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ProEXPLO 2013

Instituto de Ingenieros de Minas del Perú

Lima, Perú

17 y 18 de Mayo, 2013

Mario E. Rossi, MSc. Geoestadística, Ing. de Minas.

GeoSystems International, Inc.

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� A deterministic model does not carry any uncertainty, e.g. v=x/t. A probabilistic estimate aims at providing a range of possible values, which we call an uncertainty model.

� A typical resource estimate does not provide an uncertainty model, just the most likely value based on very specific optimization criteria. In the case of kriging, minimum error variance.

� Since we know our estimates are seldom (never?) the true values, then we need an idea of how far off we may be. In fact, we rarely need to estimate true values, only to within “acceptable” ranges. How good is the block model?

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� Regardless of the stage the Project is at (pre-feasibility, feasibility, operations), there is always uncertainty, sometimes more significant that others.

� Predictions are always inaccurate; errors are implicit and sometimes explicit errors. These stem from lack of knowledge, approximations, limitations of the models used, etc.

� Thus, we need probabilistic models.

� It is known that as the volume considered increases, all distributions tend to become more symmetric, with a corresponding decrease in variance. This is a consequence of the Central Limit Theorem.

� In mining we deal (mostly) with variables that average up arithmetically, and thus the “traditional” methods for Change of Support require this property.

� The basic idea is to reduce the point distribution variance but keep the overall mean the same. In practice, we model a “variance correction factor”.

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� The variance correction factor can be obtained from variogram models and dispersion variances, from averaging up detailed information (as in blast holes), or from conditional simulations.

� The variogram provides an indication of how fast the symmetrization of the point distribution will occur. Lack of spatial correlation entails that the distribution will symmetrize fastest.

� If blast holes or production drilling are available, could average them up to “blast blocks” or within stopes, and obtain an “experimental” correction factor. Again, assumes that there is a single geologic domain within the block.

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� We typically use variogram or correlogram models to characterize dispersion variances.

� Change of support models are very sensitive to the model chosen, specially nugget effect and short-scale ranges of the model. Other aspects to consider are spatial clustering and outliers.

� This is key when setting up change of support studies.

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� Instead of estimating average grades of blocks, we attempt to estimate a distribution of possible values for each block. This is an estimated conditional probability (density) function, or ccdf.

� There is always the issue of upscaling! “Non-linear” geostatistics typically rely on parametric models to provide a ccdf for blocks. Estimates are almost always point estimates, then we need to upscale them to obtain the corresponding ccdf for the blocks.

� This is not trivial, and sometimes ignored, when using kriging (IK). Other times it is embedded in the kriging method used (DK, MG, UC), all based on the assumption of bivariate or multivariate Gaussianity.

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� Uncertainty models can be classified as local or global

� Global uncertainty models include resource classification schemes and global probability intervals (usually through the Kriging Variance, see Journel & Huijbregts, pp. 410).

� There are a number of attempts to obtain models of local uncertainty (block by block), but do not fully describe the uncertainties involved.

� Remember, never attempt to use the kriging variance for small volumes (blocks or groups of blocks)!

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� We are interested in evaluating a truncated statistic.

� If the block distribution is known, we define an indicator:

� The tonnage can be calculated as:

� The quantity of metal and mean grade:

� How to find the block distribution?

� Uniform Conditioning is used to

estimate the recoverable reserves of a

mining panel using the panel estimate

and a change of support model

� It is useful when data are sparse and

you cannot reliably estimate the smu

sized blocks in the deposit

� It is a faster method than simulation,

but may not give results that are as good

as simulation

� Earliest reference is Matheron 1974.

� Remacre and Guibal did some work on UC in the mid

to late 1980’s

� Rivoirard included UC in his Non-linear geostats

book

� Isatis is the main commercial software that

implements Uniform Conditioning

� The first step in uniform conditioning is to estimate

the panel grades using kriging

� It is quite popular to use block ordinary kriging in

the original grade units

� Steps:� Model the 2- or 3-D variogram

� Choose the panel and smu size

� Run the block kriging with the block discretization set at the

smu resolution within the panel

1

1

( ) ( , ) ( ( , ) , 1,....,

1( , )

n

N

jj

C C V n

CN

β β α αβ

α

λ α=

=

= =

′≈

∑

∑

u u u u) u

u u

� The anamorphosisfunction defines how the real grade units correspond to the normal score units� We use an experimental anamorphosis by using the nscoreprogram

� Calculate the value of the φ coefficients:

� The above formula can be approximated as:

�

but this do not give very good results

� The following equality should be met:

� We can calculate

the modelled

anamorphosis

function

� We now need the histogram of the grade at the

smu block support

� We get this by using the anamorphosis function

from the previous step and the variogram from step 1

� Calculate the variance of the smu’s using the gammabarprogram

� Solve the following equation to get the change of support coefficient for the smu blocks

� The new anamophosis function is valid for the smu sized blocks

� Repeat the

same procedure in

step 3 to calculate

the change of

support

coefficient for the

panel sized blocks

� Need to

transform the

estimated panel

grade to Y

� Use the

modelled panel

anamorphosis

� Need to transform

the cutoff grades to

Y

� Use the modelled

smu anamorphosis

� The assumption made in uniform conditioning is that the bivariate relationship between the SMU grades and the panel grades is bivariate normal

� Therefore, if we know the normal score panel grade, we know the mean and variance of the conditional normal score smu distribution

� We calculate the proportion and the grade above cutoff for the conditional smu distribution

� When the entire panel is above cutoff, the average grade should be the estimated panel grade

From the

Isatis

Manual

� There are two methods that can be used for

determining the grade and proportion above cutoff.

The first uses hermite polynomials:

� An analytical solution can be used:

� Both methods tend to give similar results.

� Uniform conditioning only tells us the distribution of the smu block grades

� It does not tell us where the high and low grade blocks are located

� Panels with the same estimate will have the same grade and proportion curves irrespective of the surrounding data