digital.library.adelaide.edu.au · Figure 1). This channel iron deposit (CID) infills the meandering - Tertiary palaeochannels of Marillana Creek, which are 450m-750m wide and …

Chris De-Vitry – MSc Thesis

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SIMULATION OF CORRELATED VARIABLES

A Comparison of Approaches with a Case Study from the Yandi Channel Iron Deposit

Chris De-Vitry

School of Mathematical Sciences Faculty of Engineering, Computer and Mathematical Studies

University of Adelaide Submitted: 12th of February 2010


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1 INTRODUCTION

1.1 Introduction and location BHP Billiton’s (BHPB) Yandi CID is 150 km by road North of the town of Newman in the Pilbara region o f Western A ustralia ( Figure 1). T his channel i ron deposi t ( CID) i n-fills the m eandering Tertiary palaeochannels of Marillana Creek, which are 450m-750m wide and around 100 m deep in the centre. The deposits outcrop as a series of low mesas beside the present day creeks and were divided into three areas (Western, Central and Eastern) within which individual mesas are numbered (Figure 2). A North-South trending part of the channel (Mesa C4, and part of Mesa C5) is the focus of this study.

Drilling of the C4 and C 5 mesas is by a combination of diamond and r everse circulation on grids varying from 25m x 50m to 100 m x 100m . Although the Yandi C ID is mined for i ron or e, the concentrations of t he contaminants Al2O3, S iO2 and P 2O5 are cr itical. The l oss on i gnition (LOI), while not strictly a contaminant, is also important to the customer buying the iron ore. Although the oxides TiO2, K2O, MgO and MnO are not problematic contaminants at Yandi, their distributions are useful f or characterising g eological domains1

This thesis used Minesight™ software f or t he visualisation and coding of data, building bl ock models and building a simulation grid. Statistical and g eostatistical investigations were completed using Isatis™. Statistica™ was used for Principal Component Analysis, Discriminant Analysis and Ternary Diagrams.

and under standing t he behav iour o f the major contaminants.

1 A domain is an area that has similar geological, statistical and geostatistical characteristics.


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Figure 1. Map of the Pilbara region showing the location of the Yandi CID (BHP Billiton, 1999).

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Text Box

NOTE: This figure is included on page 19 of the print copy of the thesis held in the University of Adelaide Library.


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Figure 2. Map of the Yandi mine lease and individual deposits (BHP Billiton, 2002).

a1172507

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1.2 Background and objectives Drilling a t Y andi is widely spaced and current est imates are mostly l inear av erages of the surrounding local samples (Figure 3). L inear est imates such as Ordinary Kriging (OK) based on block sizes that are significantly smaller than the dimensions of the drilling grid are less variable (Figure 4) than the unknown reality (Armstrong and Champigny, 1989).

Conditional2

1. Predicting the tonnes and grade from drill hole data that will be extracted from mining in the future;

Simulation provides a series of plausible realisations that represent the actual grade variability a t a small scale (e.g. 1mx1mx1m). The simulations also reproduce the histogram and variogram (the variogram is a measure of spatial grade continuity) of the data. These simulations are very useful for a number of purposes, e.g.:

2. Optimising the size and selectivity of mining equipment;

3. Optimising the way mined material is blended to form a saleable product;

4. Selection of optimal drill hole spacing; and

5. Assessing risk via pit optimisation of multiple realisations.

At Yandi the major constituents of the ore (Fe, P2O5, SiO2, Al2O3 and LOI) are correlated to varying degrees. For ex ample, Fe and S iO2 have a c orrelation co efficient o f about -0.8. I ndependent Conditional Simulation of each assay variable will not reproduce these correlations. Reproducing the correlations (and the shape of scatterplots) is important if the simulations are used for the abovementioned pur poses. T his thesis examines and co mpares various methods of Conditional Simulation t hat aim to reproduce t he co rrelations between t he assay variables. The ob jective o f this comparison is to determine which Conditional Simulation method is both practical to implement for mining companies and best reproduces the characteristics of the assay variables.

2 Conditional is a term used to indicate that the simulations are conditional to assay data. With Conditional Simulation, the simulated values are always the same at the data locations however; each simulation can differ away from the data points.


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Figure 3. Diagram explaining linear estimation.

Figure 4. Ordinary Kriging estimate (left) and Conditional Simulation (right) of Fe.

1.3 Why choose Yandi for this study? Yandi is a large channel iron deposit (CID), which has an extremely high economic value to BHP Billiton. Channel iron deposits are also becoming increasingly important for the extraction of iron


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ore in Western Australia. In bedrock iron ore deposits (the predominant style of iron ore deposit mined in Western Australia) the stratigraphy and structure tightly control t he co ncentration and spatial distribution of the iron ore. Although geostatistics is useful in these bedrock deposits, the interpretations created by t he g eologist hav e t he bi ggest i mpact o n t he final est imates of the tonnes and grade of the ore. In CID’s the geology is still cr itical however, the geological domains are often larger and the stratigraphic boundaries more diffuse and di fficult for geologists to locate. Thus, geostatistics is more likely to have a large impact on the final estimate of tonnes and grade above economic cut-offs.

The Yandi CID contains clay pods that have dimensions smaller than the drill hole spacing. Thus drilling ca nnot pr edict the exact location o f t hese cl ay pods. These cl ay pods have hi gh concentrations of Al2O3 and SiO2 (i.e., contaminants to the iron ore). Geostatistics can potentially be very useful in making predictions of ore tonnes and grade that incorporate the uncertain locations and proportions of these clay pods.

1.4 Previous work at Yandi

1.4.1 Introduction Yandi (Figure 2: E2 Mesa) was first developed in 1991 (Kneeshaw, 2004) and has been continuously mined since. There have been numerous geological studies (Chapter 4) and resource estimation studies by BHP Billiton staff and consultants. The Yandi CID has also been the subject of g eostatistical research. The author has completed a resource estimate (De-Vitry, 2003) and written a paper on geological domaining at Yandi (De-Vitry, 2005) before commencing this MSc. Some pr eliminary obse rvations from a l iterature r eview f or t his thesis were publ ished ( De-Vitry, Vann and Arvidson, 2007).

1.4.2 BHPB resource estimates Resource es timates for Yandi ar e completed for di fferent pa rts of t he deposit at di fferent t imes. Kentwell (2006) completed the latest resource report for the study area. This report documents the data validation, q uality assurance/quality co ntrol (QA/QC), dom aining, and t he k riged r esource estimate. The same drill hole data used by Kentwell were also used for this study.

1.4.3 BRC Conditional Simulation and scheduling The WH B ryan M ining, G eology R esearch C entre (BRC) at t he U niversity of Q ueensland completed two research studies in 2005:

1. “Uncertainty and risk quantifying optimisation f or open pit mine design and production scheduling – Australian R esearch C ouncil ( ARC) Linkage G rant LP 0211446” (see Dimitrakopoulos, 2005 and Li, D imitrakopoulos and Boucher, 2005 for further information on this project); and

2. “Development of a new stochastic short-term production scheduling optimisation approach for open pit metal mines - ARC Linkage Grant LP0348798” (see Dimitrakopoulos, 2005 and Benndorf, Dimitrakopoulos and Dagbert 2005 for further information on this project).

These studies were sp onsored by X strata, B HP Billiton, A nglo G old and R io T into. The A RC Linkage Grant LP0348798 included a section on the application of production scheduling at the C1 Mesa of t he Yandi C ID. T his scheduling was based on si mulations generated using Conditional Simulation of Min/Max Autocorrelation Factors (MAF). The simulations and the methodology used were not docu mented in Benndorf, D imitrakopoulos and D agbert ( 2005); how ever, a se parate users manual w as produced. This users manual (Li, Dimitrakopoulos and B oucher, 2005 ) describes the implementation of the MAF simulation in Datamine™ software.

1.4.4 BRC Multipoint Conditional Simulation Multiple-Point Conditional Simulation is a method in which the simulations not only reproduce the histogram and v ariogram o f the da ta but the spatial characteristics of a t raining image (Journel, 2007). This technique can reproduce complex non-linear geological patterns. Thus, Multiple-Point Conditional S imulation is potentially use ful for reproducing t he sh apes of cu rved se dimentary


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channel facies such as those present at Yandi. Volker Osterholt completed an MSc by research on this topic at the University of Queensland at the BRC (Osterholt, 2006). The research applied the multipoint method to simulating the shape of the major geological domains at Yandi (The following papers were al so publ ished from t his MSc: Osterholt and D imitrakopoulos 2007a; Osterholt and Dimitrakopoulos 2007b).

1.4.5 BRC Min-Max Autocorrelation Factors case study As discussed in Section 1.2, to be useful conditional simulations must reproduce the correlations between assay variables. T o simplify the process the c orrelations can be removed via Min-Max Autocorrelation Factors and the variables simulated independently. The simulated factors are then back-transformed to reinstate the correlations. Cameron Boyle completed an MSc by research on this method at t he U niversity of Queensland (Boyle 2007a) . B oyle ( 2007a and B oyle 2007b ) compared simulation using Min-Max Autocorrelation Factors to the more traditional Conditional Co-Simulation using the turning bands method.

1.5 Abbreviations This thesis makes use of abbreviations. The meaning of these abbreviations are summarised in Table 1.


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Al2O3 Alumina

BHPB BHP Billiton

BHPBIO BHP Billiton Iron Ore

BIF Banded Iron Formation

BRC WH Bryan Mining, Geology Research Centre, University of Queensland

CaO Calcium Oxide

CS Conditional Simulation

CCS Conditional Co-Simulation

CID Channel Iron Deposit

DD Diamond Drilling

E1, C4 and W1 Mesas Eastern One, Central Four and Western One Mesas

EDA Exploratory Data Analysis

Fe Iron

K2O Potassium Oxide

MAF Min/Max Autocorrelation Factors

MgO Magnesium Oxide

MnO Manganese Oxide

LMC Linear Model of Co-Regionalisation

LOI Loss on Ignition

OK Ordinary Kriging, also Ocherous Clay at Yandi

P Phosphorous

PCA Principal Component Analysis

QA/QC Quality Assurance and Quality Control

QKNA Quantitative Kriging Neighbourhood Analysis

RC Reverse Circulation Drilling

SCT Stepwise Conditional Transform

SGS Sequential Gaussian Simulation

SiO2 Silica

SK Simple Kriging

TB Turning Bands

TD Tertiary Detrital

TiO2 Titanium Oxide

UC Uniform Conditioning Table 1. Description of abbreviations.


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1.6 Methodology and outline of this study The Yandi CID is very large and a smaller length of the channel was selected for this study. Within this smaller length of the channel, there are zones with different geological and statistical characteristics (domains). O ne su ch dom ain was selected f or t he ap plication of t he v arious simulation methods tested in this thesis.

As shown in Figure 5, this thesis can be divided into five main areas:

1. Chapters 1 to 3: These sections provide the reader with general and geostatistical background.

2. Chapters 4 t o 6 : These se ctions deal with d etermining w hether t he geological dom ain selected for further study and the drill hole data within this domain form an appropriate basis for further s tudy. G eology and ex ploratory dat a a nalysis (EDA) are al so use d t o characterise this domain;

3. Chapters 7 to 9: Each of the four simulation methods used in this thesis have underlying geostatistical assumptions. Each si mulation m ethod and t he asso ciated assumptions are discussed. The methodology (simulation grids, search parameters etc) for implementing each simulation technique is also discussed;

4. Chapters 10 to 14: In these sections, the generation and validation of simulations for each of the four techniques is outlined;

5. Chapter 15: The strengths and weaknesses of the results from the four simulation methods are assessed and compared; and

6. Chapter 16: Recommendations are made regarding the preferred method of simulation at Yandi and directions for future research discussed.


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CHAPTER 4: GEOLOGY AND DATA DESCRIPTION

CHAPTER 5: EDA AND STATISTICAL VALIDAITON OF DOMAINING

CHAPTER 6: VALIDATION AND EDA OF THE M3 DOMAIN

DOES THE ASSAY DATA AND GEOLOGICAL INTERPRETATION FORM AN ACCEPTABLE BASIS FOR THIS THESIS

ARE THE ASSUMPTIONS BEHIND EACH SIMULATION METHOD VALID AT YANDI AND WHAT PARAMETERS WILL BE USED FOR CONDITIONAL SIMULATION

CHAPTER 7: TESTING OF GEOSTATISTICAL MODELS AND ASSUMPTIONS

CHAPTER 8: BLOCK SIZE A POTENTIAL ISSUE FOR ORDINARY KRIGING AND UNIFORM CONDITIONING

CHAPTER 9: CHOOSING PARAMETERS FOR CONDITIONAL SIMULATION

CHAPTER 10: CONDITIONAL CO-SIMULATION

CHAPTER 11: SIMULATION USING MIN/MAX AUTOCORRELATION FACTORS

PRODUCING VALID SIMULATIONS USING A VARIETY OF METHODS AND VALIDATING THOSE SIMULATIONS CHAPTER 12: SIMULATION

USING THE STEPWISE CONDITIONAL TRANSFORM

CHAPTER 14: COMBINING GEOLOGY AND ASSAY VARIABLE SIMULATION

CHAPTER 15: DISCUSSION AND COMPARISON OF SIMULATION METHODS

CHAPTER 16: RECOMMENDATIONS

BACKGROUND CHAPTER 1: INTRODUCTIONCHAPTER 2: GEOSTATISTICALBACKGROUND

CHAPTER 3: BACKGROUND TO CONDITIONAL SIMULATION

CHAPTER 13: COMPARISONOF SIMULATIONS TO BLAST-HOLE DRILLING

Figure 5. Diagrammatic thesis outline.


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2 GEOSTATISTICAL BACKGROUND

2.1 Introduction Geostatistics deals with making predictions at locations for which we have no data. To account for uncertainty in t his process geostatistics uses probability t heory. T he v alue of a v ariable at a n unknown location x can be estimated by combining a deterministic (trend) component (m(x)) and a random component (R(x)) i.e., Equation 1. Z(x) = m(x) + R(x) This is known as a random function model. We view our assays e.g. Fe, P etc as the outcome of a random function at a given location (x) i.e., z3

(x).

In t his chapter, we di scuss some fundamental g eostatistical concepts. Numerous books discuss these co ncepts in det ail e. g. Chiles and D elfiner ( 1999), D avid ( 1977), I saaks and S rivastava (1989), Journel and Huijbregts (1978) and Lloyd (2007).

2.2 Stationarity In geostatistics, we consider an assay as a single realisation from a random function. This assay could theoretically have t aken any v alue within t he cu mulative di stribution function ( cdf). T he cumulative distribution function is defined as: Equation 2. F(x;z) = P{Z(x) ≤ z} for all z Where P is the probability that Z(x) is no greater than a g iven value of (z). Because we only have one realisation of the random function (i.e., one value at each location), we cannot define it exactly. Thus, in order to make predictions we must make assumptions about the random function. These assumptions are known as the ‘stationarity assumption’. Strict stationarity involves the assumption that the random function does not change as a function of (x). In m ining appl ications, t his assumption i s too s trong and the assumption o f second or der stationarity i s used. S econd or der st ationarity assu mes that t he expected value of t he mean (E{Z(x)}) and the co-variance (C(h)) do not vary as a function of (x) i.e., Equation 3. E{Z(x)} = m for all x Equation 4. C(h) = E[{Z(x) - m}{Z(x+h) – m}] The covariance only depends on the distance between samples (h) and not the actual location of these samples. It i s important to note that because we onl y ha ve one r ealisation o f t he r andom function that the as sumption o f s tationarity ca n nev er be pr oven. However, t he as sumption of stationarity could for example be considered ‘acceptable for practical purposes’. Dividing a dat aset into areas that are acceptable form the point of view of stationarity is of critical importance in geostatistics. In this thesis, domain validation and stationarity rightfully receive significant attention.

2.3 Gaussian transformation As discussed above, second order stationarity assumes that the first two moments of the distribution (mean and variance or covariance) are known. With a Gaussian distribution, these first two m oments are su fficient t o descr ibe t he complete di stribution. This property of t he Gaussian distribution makes it useful for estimating complete distributions; i.e., cdf’s at unknown locations. Gaussian transformations are used in all the Conditional Simulation methods applied in this thesis. 3 We utilise uppercase Z for the random function while the lower case z is used to denote an actual realisation (a value) from that random function.


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The m ethod use d i n t his thesis for Gaussian t ransformation i s known as Frequency Inversion (Geovariances, 2005) . This simply i nvolves sorting the assa ys from l owest t o hi ghest an d transforming each value to the corresponding value on the Gaussian cdf.

2.4 Kriging Ordinary Kriging is known as a linear estimator because it involves the estimation of the unknown value (Z) via a weighted linear combination of samples around Z4

i.e.,

Equation 55 ∑==

n

1iii zλZ*.

Intuitively a good estimator will be unbiased; i.e., on average the estimated values will equal the unknown values and t he est imation e rrors (Z-Z*) will be as small as possible. It ca n be demonstrated (e.g. Isaaks and Srivastava, 1989) that the unbiasedness condition can be met when the weights sum to one i.e.,

Equation 6. 11

=∑=

n

iiλ

The second condition requires that the estimation variance; i.e., the variance of (Z-Z*) is as small as possible. Of course, the true value (Z) is not known and thus we must introduce a model if we are to calculate the estimation variance. This model is obtained from fitting a function to either the experimental variogram ( )(ˆ hγ )6

or the experimental spatial co-variance function i.e.,

Equation 7. ∑ +−==

n

iii hxzxz

nh

1

2)}()({21)(γ̂

Equation 8. ∑ −+−==

n

iii mhxzmxz

nhC

1})()}{)({1)(ˆ

In simple terms the variogram and spatial co-variance function measure the similarity between one assay at location x and a second assay at location x+h. These measures of spatial variability can be extended to the calculation of the error obtained when estimating the grade of a block by the surrounding samples i.e.,

Equation 9. )()(),(2*)(

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

nvVnvVZZVar

ZVarZZCovZVarZZVar

γγγ −−=−

+−=−

4 Z* denotes and estimate where Z represents the actual unknown value. 5 Ordinary Kriging assumes that the mean of the domain is unknown and thus it cannot be used to estimate Z*. An alternative to Ordinary Kriging is Simple Kriging, which assumes that the mean of the domain is known and it can be used in estimation. Thus, Simple Kriging implies more stringent assumptions of stationarity than does Ordinary Kriging. For Simple Kriging Equation 5

becomes: [ ]∑ +==

n

iiii zZ

1* ωλλ , where iω is the weight of the mean and is equal to

∑−=

n

ii

11 λ . Both Ordinary Kriging and Simple Kriging can be used as a basis for conditional simulation. 6 We use the ∧ symbol to denote that both the variogram and cross-variogram are estimated from our data.


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Where:

)(Vγ = Average value of the variogram within block V.

),( nvVγ = Average variogram value between the samples (nv) and the block V.

)(nvγ = Average value of the variogram between all samples (including the variogram of a value with itself).

Now we must add the above estimation variance7

formula to the unbiasedness constraint (the sample weights ar e ex cluded from t he abov e for si mplicity). In o rder to do t his we m ultiply t he unbiasedness constraint by a term known as the lagrange parameter (µ). The lagrange parameter is required to obtain n simultaneous equations and n unknowns (without µ the kriging equations would not be so lvable). In order to find the weights for each sample in the linear estimate that will minimise the estimation variance and thus obtain the kriging variance we must use differentiation. Firstly, we differentiate the above equation with respect to the weight for sample one and se t the resulting equation to zero. We then repeat the procedure for sample two and so on. We must also differentiate w ith r espect t o the l agrange par ameter and se t t he r esulting equation to zero. This results in the completely general kriging equations:

Equation 108 niforn

jivijj ,...,1

1=∑ =+

=γµγλ.

∑ ==

n

ii

11λ

∑ −+==

n

iiviK V

1

2 )(γµγλσ

Where 2Kσ is the optimal (i.e., minimised) kriging variance9

0≥γ

. For the above equations to be valid t he v ariances obtained f rom t he v ariogram m odel must be ei ther z ero or posi tive i .e.,

. This is known as ‘positive definiteness’.

2.5 Co-Kriging and the Linear Model of Co-Regionalisation

When a second variable is available and is correlated with the primary variable, both variables can be est imated t ogether using Co-Kriging. With Co-Kriging, bot h t he variograms (auto-variograms) and the cross-variograms must be known. The cross-variogram is calculated using the following formula: Equation 11.

)}()({)}()({21)(ˆ

1hxzxzhxzxz

nh iviv

n

iiuiuuv +−∑ +−=

=γ

7 The estimation variance formula assumes that all the weights applied to the samples are equal. The kriging variance (i.e. the optimal or minimised variance) is discussed below. 8 Note this equation is for the kriging of a block i.e. Block Kriging. Point Kriging occurs when the samples and the point being estimated have the same support. To change this Block Kriging equation to Point Kriging we

replace ivλ with 0iλ because we calculating variogram values between points (not between a point

and a block) and )(Vγ becomes zero and can be removed. Conditional simulation within this thesis utilises Point Kriging. 9 Note that the Ordinary Kriging variance is also equal to the Simple Kriging variance + the variance due to uncertainty in the mean. Simple Kriging produces identical kriged values to Ordinary Kriging when Simple Kriging utilises the domain mean calculated from Ordinary Kriging.


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There needs to be Nv auto-variograms and Nv(Nv-1)/2 cross-variograms, where Nv is the number of variables. Positive def initeness (as described a bove) i s not su fficient for Co-Kriging. W ith Co-Kriging, linear combinations of variables are themselves treated as regionalised variables and their variances must be greater than or equal to zero. The Linear Model of Co-Regionalisation insures that this condition is met.

A variogram can be defined as the combination of the variograms from any two variables u and v, i.e.,

Equation 12. ∑==

L

ll

luvuv hbh

1

)( )()( γγ

Where )(l

uvb (l is an index not an exponential term) nugget and structural variances for all l must be positive def initive. For this condition to be met any structures present in the cross-variograms must be present in each of the autovariograms. In practice this limitation can be overcome by using a near zero variance for one of the variogram structures.

Various co-kriging equations can be utilised for co-simulation. The co-kriging system utilised within Isatis is discussed in Appendix 1.

2.6 Introduction to diffusivity and multivariate Gaussianity All the geostatistical estimation and simulation methods (besides Indicator Conditional Simulation) used i n t his thesis rely on t he under lying assu mption t hat t he sp atial di stribution o f t he assa y variables can be best descr ibed by a ‘diffusion model’. With t he di ffusion m odel, assay variable concentrations tend to move from lower to higher values and vice versa in a r elatively continuous way (Vann, Guibal and Harley, 2000; Rivoirard, 1994). Intuitively this assumption must be true for methods that directly utilise kriging. This is because in kriging we utilise a linear combination of the surrounding samples to produce the estimated grades. If these samples were not related (i.e., the diffusion model were invalid) but belong to an adjacent domain there would be no reason to include these samples in the kriging. In Section 7.1, the assumption of diffusivity in the Final M3 Domain is tested.

Gaussian simulation methods also rely on the assumption of bi-Gaussianity; i.e., that the values of pairs of sa mples separated by a di stance h ar e bi variate G aussian. B y def inition, a G aussian transform results in a uni variate Gaussian distribution of the transformed values but this does not guarantee that t he v alues of pai rs of sa mples separated by a di stance h follow a bi variate Gaussian distribution. In other words, for the diffusion model, given pairs of samples separated by a distance h, the conditional distribution of one variable on the other should be Gaussian.

For multivariate Conditional Simulation (i.e., Conditional Co-Simulation) the assumption of bi-Gaussianity i s necessary but not su fficient. There i s an addi tional assumption o f multivariate bi -Gaussianity. T hat i s, bi variate sca tterplots and bi variate h -scatterplots ( for ex ample Fe versus SiO2) should also be bivariate Gaussian.

At point level, the assumptions of bi-Gaussianity and multivariate bi-Gaussianity can be tested but cannot be pr oven. For example, in t he case of the multivariate-Gaussianity assumption, there i s insufficient dat a ( or t ime) t o test ev ery assay v ariable or assa y variable co mbination at ev ery conditional cut-off and every h-separation. For more information on bi-Gaussianity and multivariate bi-Gaussianity refer to Chiles and Delfiner (1999) and Emery (2005). Tests for bi-Gaussianity and multivariate bi-Gaussianity are discussed further in Chapter 7.


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3 BACKGROUND TO CONDITIONAL SIMULATION

3.1 Introduction This chapter provides descriptions of estimation and simulation methodologies that are applicable for correlated assay variables and are used in this thesis. Discussed below are three Conditional Simulation (CS) methods. Firstly, Conditional Co-Simulation (CCS) involves simulating while taking into account the correlation between assay variables. The remaining two simulation methods involve removing the correlation between assay variables, using Min/Max Autocorrelation Factors (MAF) or the Stepwise Conditional Transform (SCT) before performing independent simulation of the transformed assay variables. Also discussed in this chapter is Indicator Conditional Simulation, which is a technique used to simulate the distribution of clay pods within the Yandi CID. This chapter is substantially based on a pape r (De-Vitry, V ann and A rvidson, 2007) w hich w as published before completion of this thesis, but based on work completed for this thesis.

3.2 Conditional Simulation

3.2.1 Overview of Conditional Simulation Geostatistical simulation is a spatial extension of the concept of Monte Carlo simulation. In addition to reproducing the data histogram, geostatistical simulations also honour the spatial variability of data, usually characterised by a variogram model. If the simulations also honour the data values themselves, they are called ‘Conditional Simulations’.

Geostatistical simulations generate a series of images, or ‘realisations’ as opposed to estimates, which out put a uni que result. A se ries of realisations presents a r ange of pl ausible possi bilities about the spatial distribution of an assay variable that are consistent with t he known statistical moments (variogram a nd hi stogram) and i n t he ca se o f C onditional S imulations, t he da ta themselves. Simulation thus has different objectives to estimation. The point is to characterise and then r eproduce t he variance o f t he i nput data. Geostatistical si mulations provide an appr opriate means to s tudy pr oblems relating t o v ariability i n a way t hat es timates cannot. For ex ample, simulations provide a pat h to evaluation and an alysis of such issues as dr ill spacing, selectivity, blending, equipment se lection, and se nsitivity t o di fferent m ine scheduling approaches (for more information and ex amples refer t o H umphreys and S hrivastava, 1997; Jackson et al., 2003 and Sanguinetti et al., 1994).

To date, the two most commonly used methods for Conditional Simulation in the mining industry are:

1. Turning Bands (TB), which was the first large-scale three-dimensional Gaussian simulation algorithm implemented (Journel, 1974). The method works by simulating one-dimensional processes on l ines regularly sp aced i n t hree di mensions and then su mming t hese processes in the three-dimensional space; and

2. Sequential Gaussian S imulation ( SGS), w hich i s efficient and w idely use d (Lantuejoul, 2002). This algorithm de fines a r andom pa th t hrough al l g rid node s (including t he conditioning samples). Simple Kriging (SK)10

Both o f t hese si mulation m ethods rely on t he data hav ing Gaussian distributions. B ecause i n practice data never meet this criterion, the data must first be transformed into a Gaussian assay variable with a mean of zero and a standard deviation of one. After simulation of Gaussian values, the simulated values at the nodes are back-transformed into the raw data distribution.

is used to estimate the mean and variance of the conditional Gaussian distribution at each node in the path. A simulated value is drawn from this local distribution and added t o the node in the random path and the next node is simulated, and so on.

10 Simple Kriging is kriging without the unbiasedness constraint. Unbiasedness is assured by assuming that the mean is known.


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3.2.2 Conditional Co-Simulation Conditional Simulation, as outlined above, deals with a single assay variable. If independent simulation o f a m ultivariate dat aset i s performed ( e.g. co ntaminants in an i ron or e depo sit), the resulting si mulations will not reproduce any co rrelations among t he ass ay variables. C onditional Co-Simulation (CCS) is Conditional Simulation of more than one assay variable where inter-variable co rrelations are pr eserved i n t he out put r ealisations. C onditional C o-Simulation incorporates Co-Kriging11

3.2.3 Strengths and limitations of Conditional Co-Simulation

and t hus cross-variogram m odels of t he assa y variables as well a s individual variogram m odels to characterise t he spatial co rrelation of and bet ween al l t he assay variables.

For C CS, ex perimental v ariograms and c ross-variograms are generated and m odelled. The experimental variograms and cross-variograms are more difficult to fit with coherent models than independent experimental variograms. This is because a change to one of the variogram or cross-variogram models affects the fit of all the remaining models. The constraint that variograms and cross-variograms models must be posi tive def inite ( Armstrong, and Ja bin, 1981) i s also q uite restrictive t o obt aining models that ar e a good fit to t he ex perimental v ariograms and cr oss-variograms.

Conditional C o-Simulation assu mes, v ia t he Linear M odel of C o-Regionalisation ( LMC), t hat correlations are linear. The Linear Model of Co-Regionalisation is a method used to ensure that all the v ariances obtained from co -kriged est imates are z ero or posi tive ( Dowd, 1985 and S hibli, 2003). In some datasets, the correlations between assay variables are not linear (Figure 7) which is a limitation of CCS.

Once variogram and cross-variograms models are generated, the co-simulations take significantly more computer time than simulating the assay variables independently. Validation of the simulations is also m ore t ime co nsuming beca use co rrelations must be adeq uately r eproduced. These co mplications have l imited t he use o f simulation i n t he i ron or e i ndustry, w here t he correlations between assay variables are usually very important, and have led to the development of the alternatives discussed below. Conversely, CCS is well tested for mining applications.

3.3 Conditional Simulation using the Stepwise Conditional Transform

3.3.1 Overview of Conditional Simulation using the Stepwise Conditional Transform

As described by Leuangthong and Deutsch (2003), this technique is implemented in the same way as a typical univariate Conditional Simulation with the exception of the Stepwise Conditional Transform (SCT). This transformation has the objective of removing the correlation between assay variables. This allows the assay variables to be simulated independently, w ith the back-transformation of the simulations reinstating the correlation between the assay variables.

The order of the SCT of assay variables is important because the reproduction of the variogram and the correlations between assay variables (for the output simulations) will be most accurate for the primary assay variable and the reproduction quality decreases for the second assay variable and subsequent assay variables.

The steps as described by Leuangthong et al. (2006) are:

1. Decluster data (Section 8.2.2) to obtain representative distributions;

2. Determine the order of assay variables for transformation;

3. Apply the SCT to each assay variable, in the d etermined order, i n order to remove t he correlation between assay variables;

11 Co-Kriging is used to estimate the Gaussian mean and variance before randomly selecting a value from this estimated distribution and then adding this value to the sample data before continuing with the Co-Simulation.


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4. Calculate and model experimental variograms of the transformed data;

5. Independently simulate the transformed assay variables;

6. Back transform the simulated values by performing the SCT in reverse; and

7. Validate the simulation results.

The SCT transform is described below, with an example in which Fe is the primary assay variable, SiO2 the secondary and Al2O3 the tertiary. Fe is transformed (Figure 6a. and 6b.) simulated and back-transformed in the same way as a standard CS as described above.

The transformation of SiO2 depends (conditionally) on the distribution of Fe. Take the scatterplot between Fe and SiO2 represented in Figure 6c where Fe is divided into quartiles. Leuangthong and Deutsch, (2003) suggest that 10 classes are generally sufficient; however in this case, for the purpose o f i llustration, q uartiles were ch osen t o si mplify t he graphical r epresentation. The secondary assa y v ariable ( SiO2) ca n then be cl assified acco rding t o Fe q uartiles. A ll t he S iO2 samples within Fe quartile 1 (Figure 6c. Q1) can be transformed to a Gaussian distribution (Figure 6d and 6e). This step is repeated for the SiO2 samples within Fe quartile two and so on until there are four Gaussian distributions of SiO2. These four distributions are then recombined to obtain a single Gaussian distribution and this single Gaussian distribution is used as the conditioning data for t he co nditional si mulation o f S iO2. This single r ecombined Gaussian di stribution o f S iO2 is uncorrelated with the Gaussian distribution for Fe. The assay variables transformed in this manner can thus be simulated independently.

To t ransform Al2O3, the samples from Figure 6c. quartile one ar e t aken and the pr ocedure described above is repeated (Figure 6f, 6g and 4h). This is again repeated for quartiles two, three and four and then al l the Gaussian distributions ( in this case 4*4=16 compared to four above for SiO2) are recombined into a single Gaussian distribution, which is used as the conditioning data for the co nditional si mulation of Al2O3. T he key poi nt i s that A l2O3 has fewer ( compared t o S iO2) samples in each class when Gaussian transforms are made. Thus, the total number of samples for the pr imary assa y v ariable l imits the nu mber o f as say v ariables that ca n be t ransformed. Leuangthong (pers. comm., 2007) recommends a minimum of 10-20 samples per class, which will permit distribution inference, albeit a coarse and inaccurate one.


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Figure 6. An example of the SCT for Fe, SiO2 and Al2O3: Transformation of the raw primary assay variable a. to Gaussian b.; c. dividing the primary assay variable into quartiles; transforming the SiO2 distribution (Q1 c.) from raw d. to Gaussian e.; f. dividing the secondary assay variable into quartiles; and transforming the Al2O3 distribution (Q1 f.) from raw g. to Gaussian h.

3.3.2 Strengths and limitations of Conditional Simulation using the Stepwise Conditional Transform

The SCT process, whereby correlations are removed, has the advantage of being relatively simple to i mplement and under stand. U nlike co nventional si mulation, t he S CT r emoves all co rrelations between assay variables before simulation making the technique better at handling the problematic correlations that are not well summarised by correlation coefficients (Figure 7, Leuangthong and Deutsch, 2003). This approach (de-correlation) also makes modelling of experimental variograms and simulation faster than conventional CCS (i.e., because cross-variograms and Co-Kriging are not required).

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The S CT r emoves any correlation at a l ag o f z ero; how ever, t here i s no g uarantee t hat assa y variables are de-correlated at lags greater than zero (Leuangthong, pers. comm., 2007). Consequently, before commencing simulation, the cross-variograms of the transformed assay variables need to be examined to verify that there is no correlation between assay variables at lags greater than zero. If the transformed variables are still correlated, CCS must be performed.

As discussed abov e, t he r eproduction o f t he v ariogram and the co rrelations between assa y variables is best for the pr imary assa y v ariable, and t he reproduction quality decr eases for t he second assay variable and subsequent assay variables. An additional drawback of the method is that there is no guarantee that cross-variograms will be reproduced (Leuangthong, pers. comm., 2007). Finally, the total number of samples in a dom ain limits the number of assay variables that can be t ransformed (Leuangthong and D eutsch (2003). This would be a l imitation f or many i ron deposits, which although large have numerous domains which results in many domains having few samples. This problem is further co mpounded because i ron or e depos its typically hav e m any correlated assay variables that must be simulated.

Unlike CCS, this method has not y et been widely used for mining applications and more case studies are required to build industry confidence in the method.

Figure 7. Examples of problematic bivariate distributions for geostatistical simulation: nonlinear relations (left), mineralogical constraints (centre), and heteroscedasticity (right), from Leuangthong and Deutsch, 2003.

3.4 Conditional Simulation using Min/Max Autocorrelation Factors

3.4.1 Outline of Conditional Simulation using Min/Max Autocorrelation Factors

Except for the factor transformation, t his technique ( Desbarats and D imitrakopoulos, 2000 ) i s implemented in the same way as a typical geostatistical Gaussian simulation12

1. Decluster data (

; that is:

Section 8.2.2) to obtain representative distributions;

2. Independently transform the raw assay variables to Gaussian;

3. Perform the Min/Max Autocorrelation Factors (MAF) transformation of the Gaussian assay variables (as described bel ow) in or der t o obt ain t he unco rrelated factors (transformed assay variables);

4. Calculate and model experimental variograms of the transformed assay variables;

5. Independently simulate the transformed assay variables;

6. Back transform the simulated variables into Gaussian values;

7. Back transform the Gaussian values into the raw assay variables; and

8. Validate the simulation results.

The MAF transform removes the correlation between assay variables using an approach based on Principal Component Analysis (PCA). The MAF transformation process was developed by Switzer

12 A FORTRAN program, written and supplied by the Bryan Mining Centre at the University of Queensland, was used for the generation of MAF (Version 1.4).

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and Green (1984). For further de tails of Principal Component Analysis, t he reader i s referred to Davis (1986) or Hair et al., (1998).

PCA is an exploratory statistical approach which has the objective to take a large set of (possibly) correlated assay variables and postulate a smaller set of variables (factors) that account for a (hopefully l arge) p roportion o f the m ultivariate v ariability. P CA f actors quantify ‘ grouped correlations’ and thus indicate which variables ‘belong together’; or, equivalently ‘seem to measure the same thing’ (i.e., the factors identify redundancy). In summary, PCA aims to identify the least number of ‘combined variables’ (factors) that can explain the data set.

PCA and t he SCT decorrelate assay variables that occur at the same location. Pairs of variables separated by a v ector, however are not necessarily decorrelated. For example, at a lag of zero a cross-variogram may show zero correlation, but at a lag of 50m assay variables may be correlated. In MAF, the de-correlation not only occurs at lag zero but also at a distance input by the user. This distance should be small and roughly correspond to the data sampling (Didier, pers. comm., 2006). According to Vargus-Guzman and Dimitrakopoulos (2003) when the assay variables estimated can be represented by a two-structure Linear Model of Co-Regionalisation (LMC: Section 2.5), the MAF procedure decorrelates the transformed assay variables at all lags, not just the distance input by the user.

If there are, for example, five assay variables, five factors are generated. However, each factor is a different linear combination of the five assay variables.

3.4.2 Strengths and limitations of Conditional Simulation using Min/Max Autocorrelation Factors

Unlike co nventional si mulation, t he M AF t ransformation r emoves all co rrelations between assa y variables before si mulation. A s for the S CT approach, this de-correlation simplifies modelling o f experimental v ariograms and m akes simulation faster t han co nventional C CS ( because cr oss-variograms and Co-Kriging are not required). The MAF transform (like the SCT approach) has not yet been widely used for mining applications.

According to Rondon and T ran (2008), the MAF approach performs poorly with variables that do not demonstrate a linear correlation. Rondon and Tran (2008) also mention that including variables that are only weekly correlated13

According to D esbarats and D imitrakopoulos (2000) for a t wo-structure Li near M odel of C o-Regionalisation, t he M in/Max A utocorrelation Fact ors are r anked i n or der o f i ncreasing sp atial correlation. E xperimental v ariograms of factors t hat ar e 100% nu gget v ariance need be n ot modelled and simulated, which can result in a significant time saving.

can significantly reduce the quality of the simulations. This can in particular impact on the variograms and cross-variograms.

3.5 Indicator Simulation Sequential Indicator Simulation (SIS) is a Conditional Simulation method that works with indicator variables (see Alabert, 1987 and Chilès and Delfiner, 1999 for further information). It is used in this thesis to simulate the distribution of clay pods (i.e., not assays) within the Final M3 Domain of the Yandi CID (Chapter 14). Implementing this technique involves the following steps:

1. Transform the data to categorical indicators. For example, in our case the clay pod samples could be set to 1 and the surrounding CID values to 0;

2. Create a three-dimensional simulation grid;

3. The indicators are migrated to the nearest grid node. A small grid must be used to avoid migrating samples too far;

4. A r andom pat h i s defined t hat passe s t hrough al l t he grid nodes except for those t hat contain one of the migrated samples;

13 They have found that only using the MAF approach for variables with moderate to strong correlations can significantly improve results.


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5. Using a v ariogram o f t he i ndicators and a use r-defined se arch neighbourhood t he probability of the first node being 0 or 1 is estimated via Ordinary Kriging (OK);

6. A random number between 0 and 1 is selected. If this number is less than the estimated probability the node is set to 0 or if greater than the estimated probability it is set to 1; and

7. The simulated node i s then added t o the migrated drill hole data and t he next node in the random path is simulated. This process is repeated until all the nodes are simulated. A new simulation can be generated by repeating the process with a new random path.

3.6 Concluding comments on stationarity and assumptions

Stationarity is the essential assumption underlying most geostatistical methods such as kriging and simulation. There is generally a fundamental link between geology and stationarity. For example, a geologist will typically group samples into domains of similar geology (e.g. similar rock type, alteration, structure etc). The assumption of s tationarity w ithin these domains is then tested and the findings discussed with t he g eologist and i f nece ssary, t he do maining adj usted. Fr om t he author’s experience, this approach proves more fruitful than the g eologist and g eostatistician working independently.

In practice, many geostatistical techniques and studies are completed without adequate testing of stationarity and t he various assumptions that underlie the simulation algorithms. These areas are sometimes considered as minor compared to the perceived main objectives of the s tudy. I n this thesis, a significant effort i s dedicated t o geology, dom aining, st ationarity and t he t esting of assumptions. U ltimately, i t will be shown here that i mprovements in domaining and t hus improvements in stationarity are the key to improving the simulations. Stationarity is the focus of the next three chapters.


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4 GEOLOGY AND DATA DESCRIPTION

4.1 Geological setting and sedimentology The Hamersley Province (Figure 8) is Archaean to Lower Proterozoic in age (2800-2300 Ma) and contains the thickest and most extensive Pre-Cambrian Banded Iron Formations (BIF) in the world (Trendall, 2002). Deep chemical weathering of heavily lateritised BIF, followed by Tertiary erosion into r iver ch annels formed the Yandi C hanel I ron deposi t ( Yandi CID; M orris et al., 1993 ; Ramanaidou et al., 2003). The Yandi CID is dominantly composed of millimetre-size, subangular to rounded accr etionary cl asts (ooids) of goethite, hem atite and m aghemite (Morris et al ., 1993 ; Stone et al., 2002). The Yandi CID has a variety of sedimentary features. These are described by Stone et al. (2002, pg 137):

“Well-defined se dimentary features are co mmon at Y andi bu t have no t been pr eviously described. Lateral accretion surfaces, channel scours and fills, graded bedding, cross bedding, reworked horizons, and slumped strata are examples of the sedimentary features identified f rom pi t mapping. These f eatures are generally present on the centimetre (e.g. graded bedding) to 10s of metre scale (e.g. scours up to 20 m wide and 2 to 10 m deep).

Bedding is well preserved despite modification of original grain shapes and textures by introduction of alumina and silica in the upper part of the deposit and silica in the lower part of the deposit. Bedding surfaces are generally defined by small changes in grain size, scour surfaces, and clay pods and drapes along bed contacts. Bed thickness ranges from 10s of centimetres to >2 m ( generally ~10, 30 or 50 cm). M any mining faces exhibit exce ssive fracturing f rom bl ast da mage m aking i t di fficult to det ermine bed t hickness. H owever, i n these cases up to six prominent beds are typically discernible in a 12 m high face.”

4.2 Domaining at the Yandi Channel Iron Deposit Domains are r equired for r esource es timation in or der to gr oup ar eas of si milar geology and geochemistry. The term ‘domain’ refers to a volume that, as much as possible (given the available data), can be assumed to have homogeneous statistical properties. This statistical homogeneity is also known as stationarity (Olea, 1991).

Domaining at Yandi by BHPBilliton geologists (Table 2 and Figure 9) is based on stratigraphic units and by chemically different overprints caused by weathering and ground water movement. When visual domain identification is ambiguous, geochemistry is used as a guide (Table 2). Not all the domains described below are in the area covered by this thesis.

The geology of the Yandi Channel (summarised from Whitehouse, 2007) is outlined below.

The Weeli Wolli I ron For mation ( HJ) forms the base ment o f the Y andi c hannel and t ypically consists of BIF, dolerite and shale. The Munjina Member is the lowermost member of the Marillana Formation and co mprises two units, the Basal Conglomerate (BG), and t he Basal Clay (BK). The contact of the BK with the overlying Lower CID (M1 or M2) of the Barimunya Member is gradational and irregular. The M1 Domain is a basal zone with consistently harder CID, and the M2 Domain is denatured and of variable hardness. The M2 is also characterised by increased ochreous goethite (limonite).

Above the M2 domain is the relatively consistent Ochreous Clay ‘marker horizon’ (OK) deposited from su spension dur ing w aning f low ( Stone, 2 005). In areas where t he m arker hor izon i s not present, it can be di fficult to distinguish the boundary between the M2 and the overlying Main Ore Zone (Upper CID; M3) and the position of this boundary is based on the extent of denaturing (i.e., removal of original rock textures).

The M3 is the main ore-bearing horizon and i t consists of largely unal tered red-brown hematite-goethite gr anules within a g oethite-cemented matrix. Detailed mapping of the M3 has revealed sedimentary features such as lateral accretion surfaces, channel scours and fills, graded bedding, cross-bedding, reworked horizons, and slumped strata, many of which are indicative of deposition in a high-energy fluvial system (Stone et al., 2002). Thin (up to 5m), inconsistent clay horizons are present at depths of about 10-20m. In the Central Four deposit Kentwell (2006) identified a


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spatially coherent ar ea o f high Fe co ncentrations and l ow LO I ( C4) l ying at t he base o f t he M3 stratigraphy.

The Iowa Eastern Member is the top member of the Marillana Formation and is subdivided into the Eastern Clay (EK) and Eastern CID (M4 and M4W). The Eastern CID is divided into the M4 and M4W domains with the latter being more affected by weathering which has reduced the Fe content.

Overprinted on t hese st ratigraphic units are four chemically i dentifiable domains, t hese dom ains are de fined by el evated S iO2 and A l2O3 (generally g reater t han 8 % S iO2 and 1. 3% A l2O3). T he ‘Upper CID High Silica and High Alumina’ (M3SA) and the ‘Upper CID Weathered’ (M3W) domains equate to a surface weathering profiles. There are two zones, which follow the channels northern and so uthern margins, t he ‘ Northern M arginal Zone’ ( M3MN) and ‘ Southern M arginal Zone’ (M3MS). These domains are believed to be caused by meteoric water flow along the BK and BG units.

Overlying al l of the abo vementioned dom ains are ‘T ertiary A lluvial S ediments (Z) w hich lie in present day creeks.


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Figure 8. Geological sketch map of the Pilbara region showing the location of the Yandi CID (Stone et al., 2002).

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Domain Code Domain Name Fe % SiO2 % Al2O3 % Z Tertiary Alluvial Sediments < 55 > 8 - M4W Eastern CID Weathered < 55 > 8 - M4 Eastern CID > 55 < 8 < 1.3 EK Eastern Clay < 45 - > 8 M3W Upper CID Weathered < 55 > 8 - M3SA Upper CID High Silica and High Alumina > 55 > 8 > 1.3 M3MN Northern Marginal Zone > 55 > 8 > 1.3 M3MS Southern Marginal Zone > 55 > 8 > 1.3 M3 Upper CID (Main Ore Zone) > 56 < 8 < 1.3 C4 C4 High Grade > 60 LOI < 7.5 OK Ochreous Clay < 45 - > 8 M2 Lower CID (Denatured Zone) > 50 - 1.3-5 M1 Lower CID > 56 < 8 < 1.3 BK Basal Clay > 35 - > 10 BG Basal Conglomerate < 40 > 10 > 5 HJ Weeli Woli Iron Formation > 25 > 20 > 20

Table 2. Domain codes with indicative chemistry (Whitehouse, pers. comm., 2007).

4.3 The simplifying and grouping nature of domains As mentioned above ( Section 4. 2) by S tone et al . (2002), many o f t he sedimentary f eatures at Yandi had not previously been descr ibed. This is partly because these features are not obv ious and r equire so me sk ill t o i dentify. Sedimentology has a si gnificant i mpact on g eochemistry and should be considered in the definition of domains, however, identifying and correlating most of the sedimentary features in CID from drill hole data is generally impossible because:

1. RC drilling breaks the rock into small chips and completely mixes the sample over a t wo-metre interval. This makes identification of many features impossible; and

2. The drill spacing is wider than many of the features making it impossible to correlate many features between drill holes.

Domains, and even individual samples, by necessity group material with different geological and geochemical characteristics. Although the domains outlined above are necessary, in most cases they group areas of broadly similar geology and geochemistry.

4.4 Wireframing of domain boundaries To define the locations of the domain boundaries in three dimensions, polygons were created on cross-sections and linked via wireframes14

The wireframe solids were intersected by the author to remove any overlaps between domains and validated t o ensure that i ndividual wireframes had no i ntersections, hol es or c rossovers. These wireframes were imported into Minesight™ software for use in this study.

. These cross-sectional polygons were created on East-West oriented vertical cross-sections (separated by 50m) using information from drill hole data.

14 These wireframes were created in Vulcan™ by BHPBIO geologists however; the C4 Domain (Table 2) was created by Kentwell (2006) using Gemcom™ software.


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Figure 9. Schematic cross section of the Yandi CID (Whitehouse, pers. comm., 2007).


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4.5 Data description and preparation

4.5.1 Selection of the study area The drill hole data provided by BHPBIO extend from the C1 to E2 mesas15 Figure 10 ( ). This area was selected by BHPBIO because it is an active mining area and contains some densely drilled areas. Before resource estimation, this area was divided by Kentwell (2006) into various orientation domains16

Figure 10

. These orientation domains divide the channel into relatively straight segments that can be use d for v ariography and est imation. This appr oach w as chosen by K entwell (2006) i n preference to unfolding the channel before estimation. Since only one domain was required for this study, the ‘Vardom Two Orientation Domain’ or ‘Vardom 2’ ( ) was selected. The Vardom Two Orientation Domain covers all of Mesa C4 and part of Mesa C5.

Figure 10. Drill hole location map showing the location or individual mesas and the Vardom Two Orientation Domain (orange drill holes).

4.5.2 Overview of the data The C1 to E2 mesas contain Reverse Circulation (RC) and Diamond Drilling (DD). Typically, holes have RC co llars and D D t ails17

15 This area was also determined to be suitable for this study by the author.

but t here a re al so holes drilled co mpletely with RC. It would be

16 The various mesas are arbitrary subdivisions based on the current topography and are not believed to represent different geological or geochemical domain boundaries, hence they are combined here. This assumption is discussed further in this section. 17 I.e. the start of the hole is RC and the end of the hole is DD.


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preferable not t o m ix drilling t ypes for est imation as they ha ve d ifferent v olumes (supports) however; this would r esult i n i nsufficient sa mples for a m eaningful est imate. Thus, a ll historic estimates at Y andi co mbine t he R C and D D for st atistical anal ysis (besides the QA/QC stage where dr ill t ypes are gener ally se parated) and estimation. In addi tion, a twin hol e st udy b y Sylvester (2003) indicated that there were no significant differences in the statistical characteristics of t he two t ypes of d rilling. The strategy of combining t he da ta t ypes has been followed i n t his thesis.

Sample i ntervals for R C and D D dr illing ar e g enerally 2m i n l ength and ar e assa yed f or Fe, P , SiO2, Al 2O3, L OI, C aO, K2O, S, T iO2, M gO and MnO. H oles are l ogged f or l ithology, al teration, hardness and colour. Geophysical logging is also performed to obtain densities for resource estimation.

4.5.3 Sample coding and compositing Wireframes of the domains (Table 2 and Section 4.4) were used by Kentwell (2006) to code the drill hole samples according to the domain within which they are located. The coded drill hole data were visually checked against the wireframes on cross-sections by the author and no coding errors were detected. Thus, for this study the sample coding of Kentwell (2006) was accepted and used in this thesis.

Compositing is a process whereby samples of different lengths are regularised to the same length. This is so that the composites represent material of approximately regular support and are thus in principle addi tive. Compositing to a no minal l ength o f 2m w as performed by Kentwell ( 2006). Compositing was done within the domains outlined in Table 2 and these composites were used in this thesis.


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5 EXPLORATORY DATA ANALYSIS AND STATISTICAL VALIDATION OF DOMAINING

5.1 Introduction As discussed in Chapter 4, the domains at Yandi are based on geology, location within the channel and concentrations of Fe, A l2O3 and S iO2. This chapter of the thesis documents investigation of whether the existing domains:

1. Are acceptable with respect to the assumption of stationarity18. As discussed in Section 2.2, stationarity is one of the key assumptions required for valid kriging and simulation; and

2. Could have been defined by univariate, bivariate and multivariate exploratory data analysis (EDA) of assay variables alone; i.e., without using geology.

There are two main EDA methods used in this chapter. Firstly, scatterplots are used to separate three geochemically distinct populations. Discriminant Analysis is then used on one of these populations to examine how well geochemistry is able to classify samples into domains.

5.2 Overview of Discriminant Analysis Discriminant Analysis starts with a data set in which the samples are already classified into domains. If the mean of an assay variable for a domain varies significantly from the mean of all domains combined then that assay variable could be used to classify new samples into domains (Davis, 1986; Hair et al., 1998; Lewicki and Hill, 2005). A set of linear functions (a model)19

18 As discussed in Section 2.2, there are several types of stationarity e.g. strict, second order etc. It is examined in this section what if any type of stationarity assumption is appropriate.

is built from the assay variables that classify the samples into the predefined domains. Not all the assay variables are necessarily useful in classifying the samples into domains thus a method of ‘backward st epwise anal ysis’ i s often used, i n which all t he assay variables are i ncluded and a t each step the assay variable that has the lowest predictive ability is removed. Only those assay variables that are deemed to have a ‘significant’ predictive ability are kept in the model. Used in this way Discriminant Analysis can not only classify samples into domains but rank assay variables on their relative importance for domain identification. For more information on Discriminant Analysis, refer to Hair et al., (1998) and Lewicki and Hill (2005).

19 I.e. Discriminant Analysis is not suitable for non-linear correlations between assay variables.


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5.3 Using scatterplots of assay variables to separate geochemically distinct populations

Scatterplots of Fe, P, SiO2, Al2O3, LOI, MnO, MgO, S, CaO, K2O and TiO2 reveal three populations (e.g. Figure 11)20

Table 2

. ‘Population One’ i s composed m ostly o f t he B asal C lay ( BK) and U pper C ID Weathered ( M3W) do mains while ‘ Population T wo’ i s dominantly t he S urface A lluvials ( Z) and Basal Clay ( BK) dom ains. P opulations One an d T wo do not co ntain economic mineralisation. ‘Population Three’ has data from all the domains in and represents the majority (8,781 out of 9,324 or 94%) of the composites. Population Three is the focus of remainder of this chapter.

Figure 11. Scatterplot of SiO2 versus Fe with three different geochemical populations highlighted.

5.4 Data preparation and exploratory data analysis of Population Three The univariate statistics of population three are included in Table 3. The distribution of P is positively sk ewed, l oss on i gnition ( LOI) i s nearly sy mmetrical and Fe i s strongly neg atively skewed. T he remaining variables are all strongly pos itively skewed. Most K2O and CaO assa y values are at detection limits.

Table 4 contains a ‘ correlation co efficient m atrix’ f or al l v ariables and composites in P opulation Three, Fe has negative correlations of less than -0.9 with SiO2, Al2O3 and TiO2. The assay pairs Al2O3-TiO2, Al2O3-SiO2 and SiO2-TiO2 have positive correlations greater than 0.8. MgO has lower correlations of about 0.55 with SiO2, Al2O3, TiO2 and CaO with MgO and Fe having a correlation of -0.64.

20 These three populations were separated by manually selecting the groups of points on screen from the scatterplot using Isatis software. Scatterplots of all the other possible combinations of variables were also examined and when obvious trends existed, these were utilised to assist in separating the three populations. When examining multiple scatterplots there is some overlap in the populations and this is unavoidable.


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Variable Number Mean Mode Frequency Minimum Maximum Std. Dev. Fe 8781 55.80 Multiple 23 20 64.5 5.96 P 8781 0.04 0.035 386 0.004 0.13 0.02

SiO2 8781 6.99 Multiple 26 1.9 40 4.89 Al2O3 8781 3.00 0.7 65 0.09 22 3.40 LOI 8781 9.75 9.71 61 2.79 13 1.21 MnO 8781 0.02 0.01 1254 0.001 0.15 0.02 MgO 8781 0.09 0.005 428 0.001 0.8 0.08

S 8781 0.01 0.006 905 0.001 0.07 0.01 CaO 8781 0.05 0.02 585 0.001 1 0.07 K2O 8781 0.01 0.005 2173 0.001 0.2 0.02 TiO2 8781 0.16 0.02 358 0.001 1.5 0.22

Table 3. Univariate statistics for two-metre composites in Population Three.

Variable Fe P SiO2 Al2O3 LOI MnO MgO S CaO K2O TiO2 Fe 1 0.03 -0.95 -0.94 -0.27 0.2 -0.64 0.01 -0.31 -0.3 -0.91 P 0.03 1 -0.06 -0.01 -0.01 0.2 0.15 -0.29 0.04 -0.05 -0.04

SiO2 -0.95 -0.06 1 0.82 0.06 -0.16 0.6 -0.02 0.29 0.3 0.82 Al2O3 -0.94 -0.01 0.82 1 0.21 -0.18 0.57 0.04 0.29 0.25 0.94 LOI -0.27 -0.01 0.06 0.21 1 -0.25 0.23 -0.04 0.06 0.06 0.19 MnO 0.2 0.2 -0.16 -0.18 -0.25 1 0.01 -0.3 0.06 0.03 -0.18 MgO -0.64 0.15 0.6 0.57 0.23 0.01 1 -0.16 0.48 0.29 0.51

S 0.01 -0.29 -0.02 0.04 -0.04 -0.3 -0.16 1 -0.1 -0.01 0.05 CaO -0.31 0.04 0.29 0.29 0.06 0.06 0.48 -0.1 1 0.27 0.28 K2O -0.3 -0.05 0.3 0.25 0.06 0.03 0.29 -0.01 0.27 1 0.25 TiO2 -0.91 -0.04 0.82 0.94 0.19 -0.18 0.51 0.05 0.28 0.25 1

Table 4. Correlation coefficients of variables for two-metre composites in Population Three. Correlation coefficients < -0.4 or > 0.4 are coloured.

5.5 Discriminant Analysis to determine data structure and classify samples into domains

In this section, Discriminant Analysis is used to determine which assay variables and assay variable ratios21

Table 2 have the maximum ability to differentiate between the previously defined domains

( ). In doing this, it is possible to obtain an indication of each variables informational content for domain definition (Hair et al., 1998; Lewicki and Hill, 2005).

Table 5 gives a r anking of assay variables and assay variable ratios according to their ability t o discriminate bet ween d omains. This table al so lists the per centage of co rrect cl assifications, indicating t he percentage o f t imes the Discriminant Analysis method has classified samples into the co rrect do mains based on t heir geochemistry. V arious combinations o f assay variables and assay variable ratios were used in the Discriminant Analysis and these are referred to as different ‘case numbers’ in Table 5

Case One in Table 5 includes all the available variables in the Discriminant Analysis. This indicates that A l2O3 has the gr eatest and C aO t he l east ability to di scriminate bet ween dom ains. T his is appropriate because Al2O3 assists in the identification of clay content and clay content is one of the main di fferences between m any dom ains. The co ncentration o f CaO for many sa mples is at detection limits and t hus often does not input into discrimination. The low ranking of SiO2 and the higher r ankings of P , MnO and K 2O ar e opposi te t o t hat ex pected from cu rrent geological

21 I.e. one assay variable divided by another. The assay variable ratios selected all have correlation coefficients greater than an absolute value of 0.45. Assay variable ratios were used as they may provide some useful information e.g. the ratio between Fe and SiO2 can provide an indication of the intensity of the mineralising process. This is because SiO2 is often replaced by Fe.


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understanding. Perhaps these relationships are coincidental and different results would be evident elsewhere in the Yandi CID.

For C ase T wo P , MnO and K 2O were r emoved22

In the Case Three Discriminant Analysis, ratios for all variables that had co rrelations greater 0.45 were added. I t w as found t hat t he ratios SiO2/TiO2, S iO2/MgO, A l2O3/MgO and Fe/ MgO di d not contribute si gnificantly t o dom ain di scrimination and t hey were e xcluded from t he anal ysis. T he Case Three discriminant function resulted in a significant improvement in the percentage of correct classifications (65.7%).

before per forming Discriminant Analysis, as a result, t he importance o f SiO2 increased, and Fe decr eased. Only a m inor decr ease i n t he percentage o f co rrect c lassifications occurred in C ase T wo (55.7%) compared t o C ase O ne (59.0%).

As discussed in Section 5.2, Discriminant Analysis is optimal for Gaussian distributions however; it is generally robust for non-Gaussian distributions. The raw variables were transformed to standard Gaussian; i.e., with a mean of zero and a st andard deviation of one. Discriminant Analysis on the Gaussian v ariables (Case 4) resulted i n a sl ight i ncrease i n t he hi ghest pr oportion of co rrect classifications ( 68% as opposed t o 65 .7%). The r anking o f the variables importance for domain differentiation is broadly similar to the other cases outlined in Table 5. One notable difference however i s that t he r elative i mportance o f Fe for do main di fferentiation has increased. This probably relates to the transformation of the negatively skewed Fe di stribution to Gaussian and a resultant improvement in the Discriminant Analysis. The two assay variables most heavily relied on by BHBBIO geologists when domaining are Fe and Al2O3; i.e., those defined as the most important assay variables in Case Four.

In addition to assessing the proportion of times geochemistry can correctly classify composites into their interpreted domains, Discriminant Analysis also provides information on the nature of misclassifications. Table 6 is a representation of how well the Discriminant Function has classified individual samples. The first column represents the “correct” domains as defined in Table 2. The second column is the percentage of samples correctly classified by Discriminant Analysis. The remaining co lumns contain t he num ber o f samples classified i nto a given dom ain. For example, Domain M3 has 3186 sa mples correctly cl assified, how ever 184 sa mples were i ncorrectly classified as M3SA, 50 as M3W etc. This is very useful information and the following questions can be investigated:

1. Which domains are well identified by geochemistry and which ones are not; and

2. When m isclassification occurs, i s the dom aining i ncorrect o r i s the Discriminant Analysis unable to correctly classify the composites?

Percentages of correct classification range from 14% in Domain Z t o 86% for Domain C4 with an average of 65.7%. As discussed previously these percentages are optimistic and are likely to be lower if the Discriminant Function is used on another area of the Yandi CID with unknown domains. The low average score of 65.7% supports the observation in the previous sections; i.e., it is not practical to differentiate the existing domains within Population Three on geochemistry.

In su mmary, D iscriminant A nalysis was found to be a pow erful tool t o i dentify w hich assa y variables and assay variable ratios contribute the most to domain identification. Most of the tested scenarios for D iscriminant Analysis produced broadly similar rankings of the importance of each variable for domain identification. In all cases, Al2O3 had the highest ranking. This is consistent with the known importance of the proportion o f A l2O3 rich clays for domain identification. Conversely, SiO2 had less importance in the Discriminant Analysis than is placed on it by BHBBIO geologists. The highest proportion of correct classifications of composites (samples) into domains using their geochemistry w as obtained from G aussian t ransformed as say variables. This is not su rprising given that Discriminant Analysis assumes that distributions are Gaussian.

22 There is no geological reason why these assay variables should be able to assist with the differentiation of domains.


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Ranking Case 1 Case 2 Case 3 Case 4 (Gaussian)

1 (most impact) Al2O3 Al2O3 Al2O3 Al2O3 2 P LOI LOI Fe 3 LOI SiO2 SiO2/Al2O3 LOI 4 Fe S Fe MgO 5 MnO MgO Fe/Al2O3 MnO 6 TiO2 TiO2 S P 7 MgO CaO MgO S 8 K2O Fe Al2O3/TiO2 SiO2 9 S Fe/SiO2 TiO2 10 SiO2 SiO2 11 CaO TiO2 12 (least impact) Fe/TiO2 % of correct classification 59.0% 55.7% 65.7% 68.0%

Table 5. Ranking of each variables ability to discriminate between domains.

Domain

Percent Correct

Classification M3 M3SA M3W BG M2 C4 Z BK MSMA M3 86 3186 184 50 6 94 4 2 6 181

M3SA 49 437 657 135 10 12 23 0 9 47 M3W 61 77 266 839 104 15 4 6 34 24 BG 58 1 2 81 259 18 1 6 56 26 M2 31 107 14 48 11 198 69 0 55 134 C4 86 7 10 5 0 1 215 0 1 11 Z 14 4 1 7 1 4 0 3 1 0

BK 53 1 2 48 10 69 1 0 150 4 MSMA 36 185 99 93 16 52 7 0 11 264 Total 66 4005 1235 1306 417 463 324 17 323 691

Table 6. Domain classification matrix from Discriminant Analysis (Case Three, Table 5). Domain codes are explained in Table 2 and further details on the Discriminant Analysis are included in Appendix 2.

5.6 Exploratory data analysis to identify domains within Population Three

5.6.1 Scatterplots of SiO2/Al2O3 versus Fe for domain identification The two major contaminants in the iron ore at Yandi are SiO2 and Al2O3. The ratio of SiO2 divided by Al2O3 is potentially useful because the SiO2 indicates higher proportions of original BIF or SiO2 introduced during alteration while the Al2O3 indicates the presence of clay. In addition, the previous Discriminant Analysis suggested that SiO2 divided by Al2O3 was the ratio which had the greatest ability to differentiate between domains. A scatterplot of SiO2/Al2O3 versus Fe (Figure 12) reveals two trends. The first trend has low SiO2/Al2O3 ratios and the samples originate from every domain. The se cond t rend has high S iO2/Al2O3 ratios and el evated Fe and t he sa mples mostly or iginate from the M3 and M3W domains. In Section 5.7.2, the two trends in SiO2/Al2O3 ratios are visualised on cross-sections and the usefulness of this ratio discussed.


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Figure 12. A scatterplot of SiO2/Al2O3 versus Fe. Note the two trends defined by high (blue) and low (green) SiO2/Al2O3 ratios.

5.6.2 Discriminant Analysis for domain identification One o f the s teps in Discriminant Analysis is the g eneration o f factors (also called ‘roots’, se e Lewicki and Hill, 2 005), which ar e desi gned t o pr ovide t he m aximum di scrimination abi lity o f domains based on their geochemistry. If there are 11 variables, there will be 11 factors (roots) with most of the discrimination ability going into the first factor then the second and so on. By examining these factors on sca tterplots, it i s possible t o determine how well t he Discriminant Analysis can differentiate between domains. For example, a scatterplot of Root One versus Root Two coded by domain may provide a good se paration between the clusters of points (Figure 13). If this were the case, it would be l ikely that Discriminant Analysis and some of the other techniques discussed in this section could use geochemistry to distinguish domains.

A p lot o f R oot 1 versus R oot 2 coded by dom ain ( Figure 13) i ndicates a cl ustering and par tial separation of domains has been achieved by the Discriminant Analysis. This clustering and partial separation o f do mains is improved w hen R oot 1, R oot and R oot 3 ar e v isualised on a t hree dimensional scatterplot23

Table 6

. However, the cl usters still could not be separated by using the roots alone. If i t w ere possi ble t o v isualise t he el even r oots on a sca tterplot ( i.e., visualise a n 11 -dimensional scatterplot) an even better domain clustering and se paration could be ach ieved. It is not possible to visualise directly a scatterplot that has more than three dimensions and thus, we use the results of Discriminant Analysis. To continue with the graphical analogy discussed above, Discriminant Analysis uses the 11 roots to separate the samples into clusters. Each cluster should be mostly composed of a single domain. If a sample, f or example from the M2 domain, were located within the M3 cluster, Discriminant Analysis would classify the M2 sample as M3. In simple terms, this is the way was created. In summary, rather than physically separate clusters on a multidimensional scatterplot, we use Discriminant Analysis.

Effectively, this complete pr ocess deals with i dentifying sa mples that, based on av erage geochemistry, are more like another domain than that to which they are currently allocated (Table 6). These misclassified samples are then visualised on cross-sections (Section 5.7.3) to determine whether there is an error in the domains or a new domain could be created.

23 The roots form the three graph-axes. The graph can be spun around in three dimensions and examined from different angles and scales. It is not possible to simply convey such information on a two dimensional plot.


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Figure 13. Scatterplot of Root 1 versus Root 2 coloured by domain. For a description of the domain codes (i.e. M3, M3SA etc) refer to Table 2.

5.7 Visualisation of populations on cross sections

5.7.1 Introduction This section involves visualising the populations (or clusters) identified from the above analysis on cross-sections. T hese populations are co mpared to t he i nterpreted domains and t heir co ntinuity and de gree o f geological pl ausibility is a ssessed. T he aim of t his se ction is t o a ssist in understanding, and if possible improving, the shape of the Main Ore Zone Domain (M3). This domain was selected for further analysis as it is the primary focus of economic interest.

5.7.2 Visualisation o f populations ide ntified fr om the scatterplot of SiO 2/Al2O3 versus Fe

The scatterplot o f S iO2/Al2O3 versus Fe ( Figure 12) revealed useful i nformation r egarding the geochemistry o f the M 3 Domain, w hich w as not di rectly use d by t he geologists in creating t his domain. There are two main trends evident in this scatterplot. The first trend contains samples with elevated S iO2/Al2O3 ratios and i s predominantly composed o f M 3 and M 3W rocks. The se cond trend is of low SiO2/Al2O3 ratios and is composed of multiple domains.

Samples within the high SiO2/Al2O3 trend not domained as M3 and samples with low SiO2/Al2O3 ratios domained as M3 were visualised on cr oss-sections to determine their location and sp atial continuity.

Almost all the samples with high SiO2/Al2O3 ratios not coded as M3, M3MN or M3MS are located above these domains in continuous lenses (predominantly M3W). This observation is useful, because it appears that high SiO2/Al2O3 ratios are a geochemical indicator to differentiate between some of t he domains in t he upper and l ower par ts of t he s tratigraphy. At so me poi nt i n t he formation (or subsequent alteration) of the Yandi CID a significant change in SiO2/Al2O3 ratios occurred. The high SiO2/Al2O3 ratios are produced by either low concentrations of Al2O3 rich clay or high concentrations of SiO2 (caused either by the primary constituents or subsequent alteration). Further geological investigation is warranted to investigate the underlying reason for these


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changes in SiO2/Al2O3 ratios. In summary, while these observations do not contradict the current domaining approach, the use of SiO2/Al2O3 ratios may improve and simplify the definition of the M3 and M3W domain boundaries. The magnitude of l ikely changes to the M3 Domain24

The samples with low SiO2/Al2O3 ratios tend to be more common towards the edges of M3. This is due to higher concentrations of clay pods towards the edges of the M3 Domain. There i s some spatial continuity in the location of the samples with low SiO2/Al2O3 ratios but interpreting this on all sections is difficult and it is likely that this group of samples mostly represent lenses that are tens of metres across rather than hundreds of metres across.

because of using these ratios is likely to be small and no changes were made to the M3 domain boundary.

5.7.3 Visualisation of the results from Discriminant Analysis As discussed pr eviously, Discriminant Analysis can be use d t o classify samples into dom ains based on their geochemistry. In the Discriminant Analysis performed for this study some non-M3 samples were cl assified as M3 a nd co nversely so me sa mples within t he M 3 Domain were classified as a non -M3 samples. T hese samples, identified as anomalous in t he Discriminant Analysis, were displayed on cr oss-sections. V isualisation on cr oss-sections of these ano malous samples identified numerous sub-domains within the interpreted domains. For example, there are lenses of material with M3SA-like geochemistry within the M3 Domain. The area surrounding these lenses is still dominantly made up o f the M3 geochemistry and t hus the broad location of domain boundaries is still correct. The sub-domains appear to be present as lenses of tens of metres across but i n pl aces are up t o se veral hundr ed m etres across. This observation appear s to be consistent with known sedimentology of the Yandi CID (Figure 14; Stone, 2005). Given an average drill hole spacing of 50m x 50m, these lenses are difficult to interpret on many sections. For this reason, the current broad yet spatially continuous domains are acceptable for linear estimates such as Ordinary Kriging (OK). Such linear estimates will inevitably result in some smoothing of the grade t onnage cu rves but from a B HPBIO pe rspective, they have proved adeq uate for m ost practical uses. If a less-smoothed grade-tonnage cu rve i s required or the t rue complexity of t he geology i s to be r epresented, Plurigaussian Conditional S imulation of facies (Armstrong e t a l., 2003) could be a viable option. For example, if multiple simulations of the facies were generated then kr iging of t he assay variables within f acies could be pe rformed i ndependently i .e. onl y samples from a par ticular facies would be use d t o kr ige t he grades for t hat facies. U sing har d boundaries between facies during estimation would reduce the degree of smoothing in the grade tonnage curve.

Figure 14. Scour and fill channel within the Yandi CID (Stone et al., 2002).

24 The M3 domain is the focus of this study. Potential changes would probably be greater for the M3W domain.

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6 VALIDATION AND EXPLORATORY DATA ANALYSIS OF THE M3 DOMAIN

6.1 Introduction Chapter 5 used multivariate exploratory data analysis (EDA) to examine all the domains at Yandi. Although t he M3 Domain has been se lected f or t his study, i t was necessary t o examine al l t he domains to test the validity of the stationarity assumption for the M3 Domain. For example, there could have been M3 samples that have been incorrectly included in another domain thus reducing the representativity of the M3 Domain. This was not the case and the Chapter 5 EDA indicated that although the M3 Domain necessarily groups samples with differing geology and geochemistry that the domain is in general acceptable. In this chapter, the focus is the M3 Domain and EDA is performed to further test and characterise the M3 Domain.

6.2 Stationarity assessment via moving window statistics Moving window statistics involve calculating statistics within rectangular volumes. Although these volumes can be overlapping in this case the volumes do not overlap25

Graphs of the mean concentrations of assay variables within moving windows versus Northing indicate t hat North o f 8 6,130mN t he g eochemistry of t he M3 Domain is significantly di fferent (

. These m oving window statistics, when visualised i n pl an and section or pl otted on gr aphs, can be use ful ai ds for identifying departures from stationarity.

Figure 15 to Figure 17). Phosphorous, LOI, A l2O3 and T iO2 are al l higher while S iO2, Fe, MgO, CaO and MnO are lower. This is where the M3 Domain breaks into the C4 and M3; i.e., the lower part of the M3 Domain changes into the C4 Domain along strike (Figure 18). It is possible that, if the moving window statistics were recalculated with the C4 and M3 domains combined, the abovementioned geochemical differences would disappear. To test this the moving window statistics were regenerated (Figure 127 to Figure 129 in Appendix 3), however North of 86,130mN the geochemistry is still si gnificantly di fferent. This suggests there i s a r eal ch ange i n t he geochemistry of the CID North of 86,130mN. It is probable that the change in geochemistry is due to a paleo tributary o f t he m ain Yandi C ID entering at t his point. P resumably, t his tributary has deposited the sediments of different geochemistry. This could be investigated further by detailed geological pit mapping.

In order to generate a domain that is acceptable on stationarity grounds for this study, all samples North of 86,130mN were excluded from the M3 Domain. This domain is used for the remainder of the studies in this thesis and is named the ‘Final M3 Domain’.

25 50m slices were used for Northing and Easting while 4m slices were used in the vertical direction.


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Trace element concentrations vs. Northing

0

0.01

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84700 84900 85100 85300 85500 85700 85900 86100 86300 86500 86700Northing (m)

Con

cent

ratio

n %

CAO Mean

K2O Mean

MGO Mean

MNO Mean

P Mean

S Mean

Figure 15. Mean trace element concentrations versus Northing for 2m composites: M3 Domain and Vardom Two Orientation Domain.

Concentrations of LOI, SiO2 and Al2O3 vs. Northing

0

2

4

6

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and

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2 %

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0.5

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1.5

2

2.5

Al2O

3 %

LOI Mean

SIO2 Mean

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Figure 16. Mean concentrations of LOI, SiO2 and Al2O3 versus Northing for 2m composites: M3 Domain and Vardom Two Orientation Domain.


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Concentrations of Fe and TiO2 vs Northing

57.8

58

58.2

58.4

58.6

58.8

59

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84700 84900 85100 85300 85500 85700 85900 86100 86300 86500 86700Northing (m)

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2 %

FE Mean

TIO2 Mean

Figure 17. Mean concentrations of Fe and TiO22 versus Northing for 2m composites: M3 Domain and Vardom Two Orientation Domain.

Figure 18. Plan view of samples in the M3 (crosses) and C4 (red circle with a black cross) domains in the Vardom Two Orientation Domain.


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6.3 Exploratory data analysis of the Final M3 Domain

6.3.1 Univariate exploratory data analysis The following univariate statistics were generated for the Final M3 Domain:

1. Tabulated statistics (Table 7);

2. Histograms (Figure 19);

3. Moving window statistics of means and correlation coeficients (Figure 130 to Figure 138 in Appendix 4); and

4. Moving window statistics of mean versus standard deviation (Figure 20).

Fe SiO2, Al 2O3 and LO I ar e t he m ajor co nstituents of m ost sa mples and t he r emaining assay variables are generally only present in trace amounts (Table 7). K2O and CaO are only present in concentrations at or near analytical detection levels.

LOI has a near normal distribution and P has an approximately normal distribution (Figure 19). Fe has an approximately inverse lognormal di stribution w hile t he r emaining assay v ariables are positively skewed.

Moving window statistics indicate that Fe and T iO2 concentrations increase northwards while SiO2 decreases (Figure 131 and Figure 132). Most assay variable concentrations are relatively constant with varying Easting but MnO decreases with increasing Easting (Figure 133). All assay variables have obvious trends in their mean concentrations with varying depths except f or A l2O3, which is relatively constant (Figure 136 to Figure 138).

The p roportional e ffect occurs when t he mean i s correllated to the standard d eviation. T he presence o f t he pr oportional e ffect ca n be t ested for by ca lculating t he m ean and st andard devation within panels (3D volumes) and plotting these on scatterplots (Figure 20). The proportional effect appears to be present for Fe, SiO2 and Al2O3 but not for P and LOI.

Variable Count Minimum Maximum Mean Std. Dev. Fe

3320

39.91 62.18 58.79 2.01 P 0.019 0.117 0.039 0.010 SiO2 1.904 20.96 4.81 1.97 Al2O3 0.095 15.44 1.12 1.10 LOI 6.723 12.36 9.74 0.77 MgO 0.001 0.889 0.068 0.054 MnO 0.001 0.174 0.028 0.016 K2O 0.001 0.237 0.011 0.017 CaO 0.002 0.81 0.042 0.034 S 0.001 0.057 0.012 0.009 TiO2 0.001 1.067 0.041 0.063 Table 7. Univariate statistics of the Final M3 Domain.


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40 45 50 55 60

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Figure 19. Histograms of assay variables from the Final M3 Domain.


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0 1 2 3 4 5

AL2O3 MEAN

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53 54 55 56 57 58 59 60 61

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Figure 20. Scatterplots of mean versus standard deviation within 100m x 100m x 8m panels. The solid black line represents the average standard deviations conditional to the assay variable grades.


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6.3.2 Bivariate exploratory data analysis

Fe is strongly negatively correlated with SiO2, Al2O3 and TiO2 and moderately negatively correlated with M gO (Figure 21). Sulphur and M nO ar e a lso m oderately neg atively co rrelated (Figure 22). Silica Oxide is moderately positively correlated with Al2O3 and TiO2 (Figure 23). Aluminium Oxide and TiO2 (Figure 23) are very strongly correlated. These correlations are relatively linear (except for S versus MnO) and there tends to be a weak to moderate increase in variability with increasing grade. The pr oportional ef fect i s common t o many m ineral deposi ts (see D avid, 1977 f or a discussion on t he proportional effect). Many scatterplots are indicative of separate populations as they contain two slightly different trends. As discussed in the previous chapter, seperating these populations is not possible with deterministic domains.

Moving w indow st atistics indicate t hat co rrelations coefficients vary si gnificantly with N orthing, Easting and RL (Figure 139 to Figure 141 in Appendix 4). These variations are generally random and do not form obvious trends.

Variable Fe P SiO2 Al2O3 LOI MgO MnO K2O CaO S TIO2 Fe 1 -0.2 -0.8 -0.8 -0.2 -0.5 0 -0.2 -0.3 0 -0.8 P -0.2 1 0 0.2 0.3 0.3 0.2 0.1 0.2 -0.3 0.1 SiO2 -0.8 0 1 0.5 -0.2 0.3 -0.1 0.1 0.1 0.1 0.5 Al2O3 -0.8 0.2 0.5 1 0.2 0.5 -0.1 0.2 0.3 0 0.9 LOI -0.2 0.3 -0.2 0.2 1 0.3 0.2 0.2 0.2 -0.3 0.2 MgO -0.5 0.3 0.3 0.5 0.3 1 0.1 0.2 0.6 -0.2 0.3 MnO 0 0.2 -0.1 -0.1 0.2 0.1 1 0 0.3 -0.4 -0.1 K2O -0.2 0.1 0.1 0.2 0.2 0.2 0 1 0.2 -0.1 0.1 CaO -0.3 0.2 0.1 0.3 0.2 0.6 0.3 0.2 1 -0.3 0.2 S 0 -0.3 0.1 0 -0.3 -0.2 -0.4 -0.1 -0.3 1 0 TiO2 -0.8 0.1 0.5 0.9 0.2 0.3 -0.1 0.1 0.2 0 1

Table 8. Correlation coefficients for the Final M3 Domain. Correlation coefficients < -0.4 or > 0.4 are coloured.


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FE

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20 SIO2

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Figure 21. Scatterplots of Fe versus SiO2, TiO2, Al2O3 and MgO for the Final M3 Domain.


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0.00 0.01 0.02 0.03 0.04 0.05 0.06 S

0.00

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Figure 22. Scatterplots of MnO versus S for the Final M3 Domain.

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Figure 23. Scatterplots of SiO2 versus Al2O3, SiO2 versus TiO2 and Al2O3 versus TiO2 for the Final M3 Domain.


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6.3.3 Implication of exploratory data analysis for resource estimation The Fi nal M 3 Domain has significant t rends in assay variable co ncentrations with dept h. While there are no obviously bimodal populations represented in histograms there are mixed populations indicated by two similar but different linear trends in the scatterplots. These populations are insufficiently spatially continuous to domain separately (Chapter 5). There is also significant spatial variability in the correlation coefficients. Finally, the proportional effect is present for Fe, SiO2 and Al2O3.

As discussed i n Section 2 .2, the m inimum r equired stationarity f or robust linear est imation ( e.g. Ordinary Kriging) is q uasi-stationarity. This requires that the expected values of the mean and variance are constant within a restricted search neighbourhood. If however we are co-kriging i.e. estimating two or m ore variables, then t he expected cross-variogram should al so be translation invariant.

For co nditional si mulation ut ilising O rdinary K riging o r Ordinary C o-Kriging we ca n al so assume quasi-stationarity. D espite t his, there i s an added co nstraint for Conditional Simulation on t he shape of the histogram. We assume in Conditional Simulation that the shape of the histogram is translation i nvariant. This is because for Gaussian based simulation m ethods we m ake a si ngle Gaussian transform for each variable and a si ngle backtransformation of the simulated vales for each variable i.e. we assume that the expected histogram shape does not vary spatially within the domain. The further our local histogram shape varies from the average histogram shape for t he domain the less valid our transform and backtransform of the Gaussian simulations will be.

In summary, univariate and multivariate non-stationarity are unlikely to be present because there are t rends in t he mean gr ades (especially v ertically were t he t rends a re st rong) of t he assa y variables26

and a single linear correlation does not adequately describe the correlations between some of the key variables. This will make robust co-kriging and co-simulation more difficult.

26 Because the proportional effect exists, the variance will also vary as the mean varies.


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7 TESTING OF GEOSTATISTICAL MODELS AND ASSUMPTIONS

7.1 Testing the diffusivity assumption The ratios of cr oss-variograms to v ariograms (for i ndicators27

Downhole experimental variograms and cross-variograms were generated for the abovementioned ratios for Fe, P, SiO2, Al2O3 and LOI with four indicators generated for each assay variable. These experimental variograms were omnidirectional with a lag 2m. Diffusivity appears to be a reasonable assumption because most of the ratios increase with distance (e.g.

) is a use ful check f or diffusivity (Rivoirard, 1994 ). This check t ests whether i ndicators are co rrelated with distance. I f v arious indicators are sp atially correlated, t he ratio i ncreases with di stance. I f this is observed, spatial assay variable concentrations go from high to low in a relatively continuous way; i.e., the diffusion model is acceptable. If the ratio remains relatively constant then the diffusivity model is likely to be invalid. Typically, the required indicator variograms are generated downhole. This is because the assumption is most important at short distances such as at the SMU to panel scale at which most change of support calculations occur.

Figure 24 to Figure 26)28. The validity of t he diffusivity assumption may be due t o the strong vertical t rends29 in the concentrations of assay variables (Section 6.3. ), rather than to spatially correlated grades. To test this, the ratios were also generated for horizontal variograms (e.g.

1Figure 27). The ratios for most

assay variable and indicator combinations increase with distance suggesting t hat t he diffusivity assumption is acceptable.

0 5 10 15

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Figure 24. Ratio of cross-variogram to variogram (downhole): Fe indicators 57% and 59% (left) and P indicators 0.03% and 0.04% (right).

27 An indicator for example is defined as 1 when a variable is above a given cut-off and 0 otherwise. 28 Only selected examples for the ratio of the cross-variogram to the simple variogram are represented here. This is because there are 6 cross-variograms for each variable and five variables i.e., 30 cross variograms. 29 The variograms were generated downhole and all the holes are vertical.


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0 10 20 30 40

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Figure 25. Ratio of cross-variogram to variogram (downhole): SiO2 indicators 3% and 5% (left) and Al2O3 indicators 0. 5% and 1% (right).

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Variogram LOI IND 9 &

Figure 26. Ratio of cross-variogram to variogram (downhole): LOI indicators 9% and 10%.


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N0

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11488 22072

2032434267

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Figure 27. Ratio of cross-variogram to variogram (horizontal): Fe indicators 57% and 59% (left) and Fe Indicators 57% and 58% (right).

7.2 Testing the bi-Gaussianity and multivariate-Gaussianity assumptions via scatterplots

Scatterplots and h-scatterplots30 of Gaussian transformed variables should be elliptical in shape to satisfy the bi-Gaussianity and multivariate bi-Gaussianity assumptions (Section 2.6). Scatterplots are generally elliptical for the Gaussian transformed assay variables (Fe, P, SiO2, Al2O3 and LOI, Figure 142 in Appendix 5), except for Fe versus SiO2 (Figure 28), Fe versus Al2O3 (Figure 28), P versus LOI (Figure 29) and SiO2 versus Al2O3 (Figure 29). These are important correlations and the tests indicate that there may be pr oblems reproducing these correlations in Gaussian based Conditional Simulation.

H-scatterplots of the Gaussian transformed variables separated by 2m are elliptical or very close to elliptical (e.g. Figure 30 to Figure 32: c ompared with the abov ementioned scatterplots). H -scatterplots at a separation of 10m are all elliptical but there are no strong correlations between the assay variables; i.e., the ellipses are all circular (Figure 143 in Appendix 5).

30 H-scatterplots are scatterplots that have the plotted points separated by a distance h. In order to select points at a given h, angular and distance tolerances can be applied.


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Figure 28. Scatterplot of Fe versus SiO2 (left) and Fe versus Al2O3 for samples separated by 0m. Points are coloured on frequency.

Figure 29. Scatterplot of P versus LOI (left) and SiO2 versus Al2O3 (right) for samples separated by 0m. Points are coloured on frequency.

Figure 30. Fe versus SiO2 (left) and Fe versus Al2O3 (right) downhole h-scatterplots for samples separated by 2m (± 0.5m).


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Figure 31. P versus SiO2 and SiO2 versus Al2O3 downhole h-scatterplots for samples separated by 2m (± 0.5m).

Figure 32. SiO2 versus SiO2 downhole h-scatterplot for samples separated by 2m (± 0.5m).

7.3 Testing the bi-Gaussianity and multivariate Gaussianity assumptions via madograms

If the bi-Gaussian assumption is potentially valid, the ratio of the square root of the variogram to the m adogram31

Figure 33 should be co nstant (Emery, 2 005). These ratios were plotted bo th dow nhole

( ) and horizontally (Figure 144 to Figure 148 in Appendix 5). In this case, the ratios downhole ar e not co nstant bu t decr ease w ith i ncreasing l ag di stances. For t he ho rizontal variograms, the ratios also decrease but only up to 50m after which they are relatively constant except for LOI, which is constant for all lags.

In summary, it appears that the bi-Gaussian assumption is not strictly valid for all assay variables.

31 Isatis (2005a) define the madogram as ||

21 ZbZan n

−∑ with Za and Zb separated by a vector h.


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7.4 Testing the bi-Gaussianity and multivariate Gaussianity assumptions via back-transformed variograms

Another t est f or the bi-Gaussianity assu mption i s to co mpare t he ba ck-transformed G aussian variogram32 Figure 34 to t he r aw ex perimental v ariogram ( : E mery, 2005 ). I f th e two ar e comparable, the bi-Gaussianity assumption is not contradicted. The shape of the raw experimental variograms and bac k-transformed m odels compare moderately well. T he ba ck-transformed variance for P and LOI are higher than the raw variance. This relates to the fact that weights were used in the Gaussian transformation process but no weights were applied to the raw experimental variograms. When no w eights were used i n t he G aussian t ransformation, t he t wo variances are very similar (Figure 35).

In summary, the raw experimental variograms compare moderately well to the back-transformed variograms. This appears to be co nsistent w ith t he co nclusion o f t he pr evious section; i.e., the assumption o f bi -Gaussianity is not strictly valid but i t appear s to be a r easonable w orking assumption.

32 The interpretability of experimental variograms tends to be better for normally distributed data than for highly skewed distributions. Thus, a distribution can be transformed to Gaussian for variography. However, the variogram model fitted to this experimental variogram must be back-transformed from Gaussian to raw before it is used for Ordinary Kriging. This back-transformation of the variogram is based on a simple relationship between the covariance of the Gaussian transformed variable and the covariance of the raw variable (Geovariances, 2005).


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0 5 10 15 20 25

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Figure 33. Ratio of square root of variogram to madogram for Gaussian transformed variables: Downhole variogram. The dotted horizontal line represents the average ratio for all data.


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N0

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Figure 34. Fe, SiO2, Al2O3 and LOI back-transformed variogram models and experimental variograms: Horizontal variogram fan and downhole variogram. The azimuths of the horizontal variograms are annotated as N0, N30 etc while the vertical variograms are annotated a D-90.


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N0

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Figure 35. P back-transformed variogram model and experimental variogram: Horizontal variogram fan and downhole variogram. For the bottom figure, no weights were used in the Gaussian transformation while in the upper figure weights were utilised. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated a D-90.


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7.5 Testing the bi-Gaussianity and multivariate-Gaussianity assumptions via rodograms

The final test of bi-Gaussianity is to compare the experimental rodogram33

to t he t heoretical rodogram via the relationship: The rodogram is equal to the variogram of order ½ i.e.,

[ ] 25.05.0 )()( h Constanth γγ = (Emery, 2005).

This relationship was checked downhole (Figure 36 to Figure 38) and it is evident that the trends are approximately linear but biased; i.e., the line of best fit does not have an intercept of zero and a slope of 45º. This bias takes different forms for different assay variables (Figure 36 to Figure 38).

In summary, t he t heoretical r odograms compare m oderately well t o t he experimental r odograms and although the assumption of bi-Gaussianity is not strictly valid, it is at least partially valid. Thus, the assumption of bi-Gaussianity can be maintained as a reasonable working assumption.

Fe Experimental Rodogram vs Theoretical Rodogram

0.35

0.38

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Figure 36. Experimental rodogram versus the theoretical rodogram for Fe (left) and P (right).

SiO2 Experimental Rodogram vs Theoretical Rodogram

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Figure 37. Experimental rodogram versus the theoretical rodogram for SiO2 (left) and Al2O3 (right).

33 Isatis (2005a) define the rodogram as ||

21 ZbZan n

−∑ .


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LOI Experimental Rodogram vs Theoretical Rodogram

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Figure 38. Experimental rodogram versus the theoretical rodogram for LOI.

7.6 Discussion on the acceptability of assumptions The test for diffusivity (Section 7.1) appears to confirm that diffusivity is the dominant behaviour of the assay variables in the Final M3 Domain. From a geological point of view and simply visualising grades in three dimensions, the diffusion model is not valid in all instances, but describes the dominant behaviour of the mineralisation. An instance where the diffusion model is not valid is at the boundar ies of cl ay pods. S tep changes in t he concentration of assay variables occu r at t he margins of clay pods and presumably at some sedimentary boundaries (e.g. the base of a s cour channel). Diffusivity ap pears to be su fficiently valid t o co ntinue w ith the G aussian base d Conditional Simulation methods utilised in this thesis.

Numerous tests of the assu mptions of bi -Gaussianity and m ultivariate bi-Gaussianity w ere discussed above (Section 7.2 to Section 7.5). All of these tests are qualitative and they show that the required assumptions are not strictly valid however; they appear to be sufficiently acceptable to continue with the Gaussian based Conditional Simulation methods utilised in this thesis.

Investigated i n l ater se ctions are t he i mpacts o f appl ying simulation al gorithms that a ssume diffusivity, bi-Gaussianity and multivariate bi-Gaussianity to data sets where these assumptions are not strictly valid.


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8 BLOCK SIZE, A POTENTIAL PROBLEM FOR ORDINARY KRIGING AND UNIFORM CONDITIONING

8.1 Introduction In this chapter, multiple kriged estimates are gener ated using various search ellipse sizes and block m odel dimensions. For each combination of se arch and block size, key Ordinary Kriging (OK) performance parameters are recorded. This approach has been c alled Quantitative Kriging Neighbourhood Analysis (QKNA: Vann et al., 2003).

This section i s not utilised t o se lect block si zes or searches for t his thesis. The pur pose o f this analysis is to demonstrate some of the potential drawbacks with Uniform Conditioning compared to Conditional Simulation at Yandi. This is because Uniform Conditioning is the most commonly used non-linear est imation m ethod for co rrelated v ariables in m ining appl ications and i s sometimes employed as an alternative to Conditional Simulation.

8.2 Experimental variography and variogram modelling

8.2.1 Introduction Variography is sensitive to outliers and highly skewed distributions. Variograms are often improved when t hey ar e g enerated on G aussian t ransformed dat a. The G aussian t ransformations used below were made usi ng decl ustering w eights in or der to obt ain r epresentative hi stograms. This was necessary beca use t he dr illing is closer sp aced i n t he N orth o f t he st udy ar ea t han i n t he South. Thus, before co mmencing experimental variography, declustering and G aussian transformations are di scussed (Section 8. 2.2). T hese declustering weights and the G aussian transforms are also used in the subsequent Conditional Simulation sections of this thesis.

8.2.2 Declustering and Gaussian transformation Cell decl ustering i nvolves laying a three-dimensional grid ov er t he r aw dat a and co unting t he number of samples within each grid cell. The inverse of the number of samples in each cell is then used as the declustering weight. Often the origin of the grid is changed and the declustering results for several such exercises examined. This results in multiple possible weights for every data point from which one se t of weights from one grid origin offset must be chosen. In the cell declustering used here, rather than moving the origin of the grid, the centre of every cell is moved in sequence to every dat a poi nt. This provides a si ngle w eight for ev ery dat a poi nt34

Cell declustering was done for various grid sizes and t he results tabulated (

. T his is more r obust because a somewhat arbitrary decision on which grid offset to select for the weights does not have to be made.

Table 9). The various declustering grid dimensions were also visually examined against the raw data. A 100m x 100m declustering g rid (Figure 39) was selected, because t his is the approximate dat a spacing o f the less densely drilled areas.

Using the abovementioned weights, the frequency inversion technique was used to transform the raw distributions to Gaussian (Section 2.3; Geovariances, 2005).

Variable

Grid size

Clustered 50m 50m 100m 150m 50m 100m 100m 150m 2m 2m 2m 2m

Fe 58.8 58.8 58.8 58.7 58.7 P 0.039 0.039 0.039 0.040 0.041 SiO2 4.81 4.71 4.78 4.81 4.84 Al2O3 1.12 1.11 1.12 1.15 1.13 LOI 9.74 9.79 9.78 9.81 9.83

Table 9. Clustered and declustered means for various declustering cell sizes.

34 This is also the way declustering is typically implemented within the Isatis™ software.


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Figure 39. Raw data from the study area with a 100m x 100m grid overlain.

8.2.3 Experimental variograms and variogram modelling Experimental variograms of Fe, P, SiO2, Al2O3 and LOI were generated and f itted with variogram models. T hese m odels are only used f or Q KNA and f or f urther details on t hese experimental variograms and the variogram models see Table 10 and Appendix 6. Structure and Range Fe P SiO2 Al2O3 LOI Nugget 1.69 0.000025 1.9 0.5 0.21 Relative Nugget (%) 42 27 49 42 36 Structural Component 1 1.6 0.00002 1.17 0.4 0.125 Max Range 1 (m) 6 100 118.4 6 110.2 Int. Range 1 (m) 6 100 118.4 6 110.2 Min Range 1 (m) 6 38 19 6 33.7 Structural Component 2 0.75 0.000048 0.8 0.15 0.25 Max Range 2 (m) 80 900 229.4 38.5 2300 Int. Range 2 (m) 80 900 229.4 38.5 2300 Min Range 2 (m) 50 38 40 50 34 Structural Component 3 0.15 Max Range 3 (m) 135.1 Int. Range 3 (m) 135.1 Min Range 3 (m) 50

Table 10. Variogram model parameters for directional variograms.


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8.3 Block model creation and assessment of block percentage fill for estimation by Ordinary Kriging

Sixteen different sized block models were created for QKNA (Table 11). No ‘subcelling’ was used for these block models; i.e., all blocks had the same dimensions. Only 1% of each block had to lie within the M3 wireframe solid to be included in this domain. The percentage of each block lying within t he wireframe was recorded and t he co ded m odels were ch ecked vi sually t o e nsure t hat blocks had been correctly coded. The average percentage fill for the blocks (Figure 40) varies from over 90% for 10m x 10m x 6m blocks down to about 45% for 200m x 200m x 12m blocks. Low average block fill percentages are problematic, which is discussed further in Section 8.5. X (m) Y (m) Z (m) 10 10 6 10 10 12 25 25 6 25 25 12 50 50 6 50 50 12 50 100 6 50 100 12 100 100 6 100 100 12 100 150 6 100 150 12 150 150 6 150 150 12 200 200 6 200 200 12 Table 11. Block model dimensions used for QKNA.


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Average Block Fill Percentages

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Figure 40. Average block fill percentages for various block sizes.

8.4 Background to QKNA Quantitative K riging N eighbourhood A nalysis (QKNA) i s a method for generating multiple kriged estimates with varying numbers of samples used in the searches for each estimate and recording and analysing some or all of the following outputs from the kriging i.e.:

1. Kriging s tandard dev iation – The standard dev iation o f t he est imation error for the kriged grade;

2. The slope of regression – The slope of regression between actual versus estimated grades which is a measure of conditional bias. The slope is between 0 and 1 (or only slightly below 0 or above 1) with slopes closer to 1 indicating decreasing conditional bias;

3. Negative kr iging w eights – Negative k riging w eights often occu r w hen the s earch i s too large and some samples are located behind others and are therefore redundant; and

4. Weight on the mean for simple kriging – During Simple Kriging, the mean of the domain is assumed to be known. With Simple Kriging weights are attributed to both the samples and the mean which is different to Ordinary Kriging in which the mean is assumed unknown and all the weights are attributed to the samples. Less weight attributed to the mean indicates a higher confidence estimate.

For further details on QKNA see Vann et al. (2003).

8.5 QKNA For Q KNA, Fe and LOI were se lected as pr oxies for t he full su ite o f variables. T hese variables provided a good coverage of the range of outcomes because Fe has a short-range variogram and LOI a long-range variogram35

Table 11. The QKNA involved generating 25 OK estimates for each of the 16

different block models outlined in .

35 The QKNA is largely dependent on the variogram.


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Each of the 25 estimates used a different number of samples; however, the following estimation parameters remained constant:

1. Quadrant searching;

2. Minimum of four samples to estimate a block;

3. All samples laying within a block are selected;

4. Search ellipse is 1000m x 1000m x 250m (X, Y and Z);

5. Anisotropic searching36

6. Discretisation of 10 x 10 x 6.

; and

By analysing each estimate, it is possible to understand how estimation quality varies with block size and t he num ber of sa mples used t o est imate each bl ock. Ideally, a co nditionally unb iased estimate will have a slope close to one and a low OK standard deviation. In addition, it is preferable for there not to be a high proportion of negative weights, which can cause negative estimates and indicate t hat too m any sa mples are bei ng use d for es timation. When t he assu mption of strict stationarity appears valid, there is no theoretical problem with using a large number of samples for Ordinary Kriging. However, as discussed in Chapter 6, the Final M3 Domain is unlikely to meet the assumption of stationarity. Thus, it is preferable not to use an extremely large search.

For Fe the slope rapidly increases (Figure 41) up to about eight samples per quadrant. The slope increases, but less rapidly f or increasing sample numbers. The highest slopes of regression are obtained for blocks up to 25m x 25m x 12m. Blocks larger than this have significantly lower slopes of regression. This is because:

1. A large proportion of the blocks lie outside the Final M3 Domain which results in additional conditional bias; and

2. As the block size increases, more samples are required to minimise conditional bias.

For LOI the slopes are much higher (i.e., closer to one) than for Fe. This is because t he LOI variogram is much more continuous than that of Fe. For LOI the slope rapidly increases (Figure 42) up to about four samples per quadrant. The slope increases less rapidly (but still significantly) for increasing sample numbers. The highest slopes of regression are obtained for blocks up to 50m x 50m x 12m. Blocks larger than this have lower slopes of regression. The differences are less pronounced than for Fe above.

The proportion of negative OK weights is very low (typically zero) for estimates of Fe; i.e., for the range of searches tested negative kriging weights are considered insignificant. The proportion of negative OK weights for LOI is high when using small blocks and large searches have some blocks estimated with greater than 50% negative OK weights (Figure 43). This is expected because the LOI variogram is more continuous than the Fe v ariogram however; 50% is still an e xtremely high proportion of negative OK weights. Such a high proportion of negative weights indicates that too many samples are included in the search ellipse.

36 In anisotropic searching, the search ellipse is reduced in size until the maximum number of samples are found. When the search ellipse dimensions are reduced, the relative dimensions of the ellipse stay the same.


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Mean Slope vs. Number of Samples Per Quadrant

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Figure 41. Mean slope for Fe versus number of samples per quadrant for selected block sizes.

Mean Slope vs. Number of Samples Per Quadrant

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Figure 42. Mean slope for LOI versus number of samples per quadrant for selected block sizes.


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Maximum Positive Weights vs. Number of Samples Per Quadrant

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150x150x6 150x150x12 200x200x6 200x200x12

Figure 43. Mean of maximum positive weights for LOI versus number of samples per quadrant for selected block sizes.

8.6 Problems with block Ordinary Kriging and Uniform Conditioning For OK or Uniform Conditioning (UC) estimates (which are typically based on OK) to be non-biased, the block size needs to large relative to the drilling grid. Typically, the block size would be close to the drill grid spacing. The problem in this dataset is that for block sizes of 50m x 50m and 100m x 100m (the typical drill grid spacings) a significant proportion of the blocks lie partly outside the Final M3 Domain. These low average block fill percentages are problematic because:

1. They poorly represent the shape and volume of the domain; and

2. For block OK (and UC), a significant number of discretisation points lie outside the domain to be est imated. This is particularly problematic in the Final M3 Domain where the vertical variogram range is short and there are strong vertical trends in the concentrations of assay variables. This results in t he g rades of blocks located on t he upper or lower edge of t he domain being very smoothed. The significant conditional bias (smoothing) for large blocks was demonstrated by the QKNA in Section 8.4.

Conditional S imulation d oes not su ffer from the abov ementioned i ssues because t he grid node spacing is very small compared to the Final M3 Domain. This is a key reason for using conditional simulation at Yandi instead of Uniform Conditioning.


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9 CHOOSING PARAMETERS FOR CONDITIONAL SIMULATION

9.1 Introduction In this chapter, a grid of points (simulation grid) suitable for Conditional Simulation (CS) is created and parameters for the search neighbourhood are discussed. Other details for the implementation of the conditional simulations are also discussed. T his chapter covers the elements that are common between al l the simulation methods used here. To enable the simulation methods to be compared, a s many asp ects as possible were kept i dentical bet ween t he v arious Conditional Simulation methods.

Declustering i s used here in v arious ways such as transforming d ata, w eighting st atistics, weighting variograms etc. For t he r emainder o f t his thesis, u nless otherwise st ated, the declustering weights used are those discussed in Section 8.2.2.

9.2 Simulation grid creation A w ireframe o f the Fi nal M3 D omain was filled w ith 3m x 3m x 2m spaced poi nts (nodes) t o produce a grid for simulation (Table 12). This grid was created using Minesight™ software. The coded simulation grid was checked visually to ensure that points had been coded correctly.

Minimum (m) Maximum (m) Size (m) Number X 14,800 16,300 3 500 Y 83,300 86,300 3 1,000 Z 480 590 2 55 Table 12. Simulation grid parameters.

9.3 Selecting the search neighbourhood parameters for Conditional Simulation All Conditional Simulation, unless otherwise stated, used the search parameters discussed below. This is important to allow direct comparison between the results of the various simulation methods. The fixed parameters for the Conditional Simulation search neighbourhood are:

1. No quadrant searching;

2. Minimum of 12 samples to estimate a node;

3. Search ellipse is 500m x 500m x 10m (X, Y and Z);

4. Anisotropic searching; and

5. No discretisation; i.e., point CS was used.

Quadrant searching was not used in order to avoid artefacts with sample selection because North-South and East-West nodes will lie exactly on the boundary between quadrants. Ordinary Kriging (OK) by construction declusters (David, 1977) and although the drill grid density varies, the drilling is on a fairly regular grid and it is considered that locally, clustering should not be a significant issue.

The search ellipse used for Conditional Simulation is extremely anisotropic. This is to restrict the number of samples used downhole within individual drill holes. This forces the search to look for more samples in t he horizontal plane. Without a highly ani sotropic search, m ost samples would come from three or four holes37

1. The variograms demonstrate zonal anisotropy

. Selecting too many samples vertically is not advisable because: 38

2. There are strong vertical trends in the concentrations of assay variables;

;

3. Geology (Chapter 4) indicates that a typical thickness for a sedimentary bed is two metres; and

37 There is no option in Isatis™ to limit the number of samples used from individual drillholes. 38 The sill of the variogram depends on direction.


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4. According t o Kentwell (2006), when composited t o 6m t he variograms a re 100% nugget. This is probably because too many sedimentary beds are grouped together, obscuring any continuity.

The search was limited to a maximum of 24 values. These values can be either simulated nodes or drill hole samples. Not limiting the number of simulated nodes to, for example, half the total number of v alues, will i mprove r eproduction o f t he v ariogram but w ill also reduce t he abi lity of the simulations to reproduce grade trends in the drill hole data. These issues are discussed further in the following sections.

The num ber of values was limited t o 24 , partially due t o time c onstraints in running t he simulations39

9.4 Method of Conditional Simulation and implementation details

. In addition, if a significantly larger search were chosen, e.g. 40 samples, there would be a large proportion of negative weights as the simulation fills with simulated nodes. For a 50m x 50m grid, the search used will select two levels of data ( i.e., a 4m sl ice) for the nearest four drill holes in the grid and only some of the holes that are about 100m away would be used.

Although each of the various simulation methods tested used different data transformation techniques, the underlying simulation method was the same. This method is known as Sequential Gaussian Simulation and has been discussed in Section 3.2.

Unless otherwise stated, the each simulation method has the following implementation details in common:

1. 20 simulations were generated for each method;

2. Each method had the first simulation starting with a random number seed equal to 423141. Using this random number seed, the simulations can be exactly reproduced if required40

3. The simulations all used Simple Kriging (SK) with a mean of zero;

;

4. Each simulation had a different random path; and

5. In the case of Conditional Co-Simulation, nodes were simulated sequentially. This maintains the co rrelation be tween assay variables at t he same node. For example, if Fe was simulated first when the second assay variable, P was simulated it took into to account the already simulated value of Fe; i.e., the correlation between Fe and P was honoured.

39 To run 20 simulations for five variables with a maximum of 24 values takes about 150 hours. Adding to the maximum number of values adds significantly to the time take to run a simulation. 40 Mueller (pers. comm., 2009, the external marker for this thesis) indicated that for the simulation of MAF it is preferable to simulate each factor with a different random number seed otherwise, correlations between factors can be introduced into the simulations.


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9.5 Variance reproduction in Conditional Simulation Gaussian simulations should (on average), have a mean of zero and a variance of one. When the range of the variogram in the vertical direction is short relative to the thickness of the domain, the variance of the simulations often av erages less than one. This is not an uncommon feature in Conditional Simulation and is explained further in Nowak and Verly (2004).

A m ethod recommended by Nowak and V erly ( 2004) to i ncrease t he variance of the Gaussian simulations closer to one, is to increase the sill of the variogram models. This method was implemented here for a ll t he v ariogram models use d for C onditional S imulation. A s much a s possible, the variogram models were left unchanged at the shorter lags and only changes at the longer di stances were m ade. I n ot her words, t he i mpact o f ch anging t he si ll m ost a ffects those simulated nodes that are relatively distal from sample data (i.e., wide spaced drilling). If a change is made to the simulations, it is preferable to make such a change in these areas of wider spaced drilling41. If the variances were not increased as described here42

One drawback with this method is that the variance of the data can only be increased by trial and error; i.e., the si mulation m ust be r un be fore seeing t he r elative i ncrease i n t he variance of t he data. In addition, it is difficult to modify the sills of variogram and cross-variograms that rely on the Linear Model of Co-Regionalisation (LMC). This is because changing one model parameter affects all the models. Despite this method involving some trial and er ror it is not subjective rather it is a matter of increasing or reducing the sill of the variograms until the variance of the simulations are close to one. The method is reproducible because the variogram model parameters are recorded. The validity of the method is also verifiable because the simulations are assessed on how well they reproduce t he characteristics of the decl ustered i nput dat a ( univariate st atistics, hi stograms, variograms etc). For ex ample, increasing t he sill of the v ariogram model actually results in the variograms of t he si mulations more cl osely r eproducing the v ariogram of t he da ta. Without increasing t he sills of the variogram models, the si lls of the v ariograms of the si mulations are underestimated.

, the variance of the simulations would have been s ignificantly and unacceptably underestimated. This was not assumed to be the case but every simulation was first run with a variogram sill of one. Only after it was identified that the variance was significantly under estimated were t he simulations re-run w ith a ne w variogram model.

A second method for variance co rrection described by Nowak and V erly ( 2004) i s to ad just t he variance of the simulations according to distance from the nearest drill hole sample. As the distance from t he near est drill h ole sample increases the si ze of t he c orrection i ncreases. This method i s uni variate and t hus while i t i s applicable t o m ethods such as MAF ( Factors are uncorrelated and si mulation i s univariate) i t is not su itable for co rrelated assay v ariables; i.e., Conditional Co-Simulation. To maintain consistency between the various simulation methods this method of variance adjustment was not used.

41 Simulated values closer to the data are more likely to be similar to that data value and it is preferable to limit changes to these ‘higher confidence’ simulated values. 42 The sills are increased in the order of 10%. The amount of increase can be seen in the various images showing the experimental variogram and the variogram model as well as by adding up the variances recorded in the variogram model parameter tables.


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10 CONDITIONAL CO-SIMULATION 10.1 Experimental variograms and cross-variograms Experimental variograms and cross-variograms were generated for the Gaussian transformed assay variables Fe, P, SiO2, Al2O3 and LO I. These experimental variograms were fitted with a model that consisted of a nugget and two spherical structures (Table 13). Selected experimental variograms, cr oss-variograms and their modes are presented in (Figure 44 to Figure 45). The direction o f m aximum continuity was horizontal w ith the r ange m uch sh orter i n the v ertical direction. I t might be expected that t he continuity would be g reater in t he North-South direction (along the channel) than the E-W direction (across the channel), but this was not obvious from the experimental variography. The sills of the variogram models were fitted above the variance of the data. The reason for this has been discussed in Section 9.5.

Nugget Effect Fe P SiO2 Al2O3 LOI Fe 0.44 -0.03 -0.34 -0.19 -0.18 P -0.03 0.19 -0.01 0.15 0.00 SiO2 -0.34 -0.01 0.34 0.11 0.00 Al2O3 -0.19 0.15 0.11 0.49 0.10 LOI -0.18 0.00 0.00 0.10 0.37 Maximum, Intermediate, Minimum continuity = 50m, 50m, 20m Structural Component 1 (spherical) Fe P SiO2 Al2O3 LOI Fe 0.47 -0.12 -0.33 -0.31 -0.05 P -0.12 0.21 0.10 0.08 0.01 SiO2 -0.33 0.10 0.35 0.05 0.01 Al2O3 -0.31 0.08 0.05 0.45 0.06 LOI -0.05 0.01 0.01 0.06 0.09 Maximum, Intermediate, Minimum continuity = 600m, 600m, 50m Structural Component 2 (spherical) Fe P SiO2 Al2O3 LOI Fe 0.16 0.01 -0.12 0.02 -0.02 P 0.01 0.69 -0.33 0.18 0.36 SiO2 -0.12 -0.33 0.36 -0.20 -0.38 Al2O3 0.02 0.18 -0.20 0.15 0.27 LOI -0.02 0.36 -0.38 0.27 0.59

Table 13. Variance-Covariance Matrix for variogram models of Gaussian transformed Fe, P, SiO2, Al2O3 and LOI.


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N0

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Figure 44. Experimental variograms, cross-variograms and models of selected Gaussian transformed assay variables. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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N0

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Variogram : AL2O3 GAU

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2 Variogram : AL2O3 GAU

Figure 45. Experimental variograms and models of selected Gaussian transformed assay variables. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

10.2 Statistical validation of the Gaussian co-simulations Gaussian univariate statistics (i.e., before back-transformation) were generated for each simulation (Table 42 to Table 46 in Appendix 7). The uni variate st atistics of the i ndividual si mulations are summarised i n Table 14. The si mulations have (on av erage) a mean cl ose t o z ero, w hich i s expected. The variance, while close to one , was always less than one. As discussed in Section 9.5, this related to the range of the vertical variogram relative to the thickness of the domain being simulated. These Gaussian simulations were acceptably close for back-transformation to raw assay variables.

Variable Count Minimum Maximum Mean Std. Dev. Variance Fe 1749699 -4.91 4.88 -0.01 0.99 0.98 P 1749699 -4.22 4.47 0.05 0.95 0.90 SiO2 1749699 -4.70 4.74 -0.06 0.97 0.93 Al2O3 1749699 -4.87 5.07 0.04 0.99 0.99 LOI 1749699 -4.51 4.52 0.08 0.93 0.86

Table 14. Average Gaussian univariate statistics of 20 co-simulations.

10.3 Statistical validation of the back-transformed co-simulations Univariate statistics ( i.e., after back-transformation) were generated for each simulation (Table 47 to Table 52 in Appendix 8). The univariate statistics of the individual simulations are summarised in Table 15. The simulations have (on average) means that are almost identical to the declustered drill hole data. The variances of the simulations, while close to the variance of the declustered drill hole data, are generally lower than the variance of the drill hole data. The shapes of the histograms of the simulations were similar to the weighted histograms of the drill hole data (Figure 46; Figure 152 to Figure 154 in Appendix 8). These simulation statistics including the shape of the histograms were acceptably close to the declustered drill hole data.

The average correlation coefficients of the 20 Co-Simulations (Table 16) were generally similar to the drill hole correlation coefficients. The exceptions to this were the correlation coefficients of Fe versus Al2O3 and SiO2 versus Al2O3, which were not well reproduced in the Conditional Simulation. This is discussed further in Section 10.4.


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Dataset Variable Count Minimum Maximum Mean Std. Dev. Variance Simulations Fe 1749699 39.9 62.2 58.7 2.0 4.0 Drill Holes Fe 3320 39.9 62.2 58.7 2.0 4.2 Simulations P 1749699 0.019 0.117 0.041 0.010 0.0001 Drill Holes P 3320 0.019 0.117 0.040 0.011 0.0001 Simulations SiO2 1749699 1.9 21.0 4.7 1.9 3.5 Drill Holes SiO2 3320 1.9 21.0 4.8 2.0 4.1 Simulations Al2O3 1749699 0.1 15.4 1.2 1.1 1.3 Drill Holes Al2O3 3320 0.1 15.4 1.2 1.1 1.2 Simulations LOI 1749699 6.7 12.4 9.9 0.7 0.5 Drill Holes LOI 3320 6.7 12.4 9.8 0.8 0.6 Simulations Total 1749699 79.1 114.7 99.7 1.6 2.5 Drill Holes Total 3320 97.9 100.4 99.8 0.2 0.1

Table 15. Average univariate statistics of 20 co-simulations.

Drill Hole Data Fe P SiO2 Al2O3 LOI

Fe 1 -0.19 -0.85 -0.83 -0.23

P 1 -0.01 0.22 0.35

SiO2 1 0.5 -0.21

Al2O3 1 0.25

LOI 1

Simulations Fe P SiO2 Al2O3 LOI

Fe 1 -0.15 -0.71 -0.34 -0.21

P 1 -0.08 0.28 0.31

SiO2 1 0.00 -0.27

Al2O3 1 0.27

LOI 1 Table 16. Average correlation coefficients of 20 Co-Simulations.43

Fe Histograms of Data and S imulation 1

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35 40 45 50 55 60 65Fe %

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Figure 46. Histogram of weighted Fe for drill hole data and Co-Simulation One.

43 The maximum range of the correlation coefficients for the 20 simulations was 0.07. In other words, the correlation coefficients varied little between simulations.


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10.4 Scatterplot reproduction of the back-transformed co-simulations Figure 47 to Figure 50 represent scatterplots of drill hole data and Co-Simulation One for the key assay v ariable correlations44

Figure 47

. T he scatterplots of the drill h ole data w ere reasonably w ell reproduced i n t he si mulations. A lim iting factor on this reproduction i s that t he assumption o f stationarity within the Final M3 Domain is unlikely to be valid; i.e., in the scatterplot of the drill hole data, two t rends can b e se en ( to Figure 49). Another problem evident f rom the scatterplot reproduction is that there are mineralogical constraints evident (e.g. Figure 47 to Figure 49) and these were not completely honoured in the simulations. The scatterplot reproduction for P versus LOI (Figure 50) was better than the other variables because this scatterplot did not have obvious strong mineralogical constraints or non-stationarity.

Figure 47. Scatterplot between Fe versus SiO2 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency.

Figure 48. Scatterplot between Fe and Al2O3 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency.

44 The software used to generate simulations in this thesis (Isatis™), was not able to generate scatterplots or variograms for the complete simulation due to the large size. Thus, for scatterplots and variography all the simulations in this thesis were sub-sampled on a 20m x 20m x 6m grid. This sub-sampling is unlikely to have a significant impact on quality of the scatterplots and variograms.


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0 5 10 15

AL2O3

0

10

20

SIO2

Figure 49. Scatterplot between SiO2 and Al2O3 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency.

Figure 50. Scatterplot between P and LOI for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency.

10.5 Variogram reproduction of the back-transformed co-simulations Variograms of simulated Fe, SiO2, SiO2 versus Fe and Al2O3 versus Fe (Figure 51; Figure 156 to Figure 158 in Appendix 9) have longer ranges in the downhole direction and lower sills than the drill hole data. The differences in variances are however considered minor45. T his issue of t he simulations having l ower v ariances has been di scussed i n Section 9. 5. T he sh apes of t he experimental variograms and cross-variograms are however, considered to be comparable.

The experimental variograms and cross-variograms of P (Figure 52), Al2O3, LOI, Al2O3 versus SiO2 and LOI v ersus P (Figure 159 to Figure 162 in Appendix 9) are considered t o be ac ceptably comparable both in shape and in variance.

45 The variance of the simulations could have been increased by increasing the variance of the variogram and cross-variogram models. The Gaussian experimental variograms are already fitted with sills greater than one and to increase the variances further could have caused issues with variogram reproduction at distances less than the variogram range.


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Figure 51. Fe experimental variograms of drill hole data (black) and Co-Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

Figure 52. P experimental variograms of drill hole data (black) and Co-Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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10.6 Trend reproduction of the back-transformed co-simulations Trend r eproduction for m ost assay v ariables and di rections (Easting, N orthing and R L) i s considered good (Figure 53 and Figure 163 to Figure 173 in Appendix 10). There are examples of poor trend reproduction; i.e., LOI versus Northing (Figure 54), LOI versus Easting (Figure 174 in Appendix 10) and Al2O3 versus RL ( Figure 175 in Appendix 10). A s is expected f or al l assay variables and directions the reproduction of the trends becomes worse as the number of drill hole samples in the area decreases and where there are step changes in the concentration of variable concentrations. Overall, the trend reproduction is considered good.

Northing vs. Mean P for Data and S imulations

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Figure 53. Average P versus Northing for co-simulations and drill hole data.

Northing vs. Mean LOI for Data and S imulations

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Figure 54. Average LOI versus Northing for co-simulations and drill hole data.


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11 SIMULATION USING MIN/MAX AUTOCORRELATION FACTORS

11.1 Generation and validation of MAF A FORTRAN program, written and su pplied b y the Bryan Mining C entre at the University o f Queensland, was used for the generation of MAF (Version 1.4). As discussed in Section 3.4, the MAF methodology decorrelates assay variables at a lag of zero and one other lag input by the user (Desbarats and Dimitrakopoulos, 2000). In this case, a distance of 50m (±20m) was selected since this equates to t he m ost co mmon dr ill hol e sp acing i n t he st udy ar ea. The M AF f actors were generated and t he FORTRAN program used to confirm that the MAF can be back -transformed to exactly the same raw values as those input into the program.

To confirm that the MAF factors were decorrelated, scatterplots at a lag of zero and experimental cross-variograms were generated. Decorrelation at a lag of zero is generally considered good (e.g. Figure 55, MAF1 versus MAF4). The worst example of decorrelation at zero lag (Figure 55, MAF1 versus MAF2) is still considered sufficiently acceptable for the continuation of this study because the factors are at best only weakly correlated. At lags greater than zero, the MAF are decorrelated for all factors except for MAF1 and MAF3 (Figure 56), which, have a very weak positive correlation at lags of 50 and 100m. In summary, it was considered that the MAF were acceptably decorrelated to continue with simulation.

The FORTRAN program use d for t he generation of t he M AF t ransforms the co rrelated assa y variables into Gaussian uncorrelated factors. Histograms of the MAF indicate that the assay variables were not precisely transformed to a standard Gaussian distribution (e.g. Figure 57)46. All the MAF were t ransformed to a st andard Gaussian distribution using I satis™ and the frequency inversion method of transformation (Geovariances, 2005)47.

Figure 55. Scatterplot at lag zero of MAF1 versus MAF4 (left) and MAF1 versus MAF2 (right).

46 The distributions deviated slightly from a standard Gaussian distribution. 47 These Gaussian transformed variables were used for the MAF simulations reported in this thesis.


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Figure 56. Experimental cross-variograms between MAF1 and MAF3. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variogram is annotated as D90.

-4 -3 -2 -1 0 1 2 3 4

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Figure 57. Histograms of MAF3 and MAF5.


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11.2 Experimental variograms of MAF Experimental variograms were generated for the MAF. These experimental variograms were fitted with a m odel that consisted of a nu gget and two spherical structures (Figure 58 to Figure 60 and Table 17). The direction of maximum continuity is horizontal, with the range much shorter in the vertical direction. Variograms have the maximum direction of continuity E-W for MAF3 and MAF5. However, MAF1, 2 and 4 hav e no ob vious anisotropy i n t he hor izontal pl ane. The si lls of t he variogram models were fitted above the variance of the data. The reason for this was discussed in Section 9.5.

Factor Structure Variance Max Range (m) Int Range (m) Min Range (m) MAF1 Nugget 0.5 Structural Component 1 0.45 35 35 30 Structural Component 2 0.15 200 200 65 MAF2 Nugget 0.4 Structural Component 1 0.45 40 40 15 Structural Component 2 0.32 900 900 60 MAF3 Nugget 0.4 Structural Component 1 0.38 114 40 15 Structural Component 2 0.47 680 570 57 MAF4 Nugget 0.3 Structural Component 1 0.18 200 200 80 Structural Component 2 1.6 1350 1350 80 MAF5 Nugget 0.18 Structural Component 1 0.22 92 110 3 Structural Component 2 0.85 520 300 100

Table 17. Variogram model parameters for MAF.

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Figure 58. Experimental variograms and model of MAF1 (left) and MAF2 (right). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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N0

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Figure 59. Experimental variograms and model of MAF3 (left) and MAF4 (right). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

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Figure 60. Experimental variogram and model of MAF5. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms is annotated as D90.

11.3 Statistical validation of the MAF simulations Gaussian univariate statistics (i.e., before back-transformation) were generated for each simulation (Table 53 to Table 57 in Appendix 1 ). The univariate statistics of t he individual simulations are summarised i n

1Table 18. The si mulations have (on av erage) a m ean very close t o zero and a

variance of one, which is expected.


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Variable Count Minimum Maximum Mean Std. Dev. Variance MAF 1 1749699 -4.99 4.96 -0.01 1.00 0.99 MAF 2 1749699 -5.01 4.85 -0.04 1.00 0.99 MAF 3 1749699 -4.95 4.73 0.00 0.98 0.97 MAF 4 1749699 -4.70 4.65 0.06 0.96 0.92 MAF 5 1749699 -4.51 4.38 0.00 0.97 0.94

Table 18. Average Gaussian univariate statistics of 20 MAF simulations.

11.4 Statistical validation of the back-transformed MAF simulations Univariate statistics of the back-transformed variables were generated for each simulation (Table 58 to Table 63 in Appendix 12). T hese univariate st atistics are su mmarised i n Table 19. T he simulations have (on average) means that are almost identical to the declustered drill hole data. The variance of the simulations, on the other hand, compares poorly with the declustered drill hole data and can be significantly higher (Fe) or lower (LOI). The histograms of simulations are similar to the weighted histograms of the drill hole data (Figure 61; Figure 176 to Figure 179 in Appendix 12) with the main difference being the abovementioned variance.

It can be deduced that the issue with variance reproduction occurred in the back-transformation. This is the ca se beca use t he G aussian v ariance o f t he M AF i s very close t o one ( Table 18). Despite this, the back-transform worked as expected because it was checked by transforming and back transforming the same dataset. In this test, the raw and back-transformed data are equal. The additional Gaussian transform mentioned in Section 11.1 was also removed to test if this was introducing a pr oblem. However, t here was no si gnificant di fference i n t he r esults48

Boyle ( 2007a a nd 2007 b) and B andarian et al . ( 2006) encountered similar problems to th ose discussed abov e, with both under and over e stimation of v ariance w ith t he back-transformed simulations. These authors did not identify a cause for the problem. Desbarats and Dimitrakopoulos also (2000) i dentified problems with poor reproduction o f the ex perimental correlogram; however, they attributed this to the low number of samples and er ratic nature of the experimental correlograms. In summary, the problem with variance re-production appears to be a potential weakness of t he MAF t echnique. A p otential so lution t o t his problem i s discussed i n

when t his additional Gaussian transform was removed.

Section 16.2.

The average correlation coefficients of the 20 simulations (Table 20) are generally similar to the drill hole correlation coefficients. There is however a t endency for the correlation coefficient to be higher in the simulations than the drill hole data.


Table 19. Average univariate statistics of 20 back-transformed MAF simulations. 48 The univariate statistics were essentially unchanged.


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Fe 1 -0.19 -0.85 -0.83 -0.23

P 1 -0.01 0.22 0.35

SiO2 1 0.5 -0.21

Al2O3 1 0.25

LOI 1


Fe 1 -0.58 -0.77 -0.78 0.21

P 1 0.48 0.54 0.13

SiO2 1 0.50 -0.60

Al2O3 1 -0.05

LOI 1 Table 20. Average correlation coefficients of the 20 MAF simulations49

.


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Figure 61. Histogram of weighted Fe for drill hole data and MAF Simulation One.

11.5 Scatterplot reproduction of the back-transformed MAF simulations Figure 62 to Figure 65 represent sca tterplots of drill h ole data and Simulation One f or t he k ey assay variable correlations. The scatterplots of the drill hole data were reasonably well reproduced in the simulations. A limiting factor on this reproduction is that the assumption of stationarity within the Final M3 Domain is unlikely to be valid; i.e., in the scatterplot of the drill hole data, two trends can be seen (Figure 62 to Figure 64). Another issue with the scatterplot reproduction is that there are mineralogical constraints evident (e.g. Figure 62 to Figure 64) and t hese were not completely honoured in the simulations. The scatterplot reproduction for P versus LOI (Figure 65) was better than t he ot her v ariables because t his scatterplot di d not hav e ob vious strong m ineralogical constraints or non-stationarity.



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Figure 62. Scatterplots of Fe versus SiO2 for drill hole data (left) and MAF Simulation One (right). Points are coloured on frequency.

Figure 63. Scatterplots of Fe versus Al2O3 for drill hole data (left) and MAF Simulation One (right). Points are coloured on frequency.

Figure 64. Scatterplots of SiO2 versus Al2O3 for drill hole data (left) and MAF Simulation One (right). Points are coloured on frequency.


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Figure 65. Scatterplots of P versus LOI for drill hole data (left) and MAF Simulation One (right). Points are coloured on frequency.

11.6 Experimental variogram reproduction of the back-transformed MAF simulations

Due t o t he previously m entioned problems with v ariance r eproduction, t he v ariances of t he experimental variograms of the drill hole data and simulations do not compare well. However, the shapes of the experimental variograms are always comparable between the data and si mulations (e.g. Figure 66 and Figure 67).

Figure 66. Fe experimental variograms of drill hole data (black) and MAF Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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Figure 67. LOI experimental variograms of drill hole data (black) and MAF Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

11.7 Trend reproduction of the back-transformed MAF simulations Trend r eproduction for m ost assay v ariables and di rections (Easting, N orthing and R L) i s considered r easonable to good ( see Figure 68 and Figure 180 to Figure 187 in Appendix 13). There are examples of poor trend reproduction (Figure 69; Figure 188 to Figure 192 in Appendix 13). As is expected for all assay variables and directions the reproduction of the trends is worse as the number of drill hole samples in the area decreases and where there are step changes in the concentration o f v ariable co ncentrations. Overall, the trend r eproduction ca n be co nsidered a s acceptable.


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RL vs. Mean LOI for Data and S imulations

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Figure 68. Average LOI versus RL for MAF simulations and drill hole data.

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Figure 69. Average Fe versus Northing for MAF simulations and drill hole data.


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12 SIMULATION USING THE STEPWISE CONDITIONAL TRANSFORM

12.1 Generation and validation of SCT A FORTRAN program, supplied by the University o f A lberta (Stepcon.exe V ersion 2. 0 an d Backstep.exe Version 1.0), was used for the transformation of assay variables using the Stepwise Conditional Transformation (SCT). As discussed in Section 3.3 the SCT methodology decorrelates variables at a lag of zero but does not necessarily decorrelate assay variables at non-zero lags.

There are five variables to decorrelate (Fe, P, SiO2, Al2O3 and LOI) but there were only sufficient samples to perform the SCT for up to three of these (ten divisions were used to subdivide each distribution). This is because the number of samples in individual transformation classes is less for the second v ariable and even lower f or the third variable (Section 3.3). There were simply not enough sa mples t o co nditionally t ransform a fourth v ariable. Thus, the S CT t ransform w as performed in two stages i.e:

1. Stage 1: Fe was the primary variable, SiO2 the second and Al2O3 the third variable; and

2. Stage 2: Fe was the primary variable, LOI the second and P the third variable.

This two-stage transformation resulted in the SCT not removing the correlation between all assay variables. For example, SiO2 versus LOI are still strongly correlated (Figure 70). Thus, Conditional Co-Simulation (CCS) was used for SiO2, Al2O3 and LOI (Section 12.2). The need for CCS to some degree de feats the pu rpose o f t he S CT, w hich ai ms at r emoving t he co rrelations between variables. There are however additional benefits to the SCT method such as the methods ability to deal with non-linear correlations (see Section 3.3).

Some of the bins used for transforming the second assay variable were wide and as a result the SCT did not always completely remove all the within-bin correlations between assay variables (e.g. Fe v ersus SiO2 Figure 70). If t he bi n w idth w ere r educed, there w ould not hav e been su fficient samples to transform t he third assay variable. This is a nece ssary co mpromise when using t he SCT method.

Histograms of the SCT variables indicated that some assay variables were not exactly transformed to a s tandard G aussian di stribution (Figure 71). This transformation h owever was considered sufficiently close (Deutch pers. comm., 2007) to Gaussian to continue with simulation.

A final check on the SCT was made by transforming and back-transforming assay variables. The back transformation of the secondary assay variables was exact. The back-transformation of the third assay variable however was poor (Figure 72). This is because there were insufficient samples in each bi n for reliable bac k-transformation o f t he third assay v ariable. Leuan gthong ( 2003) recommended dy namic class expansion t o deal w ith t oo f ew sa mples in each bi n of the third transformation. Dynamic class expansion works by using samples from adjacent bins to increase the number of samples per bin. This method (using version 3.004 of Stepcon_ref.exe and version 3.01 of Backstep.exe) appeared to introduce a bias in the back transformation of the data and was not used (Figure 73).


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Figure 70. Scatterplot at lag zero of SCT-SiO2 versus SCT-LOI (left) and SCT-Fe versus SCT-SiO2.

Figure 71. Histogram of Gaussian SCT-Al2O3.


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Figure 72. Scatterplot of original versus back-transformed SCT-Al2O3.

Figure 73. Scatterplot of original versus back-transformed (dynamic class expansion) SCT-Al2O3.

12.2 Experimental variograms and variogram models of the SCT Experimental variograms were generated for the SCT assay variables. These experimental variograms were fitted with models that consisted of a nugget and two spherical structures (Figure 74 and Figure 75 and Table 21 and Table 22). The direction of maximum continuity is horizontal with the range much shorter in the vertical direction. For P the direction of maximum continuity is North-South however, the remaining assay variables have no obvious anisotropy in the horizontal plane. The sills of the variogram models were fitted above the variance of the data. The reason for this has been discussed in Section 9.5.


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Factor Structure Variance Max Range (m) Int. Range (m) Min Range (m) SC-Fe Nugget 0.307625 Structural Component 1 0.35 10.27 10.27 7.33 Structural Component 2 0.45 120 120 70 SC-P Nugget 0.254909 Structural Component 1 0.33 75 50 10 Structural Component 2 0.645 850 450 75

Table 21. Experimental variogram model parameters for SCT. Nugget Effect SC-SiO2 SC-Al2O3 SC-LOI SC-SiO2 0.34 0.02 -0.14 SC-Al2O3 0.02 0.45 -0.15 SC-LOI -0.14 -0.15 0.32

Maximum, Intermediate and Minimum continuity = 60m 60m 20m Structural Component 1 SC-SiO2 SC-Al2O3 SC-LOI SC-SiO2 0.20 -0.10 -0.14 SC-Al2O3 -0.10 0.35 0.02 SC-LOI -0.14 0.02 0.11

Maximum, Intermediate and Minimum continuity = 750m 750m 55m Structural Component 2 SC-SiO2 SC-Al2O3 SC-LOI SC-SiO2 0.57 0.00 -0.65 SC-Al2O3 0.00 0.35 -0.18 SC-LOI -0.65 -0.18 0.84

Table 22. Experimental variogram and cross-variograms model parameters for SCT.

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Figure 74. Experimental variograms and models of SCT-Fe (left) and SCT-P (right). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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Figure 75. Experimental variograms, cross-variograms and models for SCT-SiO2, SCT-Al2O3 and SCT-LOI. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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12.3 Conditional Simulation of the SCT Once the assay variables have been transformed using the SCT and variogram models generated, the conditional simulation proceeds in the same manner as the other simulation methods utilised in this thesis. The simulation parameters are discussed in Section 9.

12.4 Statistical validation of the SCT simulations Gaussian univariate statistics (i.e., before back-transformation) were generated for each simulation (Table 64 to Table 68 in Appendix 1 ). The univariate statistics of t he individual simulations are summarised in

4Table 23. The simulations have (on average) a mean very close to zero, which is as

expected. The variance of Fe is one but the other assay variables have variances slightly less than one. The reason for this was discussed in Section 9.5.

Variable Count Minimum Maximum Mean Std. Dev Variance

Fe 1749699 -4.95 5.01 -0.01 1.00 1.00 P 1749699 -4.55 4.77 -0.03 0.97 0.95 SiO2 1749699 -4.49 4.55 -0.07 0.92 0.85 Al2O3 1749699 -4.68 4.71 0.00 0.96 0.92 LOI 1749699 -4.69 4.52 0.07 0.95 0.91

Table 23. Average Gaussian univariate statistics of 20 SCT simulations.

12.5 Statistical validation of the back-transformed SCT simulations Univariate st atistics after back-transformation w ere g enerated f or each simulation ( Table 69 to Table 73 in Appendix 15). The univariate statistics of the individual simulations are summarised in Table 24. The simulations have (on average) means that are almost identical to the declustered drill hole data. The variance of the simulations also compares well to the declustered drill hole data except for P. The variance of the P simulations is higher than the variance of the drill hole data. The histograms of simulation are similar to the weighted histograms of the drill hole data (Figure 76; Figure 194 to Figure 197 in Appendix 15).

The correlation coefficients for Simulation One (Table 25) are similar for the primary and secondary transformed assay variables; i.e., Fe versus SiO2 and Fe versus LOI. The correlations for assay variables of the third transformation are not reproduced at all; i.e., Fe versus Al2O3 and SiO2 versus Al2O3.

The problems with t he variance r eproduction o f P and reproduction o f the co rrelation for so me variable combinations relates to the low number of data in the tertiary transform. This problem of insufficient data was discussed in Section 12.1.


Table 24. Average univariate statistics of 20 back-transformed SCT simulations.


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DataSimulation 1

Figure 76. Histogram of weighted Fe for drill hole data and SCT Simulation One.


Fe 1 -0.19 -0.85 -0.83 -0.23

P 1 -0.01 0.22 0.35

SiO2 1 0.5 -0.21

Al2O3 1 0.25

LOI 1


Fe 1 0.42 -0.65 -0.09 -0.19

P 1 -0.36 -0.09 -0.33

SiO2 1 -0.13 -0.21

Al2O3 1 0.21

LOI 1 Table 25. Average correlation coefficients of the 20 SCT simulations50

.

12.6 Scatterplot reproduction of the back-transformed SCT simulations Figure 77 to Figure 80 represent sca tterplots of drill h ole data and Simulation One f or t he k ey assay variable correlations. The scatterplot of Fe versus SiO2 (Figure 77) was well reproduced by the simulation. The poor reproduction of the scatterplot at SiO2 concentrations above 10% relates to the small numbers of drill hole data in this part of the Scatterplot. If more bins (classes) were used i n t he t ransformation of S iO2 relative t o Fe, this issue would b e i mproved si gnificantly. However, if this were done, there would be insufficient data to transform the third assay variable.

The denser parts of the Fe versus Al2O3 (Figure 78), SiO2 versus Al2O3 (Figure 79) and LOI versus P (Figure 80) scatterplots are well r eproduced; however, the l ess dense areas do represent t he relationships seen in the drill hole data scatterplots.



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Figure 77. Scatterplots of Fe versus SiO2 for drill hole data (left) and SCT Simulation One (right). Points are coloured on frequency.

Figure 78. Scatterplots of Fe versus Al2O3 for drill hole data (left) and SCT Simulation One (right). Points are coloured on frequency.


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Figure 79. Scatterplots of SiO2 versus Al2O3 for drill hole data (left) and SCT Simulation One (right). Points are coloured on frequency.

Figure 80. Scatterplots of P versus LOI for drill hole data (left) and SCT Simulation One (right). Points are coloured on frequency.

12.7 Experimental variogram reproduction of the back-transformed SCT simulations Experimental variograms of the conditional simulations reproduced the shape and v ariance (e.g. Figure 81) of experimental v ariograms o f t he drill h ole data except f or P . T he sh apes of t he P experimental v ariograms are similar ( Figure 82) however; the variance of t he s imulations is significantly higher than the drill hole data (see discussion in Section 12.4).

The key experimental cross-variograms were not reproduced at all. For example, SiO2 and Al2O3 in drill h ole data ha ve a posi tive co rrelation o f 0 .5 w hereas in the simulations the correlations between these assay variables are actually negative. For the key correlations of Fe versus SiO2 and Fe versus Al2O3 the co rrelations have t he co rrect sign how ever; the v ariances of t he simulations versus drill holes are very different.51 This relates to there being insufficient samples for the tertiary SCT transformations (see Section 12.1).

51 There are no figures presented for these experimental variograms of the data and simulations. This is because the variances are so different that the figures are difficult to read.


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Figure 81. SCT-LOI experimental variograms of drill hole data (black) and SCT Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

Figure 82. SCT-P experimental variograms of drill hole data (black) and SCT Simulation One (coloured). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

12.8 Trend reproduction of the back-transformed SCT simulations Trend r eproduction for m ost assay v ariables and di rections (Easting, N orthing and R L) i s considered as reasonable to good (Figure 83 and Figure 198 to Figure 206 in Appendix 16). There are however examples of poor trend reproduction (Figure 84; Figure 207 to Figure 210 in Appendix


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16

). As expected, for all assay variables and directions the reproduction of the trends is worse as the number of drill hole samples in the area decreases and when there are step changes in the concentration of variables. Overall, the trend reproduction is considered good.


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Figure 83. Average LOI versus RL for SCT simulations and drill hole data.


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Figure 84. Average LOI versus Northing for SCT simulations and drill hole data.


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13 COMPARISON OF SIMULATIONS TO BLAST-HOLE DRILLING

13.1 Introduction The previous sections of this thesis have validated simulations by comparing their summary statistics, experimental variograms etc against those of the input drill hole data. This is an essential validation step; however, it is not the only way to validate simulations. It is possible that the drill hole data and the interpreted variograms are not representative of the underlying mineralisation. Alternatively, the drill hole data may be r epresentative, but the simulations may not define the full space of uncertainty. For example, the true grade uncertainty may be greater than is defined by the simulations. T o t est for t hese possi bilities, an area o f bl ast hol e dr illing (i.e., drilling f or gr ade control purposes) was selected to compare against the simulations.

13.2 Limitations with blast-hole data There are limitations with blast-hole data that make their use for validating the simulations more difficult. Blast-holes are drilled at close spacing’s for filling with explosives and blasting rock. The holes are not drilled for obtaining a representative sample. Sampling is also extremely crude which results in the blast-hole data being less reliable and often biased.

A further complication with comparing the simulations to the blast-hole data is that there is an order of magnitude more blast-holes than the resource definition drilling. The resource drilling can easily miss or hitting the Al2O3 rich clay pods and this can bias this resource definition drilling compared to the blast-hole drilling. Finally, there are significant time pressure on sample preparation and assaying for the blast-hole drilling and correspondingly lower QA/QC standards.

In summary, there are several reasons why the blast-hole and resource definition drilling datasets may not be eq uivalent ( in t erms of a ssay st atistics). H owever, t he bl ast-hole d rilling is the onl y meaningful independent data source to validate the simulations52

13.3 Selection of area for the validation of simulations

.

It was not possible (nor necessarily desi rable) to use blast holes that covered t he complete M3 Domain. There are several reasons for this:

1. Blast hole drilling does not cover the complete study area;

2. Many blast hol es intersect bot h t he Fi nal M3 Domain and t he su rrounding hi gh A l2O3 material, i.e., some blast holes crosscut major geological boundaries. Thus, only samples completely within the Final M3 Domain were used in this chapter; and

3. Although the bench heights at Yandi are 12m, the drilling varies from 0.7m to 18.2m (modal lengths of 12m and 14m ; se e Figure 85). This m akes meaningful est imation at t he 12m bench height more difficult. Thus, an area where all the holes were consistently between 11 and 13m was selected for this chapter (Figure 86).

A benefit of selecting a smaller area for validation is that the performance of the simulations can be judged for a volume that i s highly relevant t o m ining. The study area contains about five m illion tonnes, which equates to several months of production.

52 Reconciliation is difficult to trace back to its source due to the mixing of ore from multiple pits.


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0 5 10 15

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Figure 85. Histogram of blast hole lengths within the study area.

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Figure 86. Location of blast holes used for comparison with simulations.


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13.4 Exploratory data analysis of blast-hole drilling Table 26 and Table 27 contain univariate and bivariate statistics for the blast hole data. There are 3,628 bl ast hol es, w hich hav e m ean Fe and P gr ades o f 58. 5% and 0.04% r espectively. T his material is relatively typical of the ore mined at Yandi and that within the complete M3 Domain.

Variable Data Count Min Max Mean Std. Dev. Variance

FE

Blast 3628

44.28 60.80 58.48 1.33 1.76 P 0.03 0.10 0.0425 0.0097 0.0001 SIO2 2.39 18.38 4.55 1.29 1.66 AL2O3 0.45 9.45 1.33 0.73 0.53 LOI 8.00 13.36 10.23 0.47 0.22 LENGTH 11.00 13.00 12.12 0.35 0.12

Table 26. Univariate statistics of blast hole assays. Variable FE P SIO2 AL2O3 LOI FE 1 -0.3 -0.9 -0.8 -0.2 P 1 0.1 0.3 0.5 SIO2 1 0.5 -0.2 AL2O3 1 0.2 LOI 1

Table 27. Correlation coefficients of blast hole assays. Correlation coefficients above 0.4 are highlighted in red.

13.5 Creation of an SMU sized block model Before comparing the simulations to the blast holes it was necessary to determine the block size (volume) at w hich to m ake t he comparison. T he se lective m ining uni t (SMU) at Y andi i s approximately 25m x 2 5m x 12m . The se lectivity ca n be larger or smaller, but t his block si ze represents a reasonable average sized SMU. To allow comparison of the simulations and blast holes, both data sets were copied into the SMU sized block m odel. This simply consisted o f t aking the mean o f each data se t w ithin each SMU sized block. The maximum number of blast holes per cell is approximately 11 ( it varies depending on the block and blast hole locations). Only blocks with at least seven53

blast holes (i.e., a total of 297 blocks) were used for the comparison. The modal number of simulated nodes averaged into every SMU is 384. The statistics of the 2m resource definition drill hole data (i.e., the data used to create the simulations) were also interpolated (via nearest neighbour) into the SMU sized model. This will allow assessment of wether the resource definition drilling has comparable means to the blast hole data.

The resource definition drilling in the area of the SMU block model varies from 25m x 50m to 100m x 100m ( Figure 87). T hese drill densi ties are co mparable t o o ther a reas within t he Fi nal M3 Domain. Thus, this is considered to be a useful area to test the applicability of the simulations.

53 Insuring each block contains at least seven blast holes will insure that the estimation error is negligible.


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15600 15800 16000

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Figure 87. Plan of resource definition drill holes and the SMU block model.


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13.6 Comparisons of simulation to blast hole and resource definition drilling Ideally, for a successful simulation, the mean and variance of the blast-hole model assay grades would be comparable to the mean and variance of the re-blocked simulations however; this may not be the ca se due t o t he po tential di fferences between bl ast-hole dr illing and t he r esource definition drilling (see Section 13.2).

Figure 88 to Figure 92 depict mean grades (within the SMU sized block model) of:

1. Each simulation (LMC, MAF and SCT) sorted from the lowest to the highest mean grade;

2. The mean assay grades of the blast hole model; and

3. The mean assay grades of the resource definition drill holes (i.e., the data used to generate the simulations) interpolated via nearest neighbour into the block model.

The means of the blast hole model Fe and SiO2 lie within the range of the means obtained from simulation. Thus, from t his point of v iew the pe rformance o f t he si mulations for these v ariables appears satisfactory. The mean P grade of the blast hole model is slightly above the highest grade simulations. It is likely that running 100 instead of 20 simulations would result in the mean blast hole grade of P being within the range of all the simulation methodologies. This is because running more simulations would probably increase the minimum and maximum simulated P.

The Al2O3 mean is within the range of the conditional co-simulations (LMC simulations) but slightly outside the upper range of the MAF and S CT simulations. Again, i t is possible that running 100 instead of 20 simulations would also result in the mean blast hole grade of Al2O3 being within the range of the MAF and SCT simulations. Other methods to increase the space of uncertainty are discussed in the next section.

The mean LO I grade o f the bl ast bl ock model is significantly abov e (about 0.15%) the hi ghest simulation mean g rade. I n this case, more si mulations would be un likely to correct t he problem because the difference is too great. The issue is that the resource definition data is much lower in LOI than the blast hole drilling. The simulations have performed as expected and it is difficult to expect such a high concentrations of LOI in this area. While beyond the scope of this thesis, LOI is generally m ore su sceptible t o assa ys biases due t o di fferent sa mple preparation m ethods and laboratories (from the authors experience) than ot her v ariables. For ex ample, with the fast turnaround time for blast hole assays it is possible that some batches of samples were not always completely dried. This could easily explain the slightly higher LOI reading in the blast holes.

13.7 Increasing the space of uncertainty As discussed above, increasing the number of simulations from 20 to 100 would probably result in one or more of the simulations having similar assay means to the blast hole data. It should not be expected that using only 20 simulations will cover the full space of uncertainty. Using 100 to 200 or more simulations would provide a better approximation of the full space of uncertainty.

Another v alid ex planation for the abov e si mulations not al ways accounting for the full sp ace o f uncertainty is that the presence of clay pods has not been explicitly taken into account; i.e., clay pods have not been separated but included in the simulation data set. This was because the drill hole data was too widely spaced to interpret their locations, shapes and sizes. There are numerous small clay pods (e.g. less than 10m across and perhaps 0.1-2m thick) and the rare clay pod that is 50-100m ac ross. I ntersecting one o f these l arger cl ay pods could ea sily change t he sp ace of uncertainty, making it greater than that predicted by the drill hole data and simulations (drilling can easily miss even the larger clay pods).

In summary, the presence of Clay pods result in the domain not meeting the assumption of strict or second order stationarity (see Section 2.2). Clay pods and their simulation is the subject of the next chapter.


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Mean Fe for simulations, blast hole and resource estimates

57.4

57.6

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58

58.2

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59

1 3 5 7 9 11 13 15 17 19Count

Fe %

LMC FeMAF FeSCT FeResource FeBlasthole Fe

Figure 88. Mean Fe for simulations, blast hole and resource estimates. The simulations have been sorted from lowest to highest Fe grade.

Mean P for simulations, blast hole and resource estimates

0.037

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Figure 89. Mean P for simulations, blast hole and resource estimates. The simulations have been sorted from lowest to highest P grade.


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Mean SiO2 for simulations, blast hole and resource estimates

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2 %

LMC SIO2MAF SIO2SCT SIO2Resource SiO2Blasthole SiO2

Figure 90. Mean SiO2 for simulations, blast hole and resource estimates. The simulations have been sorted from lowest to highest SiO2 grade.

Mean Al2O3 for simulations, blast hole and resource estimates

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O3

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Figure 91. Mean Al2O3 for simulations, blast hole and resource estimates. The simulations have been sorted from lowest to highest Al2O3 grade.


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Mean LOI for simulations, blast hole and resource estimates

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10

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LMC LOIMAF LOISCT LOIResource LOIBlasthole LOI

Figure 92. Mean LOI for simulations, blast hole and resource estimates. The simulations have been sorted from lowest to highest LOI grade.

De-Vitry – MSc Thesis

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14 COMBINING GEOLOGY AND ASSAY VARIABLE SIMULATION 14.1 Introduction In key scatterplots (e.g. Figure 77) there are two trends present; i.e., the Final M3 Domain probably does not meet the assumption of stationarity. This apparent lack of stationarity is due to Al2O3 rich zones, commonly known as ‘clay pods’ (Figure 93). These clay pods do not have sufficient continuity to reliably interpret their shape; i.e., they are generally smaller than the drill hole spacing.

A single simulation of these clay pods was generated. The grades within these clay pods and the surrounding C ID were then co -simulated (using the Linear Model of Co-Regionalisation, se e Section 2.5). The two grade co-simulations were then combined i nto a si ngle si mulation. Only generating one si mulation obv iously does not represent t he full r ange of geological and grade variability. T he si ngle si mulation however, was sufficient for co mparison t o the si mulations generated using the previously discussed methodologies. This is because the approach of simulating cl ay pods followed b y g rade simulation pr oduces such markedly su perior r esults that multiple simulations are not required to demonstrate this.

Figure 93. Photographs of clay pods within CID (Stone et al. 2002).

14.2 Statistical and spatial characteristics of clay pods Two geochemically distinct populations (CID and clay pods) can be identified on many scatterplots (e.g. Figure 94). The populations were separated on the basis of scatterplots and their statistical characteristics determined (see discussion below).

a1172507

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As should be the case, relative to the surrounding CID, the clay pods contain low concentrations of Fe and high concentrations of contaminants (Table 28 and Table 29). A po sitive but no t unexpected feature of removing the clay pods is that the horizontal continuity of the experimental variograms within t he CI D increased (Figure 95). Another i mprovement is that previously undetectable but geologically sensible anisotropy was observed; i.e., the continuity is greater in the North-South direction than the E-W direction (Figure 96).

Plots of the average assay variable concentration across the boundary between clay pods and CID were g enerated ( e.g. Figure 97 to Figure 98). These pl ots suggest that t he grade bounda ries between cl ay pods and C ID are sh arp which is also co nsistent w ith geol ogical under standing. Thus, these boundaries were treated as hard during simulation.

Figure 94. Scatterplot of Fe versus SiO2 with the clay rich population highlighted as red squares. Variable Count Minimum Maximum Mean Std. Dev. Variance Fe 278 39.91 59.17 54.41 3.64 13.22 P 278 0.0190 0.1170 0.0488 0.0170 0.0003 SiO2 278 2.92 20.96 7.37 3.42 11.72 Al2O3 278 0.58 15.44 3.61 2.10 4.43 LOI 278 8.30 12.36 10.60 0.86 0.74 TOTAL 278 99.26 100.43 99.93 0.19 0.04 Total Majors 278 97.87 100.17 99.49 0.31 0.09

Table 28. Univariate statistics of clay pods.


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Variable Count Minimum Maximum Mean Std. Dev. Variance Fe 3042 54.69 62.18 59.15 1.05 1.11 P 3042 0.0200 0.0820 0.0393 0.0094 0.0001 SiO2 3042 1.90 11.30 4.55 1.58 2.51 Al2O3 3042 0.10 2.32 0.89 0.38 0.15 LOI 3042 6.72 11.95 9.72 0.75 0.56 TOTAL 3042 98.93 100.55 100.02 0.18 0.03 Total Majors 3042 98.74 100.40 99.82 0.20 0.04

Table 29. Univariate statistics of CID with clay pods removed.

N0

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Figure 95. Experimental variogram of SiO2 for all drill hole data (top left), CID (top right) and clay pods (bottom). The azimuth of the horizontal variograms are annotated as N0 while the vertical variograms are annotated as D-90.


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Figure 96. Experimental variogram and model of Al2O3 for the CID with clay pods removed. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variogram is annotated as D90.


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Figure 97. Average Al2O3 concentration across the boundary between clay pods and CID. The mean is represented by the black line while the blue squares represent the number of samples.

Figure 98. Average SiO2 concentration across the boundary between clay pods and CID. The mean is represented by the black line while the blue squares represent the number of samples.


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14.3 Indicator Conditional Simulation of clay pods The l ocation and the s hape o f the cl ay pods ca n be generated v ia t he sequential indicator simulation approach of Alabert (1987) discussed below (Also see Section 3.5). Once a realisation of the clay pods and the surrounding CID are generated these two rock types can be treated as separate domains for the conditional co-simulation simulation of the assays variables.

Before si mulating, all sa mples were co ded usi ng an i ndicator v ariable ( 0=CID and 1=clay pod) . Experimental variograms of this indicator were then generated and fitted with a nugget and two-structure model (these models are discussed below in more detail). This model was then used for Indicator Conditional Simulation within Isatis™. For more information on the Indicator Conditional Simulation method applied, refer to Section 3.5.

The dimensions of t he clay pods are generally smaller t han the drill hole spacing, however clay pods of up to about 100m diameter also occur. Due to the size of the clay pods in relation to the drilling, experimental variograms are close to 100% nugget. Six different variogram models ranging from m inimum t o m aximum pl ausible continuity were cr eated. Both sp herical and t he m ore continuous Gaussian variogram models were used. Using the continuous Gaussian model was the only way to reproduce the larger continuous clay pods that are known to occur, these larger clay pods could not be reproduced usi ng spherical models. One indicator simulation was generated using each of these variogram models (Table 30 and Figure 99 to Figure 110). A simulation with a geologically realistic shape was selected f or the co-simulation of assay variables (Figure 106)54. Before adopt ing t his method at BHPBIO, further work i s required t o i dentify t he shape and si ze distribution of clay pods (this is discussed further in Section 16.1). At present, the simulation of clay pods used here is sufficient to illustrate the potential of the method.

Note that the Indicator Conditional Simulation algorithm as implemented in Isatis™ uses a different search strategy to the Gaussian based simulation methods used elsewhere in this thesis. Instead of defining the orientation and size (in metres) of the search, the search is defined by the number of blocks in the X, Y and Z directions. The search for abovementioned six indicator simulations was created to match the search described in Section 9.3 as closely as possible. However, the selected simulation (Figure 106) was rerun using a se arch with double the maximum number of data ( i.e., 48 rather than 24). The size of the Z search was also doubled to 20m and the maximum number of simulated nodes limited t o 24 ( i.e., half the se arch ca n be si mulated nodes). This new l arger search was previously not feasible due t o t he t ime t aken t o run t he simulations. H ere only one simulation was rerun. It is expected that this new search will improve the ability of the simulations to reproduce the trends in the data. This is discussed further in Section 14.5.

54 The selection of the most geologically plausible clay pod simulation was based on the author’s opinion from two site visits to the Yandi open pits. No geological maps of clay pod shapes and distribution existed at the time this clay pod simulation was completed.


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Indicator Variogram

Number Structure

Type of Structure

Variance Maximum

Range Intermediate

Range Minimum

Range

1

Nugget 0.001 Structural Component 1

Spherical 0.047 6.16 4.16 2.16

Structural Component 2 0.028 40 20 10.13

2


Gaussian 0.047 15 10 1.16


3


Gaussian 0.038 30 15 1.16


4


Gaussian 0.038 40 20 1.16


5


Gaussian 0.038 40 20 1.16


6


Gaussian 0.028 6.16 4.16 2.16


Table 30. Indicator variogram parameters for clay pod simulation.


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D90

N12

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Distance (m)

0.00

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Variogram : m3 al flag

Figure 99. Clay pod experimental Indicator Variogram One. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated as D90.


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Figure 100. Clay pod Indicator Simulation One.

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Figure 101. Clay pod experimental Indicator Variogram Two. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated as D90.


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Figure 102. Clay pod Indicator Simulation Two.

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Figure 103. Clay pod experimental Indicator Variogram Three. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated as D90.


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Figure 104. Clay pod Indicator Simulation Three.

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Figure 105. Clay pod experimental Indicator Variogram Four. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated as D90.


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Figure 106. Clay pod Indicator Simulation Four.

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Figure 107. Clay pod experimental Indicator Variogram Five. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated a D90.


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Figure 108. Clay pod Indicator Simulation Five.

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Figure 109. Clay pod experimental Indicator Variogram six. The azimuths of the horizontal variograms are annotated as N12 and N90 while the vertical variogram is annotated as D90.


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Figure 110. Clay pod Indicator Simulation Six.

14.4 Experimental variograms and cross-variograms of assay variables within clay pods and CID

Experimental variograms and cross-variograms were generated for the Gaussian transformed assay variables Fe, P, SiO2, Al2O3 and LO I. These experimental variograms were fitted with a model that consisted of a nugget and two spherical structures. This process was identical to that used in Chapter 10 with the exception that the experimental variograms were generated and fitted for both clay pods and the CID (Table 31, Table 32 and Figure 11155). Previously in Chapter 10, the experimental variograms were only generated on the Final M3 Domain, which had clay pods and CID combined.

55 Only the experimental variograms and models from the CID are presented. The experimental variograms from the clay pods are extremely erratic due to the small dimensions of individual clay pods.


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Nugget Effect Fe P SiO2 Al2O3 LOI Fe 0.382 0.002 -0.227 -0.089 -0.177 P 0.002 0.192 -0.045 0.149 0.016 SiO2 -0.227 -0.045 0.241 -0.031 -0.011 Al2O3 -0.089 0.149 -0.031 0.403 0.013 LOI -0.177 0.016 -0.011 0.013 0.372 Maximum, Intermediate and Minimum ranges = 60m, 30m, 20m Structural Component 1 Fe P SiO2 Al2O3 LOI Fe 0.364 -0.074 -0.307 -0.094 0.016 P -0.074 0.269 0.067 0.139 -0.038 SiO2 -0.307 0.067 0.333 -0.075 -0.060 Al2O3 -0.094 0.139 -0.075 0.422 0.100 LOI 0.016 -0.038 -0.060 0.100 0.078 Maximum, Intermediate and Minimum ranges = 650m, 450m, 20m Structural Component 2 Fe P SiO2 Al2O3 LOI

Fe 0.379 -0.022 -0.205 0.150 0.061 P -0.022 0.539 -0.128 -0.006 0.238 SiO2 -0.205 -0.128 0.529 -0.215 -0.346 Al2O3 0.150 -0.006 -0.215 0.393 0.075 LOI 0.061 0.238 -0.346 0.075 0.549

Table 31. Variogram and cross-variogram model parameters of Gaussian transformed Fe, P, SiO2, Al2O3 and LOI: CID with no clay pods. The model is horizontal and strikes towards 348º. Nugget Effect Fe P SiO2 Al2O3 LOI Fe 0.581 0.037 -0.483 -0.456 -0.105 P 0.037 0.142 -0.057 -0.014 0.030 SiO2 -0.483 -0.057 0.422 0.388 0.118 Al2O3 -0.456 -0.014 0.388 0.393 0.115 LOI -0.105 0.030 0.118 0.115 0.154 Maximum, Intermediate and Minimum ranges = 50m, 50m, 20m Structural Component 1 Fe P SiO2 Al2O3 LOI Fe 0.086 -0.038 -0.086 -0.109 0.123 P -0.038 0.018 0.044 0.040 -0.058 SiO2 -0.086 0.044 0.133 0.042 -0.151 Al2O3 -0.109 0.040 0.042 0.236 -0.116 LOI 0.123 -0.058 -0.151 -0.116 0.253 Maximum, Intermediate and Minimum ranges = 50m, 50m, 20m Structural Component 2 Fe P SiO2 Al2O3 LOI Fe 0.333 0.429 -0.184 -0.103 0.163 P 0.429 1.114 -0.647 -0.085 0.940 SiO2 -0.184 -0.647 0.507 -0.100 -0.651 Al2O3 -0.103 -0.085 -0.100 0.480 -0.019 LOI 0.163 0.940 -0.651 -0.019 1.069

Table 32. Variogram and cross-variogram model parameters of Gaussian transformed Fe, P, SiO2, Al2O3 and LOI: Clay pods. The model is horizontal and strikes towards 348º.


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N0

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Figure 111. Experimental variograms and models of selected Gaussian transformed assay variables within the CID (no clay pods). The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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14.5 Parameters for Conditional Co-Simulation within simulated geological domains Compared to the searches used previously for Sequential Gaussian Simulation (Section 9.3), the maximum number of data and the Z search dimension were doubled to 48 samples and 20m respectively. The maximum number of simulated nodes was limited to 24 (i.e., half t he search could be simulated nodes). As discussed in Section 14.3, this new larger search was previously not feasible due to the time taken to run the simulations. Here only one simulation was re-run. The objective of the larger search was to improve the ability of the simulations to reproduce the trends in the data (see Section 14.10). A f inal difference between this and the previous searches is that the orientation (for CID only) of the new search now strikes towards 348º.

14.6 Statistical validation of the Gaussian co-simulations Gaussian univariate statistics (i.e., before back-transformation) were generated for the combined CID and clay pod simulations (Table 33). The simulations have (on average) a mean close to zero and variance close to one, as was expected. The Gaussian simulations were thus considered to be acceptably close to a mean of zero and a variance of one to permit legitimate back-transformation to raw assay variables. Variable Count Minimum Maximum Mean Std. Dev. Variance Fe 1749699 -5.02 4.36 -0.01 0.98 0.96 P 1749699 -4.39 5.24 0.08 1.02 1.05 SiO2 1749699 -4.38 5.16 -0.03 1.00 0.99 Al2O3 1749699 -4.57 4.94 0.04 1.00 1.00 LOI 1749699 -4.52 4.63 0.09 0.98 0.95

Table 33. Gaussian univariate statistics for Co-Simulation One: CID and clay pods combined.

14.7 Statistical validation of the back-transformed co-simulations Univariate statistics, after back-transformation, were generated for the combined CID and clay pod simulations (Table 34). The si mulations had (on average) means and variances that ar e al most identical to the declustered drill hole data. Histograms of the simulation are essentially identical to the weighted histograms of the drill hole data (Figure 112; Figure 211 to Figure 214 in Appendix 17). The correlations coefficients of the combined CID and clay pod simulations (Table 35) are very similar to the drill hole correlation coefficients.


Table 34. Univariate statistics for Co-Simulation One: CID and clay pods combined.


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Drill Hole Data Fe P SiO2 Al2O3 LOI Fe 1 -0.19 -0.85 -0.83 -0.23 P 1 -0.01 0.22 0.35 SiO2 1 0.5 -0.21 Al2O3 1 0.25 LOI 1 Simulation One Fe P SiO2 Al2O3 LOI Fe 1 -0.14 -0.80 -0.75 -0.20 P 1 -0.03 0.19 0.33 SiO2 1 0.42 -0.22 Al2O3 1 0.25 LOI 1

Table 35. Correlation coefficients of Co-Simulation One: CID and clay pods combined.


0

5

10

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20

25

45 47 49 51 53 55 57 59 61 63Fe %

Freq

uenc

y %

DataSimulation 1

Figure 112. Histogram of weighted Fe for drill hole data and Simulation One: CID and clay pods combined.

14.8 Scatterplot reproduction of the back-transformed co-simulations Figure 113 to Figure 116 represent scatterplots of drill hole data and Simulation One for the key assay v ariable correlations. The k ey drill hole data sca tterplots are ex tremely well r eproduced (Figure 113 to Figure 116). Many of these data scatterplots have two trends and these trends were reproduced in the scatterplots.


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Figure 113. Scatterplots of Fe versus SiO2 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency: CID and clay pods combined.

Figure 114. Scatterplots of Fe versus Al2O3 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency: CID and clay pods combined.


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Figure 115. Scatterplots of SiO2 versus Al2O3 for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency: CID and clay pods combined.

Figure 116. Scatterplots of P versus LOI for drill hole data (left) and Co-Simulation One (right). Points are coloured on frequency: CID and clay pods combined.

14.9 Variogram reproduction of the back-transformed co-simulations Within the clay pods, the experimental variograms and cross-variograms of the drill hole data are extremely i rregular i n shape56

Figure 117

. G iven t his limitation, t hese variograms and cr oss-variograms can still be co mpared t o t hose o f the Conditional C o-Simulation. For clay pods , the ex perimental variograms and cross-variograms of assay variables and simulations are in all instances considered as comparable both in shape and in variance ( ).

For the simulations within the CID, the experimental variograms and cross-variograms (Figure 118 and Figure 119) were well reproduced in terms of both shape and variance. The exceptions to this are the experimental variograms of P and the cross-variograms of Al2O3 versus SiO2, Al2O3 versus P and A l2O3 versus Fe (e.g. Figure 120). I n these ca ses, t he shapes ar e co mparable bu t t he variances slightly different. Some issues with variance reproduction occur w ith al l t he simulation methods tested but overall the simulations in this section have performed well with regards to variogram reproduction. 56 This was expected because the dimensions of the clay pods are generally less than the drill grid spacing.


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Figure 117. LOI experimental variograms of drill hole data (black) and Co-Simulation One (coloured): Clay pods. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

Figure 118. LOI experimental variograms of drill hole data (black) and Co-Simulation One (coloured): CID. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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Figure 119. LOI versus SiO2 experimental variograms of drill hole data (black) and Co-Simulation One (coloured): CID. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.

Figure 120. Al2O3 versus P experimental cross-variograms of drill hole data (black) and Co-Simulation One (coloured): CID. The azimuths of the horizontal variograms are annotated as N0 and N90 while the vertical variograms are annotated as D90.


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14.10 Trend reproduction of the back-transformed co-simulations In general, the simulations closely reproduce the trends of the composited assay variables for Easting, N orthing and RL (Figure 121; Figure 215 to Figure 226 in Appendix 17). LOI ve rsus Northing (Figure 122) and LOI versus Easting (Figure 227 in Appendix 17) are examples were the trend reproduction was not as good.


0

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12

500 510 520 530 540 550 560 570RL (m)

LOI %

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200

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DH MeanSim 1DH Number

Figure 121. Average LOI versus RL for co-simulations and drill hole data: CID and clay pods combined.


9

9.2

9.4

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84550 84750 84950 85150 85350 85550 85750 85950 86150Northing (m)

LOI %

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DH MeanSim 1DH Number

Figure 122. Average LOI versus Northing for co-simulations and drill hole data: CID and clay pods combined.


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15 DISCUSSION AND COMPARISON OF SIMULATION METHODS

15.1 Introduction and methodology In t his chapter, the f ollowing abbr eviations are use d t o r efer t o t he v arious simulation methodologies:

1. GLMC: Gaussian Co-Simulation usi ng the Li near M odel of C o-Regionalisation ( Chapter 10);

2. MAF: Simulations generated using Min/Max autocorrelation Factors (Chapter 11);

3. SCT: Simulations generated using the Stepwise Conditional Transform (Chapter 12); and.

4. LMC/GEOL: The GLMC using two domains generated by Indicator Conditional Simulation (Chapter 14).

In order to compare the various simulation methods, a comparison of the following aspects were considered57

1. Gaussian statistics: Ho w cl ose were the G aussian ( i.e., the si mulations before ba ck-transformation) mean and variance to one and zero respectively;

:

2. Raw univariate statistics: How close were the back-transformed mean and variance of the simulations to the drill hole data;

3. Histograms: How well did the histograms of the assay variable simulations reproduce the histograms of the drill hole data;

4. Correlation coefficients: How well did the simulations reproduce the correlation coefficients of the drill hole data? The focus being on the reproduction of the strongest correlations;

5. Scatterplots: Were the scatterplots of the simulations comparable to those of the drill hole data?;

6. Variograms: Were the experimental variograms of the simulations comparable to those of the drill hole data?;

7. Trends: Were the trends of average assay variables (in Easting, Northing and RL) in t he drill hole data reproduced in the simulations;

8. Spatial pat terns: D id t he graphical images of the si mulations represent g eologically plausible sp atial pat terns and were there any o bvious visual ar tefacts or e rrors such a s lines of elevated grades;

9. Space of uncertainty: Did the simulations reproduce the full space of uncertainty. In other words, could the true unknown grades be outside the range of grades simulated; and

10. How m uch per son-time ( as distinct from computational time) d id it t ake t o co mplete validated simulations?

Each of these points above was given a subjective score between 1 and 5 (Table 36) in order to semi-quantitatively asse ss each si mulation method. A sco re of 1 i s unsatisfactory w hile a 5 indicates excellent per formance. This methodology is considered t o be sufficient t o pr ovide a reliable relative ranking of the various simulation methodologies. In any case, there are no useful and accepted st atistical m easures for q uantitatively ranking the output q uality from simulations. This is potentially a useful area for future research especially if it was accompanied by software development.

There was only one simulation generated for the LMC/GEOL method while 20 s imulations were generated for t he other three Conditional Simulation methods. The global characteristics did not vary m uch bet ween s imulations and t his is not co nsidered a significant i ssue affecting t he comparisons. One further di fference between the LMC/GEOL and t he o ther methods is that t his

57 In each of these cases, the statistics were weighted to take account of clustering.


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method used a different search strategy (Section 14.3 and Section 14.5). Areas where this different search strategy may cause differences with the other methods are discussed below.

Performance Measure GLMC MAF SCT LMC/GEOL Gaussian Statistics 4 4 4 4 Raw Univariate Statistics 4 2 4 4 Histograms 4 2 4 4 Correlation Coefficients 3 2 1 4 Scatterplots 3 4 1 5 Variograms 4 3 1 4 Trends 4 3 4 4 Spatial patterns 3 3 3 5 Space of Uncertainty 4 4 4 5 Person Implementation Time 4 4 4 2 Table 36. Semi-quantitative ranking of the key performance measures of the various simulation methods. A 5 indicates excellent performance while a 1 is unsatisfactory performance.

15.2 Gaussian statistics and search strategy The mean of all the Gaussian simulations (collectively) was close to zero and the variance slightly less than one. The average variances for the GLMC, MAF, SCT and LMC/GEOL simulations were 0.93, 0.96, 0.92 and 0.99 respectively. As discussed in Section 9.5, these low variances were at least partially related to the range of the vertical variogram being long relative to the thickness of the domain. It is also possible that the very restricted search in the Z di rection contributed to the underestimation o f t he v ariance. This is because t he v ertical r ange o f the v ariograms was very short and a restricted vertical search may not have adequately represented the vertical variability. The restricted vertical search was necessary to provide a neighbourhood in which the assumption of stationarity was potentially valid (Section 9.3).

Although there was only one simulation for the LMC/GEOL method, the variance of this simulation was very close to 1. This method used a larger vertical search and this may support the idea that a restricted search contributed to an underestimation of variance. It is still considered most likely that the under estimation o f variance was related t o bot h t he r estricted se arch and t he r ange o f t he variogram relative to the domain thickness. This is because the LMC/GEOL variogram models had sills greater than one; however, the variance of the Gaussian simulations was still less than one.

While the above discussion is relevant for explaining t he variance underestimation, it does not necessarily represent a strength or weakness of any of the underling methods being tested.

15.3 Back-transformed univariate statistics and histograms Table 37 presents a comparison bet ween t he m ean and variance o f the bac k-transformed simulations and the declustered drill hole data. The average simulation grade was divided by the average declustered diamond drill hole assay and multiplied by 100 to give a percentage ratio. The ratios are so close to 100% as to not allow meaningful differentiation between the methods.

The variances of the simulations were generally lower than the drill hole data. The reason for this (as discussed above) relates to the range of the variogram relative to the thickness of the domain and potentially (in the case of GLMC, MAF and SCT) the restricted search in the Z direction.

The high relative variance of the SCT simulations is related purely to the simulations of P, which affects the ot herwise g ood r eproduction o f v ariance. P i s the t hird assay v ariable in t he SCT transformation and there were insufficient samples to make a reliable transformation58

58 This is a general weakness of the SCT method i.e. large numbers of samples are required to reliably simulate the third variable.

. One result of this was some unrealistically high values in the initial back-transformed P for a single bin (class). These high values were capped at 0.12 % P but the variance is still inflated due t o this problem. With some trial and er ror, the back-transformation of P for this bin could be m anually adjusted to remove the variance inflation. However, this would also reduce the mean of the simulated P below


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the m ean o f the drill h ole data. I deally, i t would be nece ssary t o a djust bot h t he m ean and t he variance of P distribution within this bin.

The av erage v ariance o f t he M AF si mulations (Table 37) hi des the fact t hat t he v ariance of individual assay v ariables was very poor ly r eproduced i n t hese si mulations (Table 38). T he reasons for this poor variance r eproduction are related to t he back-transformation of the MAF. Potential ar eas of future research to resolve t his variances inflation problems are discussed in Section 16.2.

The reproduction of the histograms for the GLMC, SCT and LMC/GEOL methods was considered to be very good. Histograms reproduction for the MAF simulations was reasonable except for LOI, which was unacceptable. T hese problems with hi stogram reproduction i n t he M AF si mulations relate to the abovementioned problem of poor variance reproduction in the MAF.

Method Mean Variance GLMC 100% 95% MAF 97% 95% SCT 100% 95% LMC/GEOL 100% 99%

Table 37. Ratio (%) between declustered drill hole and variance and the simulation means and variances for all assay variables. Variable Variance Fe 228% P 100% SiO2 67% Al2O3 37% LOI 43% Average 95%

Table 38. Ratio (%) between declustered drill hole means and variance and the simulation means and variances for all assay variables: MAF simulations.

15.4 Correlation coefficients and scatterplots The average absolute difference between the Simulation One correlation coefficients and the drill hole correlation co efficients were ca lculated. T his measures (on av erage) how cl osely t he simulations reproduced the drill h ole correlation co efficients. These av erages were 0. 14, 0 .27, 0.36, and 0.04 for the GLMC, MAF, SCT and L MC/GEOL methods respectively. The LMC/GEOL method had excellent reproduction of the correlation coefficients while the GLMC reproduction was considered satisfactory. The reproduction for the MAF and SCT methods are considered unacceptable. As discussed in the previous section, there are significant problems with variance reproduction i n t he M AF si mulations. This is negatively i nfluencing ho w t he MAF m ethodology reproduces the correlation coefficients. As discussed in the previous section, the SCT method is negatively influenced by the number of data being transformed. These issues are discussed further in Chapter 16.

Correlation co efficients are an e ffective su mmary of how well assay v ariables are co rrelated on average. However, a relatively small num ber of si mulated poi nts that ar e unr ealistic on scatterplots, can severely detract from visual assessments of how well scatterplots are reproduced. How well t he si mulation sca tterplots reproduce t he drill h ole data s catterplots from a v isual standpoint thus warrants further discussion.

As mentioned above, t he GLMC was considered to have better reproduction of the correlation coefficients than MAF. Regarding the scatterplots, this observations is reversed, with the MAF producing scatterplots that were visually more comparable to the drill hole data scatterplots59

59 In this context while the MAF produced visually better scatterplots than the GLMC, the GLMC was (on average) better as was indicated by the correlation coefficients.

. The SCT scatterplots are considered generally poor reproductions of the drill hole data. Conversely, the


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LMC/GEOL scatterplots are co nsidered excellent w ith t hese sca tterplots being t he onl y ones to reproduce the two trends present in many of the scatterplots.

If the MAF and SCT methods were used within the simulated CID and clay pods their performance would also almost certainly improve significantly60

. However, the relative performance of these methods would probably stay the same; i.e., co-simulation would still be the best method.

15.5 Experimental variograms and cross-variograms The experimental variograms and cross-variograms of the simulations were well reproduced for the GLMC and LMC/GEOL methods. In some instances, the variance was underestimated by a relatively sm all am ount however; t he shapes of experimental variograms still co mpared well. As discussed above, the MAF m ethod ha d very poor variance r eproduction and t his has adversely affected the reproduction of the experimental variograms and cross-variograms. The reproduction of the experimental variograms for the SCT simulations was good except for P, which as discussed above demonstrated poor variance reproduction. The cross-variograms for the SCT method were unacceptable with many examples of both the shape and the variance not being reproduced.

15.6 Trends The GLMC, S CT and LMC/GEOL (Section 10 .6, Section 12.7 and Section 1 4.1 respectively) methods for most assay variables and directions effectively reproduced the trends observed in the drill hole data. I t was expected that using an expanded search, fewer simulated nodes and more data points in the search for the LMC/GEOL method would have improved trend reproduction. The trend r eproduction for t he LM C/GEOL method was slightly improved, but was not si gnificantly better than the GLMC, MAF (Section 11.7) and SCT. Perhaps the use of fewer simulated nodes in the search had less impact than expected because as the grid fills in, the closer simulated nodes still had a much greater impact than the more distant data points.

The MAF trend reproduction was considered acceptable but not as good as the other methods.

15.7 Spatial patterns For t he GLMC, MAF an d S CT m ethods (Figure 123 to Figure 125), t he sp atial pat terns of t he assay variable concentrations in the simulations were very similar. This was expected because the underlying al gorithm (Sequential G aussian S imulation) and the se arches were identical. I n addition, t he v ariograms were all r elatively s imilar. There were no obv iously geologically unreasonable patterns (e.g., lines of elevated grades) in t hese simulations and while t he spatial patterns are reasonable, they do not represent the clay pods. The LMC/GEOL has reproduced the clay pods and resulted in simulations that are considered highly realistic from a geological point of view (Figure 126).

60 This is because all the non-indicator based simulation methods tested depend on the underlying assumption of stationarity and simulation of the clay pods results in a significant improvement in stationarity.


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Figure 123. Plan view of co-simulated Fe at 525 RL.


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Figure 124. Plan view of Fe at 525 RL for MAF Simulation One.

Figure 125. Plan view of Fe at 525 RL for SCT Simulation One.


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Figure 126. Plan view of co-simulated Fe at 525 RL: CID and clay pods combined.

15.8 Space of uncertainty For most variables, the results of GLMC, MAF and SCT can be considered to reproduce the full space of uncertainty (Chapter 13). In some instances; e.g., Al2O3, the simulation methods do not fully r eproduce t he sp ace o f unce rtainty. This is not a problem o f the si mulations failing t o reproduce the global assay st atistics o f drill hole data bu t rather the drill hole data being l ocally unrepresentative. The Al2O3 rich clay pods are much smaller than the drill hole spacing. Thus for areas significantly smaller than the complete study area the local drill hole data will not always be representative. Because the LMC/GEOL method involves simulation of the clay pod locations, this approach holds the m ost promise f or locally simulating t he full s pace o f uncertainty. Such an approach would involve overriding to some extent the information in the local drill hole data and thus involves strong stationarity assumptions. For example, if the drill hole data indicated that there were 2% clay pods in a local area but the global proportion within the domain was 10% then the local simulation could be made to have a higher proportion of clay pods than 2%.

15.9 Person implementation time Traditionally, for geostatisticians, the f itting of variograms and cr oss-variograms for m ore t han three assay variables is a task avoided. If there are five experimental variograms there will be a further ten experimental cross-variograms to fit. Alternative methods, e.g. MAF, were in part developed t o a void t he sometimes-tedious experimental cr oss-variogram fitting pr ocess. This avoidance of fitting cross-variograms may stem from using software without automatic variogram fitting or automatic validation that the model is both positive def initive and symmetric (Chilès and Delfiner, 1999). Without t hese t ools, fitting a m odel t o five assay v ariables would be cl ose t o impossible. Fitting models to cross-variograms using Isatis™ (the software used here) was more time consuming than fitting individual experimental variograms but not particularly onerous (it took approximately one hour). Using MAF and the SCT there were additional transformations that must both be performed and validated. It is argued here that these transforms negate the time benefit of


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not needing to fit cross-variograms. The most time consuming method is the LMC/GEOL because geology must be simulated and two simulations of the assay variables produced and validated.

15.10 Summary The LMC/GEOL conditional simulation methodology produced the best or t he e qual best simulations in all the areas assessed. The only drawback to the LMC/GEOL methodology is that it takes significantly longer personal and computation time to implement than the other methods. The next best method is GMLC followed by MAF and then the SCT.


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16 RECOMMENDATIONS FOR FURTHER RESEARCH

16.1 LMC and GLMC recommendations Based on the work performed for this thesis, the recommended approach for simulating the assay variables at Yandi is:

1. Use Indicator Conditional Simulation to simulate the distribution of CID and clay pods;

2. Use the Linear Model of Co-Regionalisation, to co-simulate assay variables separately into the clay pods and CID; and

3. Combine the simulation results from the two domains.

This method was not difficult to implement and produced excellent results in all the areas assessed in t he pr evious chapter. T he r ecommended ne xt st eps to i mprove t he si mulations require the generation of a training image (Journel, 2007). A training image in this case is a three-dimensional simulation grid showing an idealised representation of the CID and clay pods. This training image could then be used to modify the Indicator Simulations to produce even more geologically realistic patterns. A training image could be generated from a combination of:

1. In-pit photography;

2. Drilling;

3. Mapping; and

4. Geophysical information.

Once a training image and an Indicator Simulation are available, the following techniques could be tested for modifying the simulations:

1. Simulated Annealing (Chilès and Delfiner, 1999); and

2. Multiple Point Simulation (Journel, 2007; Osterholt and Dimitrakopoulos, 2007a; Osterholt and Dimitrakopoulos, 2007b).

This simulation of both geology and assay variables in this thesis was done within a single deterministic wireframe. An additional step would be to simulate the shape of this domain using for example M ultiple-Point C onditional S imulation (Journel, 2007 ; O sterholt and D imitrakopoulos, 2007a; Osterholt and Dimitrakopoulos, 2007b) or generating multiple interpretations using software such as LeapFrog (see http://www.leapfrog3d.com/).

16.2 MAF recommendations As discussed above, simulating using Min/Max Autocorrelation Factors (MAF) did not, in general, produce simulations that were as good as the GLMC. The key benef it in using MAF is that i t i s faster to f it variogram m odels because t he M AF ar e unco rrelated an d ca n t hus be si mulated independently. A s discussed in Section 15.9, f itting models to 15 ex perimental variograms and cross-variograms (i.e., five assay variables) is not particularly time consuming or difficult. Fitting models for more than five assay variables would become more difficult. However, it is unlikely that more than five assay va riables would need t o be co -simulated. This is beca use the 11 assay variables typically assayed for iron ore are not all correlated. Thus, some of the assay variables could be simulated independently or co-simulated in smaller groups.

The use of MAF does however hold some promise in terms of speed for simulation with correlated variables. I t i s possible in t his case t hat so me i ssues with t he m ethod (e.g. poor v ariance reproduction) were caused by the apparent lack of stationarity, both in terms of clay pods and in trends of the concentrations of the assay variables. This is because a single data transform and back-transform is made irrespective of stationarity assumptions. This is supported by the findings of R ondon and Tran ( 2008) w ho found that t he m ethod pe rforms poo rly w hen t wo t rends are present in the scatterplots and this is the case at Yandi where two trends exist in the scatterplots (i.e. clay pods and CID). Potential next steps for this method are:

1. Use Indicator Conditional Simulation to simulate clay pods and CID;


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2. Remove trends; and

3. Perform the MAF transform, simulation and back-transformation on residuals (i.e., the part of the assay that is left after the trend is removed).

16.3 SCT recommendations The key benefit of simulating using the Stepwise Conditional Transform (SCT) is that this method can deal with non-linear correlations and mineralogical constraints. None of the key assay variable pairs has non-linear c orrelations however; many sca tterplots do ha ve st rong m ineralogical constraints. As discussed above, simulating in to CI D and clay pods using the LMC honour s t he mineralogical co nstraints in a sa tisfactory w ay. The S CT si mulations did not reproduce t he scatterplots as well as using MAF or the SCT, especially for the third transformed assay variable. For the third assay variable, there were too few samples to produce a satisfactory t ransformation and the sca tterplot reproduction was unacceptable. Thus, t here i s no major incentive to use the SCT at Yandi. One exception may be t o use the SCT to remove trends in the data. For example, Elevation could form the primary variable and all the assay variables transformed conditionally to Elevation.

As an aside, the author has observed strong non-linear correlations in detrital iron ore deposits at Area C (A deposit about is 40 km southwest of Yandi). In such deposits, simulating using the SCT would be t he only viable alternative. The performance of the method could be i mproved by using order relations corrections, which were implemented in the latest version of the SCT transformation software


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