Non-multiGaussian Multivariate SimulationsWith Guaranteed Reproduction ofInter-variable Correlations

Alastair Cornah and John Vann

Abstract Stochastic modelling of interdependent continuous spatial attributes isnow routinely carried out in the minerals industry through multiGaussian condi-tional simulation algorithms. However, transformed conditioning data frequentlyviolate multiGaussian assumptions in practice, resulting in poor reproduction of cor-relation between variables in the resultant simulations. Furthermore, the maximumentropy property that is imposed on the multiGaussian simulations is not universallyappropriate. A new Direct Sequential Cosimulation algorithm is proposed here. Inthe proposed approach, pair-wise simulated point values are drawn directly fromthe discrete multivariate conditional distribution under an assumption of intrinsiccorrelation with local Ordinary Kriging weights used to inform the draw proba-bility. This generates multivariate simulations with two potential advantages overmultiGaussian methods: (1) inter-variable correlations are assured because the pair-wise inter-variable dependencies within the untransformed conditioning data areembedded directly into each realisation; and (2) the resultant stochastic models arenot constrained by the maximum entropy properties of multiGaussian geostatisticalsimulation tools.

Alastair CornahQuantitative Group, PO Box 1304, Fremantle, WA, Australia 6959, e-mail: ac@qgroup.net.au.

John VannQuantitative Group, PO Box 1304, Fremantle, WA, Australia 6959.Centre for Exploration Targeting, The University of Western Australia, Crawley, WA, Australia6009.School of Civil Environmental and Mining Engineering, The University of Adelaide, Adelaide,SA, Australia 5000.Cooperative Research Centre for Optimal Ore Extraction (CRC ORE), The University of Queens-land, St. Lucia, Qld, Australia, 4067.

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2 Alastair Cornah and John Vann

1 Introduction

The advantages of stochastic modeling of in-situ mineral grade attribute variabil-ity and geometric geological variability through conditional simulation are increas-ingly recognised within the mining industry. Typical applications include studiesfor recoverable resource estimation, mining selectivity, drill hole spacing, multi-variate product specification, quantification of project risk through the value chainand many others. Recent applications for conditional simulations involve feedingmultiple realizations of key attributes (grade or geometallurgical) through a virtualmining and processing sequence in order to analyse a range of different project sce-narios under real variability conditions. The development of such approaches meansthat the demand for realistic multivariate simulations is increasing.

MultiGaussian conditional simulation algorithms, mainly the Turning BandsMethod (TBM; see Journel, 1974) and Sequential Gaussian Simulation (SGS;Deutsch and Journel, 1992) are now widely used to simulate continuous attributesin the minerals industry. Both these algorithms require that conditioning data hon-our the properties of the multiGaussian model (Goovaerts, 1997) and consequently aprior, single point, univariate transformation of the data to their normal scores equiv-alents is therefore generally required. This ensures Gaussianity of the transformedpoint attribute data, but higher order properties of the multiGaussian distributionmust also be met (Deutsch and Journel, 1998; Rivoirard, 1994; Goovaerts, 1997).This is usually assumed to be the case, but in practice that assumption is often vio-lated.

For most mining projects, multiple continuous interdependent attributes are ofessential interest. Examples include bulk commodity mining (iron, manganese,coal, bauxite, phosphate, etc.); multi-element metalliferous deposits (base metals, orbase-precious metals), etc. In many mining operations deleterious components andgeometallurgical attributes are as important to financial viability and ultimate opera-tional performance as the revenue variable(s). Because of this, reproduction of inter-variable dependencies in realisations is as important as the reproduction of marginalhistograms for each individual variable. The traditional multivariate extension ofthe multiGaussian simulation approach uses the Linear Model of Coregionalisation(LMC) to enforce inter-variable correlations between simulated attributes (Dowd,1971). The n inter-related random variables z1...n are transformed into their normalscores equivalents y1...n through monotonic, univariate, single point normal scorestransforms. Multivariate pair-wise dependencies in the untransformed sample dataare also transferred into their normal scores equivalents, potentially violating themultiGaussian assumption. Because the inverse of the normal scores transform φ−1

is also monotonic, applying the back transforms (pertaining to non-multiGaussiansample normal score equivalents) to vector of n simulated multiGaussian distribu-tions φ−1(ys1...sn) will result in multivariate pairwise dependencies in the simulateddata that are not found in the untransformed sample data; often these pairings aremineralogically impossible. In addition, the limitations of modeling direct and crossvariograms through the LMC is well documented (Wackernagel et al. 1989; Goulardand Voltz, 1992; Oliver, 2003).

Non-multiGaussian Multivariate Simulations 3

Figure 1 presents the bivariate distributions between four grade variables for sam-ple data and a back transformed multiGaussian (TBM) simulation. The black poly-gons represent areas of bi- and multi-variate pairings in the simulated values that areimpossible due to mineralogical constraint. While the realisation generally honorsthe marginal distributions of each of the variables in the conditioning data and thegeneral shape of the bivariate relationships, there are clearly a significant number ofmineralogically impossible pairings present in the simulation.

Fig. 1 Comparison of pairwise dependencies in the sample dataset (blue) for Al2O3, Fe, P andSiO2 against simulated values (red) generated using the multiGaussian LMC approach. The blackpolygons denote invalid simulated bivariate pairings.

Existing solutions to this problem within the multiGaussian framework aim tofirst decorrelate the conditioning data so that it is free from multivariate non-multiGaussian behaviors. Simulation by TBM or SGS can then be carried outindependently and the original inter-variable correlations are reconstructed post-simulation. The Stepwise Conditional Transform (SCT) was first used in geostatis-tics by Leuangthong and Deutsch (2003) and involves the transformation of multiplevariables to be univariate and multivariate Gaussian, with no cross correlations, un-

4 Alastair Cornah and John Vann

der an assumption of intrinsic correlation. However, the method requires relativelylarge datasets to be effective, can be sensitive to ordering and loses efficacy forsubsequent variables. Min-Max Autocorrelation Factors (MAF; Switzer and Green,1984; Desbarats and Dimitrakopoulos 2000; Boucher and Dimitrakopoulos, 2009)is a principal components based decomposition which is extended to incorporatethe spatial nature of data. However, non-linear dependencies between attributes areproblematic (Boucher and Dimitrakopoulos, 2004) and will also result in unlikelyor impossible attribute value pairings.

Nevertheless, even overlooking this issue, a well known feature of the multi-Gaussian approach is the imposition of maximum entropy upon stochastic realiza-tions (see Journel and Alabert, 1989). This may be pragmatically compatible withthe real deposit grade architecture for some mineralisation styles, depending on thescale considered, but in many cases it is at least questionable. For example in BandedIron Formation (BIF) hosted iron ore deposits, such as those in the Pilbara regionof Australia and elsewhere, from which the example dataset is drawn, the lowerextreme of the Fe grade distribution and upper extremes of the SiO2 and Al2O3distributions generally comprise shale bands. These shale bands are thin relative tostandard mining selectivity and are cannot be practically separated from the rest ofthe material during mining, but are far from spatially disordered. In fact individualshale bands are continuously traceable over hundreds of kilometres and are used byexploration and mining geologists as marker horizons to aid geological interpreta-tions. Consequently, imposition of maximum disorder in the simulated extreme lowFe and extreme high SiO2 and Al2O3 values in these deposits will not only result inpoor spatial representations, it is also utterly inconsistent with important features ofthe real grade architecture and known geological continuities.

Sequential Indicator Simulation (SIS; Alabert, 1987; Emery, 2004; Goovaerts,1997) provides a potentially lower entropy alternative to the multiGaussian ap-proach for the simulation of a single continuous attribute but no practical multivari-ate extension is available. The requirement for a conditional simulation algorithmwhich avoids the maximum entropy property and also ensures proper reproductionof inter-variable dependencies is therefore clear. It is also clear that these two prop-erties (maximum entropy and poor inter-variable dependency reproduction) are cou-pled. In summary, whilst the multiGaussian assumption provides congenial proper-ties for the simulation of an individual variable, it imposes maximum entropy on theresulting realisations and causes difficulty in replicating non-multiGaussian inter-variable dependencies.

2 Direct Sequential Simulation

Direct Sequential Simulation (DSS: Journel, 1994; Caers, 2000; Soares, 2001; Hortaand Soares, 2010) has been proposed to avoid the requirement for a multiGaussianassumption and also therefore offers the possibility of lower entropy stochastic im-ages. DSS is based upon a concept that was introduced by Journel (1994) who sub-

Non-multiGaussian Multivariate Simulations 5

mitted that sequential simulation will honour the covariance model whatever choiceof local conditional distribution (cdf) that simulated values are drawn from, pro-vided that distribution is informed by the Simple Kriging (SK) mean z(xu)

∗ and SKvariance σ2

sk (xu), where:

z(xu)∗ = m+∑

α

λα (xu) [z(xα)−m]

and xα are the conditioning data (including sample data and previously simulatednodes). One critical drawback was that, unless the cdf is fully defined by its meanand variance (which is true only in the case of a few parametric distributions suchas the Gaussian), the realisation does not reproduce the input (target) histogram.Recognising this, further development of DSS was carried out by Soares (2001)who, instead of using z(xu)

∗ and σ2sk (xu) to define a local cdf from which to draw

zs (given a parametric assumption), zs (xu), drew directly from the (untransformed)global cdf Fz (z). The draw uses a Gaussian transform φ of the original z(x) values.The SK estimate z(xu)

∗ is converted into its Gaussian equivalent y(xu)∗ and with the

standardised estimation variance this defines the sampling interval in the Gaussianglobal conditional distribution:

G(y(xu)∗,σ2

sk(xu))

The drawn Gaussian value ys is then back transformed to a simulated value zs (xu)using the inverse of the transform φ−1:

zs(xu) = φ−1(ys)

This development allowed DSS realisations to honour the target histogram aswell as the variogram. The method was extended to incorporate a secondary vari-able, using Collocated Cokriging (Xu et al., 1992) to define the sampling of theglobal conditional distribution of the secondary variable (Soares, 2001). Horta andSoares (2010) subsequently proposed an improvement whereby the second variableis drawn conditionally on the ranked Gaussian equivalents for the first variable. Thisapparently results in improved reproduction of dependency between variables; how-ever, the method could be progressively more problematic as the number of variablesinvolved increases. Furthermore, we suggest that this method still does not capital-ize on a critical potential benefit of the direct simulation approach, which is to allowpair-wise dependencies in the drillhole data to flow through the simulation.

3 Direct Sequential Cosimulation (DSC)

By avoiding the constraints of trying to meet multiGaussian assumptions, the di-rect approach opens the possibility of circumventing the need to destroy and thensubsequently attempt to reconstruct inter-variable dependencies. The proposed Di-rect Sequential Co-simulation (DSC) algorithm, which is built upon DSS, extracts

6 Alastair Cornah and John Vann

pair-wise dependencies from the experimental data and embeds them directly intothe realization, thus, by construction, guaranteeing the reproduction of inter-variabledependencies.

The DSC algorithm follows the traditional methodology for sequential simula-tion: visiting a random sequence of nodes, estimating the local (multivariate) cu-mulative distribution function, drawing (multivariate) values and repeating until allnodes have been visited. As with other sequential simulation algorithms, the krig-ing weights that are used to determine the local (multivariate) cdf must incorporatesample data locations and previously simulated points. The drawn values are se-quentially incorporated into the simulation until all nodes have been simulated.

In keeping with the existing DSS algorithms outlined by Horta and Soares (2010)and summarised above, simulated data values are drawn directly from the (untrans-formed) global conditional distribution. However, two key differences exist: (1) thesimulated data values are drawn without an intermediate Gaussian step and (2) pair-wise multivariate simulated values for all attributes of interest are drawn directlyand simultaneously from the experimental multivariate distribution. Correlations be-tween variables inherent within the experimental data thus become embedded withinthe realizations because of the pair-wise draw and inter-variable dependencies in theresultant simulations are assured.

The proposed DSC approach firstly requires that all variables in the sampledataset are collocated. Secondly, an assumption of intrinsic correlation between theattributes of interest is required. Under this assumption, direct and cross covariancefunctions of all variables are proportional to the same basic spatial correlation func-tion. In practice this is a reasonable assumption for key variables in many iron andbase metal deposits. These two requirements permit the following: at each locationto be simulated, a single set of OK weights (based upon the intrinsic spatial covari-ance function and the geometry of data locations as well as previously simulatednodes) can be used to define a probability mass weighting for each multivariatepair-wise data location in the surrounding neighbourhood (simulated and sampledata) to be drawn into the simulation. This can then be utilized for the draw of pair-wise simulated values from the experimental multivariate distribution. OK weightsare effectively used to represent the conditional probability for a pairwise value tobe drawn from the experimental multivariate distribution. This is consistent withthe interpretation of OK as an E-type (conditional expectation) estimate (Rao andJornel 1997) and consistent with the same author’s usage of OK weights to modellocal conditional distributions.

Because OK (with unknown mean) is used instead of SK the algorithm simulatesthe discrete multivariate distribution and cannot generate a value different from theoriginal data. In other words, in their point form, the output point simulations arenot continuous.

The DSC algorithm progresses as follows.

1. Define a random path through all nodes to be simulated.2. For each node:

Non-multiGaussian Multivariate Simulations 7

a. Determine local OK weights λ OKα (u) for the surrounding experimental data

locations z(xi) and previously simulated locations zs(xi).b. Sort the OK weights for each experimental and simulated data location

λ OKα (u) by magnitude and calculate the cumulative frequency weighting value

C(0,1) for each λ OKα (u).

c. Draw a p value from a uniform distribution U(0,1) and match to the cumu-lative frequency C(0,1); assign zs1...n(xu) and add the multivariate pair-wisevalues at the drawn experimental data or simulated data location to the condi-tioning dataset.

3. Proceed to the next node along the random path and repeat steps 2a-2d until everynode has been simulated. As with other sequential simulation algorithms anotherrealisation is generated by repeating the entire procedure with a different randompath.

Note that OK is required in DSC because applying a probability weighting tothe mean as required by SK is impossible in the pair-wise draw sense; consequentlynegative weights λ OK

α (u) < 0 are a potential outcome. Because p is drawn from abounded uniform distribution any pair-wise data location within the neighbourhoodwhich is assigned a negative weight is excluded from being drawn at the simulatedlocation and the remaining weighting is rescaled to equal 1. The implication of thisis that in the presence of negative weights, unbiasedness in the expectation of thesimulation is not explicitly guaranteed; i.e.

E {zs (xu)} 6= E{

z (xu)∗}

Other approaches, such as adding a constant positive value to all OK weights ifthere is one or more negative weight, followed by restandardisation to 1 (Rao andJournel, 1997) would also fail to guarantee unbiasedness. Our testing has shown theimpact of this to be minimal provided that estimation neighbourhood parametersare chosen to minimise negative weights. Assuming this is the case; at each gridlocation simulated values for each variable of interest are centred upon the local OKestimate z∗OK(xu):

z∗OK(xu) =n(u)

∑α=1

λOKα (u)z(uα) with

n(u)

∑α=1

λOKα (u) = 1;

The range of simulated values is provided by the range of experimental and sim-ulated data values within the search neighbourhood and the variance is the weightedvariance of the data values within the neighbourhood:

σ2OK(xi) =

n

∑α=1

λOK ·[z(xi)− z(xu)

∗]In the extreme case of no spatial continuity within the underlying geological

process, and a neighbourhood covering all of the sample data, simulated data pairs at

8 Alastair Cornah and John Vann

non data locations will be drawn at random from the multivariate global conditionaldistribution.

4 Example For an Iron Deposit

A two dimensional iron ore dataset which is sourced from a single geological unitand comprises 290 exploration sample locations within a 1.2km by 1.5km miningarea is used to demonstrate the algorithm. Four grade attributes (Fe, SiO2, Al2O3and P) are fully sampled across the area at an approximate spacing of 75m x 75m.For demonstration purposes, 25 point realisations of the four attributes (Fe, SiO2,Al2O3 and P) at 19026 nodes have been generated using the proposed DSC algo-rithm. Experimental direct variograms and cross variograms were generated for thefour variables; the direct experimental variogram for Fe was modelled using a 20%nugget contribution and single spherical structure with 80% contribution and 200mrange. This model represents a reasonable fit to the direct SiO2, Al2O3 and P exper-imental variograms when the total sill is appropriately rescaled, thereby satisfyingthe intrinsic correlation requirement for the proposed DSC algorithm.

Fig. 2 A comparison between multiGaussian and DSC realisations of Fe.

A comparison between a DSC realization and a multiGaussian realization isshown in figure 2. The grade architecture generated by DSC is more like the mosaicmodel (Rivoirard, 1994; Emery, 2004) and is thus appealing in cases like the onepresented here, where maximum entropy is clearly unacceptable. A DSC realisationof the four attributes is shown in figure 3 with the bivariate distributions between

Non-multiGaussian Multivariate Simulations 9

the four simulated attributes also shown. The two key advantages of the DSC algo-rithm are apparent from the figure, specifically (1) the inter-variable relationshipsin the simulated attributes match those observed in the sample dataset; and (2) thearchitecture of the grades in the realization is driven by the sample data itself anddoes not have the appearance of maximum entropy. In addition, by construction therealisations honor the declustered sample histograms of the variables of interest.Experimental direct and cross variograms for the 25 DSC realisations of the fourgrade attributes are compared against their sample data equivalents in figure 4. Aclose comparison is noted for both the direct and cross experimental variograms.Ergodicity of the simulations around the sample data experimental variograms isapproximately consistent with that which might be expected from multiGaussianalgorithms.

Fig. 3 A DSC realisation of Al2O3, Fe, P and SiO2 with inter-simulated attribute dependencies.

10 Alastair Cornah and John Vann

Distance (m) Distance (m) Distance (m) Distance (m)

Distance (m) Distance (m)

Distance (m) Distance (m)

Distance (m)

Distance (m)

Fig. 4 Comparison of experimental direct and cross variograms for 25 DSC realisations of Al2O3,Fe, P and SiO2 and the sample dataset.

5 Limitations

Because the proposed DSC method simulates the discrete multivariate distribution,simulated values cannot deviate from those in the experimental dataset. This is botha strength of the algorithm, because it allows the pair-wise dependencies in the sam-ple data to be embedded directly into the realizations; but it is also a weaknessbecause potential for simulation of adjacent nodes with the exact same value ex-ists. This is particularly likely to be the case if the modeled spatial continuity ishigh. Consequently re-blocking to change support is a critical pre-requisite to anyper realization use of the simulations with the assumption being that the averagingprocess will reproduce a realistic continuous distribution at the support re-blockedto. Prior kernel smoothing of the input data would result in a reference distribution

Non-multiGaussian Multivariate Simulations 11

from which to draw continuous multivariate simulated values, however this couldintroduce invalid data pairings which are avoided in the algorithms’s proposed im-plementation. A further disadvantage of the DSC approach over the multiGaussianapproach is that unlike the latter (see Deraisme et al., 2008), no direct block equiv-alent is available. This is not a drawback from small simulations, but is a potentialrestriction for large scale mining problems.

6 Conclusions

Applications of conditional simulation within the minerals industry rely directlyupon the integrity of the simulated architecture of those in situ attributes. Ade-quately representing the connectivity of extremes of attribute distributions is criticalbecause very often these are the key drivers of value or causes of interaction withmining, processing, or product specification constraints; adequate representation ofinter-attribute dependencies is equally important. The authors believe that these re-quirements demand the ongoing development of conditional simulation algorithmsfor continuous multivariate attributes outside the of the multiGaussian framework.The DSC algorithm presented is intended to contribute to this effort by providing alow entropy alternative to the multiGaussian approach for the conditional simulationof multiple continuous attributes that also guarantees that inter-variable dependen-cies are honored.

The intrinsic correlation, full sampling and collocation of sampling requirementsof DSC are no more onerous than those of SCT and in general the practical imple-mentation of DSC is far more straightforward than the multivariate multiGaussianapproaches. No LMC modeling is required and the additional transformations andassociated validation steps that are required in MAF and the SCT are also avoided.

Acknowledgements Professor Julian Ortiz of the University of Chile and our Quantitative Groupcolleagues, in particular Mike Stewart, are thanked for feedback and discussions about the pro-posed method prior to the writing of this paper. Any remaining deficiencies are entirely the respon-sibility of the authors.

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