comparative study of localized block simulations and ......geostats 2012, oslo, norway and y 2v, and...

Comparative study of Localized Block Simulations

and Localized Uniform Conditioning in the

Multivariate case.

JACQUES DERAISME1, WINFRED ASSIBEY BONSU2

1 Principal Geostatistician, Geovariances, MAusIMM, 49 Ave Franklin Roosevelt, Avon,

77212 France, [email protected]. 2 Group Consultant, Geostatistics & Evaluation, Gold Fields Ltd., PO Box 628, West

Perth, 6872 WA Australia. [email protected].

1. Abstract

The general indirect estimation technique of recoverable resources during long-term

planning derives the unknown Selective Mining Unit (SMU) distribution from the

modelled distribution of large kriged blocks (panels). The Gaussian model and the

Uniform Conditioning (UC) technique provide an alternative consistent framework to

achieve this task that has been extended to the multivariate case (MUC). The MUC

method provides a practical advantage, in that no specific hypothesis on the correlation

between the respective secondary elements is required.

The Localised MUC (LMUC) technique has been developed to enhance the indirect

MUC by localising the results at the SMUs scale.

The paper investigates the possibility of improving the LMUC estimates through

multivariate block simulations which incorporate all the correlations of the secondary

and main elements.

As per the LMUC process, the tonnages and metals represented by the grade tonnage

curves simulated through several multivariate simulations, are used to derive probable

tonnages and metals, which are decomposed and distributed into the SMUs within

respective panels (referred to as Localised Multivariate Simulated Estimates or LMSE).

The paper provides a brief review of the Multivariate Uniform Conditioning, Direct

Block Simulation, LMUC and LMSE techniques and presents a comparative case study

based on a porphyry copper gold deposit in Peru.

2. Introduction

Uniform Conditioning (UC) consists of estimating the grade distribution on selective

mining unit (SMU) support within a panel, conditioned to a panel grade, usually

estimated by Ordinary Kriging to circumvent non-stationarity issues. The general

framework which forms the basis of Uniform Conditioning is the Discrete Gaussian

Model (DGM) of change of support, based in particular on the correlation between

Gaussian- transformed variables. Rivoirard (1994) and Emery (2005) discuss some of

the tests to determine if UC is a suitable methodology for a particular type of

mineralisation. They have been performed as part of the case study and confirm the

diffusive nature of the mineralisation compatible with the application of the DGM. The

DGM applies to the multivariate case, where the correlations between all variables on

2 J.DERAISME, W.ASSIBEY-BONSU

GEOSTATS 2012, Oslo, Norway

any support can be calculated after transformation into gaussian space. More precisely,

the model only requires specifying the correlations between any variable and the

primary or main variable, cut-offs being applied to the main economic element of the

deposit; while the correlations between the secondary variables are purely ignored.

Moreover when the selection of SMUs is based on a combination of different elements,

Multivariate Uniform Conditioning as applied in this paper, uses a new variable or

“equivalent” grade for the block estimate expressed by means of a “net smelter returns

(NSR). This NSR variable is then declared as the main variable. The drawback is that

the formula used to calculate the NSR variable depends on economic parameters that

may vary during the mine life. Besides, the correlations between the different element

grades are not used in the model and can only be checked after the estimates have been

derived..

In order to appreciate the impact of such a decision (ie, of cut-off being applied to the

main economic element of the deposit); the paper investigates an alternative approach

based on block co-simulations. In this latter case, the linear model of co-regionalisation

is fully incorporated in the analyses, i.e. all correlations between any pair of variables

are modelled. The expected values of tonnages and metals after applying a cut-off

within panels from a series of simulated SMUs are derived and compared with the

corresponding grade tonnage curves calculated from the MUC approach.

To compare the alternative techniques, at a local scale, the grade tonnage curves

obtained by both approaches have been assigned to each SMU by using a localization

post-processing method (See Deraisme and Assibey-Bonsu, 2011).These localised

estimates are referred to as Localized Multivariate Uniform Conditioning (LMUC) or

Localized Multivariate Simulated Estimates (LMSE) in the paper.

To optimise the simulations process a direct block simulation (DBS) method has been

used that is also based on the properties of the DGM. In the DBS method the change of

support coefficient is calculated from the multivariate variogram model of Gaussian

variables (see Emery and Ortiz, 2005), which is different from the change of support

coefficient calculated for MUC.

3. Models for non linear geostatistics

3.1. Basis of the Discrete Gaussian Model

Let v be the generic selection block (SMU) and Z(v) its grade, that will be used for the

selection at the future time of exploitation (we assume that this grade will then be

perfectly known, i.e. there is no information effect). The recoverable resources above

cutoff grade z for such blocks are:

- the ore T(z) = ( )1

Z v z≥

- the metal Q(z) = ( )( )1

Z v zZ v

≥

We use here the discrete Gaussian model for change of support (e.g. Rivoirard, 1994). A

standard Gaussian variable Y is associated to each raw variable Z. Let ( ) ( ( ))Z x Y x= Φ

be the sample point anamorphosis. The block model is defined by its block

anamorphosis ( ) ( )r v

Z v Y= Φ , given by the integral relation :

2( ) ( 1 ) ( )r

y ry r u g u duΦ = Φ + −∫ (1)

LOCALIZED UNIFORM CONDITIONING AND BLOCK SIMULATIONS 3

IX International Geostatistics Congress

where the change of support coefficient r is obtained from the variance of blocks.

Then the global resources at cutoff z are:

- ore: [ ] ( )( ) 1 1 1 ( )vZ v z Y yE T z E E G y≥ ≥

= = = − (2)

- metal: [ ] ( )( ) ( )1 1 ( ) ( ) ( )vZ v z Y y r v r

yE Q z E Z v E Y u g u du≥ ≥

= = Φ = Φ ∫ (3)

where g and G are the standard Gaussian p.d.f. and c.d.f., and y is the gaussian cutoff

related to z through ( )r

z y= Φ .

In the multivariate case indices are added to distinguish the variables. Let Z1 be the

metal grade used for the main variable, and let Z2 be one of the secondary metal grades.

In addition to the univariate case seen above, we now wish to estimate the other metals,

for instance:

zvZvZzQ ≥= )(22 11)()(

Its global estimation is given by:

[ ] [ ]

( )

1 1

1 2 1 2 1 2

2 1 2

2 2 ( ) ( ) 2 1

2, 2 1 2, 1

2,

( ) ( )1 1 ( )| ( )

1 ( ( | )) 1 ( ))

( )

v v v v

v v

Z v z Z v z

Y y r v v Y y r v

ry

E Q z E Z v E E Z v Z v

E E Y Y E Y

u g u du

ρ

ρ

≥ ≥

≥ ≥

= =

= Φ = Φ

= Φ∫

(4)

where r2 is the change of support coefficient for Z2, and 1v

Y and 2v

Y are bigaussian,

with a correlation vvvv YYcorl 2,121 ),( ρ= .

3.2. UC in the Multivariate Case

Multivariate UC (Rivoirard, 1984) consists in estimating the recoverable resources of

blocks v in panel V from the sole vector of panel estimates (Z1(V)*, Z2(V)*, …). The

problem is simplified by assuming that:

- 1( )Z v is conditionally independent of the auxiliary metal panel grades given

1( ) *Z V ,

and so the UC estimates for the selection variable correspond to the univariate case.

- similarly, Z2(v) is conditionally independent of Z1(V)* given Z2(V)*,

- 1( )Z v and

2( )Z v are, conditionally independent of the other metal panel grades given

1 2( ( )*, ( )*)Z V Z V . It follows that the multivariate case reduces to a multi-bivariate case.

In particular we have:

[ ]

=

≥

*2

*1)(2

*2 )(,)(1)()(

1VZVZvZEzQ

zvZV (5)

We further impose, for the metal at cut-off 0:

[ ] [ ]2 1 2 2 2 2( )| ( )*, ( )* ( )* ( )| ( )*E Z v Z V Z V Z V E Z v Z V= = (6)

This is similar to the univariate case, so that 2( )*Z V must be conditionally

unbiased. The model is entirely specified by the anamorphosis, the

different change of support coefficients, and the correlations between the

Gaussian variables (Y1v, Y1V*, Y2v, Y2V*). The correlation between Y1v



and Y2v, and that between Y1V* and Y2V* allow completing the

correlations by using the conditional independence relationships. For

more details on the equations see Deraisme, 2008.

3.3. Direct Block Simulations (‘DBS’)

The DGM relies on the partition of the domain into small blocks v. Then each sample

point is considered as random within its block, and conditioned to its block value (here

the multivariate value of the different elements), the point (multivariate) value does not

depend on any other variable, whether they are values of other blocks or other points,

even in the same block. This allows the deduction of the point-point and point-block

covariances from the block-block covariances. After anamorphosis, all Gaussian values

are considered as multi-gaussian, allowing conditional simulation to be implemented.

The methodology (Emery and Ortiz 2005) considers that the Gaussian transform on

block support is nothing but the regularized point Gaussian variable, normalized by its

variance, i.e. the square of the change of support coefficient r. It leads to another

determination of the change of support coefficient, using the variogram and variance of

the gaussian variable, instead of the raw variable variograms used in the previous

method: 2

var ( ) var ( ) ( , ) 1 ( , )Y Y

r Y v Y x v v v vγ γ= = − −�

In the same manner one can calculate directly the block gaussian covariances and cross-

covariances from the regularized simple and cross-covariances of the gaussian data:

1 11 11 1 2 2 2

1 1 1

( , ) 1 ( , )cov( ( ), ( ))cov( , )

h

Y h Y hhv v

v v v vY v Y vY Y

r r r

ρ γ−= = = (7)

[ ] [ ]1 2 1 21 21 2

1 2

1 2 1 2 1 2

( , ) cov ( ), ( ) ( , )cov ( ), ( )cov( , )

h

Y Y h Y Y hh

v v

v v Y x Y x v vY v Y vY Y

r r r r r r

ρ γ−= = = (8)

Developing the model above leads to a consistent method for simulating directly the

block grades conditioned to “point” data. The method is based on the use of the turning

bands simulation technique, characterized by the fact that the simulations are achieved

in two steps: first non conditional simulations to reproduce the variogram model and

second conditioning of the simulations by co-kriging from the data.

The co-kriging is based on the DGM that derives the covariances used in the co-kriging

system from the block covariance model that has been used to generate the non

conditional simulations.

Hence the covariances between elements 1 and 2 are linked by the following

relationships with the input block-block cross-covariance (Cov[Y1,vi , Y2,vj]):

- point-block cross-covariance: Cov[Y1(xi), Y2,vj] = r1 Cov[Y

1,vi , Y2,vj]

- point-point cross-covariance: Cov[Y1(xi), Y2

(xj)] = r1 r2 Cov[Y1,vi , Y2,vj] except

between the point and itself where the covariance is derived from the statistics on

the data Cov[Y1(xi), Y2

(xi)] (for a single point element, the variance is 1).

These relationships hold by considering the points as randomly located within the block

the point belongs to. The actual data location is then lost, which is acceptable when the

block is small regarding the ranges of the variograms.



4. Case study

4.1 Geology of the Deposit

The case study is based on a porphyry copper gold deposit in Peru. The mineralization

is found in intrusive rocks within sedimentary rocks. Oxidation, weathering, leaching

and subsequent secondary enrichment has led to the formation of four mineral domains

with distinct metallurgical behaviours. The Oxide Domain, is characterised by the

complete removal of copper mineralization through the action of oxidation and

leaching. Gold mineralization within the Oxide Domain is characterised by some

improvement in grade and is free milling due to the complete breakdown of primary

sulphide minerals. All of the ore beneath the Oxide Domain comprises parts of the

sulphide zone, which is separated into three domains on the basis of degree of oxidation

and consequent change in sulphide mineralogical composition. The sulphide zone has

three main domains, which from top to bottom are the Mixed Domain, the Supergene

Domain and the Hypogene Domain. The Supergene Domain is an enriched copper

blanket comprising chalcocite-covellite-chalcopyrite. The study presented in this paper

was conducted in one of the hypogene Domains. The variables studied are total gold

(AUTOT), total copper ( CUTOT) and a combination of gold and copper grades giving

an economic value, Net Smelter Return (NSR).

Figure 1: Example of West East and plan sections with coloured estimation domains.

4.2 Data Analysis

The composited data on 2m have been used to perform LMUC and LMSE on 10m

x10mx10m SMU support from:

• MUC calculated on 50m x50mx10m panels with NSR as the main variable

X

Z

X

Y



• 50 blocks co-simulations of CUTOT and AUTOT.

The three variables have a positively skewed distribution (Figure 2Figure 2) with

coefficients of variation from 0.7 to 0.9. Besides the correlations are highly significant

(see Table 1)

Figure 2: Histograms of the 2m composites for CUTOT, AUTOT and NSR.

Table 1: Matrix of coefficients of correlation between 3 variables on 2m composites.

CUTOT AUTOT NSR

CUTOT 1 0.68 0.90

AUTOT 0.68 1 0.94

NSR 0.90 0.94 1

4.3 Variographic analysis

Two different variographic analyses were performed in order to carry out the analyses

for both approaches.

To sum up there are two important differences:

• Firstly, for MUC a variogram model of raw grades is used while a variogram

model of Gaussian grades is used for the simulations.

• Secondly, for MUC the variogram model concerns the 3 variables NSR-

CUTOT-AUTOT, while for the simulations only the 2 original grades

variables CUTOT-AUTOT have to be modeled. The Net Smelter return is

calculated only after having obtained the simulated block values of grades..

MUC requires the calculation of change of support coefficients on the SMU support

and cokriging of the panels for the 3 variables NSR-CUTOT-AUTOT, NSR being the

main variable. The variograms have been calculated from the raw variables and

modelled by a small nugget effect and two spherical variograms with longer ranges

vertically than horizontally.

CUTOT

0.000

0.025

0.050

0.075

0.100

Frequencies

AUTOT

0.00

0.05

0.10

0.15

Frequencies

NSR

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequencies

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Mis en forme : Police :10 pt, NonGras, Vérifier l'orthographe et lagrammaire



Figure 3: Experimental and modelled variograms for NSR, CUTOT and AUTOT.

For simulating CUTOT and AUTOT the process is the following:

• normal score transforms of both variables,

• variograms of the normal variables are calculated and modelled,

• the variogram model is then regularized on the SMU support,

• a variogram model on the Gaussian variables regularized on the SMU support

was then fitted. On the block support the nugget effect is almost non-existent .

Figure 4: Experimental and modeled variograms on the normal score transforms of

CUTOT and AUTOT regularized to SMU support.

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

0

100

200

300

400

Variogram : NSR

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

-2.5

0.0

2.5

5.0

Variogram : CUTOT & NSR

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

0.000

0.025

0.050

0.075

0.100

Variogram : CUTOT

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

-5

0

5

10

Variogram : AUTOT & NSR

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

-0.1

0.0

0.1

Variogram : AUTOT & CUTOT

N0

D-90

0 100 200 300 400 500 600 700

Distance (m)

0.0

0.1

0.2

0.3

Variogram : AUTOT

N0

N270

D-90

0 100 200 300 400 500 600 700

Distance (m)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Variogram : Gaussian CUTOT (Block

N0

N270

D-90

0 100 200 300 400 500 600 700

Distance (m)

-0.5

0.0

0.5

Variogram : Gaussian AUTOT (Block

N0

N270

D-90

0 100 200 300 400 500 600 700

Distance (m)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Variogram : Gaussian AUTOT (Block



4.4 Change of support

The change of support coefficients are calculated in two different ways for MUC and

DBS.

The change of support used in MUC is calculated:

• for the SMU support, from the variance computed from each variogram model

of raw data.

• for the kriged panels, from the theoretical dispersion variance of the cokriging

of the panels. In order to account for the heterogeneity of the cokriging

configurations, the panels can be classified according to the variance of the

main variable, with a value of the change of support coefficient that depends

on the class.

Table 2: Change of support coefficients of SMUs, calculated from the variogram of raw

data and used for MUC

NSR CUTOT AUTOT

Punctual Variance (Anamorphosis) 307.934 0.061 0.277

Variogram Sill 300.000 0.062 0.295

Gamma(v,v) 36.385 0.017 0.062

Real Block Variance 271.549 0.045 0.214

Real Block Support Correction (r) 0.951 0.876 0.908

Main-Secondary Block Support Correction --- 0.978 0.985

For the simulations the change of support coefficients result from the regularization of

the Gaussian variogram model on the SMU support. The change of support coefficients

are rather close to the ones obtained in the previous calculations.

Table 3: Change of support coefficients for SMUs, calculated from the variogram of

Gaussian transforms of the data.

CUTOT AUTOT

Real Block Variance 0.044 0.201

Real Block Support Correction (r) 0.874 0.886

Correlation between Gaussian variables 0.860

4.5 Results

After having performed the calculations of MUC and Block Simulations, the results are

analysed at the global then at the local scale.

4.5.1 Comparison of the global grade tonnage curves obtained by MUC and DBS

The grade tonnage curves are immediately derived from the MUC results.

For calculating the same curves from 50 simulations the following procedure was used

:



• the NSR values were calculated for each SMU and each simulation using the same

formula as used for the composites.

• for the SMUs regrouped into panels 50mx50mx10m, the tonnages and metals for each

cut-off is calculated for each SMU.

• the 50 possible tonnages and metals values are then averaged in order to get an

estimate of the recovered tonnage and metals.

Figure 5 shows that globally, and for the entire domain, both curves are similar.

Figure 5: Grade-tonnage curves on smus CUTOT calculated from MUC or simulations.

4.5.2 Comparison of the local grade tonnage curves obtained by MUC and DBS

The grade tonnage curves calculated at the scale of each panel by both methods have

been localized on the SMU by using as a guide the same kriged estimate of the NSR

variable. In other words the highest grades from the grade tonnage curves are assigned

to the SMU whose kriged NSR is the highest and so on.

An example of a bench with LMUC grades assigned from MUC or LMSE from

simulations is shown on Figure 6Figure 6.

Figure 6: Plan section of the CUTOT grades assigned to SMUs from MUC or from

simulations, the cut-off being applied on NSR.

0 10 20 30 40 50

Total Tonnage

0.00

0.05

0.10

0.15

Metal Tonnage

0 10 20 30 40 50 60 70 80 90 100

Cutoff

0

10

20

30

40

50

Total Tonnage

UC CUTOT from NSR

Simulations CUTOT

763000 763100 763200 763300 763400 763500 763600 763700 763800 763900

X (m)

9252100

9252200

9252300

9252400

9252500

9252600

Y (m)

LUC CUTOT from MUC

CUTOT

N/A

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

763000 763100 763200 763300 763400 763500 763600 763700 763800 763900

X (m)

9252100

9252200

9252300

9252400

9252500

9252600

Y (m)

LUC CUTOT from simus





Figure 7: Plan section of the AUTOT grades assigned to SMUs from MUC or from

simulations, the cut-off being applied on NSR.

The similarity between both techniques can be compared by plotting the scatter

diagrams between SMU’s grades assigned by both approaches (Figure 8Figure 8).

Figure 8: Scatter diagrams of AUTOT and CUTOT SMU’s grades assigned from MUC

and 50 simulations.

The comparison of both techniques was aimed at investigating the impact of not

incorporating the correlation between the secondary variables when using the MUC

approach. The linear coefficient of correlation resulting from both approaches is quite

similar, but as shown by Figure 9Figure 9, the scatter diagram between CUTOT and

AUTOT grades shows a correlation closer to bi-gaussian when using the simulations

approach than when using the MUC approach.

This observation is probably a consequence of the strong multi-gaussian property of the

simulations achieved in the frame work of the Gaussian model.

This interpretation is confirmed by the slightly lesser standard deviation of the grades

distribution observed for the simulations compared to the MUC results.

763000 763100 763200 763300 763400 763500 763600 763700 763800 763900

X (m)

9252100

9252200

9252300

9252400

9252500

9252600

Y (m)

LUC AUTOT from MUC

AUTOT

N/A

3.02.82.62.42.22.01.81.61.41.21.00.80.60.40.2

763000 763100 763200 763300 763400 763500 763600 763700 763800 763900

X (m)

9252100

9252200

9252300

9252400

9252500

9252600

Y (m)

LUC AUTOT from simus

0 1 2 3 4

LUC AUTOT from UC

0

1

2

3

4


rho=0.909

0 1 2

LUC CUTOT from UC

0

1

2


rho=0.920


Mis en forme : Police :9 pt, Non Gras,Vérifier l'orthographe et la grammaire





Figure 9: Scatter diagrams of AUTOT vs. CUTOT SMU’s grades assigned from MUC

and 50 simulations.

Table 4: Statistics on the average of 50 simulated panel values obtained by direct block

simulation compared with the panels cokriging values. (the variance of the variable is

put on the diagonal, and the coefficients of correlation of two variables out of the

diagonal).

VARIABLE LUC CUTOT from MUC LUC AUTOT from MUC LUC CUTOT from simus LUC AUTOT from simus

LUC CUTOT from MUC 0.175 0.893 0.920 0.859

LUC AUTOT from MUC 0.893 0.372 0.801 0.909

LUC CUTOT from simus 0.920 0.801 0.157 0.898

LUC AUTOT from simus 0.859 0.909 0.898 0.343

5. Conclusions

When the orebody mineralization follows the conceptual framework of diffusive

models, multi-gaussian models provide a practical solution for calculating non linear

quantities such as tonnages and metals after cut-off.

Two different approaches of these models are available and have been compared on a

real case study using: Multivariate Uniform Conditioning and Block co-simulations.

Preliminary conclusions can be drawn from this example, indeed other applications will

help in confirming their generality.

The MUC approach does not use explicitly the correlations between the secondary

variables. Nevertheless, these correlations are in practice well reproduced as confirmed

by the simulation approach.

Block co-simulation are developed in a full multi-gaussian approach, it results in

making the correlations between pairs of block variables tending towards bi-gaussianity.

0.0 0.5 1.0 1.5 2.0


0

1

2

3


rho=0.898

0 1 2

LUC CUTOT from MUC

0

1

2

3

LUC AUTOT from MUC

rho=0.894



The comparison has been made both on the global and local scales for grade, tonnage

and metal using LMUC and LMSE techniques The study shows that both approaches

lead to similar results.

The MUC approach has the advantage of being straightforward and less time

consuming.

Working with simulations is also practically feasible as soon as a dedicated algorithm

for block simulations is used. Two important benefits come out of the simulation

approach. The first is that all models are obtained prior to any transform by economic

parameters. The second is to have access to the quantification of the uncertainty

allowing for example the selection of scenarios corresponding to different risk levels or

the building of localised confidence intervals.

Acknowledgements

The authors are grateful to Gold Fields for permission to publish this paper based on a

case study of the Group’s Cerro Corona mine.

References

Abzalov, M Z, 2006. Localised Uniform Conditioning (LUC): A New Approach to

Direct modelling of Small Blocks. Mathematical Geology 38(4): p393-411.

Deraisme J, Rivoirard J, Carrasco P, 2008. Multivariate Uniform Conditioning and

Block Simulations with

Discrete Gaussian Model : Application to Chuquicamata deposit, in Proceedings 8th

International Geostatistics Conference, 2008, Santiago, Chile:69-78. Deraisme J, Assibey-Bonsu W., 2011. Localised Uniform Conditioning in the

Multivariate Case – An application to a Porphyry Copper Gold Deposit. Proceedings

35th

APCOM Symposium, 2011, Univ. Of Wollongong, Australia, 131-140 Emery, X. and Ortiz J. 2005, Internal Consistency and Inference of Change-of-Support

Isofactorial Models,

Geostatistics Banff 2004, Springer, 2005, p. 1057-1066.

Rivoirard, J. 1984, Une méthode d’estimation du récupérable local multivariable, Note

894, CGMM Fontainebleau, Mines Paris-Tech. 10 p. Edited under

www.cg.ensmp.fr/bibliotheque .

Rivoirard, J. 1994, Introduction to disjunctive kriging and non linear geostatistics,

Oxford, Clarendon Press. 181 p.

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