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references > conditional simulation
What is conditional simulation?
This is designed as a basic overview of conditional simulation
and co-simulation.
Spatial grade variability
Good estimation has the goal of providing the 'best' estimate
for a block. This is achieved in kriging
by reducing the estimation variance to a minimum for all possible
linear weighted averaging schemes. This a good argument to use
kriging when designing stopes, performing grade control and
so on, where the overall in-situ grade of the is the key issue.
When is simulation preferred to estimation?
However, if we wish to deal with issues of variability, the
kriging has a downside. The means of achieving minimum estimation
variance in kriging is to smooth the values (all weighted averages
do this, kriging just does the optimal job). This implies that
the resultant block values have a lower variance than the 'true
block' grades we will actually mine.
Making the blocks smaller results in the unrealistic smoothness
of the blocks getting worse! Note that this is not a problem
for predicting gross grade and tonnage, but is a real issue
if we need to predict the local variability of grades, for example
dealing with variability control issues.
If we estimate points to try to deduce what grade behaviour
is like between the known sampled locations, we smooth, by necessity.
Note that an estimate by kriging honours the known points (this
is a property of kriging), but doesn't reflect the variability
between points: because of necessary smoothing. The answer is
to simulate.
Click here
for a review paper on Simulation 
pdf (261
Kb)
Conditional simulation
Conditional Simulation (CS) is a geostatistical method that
builds many, densely informed point realisations of mineralisation.
These reproduce the histogram and variogram of the input data,
as well as honouring the known data points (hence 'conditional').
Each realisation is different to others because we always have
uncertainty away from known data points (drill samples), hence
individually such realisations are simulations not estimates.
Objectives of simulation
Simulation has different objectives to estimation. The whole
point is to reproduce the variance of the input data, both in
a univariate sense (i.e. the histogram) and spatially (i.e.
the variogram). Thus simulations provide an appropriate platform
to study any problem relating to variability in a way that estimates
cannot.
There is a price to pay for this too (no free lunch)...
When we estimate, we get minimum estimation variance (if we
krige), but lose variability
in the output as a price (i.e. smoothing). When we simulate,
we retain the variability of the input data, but the price we
pay is that an individual simulation if taken as an estimate
has higher estimation variance than a kriging. In fact the whole
spread of simulations is equivalent to the estimation variance.
Conditional simulation honours the data
In essence a simulation is an alternative grade-profile that
has the same 'character' as the true profile. To meet this criteria
the histogram and variogram of a simulation must be in agreement
with those of the data. If we stipulate that known sample data
must be honoured, then the resultant simulation is called a
conditional simulation.
Note: there are many (theoretically an infinite number) of simulations
that can meet the above conditions. These are referred to as
'equiprobable images' of the mineralisation (which has the same
meaning as 'realisations'). Hence any individual simulation
is a poor estimate. However, averaging a very large set of simulations
yields a good estimate.
Averaging a set of conditional simulations
A collection of many such simulations, when averaged over a
block volume, is equivalent to a kriged estimate. Because we
can have multiple realisations within each estimated block,
we have access to the distribution of metal within a block:
as for Uniform Conditioning (UC), Multiple Indicator Kriging
(MIK) or any other non-linear geostatistical estimate of recoverable
resources. Thus CS is a route to recoverable resource estimation.
Until recently, computers were too slow to really exploit these
methods.
A conditional simulation passes through, or honours, the known
data (hence it is said to be 'conditioned' by the data). Notes:
- The simulation and the reality have the same general character.
The degree of noise (nugget and short range variability)
and the longer-scale spatial character of these two curves
are similar.
- In fact, the variogram is the same for the simulated and
the real grade profiles (or at least reproduces the input
variogram).
- The simulation is a poor estimate of the local average
grade, compared to the kriging (smooth black curve). The
only exception is when we have a data point, then the two
agree. Elsewhere, the simulation is poor.
We can use simulations to deal with a range of problems that
involve variability, and for such problems they are a far superior
basis than any estimate (including kriging). One example of
an application is to generate confidence intervals (CI's) for
an estimated block. Other applications include modelling local
spatial variability for blending, stockpiling or mining selectivity
studies.
Extension of the idea to two variables
Conditional simulation, as outlined above, deals with a single
variable. For example, in a gold-copper deposit, we could simulate
gold or copper. We could perform simulation of gold and
copper, but these would be independent. The resulting simulations
would reproduce the histograms of gold and copper, but not the
intrinsic correlation between (correlation coefficient and cross-variograms).
This idea is interesting for a range of deposit types:
- multi-element base metal deposits, eg: lead-zinc-silver
- nickel deposits, especially dealing with cobalt credits
or penalty elements (e.g. As or Mg)
- iron deposits, bauxite deposits, manganese deposits, mineral
sand deposits, magnesium deposits or other industrial minerals
where the problems are multivariate by nature and the financial
problem involves multivariate optimisation of processes.
Conditional cosimulation
Conditional Co-Simulations (CCS) is a multivariate conditional
simulation, i.e. of more than one variable. In our example above,
Cu and Au can be 'co-simulated' so that their cross variograms
and correlation are both respected.
For many copper-gold deposits, gold and copper are spatially
correlated variables, i.e. there exists a link between the spatial
variability of the two variables. In other words, the high grade
occurrences of gold tell us something about the neighbouring
copper grades and vice versa.
This connection between spatial variabilities can be captured
by a geostatistical tool called the cross-variogram. A cross
variogram measures the spatial covariability of two grades (say
Cu and Au) in the same way that a variogram measures spatial
variability of a single grade (say Cu or Au).
Bivariate recoverable resource estimation by conditional
co-simulation [CCS]
Any conditional simulation technique which overlooks the modelling
of the 'cross-structures', seen in cross-variograms, would miss
the crux of the problem as it would fail to predict the joint
and intricate contributions of both metals to the value of the
project. This might be a trivial economic problem if one metal
contributed 95% of the value and the other 5%, but the price
of overlooking it becomes higher as the dollar contributions
of metals approach equality.
CCS can offer the ultimate answer to this problem because they
are precisely aimed at reproducing the co-variation seen in
the experimental cross-variograms. In other words CCS attempts
to capture the spatial covariability patterns of the real deposit.
Just like monovariate conditional simulations, CCS reproduces
both the gold and copper histograms. But CCS will also respect
both the variograms and cross-variograms while honouring the
both grades at informed data points.
Selectivity studies using conditional simulation
Each realisation (individual CCS image) gives us a possible
joint gold-copper distribution based on point support and is
thus amenable to a lot of post-processing, for example re-blocking
will allow us to obtain joint distributions of gold and copper
for small blocks: an ideal input to selectivity studies, for
example.
Risk analysis using conditional simulation
Each simulation provides an alternative equiprobable representation
of the joint-distribution, meaning that we can get as many equally
realistic answers to any of our questions as we want. The differences
between these answers give us a measure of the joint spatial
uncertainty and can help us manage the financial risk attached
to the project. |
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Scott Jackson (left), Scott Dunham (centre), and John Vann (right) are the Directors of Quantitative Group
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