Geoscience Reference
In-Depth Information
Simulation techniques and the resulting models of un-
certainty rely heavily on stationarity; trends and departures
from stationarity significantly affect the model of uncer-
tainty and its quality and usefulness. A model of uncertainty
based on simulation depends on the Random Function model
used. In certain deposits, a Gaussian-based model may be
appropriate, while for others a non-parametric technique
such as indicator simulations may be preferable. The model
of uncertainty will also depend on the number and statistical
characteristics of the conditioning data. Therefore, the model
of uncertainty is not unique, nor is there an objective or true
model of uncertainty: uncertainty is model-dependent. This
has been discussed in Journel and Kyriakidis (
2004
) and
Goovaerts (
1997
) among others.
Typically, simulations models cannot capture all possible
sources of uncertainty that exist in a resource model. In this
sense, they are incomplete descriptions of the space of un-
certainty, and thus it is relevant to discuss how appropriate
the conditional simulation model is with respect to the prob-
lem at hand.
A practical consequence of the dependence on a model
is that the simulation method should be simulated from the
same RF model used to obtain the resource model. It is im-
portant that they both share the same basic assumptions and
implementation parameters; otherwise, the base case re-
sources could be different and the models incompatible.
iogram model and kriging plan. In addition, a correct amount
of dilution must be included in order to predict tonnages
and grades available at the time of mining. The prediction
of recoverable resources and reserves is another significant
source of uncertainty for resource models.
The model of uncertainty can also change when differ-
ent implementation parameters of the geostatistical models
are used, as discussed in Chap. 11 and also Rossi (
2003
),
among others. Seemingly minor decisions, such as whether a
random path or a multiple grid search for simulating values
is used, can impact the resulting uncertainty model. Other
parameters typically considered are search radii, number
of original data used, number of previously simulated data
used, the number of simulations to be run, and the kriging
method to be used, among others. One alternative is to assess
the uncertainty related to implementation criteria by choos-
ing bounds or “best” and “worst” cases, although the process
is subjective and difficult to justify.
There is limited information with large, unsampled areas
between data points. There is uncertainty in the statistical
parameters such as the overall mean of the deposit. A model
of parameter uncertainty is also subjective, but may lead
to a more realistic assessment. Some possible approaches
to quantify parameter uncertainty include using an analyti-
cal model, the conventional bootstrap method or the spatial
bootstrap method.
Bootstrapping is a name generically applied to statistical
resampling schemes that allow uncertainty in data statistical
parameter to be assessed from the data used to calculate the
same parameter in the first place. The basic procedure is to
draw
n
values from the original data with replacement, cal-
culate the statistic from the bootstrapped sample, and repeat
a number of times to build up a distribution of uncertainty.
It is assumed that the input distribution is representative of
the overall distribution. If the drawing is done using Monte-
Carlo simulation (MCS), then there is an additional assump-
tion that the data are independent.
Assuming that the sample data are independent is not
realistic when they are known to be correlated. The spatial
bootstrap simulates at the data locations. The uncertainty
generally decreases as the number of drawn values (
n
) in-
creases. The spatial bootstrap requires a variogram for the
data set, simulation, and then computation of the mean for
each simulated set of data.
Sources of Uncertainty
Resource models will include
uncertainty from many sources. There are several factors that
contribute to the overall uncertainty, and they do not necessar-
ily cancel each other out. The sample values themselves have a
degree of uncertainty, partly coming from the intrinsic hetero-
geneity of the material being sampled; however, most sampling
errors are due to the sampling process itself. Sampling theory
deals with the development of procedures for minimizing sam-
pling variances, although there will always be an error that can-
not be fully eliminated. Sample collection, sample preparation,
the chemical analysis itself, and the overall data handling are
all sources of uncertainty.
The amount of drill hole information available depends
on the geology and the project's development stage. Typi-
cally, when additional data is included in the model, the un-
certainty will tend to decrease. Geologic models are also a
major source of uncertainty. Based on sparse drilling, they
are representations of mineralization controls but still car-
rying a degree of uncertainty stemming from mapping and
logging; data handling; the interpretative process itself; and
the development of the computerized model. Often, the geo-
logic model's uncertainty has the most important impact on
the resource model since it heavily conditions the estimated
tonnages above cutoff (Fig.
12.1
).
There is uncertainty related to the process of grade inter-
polation including data spacing, kriging method chosen, var-
12.2
Assessment of Risk
An uncertainty model can be used to characterize risk. It is
important to distinguish uncertainty and risk, since large un-
certainties, in some cases, may not lead to significant risks.
In other situations, small uncertainties may correspond to
unacceptable risk.
