 1-D example, see Fig.
10.4
. At each real data loca-
tion, the unconditional simulated value is taken out, and the
conditioning datum is put in. Near the data location, the krig-
ing estimator smoothes the change between the sample data
and the unconditional simulated values outside the range of
kriged values. Therefore, the conditional simulated values at
these N data locations will be the data values. Beyond the
range of correlation, the conditional simulated values will be
the unconditional simulated values.
There are some inherent limitations of the turning bands
algorithm: (1) the post processing for conditioning is cum-
bersome; (2) the 1-D covariances must be worked out sepa-
rately for each nested structure and covariance shape because
they are different in 2-D and 3-D; and (3) only isotropic co-
variances can be used; anisotropy is introduced by geometric
transformation.
