Geoscience Reference
In-Depth Information
Fig. 10.2
Comparison of high
(
left
) and low (
right
) spatial
entropy distributions
indicator-based simulations, were developed during the sec-
ond half of the 1980's. These methods were tested in several
mining scenarios and applications, but because of hardware
limitations and other practical reasons not fully implemented
within industry.
It was not until 1991 that the first, full scale implementation
and application was completed within the mining industry.
Two sequential Gaussian simulations (SGS) were developed
by H.M. Parker and E.H. Isaaks (personal communications).
The conditional simulation study was developed to support a
Feasibility Study for the Lihir Au project in Papua New Guin-
ea, at the time owned by Kennecott Mining Corporation. Soon
after that, the first conditional simulation-based grade control
method was implemented by N. Schofield (at the time with
FSS International Consultants) for the Marvel Loch mine in
Western Australia, and followed a few months later with the
implementation of a similar grade control method at the San
Cristóbal mine in Northern Chile (Aguilar and Rossi
1996
).
As computer hardware capabilities improved drastically
throughout the 1990's, more and more simulation studies
were tested and published, although full, industry-scale im-
plementation remained relatively scarce. Gradually the num-
ber of implementations grew, and by the second half of the
following decade, several mining companies began to use
conditional simulations routinely, with large companies like
BHP-Billiton adding these methods to their internal good
practices guidelines for project evaluation.
The sequential Gaussian simulation entails the following
steps:
1. Complete a full exploratory data analysis of the original
data, including variography and domain definition.
2. After defining the domains, analyze whether the data
needs to be de-trended, that is, whether the simulation
should be done on the residuals.
3. Apply the normal scores transformation to the original
data to obtain the corresponding Gaussian distribution.
4. Obtain the Gaussian variogram models for the trans-
formed variable.
5. Define a random path through each domain to be simu-
lated. The path for the simulation is defined randomly
to avoid artifacts.
6. Estimate through simple kriging the conditional distri-
bution for each node to be simulated in the Gaussian
space. The estimated simple kriging value Y*(
u
) is the
mean of the conditional distribution, and its variance
the simple kriging variance, σ
sk
(
u
). If the simulation is
done on residuals after de-trending, then the Gaussian
mean of the conditional distribution is 0.
7. Draw randomly from the conditional distribution to ob-
tain a simulated value for the node, Y
s
(
u
).
8. Incorporate the simulated value Y
s
(
u
) as conditioning
data for nodes simulated later. This is necessary to en-
sure variogram reproduction.
9. Repeat and continue the process until all nodes and all
domains have been simulated.
10. Verify that the univariate distribution (histogram) of the
simulated values is Gaussian; also check that the sim-
ulated values reproduce well the Gaussian variogram
model used in the simulation.
11. Back-transform the Gaussian simulated values to the
original variable space.
12. Add back the trend if the simulation was performed on
residuals.
13. Verify that the histogram of the back-transformed data
is similar to the original distribution; also verify that the
10.2.1
Sequential Gaussian Simulation
The sequential Gaussian simulation algorithm (Isaaks
1990
)
is based on a multiGaussian RF model assumption. It is the
simulation version of the MG algorithm, and it benefits from
all the convenient properties that the multi-Gaussian RF
offers: all conditional distributions are Gaussian, and simple
kriging is the only method that yields (exactly) the estimated
Gaussian mean and variance.
