Plurigaussian Simulation of rocktypes using data from a gold mine in Western Australia

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1 Edith Cowan University Researh Online Theses: Dotorates and Masters Theses 2011 Plurigaussian Simulation of roktypes using data from a gold mine in Western Australia Robin Dunn Edith Cowan University Reommended Citation Dunn, R. (2011). Plurigaussian Simulation of roktypes using data from a gold mine in Western Australia. Retrieved from This Thesis is posted at Researh Online.

2 EDITH COWAN UNIVERSITY SCHOOL OF ENGINEERING Plurigaussian Simulation of roktypes using data from a gold mine in Western Australia Robin Dunn ( ) May 2011 Supervisor: Assoiate Professor Ute Mueller

3 Edith Cowan University Copyright Warning You may print or download ONE opy of this doument for the purpose of your own researh or study. The University does not authorize you to opy, ommuniate or otherwise make available eletronially to any other person any opyright material ontained on this site. You are reminded of the following: Copyright owners are entitled to take legal ation against persons who infringe their opyright. A reprodution of material that is proteted by opyright may be a opyright infringement. Where the reprodution of suh material is done without attribution of authorship, with false attribution of authorship or the authorship is treated in a derogatory manner, this may be a breah of the author s moral rights ontained in Part IX of the Copyright At 1968 (Cth). Courts have the power to impose a wide range of ivil and riminal santions for infringement of opyright, infringement of moral rights and other offenes under the Copyright At 1968 (Cth). Higher penalties may apply, and higher damages may be awarded, for offenes and infringements involving the onversion of material into digital or eletroni form.

4 USE OF THESIS The Use of Thesis statement is not inluded in this version of the thesis.

5 ABSTRACT Stohasti simulation of roktypes, or the geometry of the geology, is a major area of ontinuing researh as earth sientists seek a better understanding of an orebody as a preursor to the assignment of ontinuous rok properties, allowing more eonomially appropriate deisions regarding mine planning. This thesis analyses the suitability of partiular geostatistial rok type modelling algorithms when applied to the five roktypes evident in drill hole data from the Big Bell gold mine near Cue, Western Australia. The bakground of the geostatistial theory is onsidered, in partiular the onept of the random funtion model and the link between the ategorial statistis determined from the drill hole data and the three models used for estimation and simulation. The ommonly applied indiator kriging (IK) and sequential indiator simulation (SIS) algorithms are ompared in a non-sedimentary gold deposit environment to the more omputationally demanding and more omplex plurigaussian simulation (PGS). Comparisons between the three models are made by examining global and regional roktype (lithotype) proportions of the outputs of the models, both visually and empirially. The models are validated by onsidering the ontats whih our in reality between different lithotypes and the proportion of ontats whih do not onform to this reality in eah of the models. This inadmissible ontat ratio measures the short range validity of the estimation and simulation tehniques. Finally, ores taken from the output of the models are ompared to the drill hole data in terms of transition proportions between the twenty five possible transitions for the five lithotypes. Inadmissible ontats were at a minimum with PGS, and the visual and empirial natures of the PGS output were losely linked to the reality of the drill hole data. Whilst eah model produed similar 3D images, PGS was a realisti balane between the lustering effet produed by IK and the fine mosai effet from SIS. The PGS output numerially outperformed the other two models in terms of admissible ontats and onnetivity, most losely mathing the drill hole data. All results indiate that, whilst demanding to implement, PGS produes the most adequate model of the study region. iii

6 COPYRIGHT AND ACCESS DECLARATION I ertify that this thesis does not, to the best of my knowledge and belief: (i) (ii) Inorporate without aknowledgment any material previously submitted for a degree or diploma in any institution of higher degree or diploma in any institution of higher eduation; Contain any material previously published or written by another person exept where due referene is made in the text of this thesis; or (iii) Contain any defamatory material. (iv) Contain any data that has not been olleted in a manner onsistent with ethis approval. The Ethis Committee may refer any inidents involving requests for ethis approval after data olletion to the relevant Faulty for ation. iv

7 Table of Contents Chapter 1: Introdution Bakground and signifiane Early tehniques in geostatistis Objetives Outline of Thesis Common Notation... 7 Chapter 2: Theoretial Framework Random Variable and Random Funtion The geostatistis of ategorial data Kriging Indiator kriging of ategorial variables Sequential indiator simulation The Gaussian funtion as a model for simulation Turning Band method of non-onditional simulation The trunated Gaussian and Plurigaussian models The link between the indiator semivariograms and the underlying Gaussian semivariograms Chapter 3: Aspets of Plurigaussian Simulation The rok type rule Vertial proportion urves and referene levels The Gibbs sampler Bak-transforming the onditioned values Chapter 4: Exploratory Data Analysis Bakground of Big Bell Gold Mine Data analysis Variogram models for IK and SIS Chapter 5: PluriGaussian simulation Overview of the algorithm Defining the horizontal boundary surfaes of the strutural grid Disretisation and Flattenning Analysis and estimation of lithotypes Roktype rule and Gaussian random funtions Conditional plurigaussian simulation Chapter 6: Comparison of output of models Visual omparison between simulations and estimation Global and regional rok type proportions Inadmissible ontat numbers for IK, SIS and PGS Transition matries for IK, SIS and PGS output Disussion and onlusion Referenes Appendix v

8 List of Figures Figure 1 Categorial data ontats in a 2D region Figure 2 Representation of nature of ategory ontats from Figure Figure 3 Projetions from a point 1D ovariane funtions Figure 4 Representation of four ategories where ontats are non-sequential Figure 5 Possible ontats for three lithotypes Figure 6 Plurigaussian ase for three lithotypes Figure 7 Possible ontats for four lithotypes Figure 8 Lithotype olour sale used in this study Figure 9 Proportions of lithotypes in the study region Figure 10 Drill hole trunation Figure 11 Proportions of lithotypes in the study region in the absene of L Figure 12 Projetions and 3D image of data set Figure 13 Variography for IK of the five lithotypes, L1 to L Figure 14 A flowhart of the plurigaussian algorithm Figure 15 Lines and surfaes Figure 16 Pie harts indiating regional lithotype proportions Figure 17 Five polygons and their ompleted VPCs in the study region Figure 18 Raw (right) and ompleted (left) VPCs Figure 19 3D proportion map Figure 20 Lithotype ontat map Figure 21 The roktype rule definition panel Figure 22 Underlying Gaussians and assoiated thresholds for PGS simulation Figure 23 Experimental and model horizontal variograms for lithotypes for L1 (top), L2 (middle) and L3 (bottom) Figure 24 Experimental and model horizontal variograms for lithotypes L4 (top) and L5 (bottom) Figure 25 Experimental and model vertial variograms for lithotypes 1, 2and 3 (top) and lithotypes 4 and 5 (bottom) Figure 26 3D utaway representation of the onditional PGS over the study region.. 62 Figure 27 Horizontal slies of the study region from z 110m (top left) to z 30m (bottom right) Figure 28 Six different realisations of PGS at depth z 110m Figure 29 Exavated 3D projetions of IK (top left), SIS (top right) and PGS (bottom) Figure 30 IK (left), SIS (middle) and PGS (right) horizontal slies at z 110m (top), z 100m (middle) and z 90m (bottom) Figure 31 IK (left), SIS (middle) and PGS (right) horizontal slies at z 80m (top), z 70m (middle) and z 60m (bottom) Figure 32 Lithotype proportions for 100 simulations of SIS (top) and PGS (bottom) 70 Figure 33 Relative proportions of L2, L3, L4 and L5 with L1 removed for SIS (top) and PGS (bottom) Figure 34 Lithotype proportions at z 110m (top), z 100m (middle) and z 90m (bottom) for (from left to right) IK, SIS, PGS and the data set Figure 35 Lithotype proportions at z 80m (top), z 70m (middle) and z 60m (bottom) for (from left to right) IK, SIS, PGS and the data set Figure 36 Lithotype proportionsat four grid positions in the study region for 100 simulations of PGS (red) and SIS (blue) vi

9 Figure 37 Cells onsidered for inadmissible ontat numbers Figure 38 Inadmissible ontat perentages for SIS and PGS over 100 simulations.. 77 Figure 39 Positions of 18 ores taken from IK, SIS and PGS outputs Figure 40 Boxplots of transition proportions for (left to right) L1->L1 to L5->L5 with SIS transition averages ( blue) and experimental proportions (red) Figure 41 Boxplots of transition proportions for (left to right) L1->L1 to L5->L5 with PGS transition averages ( blue) and experimental proportions (red) Figure 42 Transition proportions for (left to right) L1->L1 to L5->L5 from IK (blue) and experimental data (red) vii

10 List of Tables Table 1 Correlations and ovarianes for RFs Z 1, Z Table 2 Lithotype perentages in raw data Table 3 Variogram parameters used in IK for ategories (Mueller et. al, 2002) Table 4 Downward matrix of transition proportions Table 5 Upward matrix of transition proportions Table 6 Parameters for underlying Gaussian models Table 7 Summary lithotype perentage statistis for Data set, IK, SIS (100 realisations) and PGS(100 realisations) Table 8 Summary data of ategory perentages from SIS (top) and PGS (middle) and IK and Data Set (bottom) in the absene of L Table 9 Summary inadmissible ontat perentages for IK, SIS and PGS Table 10 Downward transition matries for (top to bottom); experimental data, IK, SIS (average of 100 realisations) and PGS (average of 100 realisations) Table 11 Advantages and disadvantages of the three algorithms viii

11 Chapter 1: Introdution 1.1 Bakground and signifiane Geostatistis is a well-established suite of mathematial and probabilisti tehniques developed in the 1960s in response to the need for mathematial tehniques that would enable evaluation of the reoverable reserves in ore deposits. It an be defined as a set of statistial tools that allows desription, interpretation and modelling of the spatial ontinuity that is a fundamental feature of many naturally ourring phenomena, taking into aount both the struture and randomness of suh phenomena. The tehniques developed have found appliation in areas suh as petroleum, soil sienes and hydrogeology, where there is a need for interpretation of spatially orrelated data and subsequent reservoir modelling. Modelling of spatial ategorial data (in this thesis: rok types) is an important, initial step in suh areas as mining viability or regional hydrology. In the mining ontext, one a three dimensional simulation model of the rok types has been produed the different ontinuous variables within those rok types an be alulated. Plurigaussian simulation (PGS) is omputationally demanding and there are a number of sequened steps needed before the simulation an be produed, however it is the inorporation of spatial variability measures in PGS that will, it is surmised, produe a more aurate simulation of the study region. PGS models provide more straightforward and realisti transitions between ategories than sequential indiator simulation (SIS) (Galli et al, 1994). There are inherent smoothing issues in indiator kriging (IK) that post proessing algorithms have attempted to address. It is eonomially important to produe realisti models that honour both the original ategorial transitions and proportions. PGS employs a onstraint on allowable ontats between rok types that is initially defined by the drill hole data set. Neither IK nor SIS has suh a onstraint and an important omparison to PGS is the proportion of IK and SIS models that have so alled illegal ontats between rok types in their output, indiating that the geometry found in the initial exploration has not been honored. 1

12 1.2 Early tehniques in geostatistis The linear least squares regression algorithms introdued by Danie Krige (1951) were an initial effort at estimation of attributes of interest at unknown loations. Krige s empirial work to evaluate mineral resoures was formalised in the 1960s by Frenh engineer Georges Matheron, who subsequently ontributed signifiantly to the disipline by the introdution of the onept of a regionalised variable (ReV). Initial tehniques in geostatistis were onerned with desribing the spatial variability of earth siene data and estimating attribute values at unsampled loations (kriging). A well-researhed aount of the history and origins of kriging an be found in Cressie (1990). The proess of kriging gives an aurate least squares regression value and is onvenient in obtaining the onditional umulative distribution funtion (df). It therefore is ommonly used to estimate the probability of distribution, or single loation unertainty. Three ommon versions of kriging are Simple Kriging (SK), where the mean of the attribute in question is onsidered to be onstant over a study region, Ordinary Kriging (OK), where loal variation in the mean of the attribute in question is aounted for, and Indiator Kriging (IK) where the attribute in question is apportioned by a series of thresholds into lasses. Kriging is used both for ontinuous data (attributes suh as mineral grades) and, in this thesis, ategorial data suh as rok types. Drawbaks of kriging are that the single realisation of an attribute does not reflet the full spatial and statistial variability of the sample data, there being a smoothing effet inherent in the proess. This smoothing effet is minimal when the loations to be estimated are nearby the observed data but inreases as the distane from the loation to be estimated to the observed data inreases. The development of geostatistial simulation tehniques was a neessary step in overoming the pereived shortomings of the estimation algorithms, and began with the work of Matheron (ited in Chiles& Delfiner, 1999, p.472) and Journel (1974). Compared to kriging, onditional simulation is an advane in haraterising the spatial variation of earth variables suh as rok types. The simulation approah takes into aount not only the spatial variation of observed data at sample loations but also the variation in estimates at unsampled loations (Deutsh, 2006). Vann, Bertoli and 2

13 Jakson (2002) provide a summary of the development of these tehniques. Numerous simulation tehniques have been developed over the years and as they are omputationally demanding their growth an be aligned with the inreased availability and power of omputers, beginning in the 1980s. Many are based on the assumption that the random funtion(s) is multigaussian and inlude methods suh as ovariane matrix (LU) deomposition, turning bands, sequential indiator simulation (SIS), sequential Gaussian simulation (sgs), trunated Gaussian and the more omplex PluriGaussian simulation. Categorial data, suh as rok types (lithotypes), an be modelled using a variety of algorithms. Indiator kriging (IK) was introdued by Journel (1983) and for a onsiderable time was one of the most widely applied grade estimation tehniques in the mining industry. The original appeal of IK was that it was non-parametri- it made no assumptions about the distribution underlying the sample data. The basis of the indiator approah is the binomial oding of data into either 1 or 0 depending upon its relationship to a ut-off value, resulting in a non-linear transformation of the data values. A benefit for ontinuous data is that values muh greater than the ut-off value will reeive the same indiator value as data lose to the ut-off, thus limiting the effet of very high values in the kriging proess. The outome of IK is a onditional umulative distribution funtion (df) whih is a distribution of loal unertainty or possible values onditional to data in the neighbourhood of the blok or ell to be estimated. This umulative distribution at a blok or ell should be non-dereasing and valued between 0 and 1. These two requirements are sometimes not met, leading to order relations violations. Many methods have been proposed to overome this issue, inluding diret orretion of the indiator values (eg. Deutsh and Journel, 1998), using nested indiator variables (Dagbert and Dimitrakopoulos (1992) and probability kriging (Isaaks, 1984) whih is indiator o-kriging between the indiator-transformed data and a uniform transform of the sample data. A ritiism of IK is the need to alulate a variogram and develop a model for eah ut-off value, whih for multiple ut-offs was initially a major issue, however omputer speed and memory apaity have removed this problem. The indiator transform lends itself to the estimation of ategorial data suh as rok types. In this ase, instead of kriging indiators at a set of thresholds, ategorial IK will produe the probability of a given rok type at a given loation produing probability maps of given rok types. This may be ombined with 3

14 indiator estimation or simulation of grade data, as desribed in Dowd (1996). Lloyd and Atkinson (2001) suggest that adaptation of the variogram for IK to loal variation would inrease the auray of the estimations by resolving the problem of hanging means and variograms aross the study region, and this approah has been formalized by the onditional simulation paradigm. Sequential indiator simulation (SIS) is a non-parametri algorithm that, as for IK, requires the binary transformation of ontinuous (eg. porosity or permeability) or ategorial (eg. rok type) data into a series of indiator variables (Goovaerts, 1997, Deutsh and Journel, 1998). It applies IK in a sequential fashion where a preise ategory is drawn by Monte Carlo simulation at eah loation following a random path through the three dimensional grid. Individual grid nodes are simulated, one after the other, using ontinually updated (and inreasing-sized) onditioning data sets, the onditioning data set inluding the original data set and all previously simulated values within a speified neighbourhood. This ensures that losely spaed values have the orret short sale orrelation. This algorithm has been employed in suh varied studies as ategorial soil variable simulation (Li, Zhang, 2007), simulation of faies for groundwater flow (Moysey et al, 2003), reservoir fluid flows (Seifert and Jensen (1999) and unertainty assessment of the spatial distribution of soil organi arbon (Delbari et al (2010). It has been desribed in detail by Deutsh and Journel (1992) and Srivastava (1994). Deutsh (1996) uses SIS in a ategorial ontext and ompares a number of SIS algorithms, but notes two ommon ritiisms of the tehnique: the models produed an appear very path and unstrutured, and SIS often leads to unontrolled and unrealisti transitions between ategories. Li and Zhang (2007) also found that SIS models (using simple or ordinary kriging) do not obey interlass relationships in simulation maps. A number of SIS variations have evolved over the years (Gomez-Hernandez and Srivastava, 1990; Goovaerts, 1994; Goovaerts, 1997), mostly relating to the use of soft seondary data arising from geologial interpretation or geophysial measurements. The above models of IK and SIS are both non-parametri, in that they have no predetermined underlying model. Many simulation algorithms make the purely mathematial assumption that suh an underlying model theoretially exists, and a popular set of these models use the Gaussian funtion as its basis. Two algorithms whih use this assumption are trunated Gaussian simulation (TGS) and plurigaussian 4

15 simulation (PGS) Both TGS and PGS were initially developed in the 1980s to model oil reservoirs in sedimentary geologial environments, where different rok types typially our in a defined sequential order. A two-step approah was developed to model quantities suh as porosity and permeability in reservoirs: firstly a model of the geometry of rok types in the reservoir was done and then this model was used to generate values of porosity. The trunated Gaussian method is based on the notion of assoiation of a region under the normal urve to a ategory proportion. The trunation of a standard Gaussian random funtion by different thresholds alulated from experimental set proportions was first applied to model the lithotype in deltai reservoirs by Matheron et al. (1987). The ovariane model of the underlying Gaussian funtion is obtained by an iterative proess of inversion of the ovariane model of the experimental indiator variables of the sets. One this Gaussian semivariogram is known, kriging weights are available and further work is possible. If the ategorial data are sequential in their ontats then trunation of one Gaussian RF is required (TGS), however PGS is an extension to the algorithm that allows for more omplex patterns and different types of ontats between rok types to be modeled. Few texts, with the exeption of Armstrong et al. (2003), are solely onerned with a suint explanation of the basis of the reent development of plurigaussian simulation tehniques, although numerous theses, journal artiles, ourse notes and onferene proeedings are available. The trunated Gaussian and plurigaussian simulation algorithms have been investigated by Dowd (2003), Le Lo h and Galli (1996) and Lantuéjoul (1994, 1997). Remare and Zapparolli (2003) employ plurigaussian tehniques to model the reservoir harateristis of an oilfield enabling the pratitioners to aurately predit oil prodution foreasts. Deutsh (1996) suggests that PGS is a more straightforward approah than SIS when handling multiple ategory interations 5

16 1.3 Objetives The objetive of this researh is to explore the theory underpinning the omputationally expensive plurigaussian simulation (PGS) tehnique and ompare the output from PGS with the previously developed and more simplisti models of indiator kriging (IK) and sequential indiator simulation (SIS), with a view to assessing overall performane of the three models. Comparisons will be undertaken in three areas: 1. Visual inspetion and empirial analysis of output models. Partiular emphasis will be made of global and regional proportions of the five roktypes present in the data set and by onsideration of the regional spatial variability displayed in eah model with omparison to the data set. 2. Analysis of the four non-bounding roktypes within the study region. 3. Calulation of the degree to whih the three models honour the speifi roktypes that appear in the data set in terms of ontat admissibility and ontinuity. 1.4 Outline of Thesis In this thesis Chapter 2 will provide a theoretial bakground for the basi geostatistial measures assoiated with the random funtion (RF), the measures assoiated with ategorial data, the kriging paradigm, its variant indiator kriging (IK), the sequential indiator simulation (SIS) algorithm, the Gaussian funtion as a model for simulations, the turning band algorithm, trunated Gaussian and PGS model overview and lastly the link between the indiator semivariogrmas and the underlying Gaussian semivariograms. An extensive disussion of aspets of the workflow of the plurigaussian algorithm follows in hapter 3, inluding the Gibbs sampler onditioning algorithm and the turning bands simulation method. Chapter 4 explores the Big Bell gold mine data set used in the modelling proesses. Chapter 5 details the steps of the plurigaussian algorithm as run on the platform of the geostatistial programme Isatis, developed by Geovarianes. Chapter 6 provides pitorial and statistial analysis of the output generated from the three algorithms onsidered in this thesis, with partiular attention on the adequay of the models to reprodue the initial global proportions of the data set and the identifiation of important differenes in the output of the three algorithms. 6

17 1.5 Common Notation The notation used throughout this doument is a ombination of that used by Goovaerts (1997) and Armstrong et al. (2003). A study area C(0) : ovariane value at separation distane h 0 C ij (h): stationary ross ovariane between the two RFs Z i (u) and C Z (h) : stationary ovariane of the random funtion : expeted value E{(Z(u);: z (n)}: onditional expetation of the RV Z(u) given the realisations of n other neighboring RVs (alled data) F( u : z) non-stationary umulative distribution funtion of the RV Z(u) F(u : z (n)) : non-stationary onditional umulative distribution funtion of the ontinuous RV Z(u) given neighboring information, suh as realisations of n other RVs (alled data). F(u, u : z, z ) : non-stationary two point umulative distribution funtion of the RV Z(u) G( y) : standard normal umulative distribution funtion G 1 ( y) : standard normal quantile funtion suh that G(G 1 ( p)) p [0,1] (u, u ) : non-stationary semivariogram between the two RVs Z(u) and Z( u ) h : separation or lag vetor I(u;s k ) : binary indiator RF at loation u and for threshold s k i(u;s k ) : SK (u) : m : N(h) : n : n(u) : p(u : s k ): p(u : s k (n)): p k : Z j (u) binary indiator value at loation u and for threshold s k simple kriging weight assoiated to z -datum at loation u for estimation of the attribute z at loation u. The same type of notation applies to other algorithms, suh as OK. stationary mean of the RF Z(u) number of pairs of data values available at lag vetor h number of data values s(u ) or z(u ) available over the study area number of data values z(u ) used for estimation of the attribute z at loation u probability for the ategory s k to prevail at loation u onditional probability for the ategory s k to prevail at loation u given the neighboring information (n) global proportion of ategory s k within the area A 7

18 R(u) : (h) : S(u) : s(u ) : residual omponent model in the deomposition Z(u) R(u) m(u) where m(u) represents the trend omponent model stationary orrelogram of the RF Z(u) for lag vetor h generi ategorial RV at loation u or a generi ategorial RF at loation u s -datum value at loation u α 2 : variane of the RV Z SK (u) : simple kriging variane assoiated with the simple kriging estimate Z SK (u) at loation u. The same type of notation applies to other algorithms, suh as OK u : oordinate vetor u : datum loation Var{}: Z(u) : variane generi ontinuous RV at loation u, or a generi ontinuous RF at loation u Z SK (u) : simple kriging estimator of Z(u) z : ontinuous attribute z(u) : true value at unsampled loation u z(u ) : z * (u) : z -datum value at loation u α an estimate of value z(u) 8

19 Chapter 2: Theoretial Framework 2.1 Random Variable and Random Funtion Before onsideration of the mathematis assoiated with the plurigaussian tehnique, a number of mathematial and geostatistial onepts must be disussed, suh as random variables, the onept of a random funtion, the Gaussian funtion and the kriging paradigm. In this setion the onept of a ontinuous random funtion (RF) is disussed. Informally, a RF is a olletion of random variables defined at eah of the loations that omprise the study area of interest. A random variable (RV) is a variable, denoted by Z( u ), that an take a series of outome values aording to some probability distribution. For the purposes of simulation of rok types, the random variable is a disrete (ategorial) variable, however in the proesses of simulation to be onsidered in this thesis, these ategorial variables are onverted into ontinuous (Gaussian) variables for the modelling proess and then bak-transformed to ategorial. The following refers to ontinuous RVs, and when ategorial attributes will be onsidered at a later time, speifi notation will be introdued. A random variable is haraterised by its umulative distribution funtion (df) whih indiates the probability that the variable Z at loation u is no greater than a given threshold z : F( u ; z) p{z( u) z} (1) A random funtion (RF) is defined as a set,{z( u) u A} of (usually dependent) random variables Z(u ), one for eah loation u in the study region A. The basi paradigm of the probabilisti approah is to model a value z (u) at an unsampled loation u as a realisation drawn from a random variable Z( u ). The problem of assessing the 9

20 unertainty about an attribute value at u thus redues to that of modelling the probability distribution of the random variable Z at that loation. The unertainty at u an be redued by aounting for (onditioning on) neighbouring sample data values z( u α ), 1,...,n. In pratie the distintion between Z(u ) and z (u) redues to one of ontext, with z( u α ) indiating a (known) sample value and z * ( u ) an estimate of Z at loationu. The univariate umulative distribution funtion (df) (1) an be extended to the multivariate or n point (df ) where: F( u 1,..., un; z1,... zn ) p{z( u1) z1,...z( un ) zn} (2) The spatial law of the RF Z( u ) is defined by the joint unertainty shown in (2). Generally only the univariate and bivariate dfs are onsidered when defining the moments of interest. These are: the one point df F( u;z)= P{Z( u) z} the expeted value m( u) E{Z( u)} the two point df F( u, u;z,z ) P{Z( u) z,z( u) z } the variane Var{Z( u)} E[Z( u) - E{Z( u)}] the two point ovariane C{ u, u' )} E{Z( u) Z( u')}- E{Z( u)} E{Z( u' )} 2 (3) (4) (5) (6) (7) the two point orrelogram C( u, u') ρ( u, u) C( u, u) C( u', u' ) (8) the semivariogram 1 γ ( u, u) Var{Z( u) Z( u)} (9) 2 10

21 The term two point refers to the value of the same attribute at two different loations as opposed to the term bivariate, whih may refer to two different attributes measured at the same loation. An important statisti is the onditional df (df) F( u ; z Z( u) z) of the df of Z( u ) where the df at loation u is alulated with referene to (onditioned on) the statistis assoiated with the neighbouring data, speifially that F( u ;z Z( u) z) P{Z( u) z Z( u) z} Z( u ) z. Using (3) and (5); P{Z( u) z, Z( u) z} P{Z( u) z} F( u, u; z, z) F( u; z) (10) A random funtion is stritly stationary if its distributions are invariant under translation of the points by an arbitrary vetor h; (for all ui, i 1,..., nand zi, i 1,..., n ) P{ Z( u 1) z1,...z( u n ) zn} P{Z( u1 h) z1,...z( un h) zn} (11) In physial terms this means that the phenomenon is homogenous in spae. When a random funtion is stationary its moments are invariant under translation. In pratie the analysis is limited to dfs involving no more than two loations at a time and their orresponding moments. The two moments (4) and (8) depend on the separation vetor h: m E{Z( u)} (12) and C( h) E{Z( u) Z( u h)} E{Z( u)} E{Z( u h)} (13) A random funtion with these two properties is said to be seond order or weakly stationary. 11

22 Stationarity is not neessarily a property of the underlying harateristi of the phenomenon under study, rather it is a deision made by analysts to failitate inferene and modelling. It may well be that non-stationarity (where the moments of the distribution vary aross the study region) is a fator to be taken into aount in the modelling proess. 2.2 The geostatistis of ategorial data The models onsidered so far relate largely to a ontinuous random variable. The ategorial data to be onsidered in the simulation proess need to be defined, and then the link between these variables and the ontinuous variables Z(u) an be made. For the purposes of explanation at a generi ategorial level, sample data will be ategorised into k ategories S i, i 1,...,k,.. S 2 S 3 S 2 S 1 Figure 1 Categorial data ontats in a 2D region. Figure 1 showing three ategories in A with an important feature being whih ategories are neighbours for others. Category S 1 is in ontat solely with S 2 whereas S 2 is in ontat with both S 1 and S 3. This sequential nature of ontats between ategories may be represented pitorially by partitioning a retangle, as in Figure 2. 12

23 S 1 S 2 S 3 Figure 2 Representation of nature of ategory ontats from Figure 1. Suppose that there are k mutually exlusive ategories will define an indiator funtion by putting i(u;s k ) The assoiated indiator RV will be denoted by 1 if s(u) s k 0 otherwise s 1,..., sk. For eah ategory we (14) I(u;s k ) 1 if s(u) s k 0 otherwise (15) The proportion of the study region that a ategory oupies will be denoted by p( u;s k) E{I(;s u k)} with 0p( u; sk) 1 (16) 2 Sine I(u;s k ) I(u;sk ) the variane of the indiator funtion is given by Var[I(u;s k )] E[{I(u;s k ) - E[I(u;s k )]} 2 ] (17) Given any two loations u and u h the non-entred ovariane C Sk (u,u h) is defined as C Sk (u,u h) E[I(u;s k ),I(u h;s k )]=P[I(u;s k )=1,I(u h;s k )=1] (18) 13

24 and measures the probability that the ategory at loations u and u h is s k. The entred ovariane Sk (h) and entred ross-ovariane Sk S j (h) between S k and S j are defined as Sk (u,u h) E{[I(u;s k ) P(I(u;s k ))][I(u h;s k ) P(I(u h;s k ))]} and C Sk (h) P(I(u;s k )P(I(u h;s k ) (19) Sk S j (u,u h) E{[I(u;s k ) P(I(u;s k ))][I(u h;s j ) P(I(u h;s j ))]} (20) The indiator semivariogram Sk (u,u h) of S k (the indiator version of (9)) is defined as Sk (u,u h) 1 2 {E[I(u;s ) I(u h;s k k )]2 (E[I(u;s k ) I(u h;s k )]) 2} (21) and measures how often one of the loations separated by a vetor h belongs to the distint ategory s k The ross semivariogram Si (u,u h) between S S j i and S j is defined as Si (u,u h) 1 S j 2 {E[I(u;s ) I((u h);s )][I(u;s ) I((u h);s )]} (22) i i j j and measures how often two loations separated by a vetor h belong to the different ategories s i and s j. In reality the theoretial models shown above are unknown and need to be inferred from the experimental semivariogram and ross semivariogram using the data points {u, 1,...,N} The experimental semivariogram for the ategory S i is: * Si (h) 1 2N [i( u ;s i ) i(u ;s i )] 2 (23) u u h 14

25 where the data loations u and u are separated by the vetor h, and N denotes the number of pairs of data that are separated by eah vetor h. Similarly, the formula for the experimental ross semivariogram is given by * Si (h) 1 S j 2N [i( u ;s i ) i(u ;s i )][i(u ;s j ) i(u ;s j )] (24) u u h 2.3 Variogram models For estimation and simulation semivariogram values need to be omputed at separation distanes other than those available experimentally. To failitate this, semivariogram models are inferred from the experiment semivariogram. In this thesis, models for indiator semivariograms as well as semivariograms of ontinuous variables are required. The following models defined in their isotropi form will be used. The nugget effet model is defined as g 0 (h) 0 if h=0 1 otherwise (25) And the spherial model (with a sill value of 1 reahed at range a) is defined as Sph a (h) 0 h=0 3 h 2 a 1 2 h a 3 h a 1 h a (26) 2.4 Kriging Kriging is a generi term for a family of generalised least squares regression algorithms named after Danie Krige s pioneering work in the field of geostatistis. The basi linear regression estimator Z * ( u ) is defined as * Z ( u) m( u) n( u) α1 λ ( u)[z( u ) m( u )] α α α (27) 15

26 where λα ( u) is the weight assigned to datum z( u α ) interpreted as a realisation of the RV Z( u α ). The quantities m(u ) and m( u α ) are the expeted values of the RVs Z(u ) and Z * ( u ). Z(u) is treated as a random funtion with a trend omponent, m(u), and a residual omponent R(u) Z(u) m(u). Kriging estimates the residual at u as a weighed sum of residuals at neighborhood data points. The goal is to determine the kriging weights,, whih are derived from ovariane funtion or semivariogram funtion, and whih minimize the variane of the estimator E 2 (u) = Var{Z*(u) - Z(u)} under the unbiasedness onstraint E{Z * (u) Z(u)} 0. The random funtion (RF) Z(u) is deomposed into residual and trend omponents, Z(u) R(u) m(u), with the residual omponent treated as an RF with stationary mean of 0 and a stationary ovariane: E{R(u)} 0 Cov{R(u),R(u h)} E{R(u),R(u h)} C R (h) The residual ovariane funtion is generally derived from the input semivariogram model, C R (h) C R (0) (h) Sill (h) (28) In Simple Kriging (SK) the expeted value m is known and onstant, and the simple kriging estimate at a loation u is given by Z SK * (u) m SK (u)[z(u ) m] This is an unbiased estimator sine E[Z(u ) m] 0. The estimation error n(u) 1 Z * SK (u) Z(u) is a linear ombination of random variables representing residuals at data points, u, and the estimation point, u : Z * SK (u) Z(u) [Z * SK (u) m][z(u) m] n(u) = SK 1 16 (u)r(u ) R(u) R * SK (u) R(u) (29) (30)

27 Using the rules for the variane of a linear ombination of random variables, the error variane is then given by 2 E (u) Var{R * SK (u)} Var{R SK (u)} 2Cov{R * SK (u),r SK (u)} n(u) 1 n(u) SK (u) SK (u)c R (u u ) C R (0) 2 SK (u)c R (u u) 1 n(u) 1 (31) To minimise the error variane, the derivative of the above expression is taken with respet to eah of the kriging weights and then equated to zero. This leads to the system of equations: n(u) SK (u)c R (u u ) C R (u u) 1 1,...,n(u) (32) For SK the mean is onstant so the ovariane funtion for Z(u) is the same as that for the residual omponent, C(h) = C R (h) so the kriging system an be written in terms of C(h) : n(u) SK (u)c(u u ) C(u u) 1 1,...,n(u) (33) 17

28 This an be written in matrix form as K SK (u) k (34) where K SK is the matrix of ovarianes between data points, with elements K i, j C(u i u j ), k is the vetor of ovarianes between the data points and the estimation point, with elements given by k i C(u i u), and SK (u) is the vetor of simple kriging weights for the surrounding data points. If the ovariane model is permitted and no two data points are o-loated, then the data ovariane matrix is positive definite and kriging weights an be determined from the equation: SK K 1 k (35) One the kriging weights are established, both the kriging estimate and the kriging variane an be alulated. In Ordinary Kriging (OK) the expeted value m(u) is onsidered to flutuate over the study region and is alulated by limiting the domain of stationarity of the mean to the loal neighbourhood entred on the loation u being estimated. In this ase the linear estimator (29) is determined as a linear ombination of the n(u) RVs Z(u ) plus the onstant but unknown loal mean m(u)=m. n(u) n(u) Z*(u) (u)z(u ) 1 (u) m(u) 1 1 The unknown loal mean m(u) is filtered from the linear estimator by foring the * kriging weights to sum to 1. The ordinary kriging estimator Z OK (u) is thus written as a linear ombination of the n(u) RVs Z(u ) : (36) * Z OK (u) n(u) OK OK (u)z(u ) with (u) 1 1 n(u) 1 In order to minimize the error variane subjet to this unit-sum onstraint on the kriging weights, a system is set up whih involves the error variane plus an additional term involving a Lagrange parameter OK (u) : (37) n(u) L 2 E (u) 2 OK (u)[1 OK (u)] 1 (38) 18

29 so that minimization with respet to the Lagrange parameter fores the onstraint to be obeyed: n(u) 1 L 2 1 OK (u) 0 In this ase, the system of equations for the kriging weights is n(u) 1 1 OK (u)c R (u u ) OK (u) C R (u u) n(u) OK (u) 1 1 1,...,n(u) (39) (40) where C R (h)is the ovariane funtion for the residual omponent of the variable. When dealing with Simple Kriging, C R (h) C(h) due to the stationarity of the mean, and although that equality does not stritly hold here, in pratie this substitution is often made on the assumption that the semivariogram, from whih C(h) is derived, effetively filters the influene of large-sale trends in the mean. This OK method is used in an indiator ontext with the ategorial variables in the following setion. 2.4 Indiator kriging of ategorial variables. Indiator kriging (IK) uses the basi kriging paradigm but in the ontext of an indiator variable. In this thesis, indiators are interpreted as hard ategories. The first step in indiator kriging is to model the unertainty about a ategory s k, k 1,...N of the ategorial variable s at the unsampled loation u. This unertainty is modelled by the onditional probability distribution funtion (pdf) of the disrete RV S(u ) : p( u;s k ( n)) p{s( u) =sk ( n)} where the onditioning information onsists of (41) n neighbourhood ategorial data s( u α ). Eah onditional probability p( u ;s ( n)) is also the onditional expetation of the k lass-indiator RV I( u ;sk) : p( u;s k ( n)) E{I( u;s k ) ( n)} (42) 19

30 The kriging estimator for the indiator RV is defined as * [I(;s u )] E{I( u;s k k )} n( u) α1 λ ( u;s )[I( u ;s ) E{I( u ;s )}] α k where λα( u ;sk ) is the weight assigned to the indiator datum i( u ;s α k) interpreted as a realization of the indiator RV I( u α;sk). As for the previous setion, there are two indiator kriging (IK) variants; simple indiator kriging (sik) where the indiator mean E{I( u ;s )} is onsidered onstant over the study region, and the version used in this thesis, ordinary indiator kriging (oik), where loal flutuations of the indiator mean E{I( u ;s )} are taken into aount. α k α α k k α k (43) Effetively indiator kriging produes a onditional probability distribution for eah ategory at eah grid loation in the study region. From this probability distribution a ategory needs to be derived and so there must be a post-proessing lassifiation algorithm applied to the output of IK. There are a number of different post-proessing lassifiation algorithms, suh as maximum likelihood. These algorithms tend to overrepresent the ategory (lithotype) with the greatest global proportion and underrepresent or ignore the ategory of least global proportion (Goovaerts, 1997 p.441) The lassifiation method of the indiator kriging output that will be used for the purposes of omparison in this thesis is the algorithm developed by Soares (1992). In this algorithm eah grid loation is alloated a ategory under the onstraint that the global proportions are reprodued. Initially the most frequently ourring ategory is alloated to those loations where it has greatest onditional probability until the global proportion of that ategory is met. This ontinues through the hierarhy of proportions of ategories until all grid loations have been alloated a ategory. This algorithm allows for ategories with small global proportions to be represented adequately and does not allow ategories with large global proportions to dominate, whih would our in a lassifiation algorithm suh as maximum likelihood. 20

31 2.5 Sequential indiator simulation. The SIS method is ommonly used for the stohasti modelling of ategorial (e.g. rok types) and also ontinuous attributes (e.g. porosity and permeability). SIS simulates the spatial distribution of the mutually exlusive ategories (lithotypes) sk onditional to the data set {s( u ), 1..., n}. A simplified work flow of the proedure, (Goovaerts, 1997) is as follows: 1. Define a random path visiting eah node of the study area grid only one. 2. At eah node u : (a) Use ordinary indiator kriging to determine the onditional probability of ourrene of eah ategorys k,[ p( u ;sk ( n)) ]. The ( n ) onditioning information onsists of the neighbourhood original indiator data and previously simulated indiator values. (b) Build a df-type funtion by adding the orresponding probabilities of ourrene of the ategories () Draw a random number p uniformly distributed in [0,1]. The simulated ategory at loation u is the one orresponding to the probability interval that inludes p. (d) Add the simulated value to the onditioning data set. (e) Proeed to the next node along the random path and repeat steps (a) to (d). Repeat the entire sequential proedure with a different random path to generate another realisation. 2.6 The Gaussian funtion as a model for simulation The SIS method disussed above is a simulation method where no assumption about the prior distribution at a node is made. It is often the ase that a model is assumed and in many ases this is a model where the spatial law (multivariate distribution) of the RV Z(u) is assumed to be Gaussian in nature. 21

32 A random variable Z has a Gaussian distribution if its probability density funtion (pdf) is: 1 zμ 2 σ 2 1 f( z ) e (44) σ 2π where μ is the mean and 2 σ the variane of the pdf. This distribution is fully haraterized by μ and 2 σ and so the problem of determining the df at any point in A redues to that of estimating these two parameters. If { Z( u ), u A} is a standard multivariate Gaussian RF with ovariane ( h ) then the following properties are true: C Z 1. All subsets of the RF Z are also multivariate Gaussian. 2. The two point distribution of any pairs of RVs Z(u ) and Z( u + h) is normal and fully determined by the ovariane funtion ( h ) 3. All onditional distributions of any subset of the RF Z( u ) are normal. In partiular the onditional distribution of a single variable given the C Z n(u) data z u ) is normal and fully haraterised by its two parameters, mean and ( variane: z E{Z( u) ( n) G( u ; z ( n)) G (45) σ{z( u) ( n) where G is the standard normal df. Under the multigaussian model, the mean and variane of the df at loation u are equal to the simple kriging estimate and simple kriging variane. The df (10) is then defined as z zsk( u) G( u : z ( n)) G SK (46) σ SK( u) where z (u SK ) and σ 2 SK ( u) are the simple kriging mean and variane. 22

33 2.7 Turning Band method of non-onditional simulation. The turning band method is a non-onditional simulation algorithm that will allow the Gaussian variables to be simulated in a three dimensional spae. It onsists of taking the Gaussian ovariane funtion C 3 (u) determined from the variogram model of the underlying Gaussian random funtion in 3 dimensional spae and determining the orresponding ovariane funtion C 1 (u) in one dimension (Armstrong et al., 2003). Next, a set of n lines with arbitrary diretions in 3D spae is determined. Along eah line a Gaussian random funtion with the determined 1D ovariane funtion C 1 (u) is simulated. The projetions x i at eah of the n lines from any point region are determined and the n simulation values grid value as s(x) = 1 s(x i ) n i=1 n s(x i ) at x in the 3D study x are used to ompute the Figure 3 shows 8 lines along whih the 1D Gaussian ovariane funtion has been simulated. The grid value for the point (7,5) shown on the plane will be determined as the sum of the simulated values obtained from the projetions of this point onto the simulated values from the eight 1D lines. One this unonditional simulation has been ompleted, the model is onditioned by the loations where data values our. The error between the data value and the unonditional simulation value at those data loations is kriged using the same model used to generate the unonditional simulation. (47) 23

34 Figure 3 Projetions from a point 1D ovariane funtions 2.8 The trunated Gaussian and Plurigaussian models The trunated Gaussian method has been widely used when data are required to be separated into a number of thresholds (indiator oding) and the Gaussian funtion is assumed to be a suitable modelling tool (see setion 2.6). In this thesis the trunated Gaussian method is used with referene to the alloation of ategorial (roktype) proportions. The method is based on the notion of trunation of a standard Gaussian random funtion by different thresholds alulated from experimental set proportions and was first applied to model the lithotype in deltai reservoirs by Matheron et al. (1987). The ovariane model of the underlying Gaussian funtion is obtained by an iterative proess of inversion of the ovariane model of the experimental indiator variables of the sets. If the ategorial data are sequential in their ontats then trunation of one Gaussian RF is required, however if two Gaussian funtions are required to be trunated then this is the plurigaussian ase, disussed in the setions below. 24

35 In trunated Gaussian simulation, a Gaussian random funtion z is simulated first and then it is transformed into the set of indiator variables by trunation. In the ase of two sets, S 1 and S 2, being simulated, a numerial threshold value t 1 is produed so that: P(z t 1 ) P(S(u) s 1 ) and P(z > t 1 ) P(S(u) s 2 ) When N sets are being simulated, (N-1) thresholds are neessary. The set assoiated with a subinterval [t i1,t i ]of the domain of the Gaussian pdf suh that S i is P(t i1 z t i ) P(S(u) s i ) (48) It is important to have the thresholds reflet the ordering of the sets ie. t1 i 1i i 1t k ttt... From (16): P Si (u) = E{I( u s; )} and I( u s;) 1 i i ti 1 Z( u ) ti These two statements an be inorporated so that: u )(P P(t i 1 u )Z( i )t (49) S i P( u )t)z( P( 1 )t)z( i u i ie u )(P S i )G(t 1 ) (50) i i where G(t)is the df of the standard normal distribution. As the proportion of eah set is known experimentally, the thresholds an be dedued from the inversion of (1.34): 1 t G[P()] 1 u 1 S t S 1 S 2 1 G [P u )( (P)] 2 u 1 t G)(P)([P u... ])(P (51) i SS u 1 2 S i u 25

36 If the ategories under onsideration are not sequential in their ontats but have a more omplex relationship, the plurigaussian model may be employed. If there were a fourth ategory, S4, added to Figure 2 whih ontated both S2 and S3 then the representation shown in Figure 2 would be modified as shown in Figure 4. This is further developed in setion 3.1. S 3 S 1 S 2 S 4 Figure 4 Representation of four ategories where ontats are non-sequential. The above model of ategory ontats would require the trunation of two Gaussian funtions for modelling purposes, whih is usual for the plurigaussian model, however the following onsiders the more general ase where N Gaussian funtions may be used. Where N Gaussian funtions may be used in the simulation a spae in N dimensions is being onsidered for simulating sets. At a loation u in this N-spae the values of the N Gaussian funtions Z 1,Z 2,...,Z N define the o-ordinates of the simulated value. Let be the subset of the Gaussian spae orresponding to the ategory S i. Then Q i I Si (u)1(z 1 (u),z 2 (u),.,z N (u)q i (52) From (1.20): P Si (u) E{I(u;s i )} and (1.36): u )(P 12 (P{(Z (Z), uu ),..., N (Z}Q)) u i S i g (z 1,z 2,...,z N ) dz 1,dz 2,...dz N (53) Q i where g (z 1,z 2,...,z N ) is the N-variate Gaussian density funtion with mean 0 and variane 1, and is its orrelation matrix. 26

37 2.9 The link between the indiator semivariograms and the underlying Gaussian semivariograms The tehnique used in this thesis relies on a simulation of Gaussian values from the underlying Gaussian semivariogram model(s). A ritial feature of this is the inferene of these Gaussian semivariograms from the experimental indiator semivariograms and ross-semivariograms desribed in the previous setion. The links between all these semivariograms are funtions of the trunation proess and of the onditioning to the set proportions. The semivariogram inferene is based on a method desribed in Armstrong et al. (2003) in whih the ranges of the underlying Gaussian semivariograms are adjusted iteratively. Firstly initial parameters of the underlying Gaussian models are defined and then these models are used to onstrut an unonditional simulation of the study area. From this simulation, indiator semivariograms of eah of the sets are omputed. A omparison of these omputed semivariograms and the previously determined experimental semivariograms is undertaken and the parameters are adjusted iteratively until an aeptable math is obtained. In the trunated Gaussian ase, if t i 1 and t i are the thresholds for the elements of set S i then the semivariogram model is alulated using: 1,(γ hu ) {P[t i 1uhuu ii ]t)z( 1 t)z( ii ]} P[t 1 t,t)z( ii 1 hu )Z( t] 2 Si ti ti ps i ρ() h ti 1 ti 1 g(,)d aba db 1 ps i 2π 1() ρ h 22 t it ab2() h ab i 2 2(1 ρ ()) h 2 ti 11 ti e dd ba (54) where p S is the proportion of the study region where set S i i exists and g ρ ( h ) is the bigaussian density funtion for the same set S i. The orrelation between the two Gaussian variables a and b is ρ (h ). A orrelation value is determined by a trial and error proess in order to improve the fitting of sample indiator variograms. 27

38 The plurigaussian ase is a diret extension of this. Considering two Gaussians Z 1 (u) and Z 2 (u) initially, the semivariogram model is given by:,(γ Si Si 1Σ 212 ii D )p hu,(g,,)dd baba (55) The two Gaussians give rise to four Gaussian variables, Z 1 (u), Z 2 ( u, ) (+ u h) and Z 1 Z 2 (+) u h. Here four variables. represents the matrix of orrelations and ovarianes between the A ovariane for a Gaussian is a fundamental parameter and an be found from the experimental data. The orrelation between the two Gaussians ( ) is a more arbitrary value. The larger the orrelation value the greater the bordering effet apparent in the outputs of the simulations, and so hoosing a zero orrelation minimises artifiially affeting the PGS output, and neither IK nor SIS onsider this border effet issue. The link between the four variables depends on the hoie of oregionalisation model between Z 1 (u) and Z 2 (u). A simple model is one where the two Gaussians RFs Z 1 and Z 2 are linear ombinations of independent Gaussian RFs, say B 1 (u) and B 2 (u), eah with a N(0,1) distribution having ovarianes B1 (h) and B2 (h) respetively. So: Z 1 (u) B 1 (u) and Z 2 (u) B 1 (u) ( 1 2 )B 2 (u) where B1 (h) Z1 (h) and B2 (h) 2 Z1 (h) (1 2 ) Z2 (h) The oregionalisation orrelations and ovarianes are summarised in Table 1: Table 1: Correlations and ovarianes for RFs Z 1, Z 2 Z 1 ( u ) Z 2 ( u ) ( u h) ( u h) Z 1 Z 2 Z 1 ( u ) 1 ) () ( h Z1 Z 2 ( u ) 1 () Z 1 Z 2 ( u h) ) () ( h Z1 ( u h) () 1Z h ) 1Z h 1Z h 1Z h ) 1 1 ( h Z2 ( h Z2 28

39 1 ρ ρ() Zh ρρ() 1 Zh 1 ρ 1 ρρ Z() h ρ() 1 Zh 2 This leads to the matrix () () 1 used in (55), ρz h ρρ 1 Zh ρ 1 ρρ Z() h ρ() 1 1 Zh ρ 2 whih is a starting point for the iterative proess of omputation of the required indiator semivariograms. In Isatis the iterative proess begins with seletion of (in the plurigaussian ase) two possible underlying Gaussian models. These models are then tested (and altered where neessary) so as to optimise the fits of the indiator variograms to the experimental data. 29

40 Chapter 3: Aspets of Plurigaussian Simulation The simulation of ategorial (set) data outlined in the preeding setions is now applied to the simulation of rok types (lithotypes) in a geologial setting. The initial appliation of this method was to simulate rok types in a sedimentary deposit (Armstrong, 2003) The fundamental onept of non-stationary proportion urves is entral to the model, where the rok type rule plays an essential role in produing geologially realisti models that represents the transitions between the different lithotypes. The advantages of this model are theoretial onsisteny and the ability to inorporate external data suh as vertial proportion urves. Trunated Gaussian simulation is an important theoretial starting point: plurigaussian simulation is an extension of the trunated Gaussian model whih allows the aquisition of two or more Gaussian variables, orrelated or not, that desribe different spatial behaviours of a lithotype or group of lithotypes as well as inorporating non-sequential ontats in the model. Depending on the number of lithotypes, there is a finite number of retangular partitions that may represent the relationships between the Gaussian random funtions and the different lithotypes. The most sensible geologially speaking is often evident from the vertial proportion urves in the layered nature of the lithotypes: one partiular lithotype may only be seen to be in ontat with another speifi lithotype. As well as this information, global proportions of eah lithotype give an indiator of proportion when seleting the rok type rule. 3.1 The rok type rule. The rok type rule splits a retangle into sub-retangles and assigns a lithotype to eah sub-retangle, without referene to proportions. The sub-retangles reated determine allowable ontats between lithotypes. To illustrate the rok type rule we will onsider the ase where three lithotypes are to be simulated. The ordering of the lithotypes in the rok type rule must respet the 30

41 sedimentary or ontat sequene in order to orretly reprodue the geology of the study area in terms of rok type. For trunated Gaussian simulation the single Gaussian funtion is partitioned into three regions onsistent with the proportions alulated in the vertial proportion urve and the sequential nature of the lithotypes. For the ase of trunated plurigaussian simulation non-sequential ontats of lithotypes are permitted and so a number of different possibilities exist, for example where both F 2 and F 3 are in ontat with F 1. These regions are diagrammatially represented by retangles partitioned into proportions as shown in Figure 5. The ase where trunated Gaussian simulation is appropriate is shown in (a), a senario where PGS is appropriate is shown in (b). F 1 F 2 F 3 F 1 F 2 F 3 (a) (b) Figure 5 Possible ontats for three lithotypes In both ases the horizontal line on the bottom of the retangle represents the axis for the first Gaussian variable z 1 (G1) and for (b) in Figure 5 the vertial line on the left represents the seond Gaussian variable z 2 (G2). The three olours represent the different lithotypes, and the equal areas of the retangles indiate that the three lithotypes have equal proportions, however proportions need not be equal. The plurigaussian ase shown in Figure 5 is further detailed in Figure 6. t 1 t 2 Figure 6 Plurigaussian ase for three lithotypes. 31

42 In this ase two thresholds, t 1 for z 1 and t 2 for z 2, are used. In Figure 6 the three lithotypes are defined within the roktype rule by the following: Dereasing the value of the threshold t 1 in the first Gaussian would redue the proportion of lithotype F 1 and inrease the proportion of lithotype F 2 and F 3. Diagrams representing possible ontats for four lithotypes are shown below. F 1 F 2 F 3 F 4 F 2 F 1 F 3 F 4 (a) (b) F 2 F 1 F 3 F 4 () F 3 F 2 F 1 F 2 F 1 F 4 F 4 F 3 (d) (e) Figure 7. Possible ontats for four lithotypes Trunated Gaussian simulation would be applied to a situation depited in (a) but the other lithotype ontats would require a plurigaussian treatment. Retangular divisions of the roktype rule are preferred as they allow thresholds to be defined in the two Gaussian funtions. For diagram () in Figure 7 the yellow region is not defined by 32

43 edges horizontally or vertially and while possible in terms of ontats is not physially able to be modelled. If lithotype proportions vary aross the study region A then rok type rules an be alulated in a grid. This would assist in realisti simulation of a horizontally non-stationary situation. 3.2 Vertial proportion urves and referene levels. A basi quantity that an be dedued from a data set is the proportion of this data set that is alloated to eah of the ategories of interest. It is unommon for the proportions of these ategories to be either vertially or horizontally onstant in A. For example, petroleum reservoirs are not stationary in the vertial diretion beause of yli hanges during their deposition and non-stationarity an also our in mining deposits (Armstrong, et al., 2003). Vertial proportion urves (VPCs) were designed to quantify these hanges and it is important to analyse this non-stationarity to enable an output model to reflet the differenes in these proportions in a regionally sensitive manner. The first step is to simply ount the number of ourrenes of eah ategory along lines parallel to the hosen referene level. The proportions are displayed as a graph showing the proportion of eah ategory at eah level. As it is likely that the VPCs vary horizontally depending on their loation in the study region, a proportion map of VPCs is alulated over the entire study region. Initially, drill holes (wells) are grouped loally and a VPC is alulated for eah set of wells. These VPCs are plaed on a regular, oarse grid whih overs the study region. At grid points where a VPC is not evident, an interpolation method (usually a kriging proedure) is used to generate a vertial proportion urve (with ategory proportions transformed where neessary so that they sum to 1). This proportion grid is used to identify loal variations in ategory distributions. Determination of the vertial proportion urve from the sample data is highly dependent on the hoie of a referene level. This level an be defined as a guide to the system of deposition of the different ategories. The sample data will be transformed into a flattened spae where the referene level represents the horizontal surfae at zero elevation. The simulation will be done in this artifiial volume and then the simulated values transferred to real stratigraphi spae. 33

44 3.3 The Gibbs sampler This iterative algorithm (desribed in detail in Armstrong et al (2003) allows for loal inequality onstraints to be imposed on a simulation of a Gaussian random funtion. It is based on simulation of the underlying Gaussian funtion and not on the simulation of the indiator variables. Eah sample data point in the study region must therefore have a Gaussian value within a predetermined area of the Gaussian funtion(s) that has been alulated from the lithotype proportions. Using the multivariate Gaussian approah, eah lithotype data point is assigned an arbitrary starting value that is within its Gaussian thresholds. The Gibbs sampler is a proess whereby these arbitrary values are then iteratively onditioned by all other regional (arbitrary) data values assigned to lithotypes until a steady state value is determined and the sores remain within their Gaussian threshold and also reflet the ovariane model of the underlying Gaussian funtions. Given vetors u 1,...,u n, and orresponding initial values z 1,...,z n : For 1 1. Calulate the SK estimate z * SK (u ) and SK standard deviation * SK (u ) from {u 1,...,u n }\{u }. 2. Put z(u ) z * SK (u ) * SK (u )r where r ~ N(0,1) 3. If t 1 z t aept this value, otherwise draw a new r value. Next. This onditioning step is repeated until a steady state is reahed for eah of the Gaussian values. More than 5000 iterations may be needed for the updated data to reah a steady state (Walsh, 2004). 34

45 The key issue in the suessful implementation of the Gibbs sampler is the number of iterations until the values eah approah their own steady state, whih is the length of the burn in period. Typially the first 1000 iterations are ignored and after this a test for onvergene is used to see if a steady state has been reahed. A poor hoie of starting values and/or proposed distribution (as defined by the model) an greatly inrease the burn in time. Starting values lose to the entre (mode) of the proportion in question redue the burn in time. Iterations are mixed poorly if the values stay in a small region of the proportion, and well-mixed iterations explore the proportion randomly before approahing the steady state (Walsh, 2004). 3.4 Bak-transforming the onditioned values When the sample ategorial data have been given appropriate values within the Gaussian regions, the final stage of simulation is undertaken. The values are then baktransformed into ategorial data by onsideration of their loation within the thresholds of the Gaussian funtions. This is a simple proess in the ase of trunated Gaussian simulation but for trunated plurigaussian simulation the values generated may fall into different regions of the Gaussians used for trunation and so the rok type rule diagram will allow bak-transformation to ategorial data. As a final step the data must be reloated from the referene level to their true loation so that a true three-dimensional model simulation may be generated. 35

46 Chapter 4: Exploratory Data Analysis 4.1 Bakground of Big Bell Gold Mine The Big Bell gold deposit lies in the Murhison mineral field, 30km WNW of the township of Cue, whih is 650 km NNE of Perth, Western Australia. Gold was disovered at Big Bell in The Big Bell deposit is hosted by a regional volani (greenstone) and sedimentary sequene within the Murhison Provine of the Arhean Yilgarn ratoni blok. At Big Bell the greenstone belt is narrow, steeply dipping, strongly attenuated and loally overturned. It forms the west limb of a north plunging struture. An open fold style is harateristi of the Big Bell regional struture. At least two generations of graniti intrusive ativity are reognized in the region. In the region of the mine, there are three major subdivisions within the volanosedimentary sequene: a lower mafi (roks and minerals ontaining abundant magnesium and iron) sequene, a felsi (roks and minerals rih in silia and other feldspar omponents suh as sodium and potassium) volani sequene, and an upper mafi sequene. The Big Bell mine greenstone belt is bounded by granites. The broad subdivisions desribed above are reognized at the Big Bell mine, where the volano-sedimentary sequene is 1500m wide, and are struturally overturned, so that the younger upper mafi sequene now lies on the footwall of the orebody. Almost all the gold mineralisation at Big Bell is onfined to the quartz-musovite- potassium feldspar rih roks of the ore zone. The main orebody, mined underground and drill-indiated to a vertial depth of 1500m (Handley and Cary, 1990) forms part of a more than 1800m long and 30-70m wide zone of intensely altered amphibolite. This study will model the spatial distribution of the gold bearing altered roks in the mined out open pit south of the main orebody, based on drill hole intersetions below the base of weathering. The rok types indiated in the data are the boundary granodiorite gneiss (Lithotype 1), magnesium olivine-alite 36

47 skarn (Lithotype 2), skarn-banded biotite (Lithotype 3), ordierite-bearing gneiss (Lithotype 4) and pyriti miroline gneiss (Lithotype 5) representing the southern tail of the main orebody. Initial geologial reports were by Woodward (1914). The staff at Big Bell Mines Limited (1953) suggested that Big Bell was a replaement deposit. More reently Chown et al. (1984) highlighted the diffiulty in determining the genesis, while Philips (1985) suggested a metamorphi replaement model. Further lassifiation and desription was undertaken by Mueller et al (1991).More reently, Mueller et al (2002) modelled ategorial data obtained from 42 drill holes using an ordinary indiator kriging algorithm and then a series of three different single step lassifiation algorithms to model the rok types. Historially, most mining at Big Bell ourred between 1937 and During that time, 5.6 megatonnes of underground ore were milled at a rate of about 30,000 tonnes/month for a reovered grade of 4 grams/tonne gold. Redrilling of the area ommened in 1980, and was inreased over a two year period when ore samples from 1974 drill holes were reassayed and found to ontain ore grades. In 1982 Australian Consolidated Minerals Ltd established 100% ownership of the area and in 1983 began a shallow drilling programme for assessment of possible strike extensions of the deposit. A reserve amenable to open pit mining was outlined, and an assessment of the gold reoverable from the old tailings dump was arried out. The ACM Plaer joint venture began mine optimisation studies in A drilling programme more fully defined the deposit and tested for further reserves within the tenement. As a result of a feasibility study, the deision to reopen the mine was made in Deember 1987, and prodution ommened in February Up to 1999, when the mine losed, the mine produed 27.8 million tonnes of ore at a reovered grade of 2.38 grams/tonne from both open pit and underground mining (Mueller et al, 2002). 37

48 4.2 Data analysis The input data for this study onsist of 1501 data from a total of 42 drill holes that have been lassified aording to the above five lithotypes. The data were rotated in spae so that the dip plane oinides with the horizontal whih will failitate modelling. The purpose of this thesis is to simulate the five lithotypes over the study region. The motivation for onsideration of lithotypes is a first step in analyzing mineral deposits in a geologial region as these minerals vary greatly in grade from lithotype to lithotype. One lithotypes are simulated mineral grades over the study region ould be alulated, however this is not the basis of this thesis. Initial data analysis here simply onsists of determining the positions and lithotypes assoiated with eah of the ore samples within the study region and proportions of eah of the five lithotypes over the study region. The lithotype olour sale used in this study is shown in Figure 8. L1 (granodiorite gneiss) L2 (magnesium olivine-alite skarn) L3 ( skarn-banded biotite) L4 ( ordierite-bearing gneiss) L5 ( pyriti miroline gneiss) Figure 8. Lithotype olour sale used in this study. Within the drill ores, the five defined lithotypes our in the following proportions. Table 2 Lithotype perentages in raw data. Lithotype Sample ount Perentage % % % % % 38

49 Figure 9. Proportions of lithotypes in the study region. When omparisons are made between the perentages of lithotypes present in the raw data and the subsequent plurigaussian simulation and ordinary kriging algorithms, lithotype 1 perentages will not be onsidered, as this lithotype envelops the study region, and will by neessity inrease to fill the perimeter of the simulation grid. Trunation of drill holes Figure 10 Drill hole trunation. The data have been reated using lithotype 1 as the bounding lithotype for the drill holes, that is eah drill hole was trunated at both ends so that only one or two ore measurements involving lithotype 1 were kept. 39

50 Figure 11. Proportions of lithotypes in the study region in the absene of L1. Figure 12 shows the lines of the ore samples in both perspetive and the three projetions. The drill holes are not vertial, although they are parallel to the z axis in the y-z projetion. The data may be rotated to make the drill holes vertial, although this is not neessary for suessful simulation in this algorithm. 40

51 Figure 12 Projetions and 3D image of data set. (Lithotype olours from Figure 8) 41

52 4.3 Variogram models for IK and SIS The variogram models for the five ategories (lithotypes) used in this thesis for the IK estimations are summarised in Table 3. Eah ategory was modelled with a nested struture onsisting of a nugget effet model and two spherial models (Mueller, 2002), the parameters of whih are listed in Table 3. Table 3 Variogram parameters used in IK for ategories (Mueller et. al, 2002) Major Category Nugget Continuity Sill1 Ranges 1 Sill 2 Ranges 2 Diretion (75,75,7) (150,75,3) (75,75,32) (50,50,20) (40,40,8) (600,150,45) (500,100,15) (500,400,35) (600,150,30) (75,75,1000) 42

53 L1 down hole m L1 major axis m L1 minor axis m L2 down hole m L2 major axis m L2 minor axis m L3 down hole m L3 major axis m L3 minor axis m L4 down hole m L5 down hole m L4 major axis L5 major axis m m L4 minor axis L5 minor axis m m Figure 13 Variogram models for IK of the five lithotypes, L1 to L5, indiating down hole (left), major axis (middle) and minor axis (right). (Mueller, 2002). 43

54 The normalised variograms of the lithotypes L1 to L5, shown in Figure 13, all display a very good fit, espeially for the first six lags, with the effetive sill being modeled for those data that display this feature. The variogram model used in for SIS in this thesis is that of the ategory with the largest proportion in the data set: L4 (45.57% of data set). This model is a nested struture model onsisting of a nugget and two spherial models defined in (54) and (55), with parameters listed in Table 3. 44

55 Chapter 5: PluriGaussian simulation 5.1 Overview of the algorithm. Simulation of the drill hole data from the Big Bell gold mine will be onduted with referene to the gold and silver grade in eah of the five lithotypes and also the distribution of lithotypes within the study region. Analysis and simulation of data are onduted using the geostatistial pakage Isatis, produed by the Frenh ompany Geovarianes. The plurigaussian algorithm requires a number of sequential steps to be performed. These steps are indiated in the flowhart shown in Figure 14. Detailed elaboration of eah of the steps outlined will be done in the following hapter. Initially, the data set from the drill holes is loaded as a lines file, with a header file indiating the loation and name of eah drill hole. The drill holes are defined within a three dimensional strutural grid of ells, and horizontal surfaes are defined as vertial boundaries of the strutural grid. Eah drill hole is disretised, or divided into equal length setions, within the strutural grid, and the ells are referened to the lower boundary surfae. A number of the disretised lines of data may pass through a single strutural grid ell, and so a single lithotype is assigned to eah disretised ell within the grid. (Isatis automatially onverts the defined strutural grid into a working grid for the purposes of alulation). The horizontal plane of the strutural grid is divided into polygons and lithotype proportions in eah of these polygons are alulated. Vertial proportion urves (VPCs) are also alulated within these polygons and these VPCs are extrapolated (ompleted to the bounding horizontal surfaes) and smoothed. These VPCs, together with the defined roktype rule, allow the seletion of underlying Gaussian random funtions that will produe a model whih will be used to simulate Gaussian values for lithotypes at unsampled loations. As a final step, the simulated Gaussian values are bak-transformed to the lithotypes to be displayed in the simulation within the strutural grid. 45

56 Figure 14 A flowhart of the plurigaussian algorithm. 46

57 The first step in the plurigaussian algorithm is the reation of a 3D strutural grid in the study region within whih the simulation will be displayed. Seondly, the line style drill hole data are disretised along their lines and referened to a plane. Next, an analysis of proportions of eah lithotype in eah ell in the 3D grid is undertaken using VPCs, alongside the reation of the roktype rule. 5.2 Defining the horizontal boundary surfaes of the strutural grid. The ranges for the o-ordinates of the drill hole data are displayed below. As an be seen from Figure 12, a large proportion of the data lies within the x o-ordinate range 570m 650m. Upper and lower horizontal surfaes are defined as a first step towards defining a three dimensional grid whih enapsulates a large proportion of the available data. The surfaes are at z 40m and at z 210m and these surfaes bound 91.6% of the drill hole data set. Figure 15 displays these surfae files with the line data. Figure 15 Lines and surfaes The strutural grid is now defined. The horizontal harateristis of the grid (xo,dx, nx, yo, dy, ny) are the same as for the surfaes, with the vertial dimensions reahing from surfae to surfae with a mesh of 2 metres. This reates a grid with ells, eah of dimensions 4m by 10m by 2m. This strutural grid will be transformed into a working grid for the simulation and then the simulation in this working grid will be bak-transformed into the strutural grid for the display of lithotypes. 47

58 5.3 Disretisation and Flattenning The drill hole data are not regular in their segmentation along eah line and so eah line is disretised into equal segments of 0.5m, whih is appropriate as it is equal to the mesh of the strutural grid. As the horizontal lag of the strutural grid ells is greater than the vertial lag, a distortion ratio is applied to the ells. The distortion ratio is alulated by a division of the produt of the horizontal extension values (dx,dy) by the vertial extension value (dz). Here, dx =dy =2 and dz = 0.5, thus the distortion ratio is 8. The working grid is now reated and is the struture within whih the analysis of lithotype proportions of the disretised data takes plae. The working grid has the same harateristis as the strutural grid. In this thesis the working grid has been transformed parallel to the lower surfae and has a new proportions Maro variable in a new disretised well file in whih the proportion of eah lithotype within eah ell of the strutural grid is stored (and updated subsequently). The next step is to hoose a method of assigning a single lithotype to eah ell of the (disretised) working grid. There are three hoies available in Isatis, entral (the lithotype in the middle of the disretized ore), most representative (the lithotype with the greatest proportion within the ore) and random (a lithotype is randomly assigned from those present in the disretised ore). In this study the entral method has been used. 48

59 5.4 Analysis and estimation of lithotypes The proportions of lithotypes in eah drill hole an be displayed in pie graphs as shown in Figure 16. Broadly, it is lear lithotypes 4 and 5 our in larger proportions in the North of the study region ( y 3000m ) and lithotype 3 ours in larger proportions in the South of the study region ( y 2900m) where lithotype 5 is absent. Figure 16 Pie harts indiating regional lithotype proportions. 49

60 In the analysis of the data displayed in the working grid in Figure 16, two proesses require onsideration: polygon definition and reation of proportion urves. Polygon definition allows the working grid to be setioned into a number of polygons that are 3D setions of the strutural grid muh larger than the horizontal mesh. The balane here is to form enough polygons within the study region so that regional sensitivity of lithotype proportions an be assessed whilst not dividing the 42 drill holes into polygons where similar VPCs are alulated. The longest axis for the data is the in the y diretion, and so five polygons were reated, splitting this axis into four equal areas in the north and one larger area the more sparsely populated south ( y 2650m), allowing eah polygon to be populated with at least five drill hole ores. The Proportions of lithotypes an be alulated from VPCs within eah of the (five) polygons defined, allowing for the indiation of speifi lithotypes in partiular areas of the working grid and also to update the lithotype Proportions Maro file in a regionally sensitive manner. The next step in analysis of the lithotype proportions is to onsider the VPC that an be generated in eah defined polygon of the study region. Eah VPC is alulated from surfae to surfae in the strutural grid, so it needs to be ompleted using a linear extrapolation of the top and/or bottom informed layers. One ompleted, the VPCs are normalised at eah layer and also smoothed by running a filtering algorithm a number of times through the data. Five passes of the filter have been run in the examples shown. Figure 17 shows the working grid split into the five polygons, eah with its normalised, ompleted and smoothed VPC, with the global VPC in the bottom right orner. Five polygons were defined to ater for the different lengths and starting heights of the drill holes in the study region. The individual VPCs are shown in Figure 18. Eah VPC is reprodued with raw proportions on the right and with normalised, ompleted and smoothed proportions on the left. Areas above and below eah raw VPC are generally ompleted with lithotype 1, the bounding lithotype of the study region. This will allow for the prodution of more realisti simulations. 50

61 Figure 17 Five polygons and their ompleted VPCs in the study region. 51

62 Figure 18 Raw (right) and ompleted (left) VPCs. 52

63 One the VPCs from eah polygon are ompleted, normalised and smoothed, they an be used in the subsequent interpolation proedures. It should be noted that normalising of the VPCs has aused possible artefats in diagrams two and four in Figure 18, where the bordering lithotype L1 is not the only lithotype at the top of the ompleted VPC. This may be the ause of inadmissible ontats ourring in some of the simulations (setion 6.3). An interim step is to alulate the 3D lithotype proportions on the working grid. Initially, these proportions were set to the global proportions. They are now alulated more aurately in the working grid using the inverse squared distane interpolation, using the VPCs as onstraining data. For eah ell at a partiular level in the working grid, the proportion of eah lithotype is a linear ombination of the proportions in all the VPCs, with the weights of the ombination being proportional to the inverse squared distane between eah VPC and the target ell. Other possible proedures for the interpolation are a user defined 2D kriging model or a layer-by-layer global proportion alulation. Displaying eah VPC over the entire working grid is diffiult, so a horizontal projetion (3D proportion map) of a oarse grid of these VPCs is shown. The VPCs are averaged layer by layer in moving windows, with extension for this moving window of 2 units. Eah VPC in this grid is an average of 10 VPCs in the x diretion with the origin of this axis at rank 5, and an average of 20 VPCs in the y diretion with the origin for this axis at rank 10. Figure 19 displays the 3D proportion map, whih again shows the absene of lithotype 3 in the North and its appearane in the South, together with the emergene of lithotype 2. 53

64 Figure 19 3D proportion map 54

65 5.5 Roktype rule and Gaussian random funtions. One lithotype proportions in eah ell have been alulated, determination of the underlying Gaussian random funtions and also the allowable transitions between lithotypes must be defined. If the lithotype ontats are purely sequential, then only one Gaussian random funtion needs to be trunated to simulate the lithotypes. In this study, whilst lithotypes 2,3, 4 and 5 are generally sequential in their ontats ( L2 ontats L3 but not L4 or L5 for example), they also all ontat lithotype 1. Transition matries (Table 4 and Table 5) indiate the proportions of ontats between the five lithotypes along the drill ores. They assist in the development of the roktype rule. The indiator variograms between lithotypes are analysed and this information allows for the underlying Gaussian random funtions to be modeled subsequently. When all the parameters have been defined, onditional plurigaussian simulations an be run. Table 4. Downward matrix of transition proportions. Number L1 L2 L3 L4 L5 L L L L L Table 5 Upward matrix of transition proportions. Number L1 L2 L3 L4 L5 L L L L L

66 Figure 20 Lithotype ontat map The ontat proportions indiatd in the above matries and the lithotype ontat map are summarised in the rok type rule shown in Figure 21 both in terms of relative proportion and physially allowable ontats. The rok type rule has three setions: ontats between lithotypes on the left, the five lithotypes in the entre and their assoiated summary ontat histograms on the right. It an be seen from the matries that L1 has ontat with all other lithotypes and this is indiated as proportions in the bottom right histogram and also in the left hand panel of Figure 21, where L1 (red) is in ontat with the other four lithoptypes. L2 (orange) only ontats L1 (red) and L3 (blue), L3 only ontats L1, L2 and L4 (green), L4 only ontats L1, L3 and L5 (pink), and L5 only ontats only L1 and L4. Figure 21 The roktype rule definition panel 56

67 The horizontal axis is modelled by one Gaussian random funtion (G1) and is trunated one to model L1 against the ombined proportion of the other four lithotypes, whereas the vertial axis defines the seond Gaussian random funtion (G2), whih is trunated at three points to model proportions of L2, L3, L4 and L5. The model for eah underlying Gaussian random funtions is determined by an iterative proess desribed in setion 2.7. Variogram fitting for the lithotypes in the Isatis software is an indiret, iterative proess. Underlying Gaussian funtion models are defined and, together with the roktype rule and the threshold parameters, theoretial variograms are tested against the experimental lithotype variograms until an appropriate fit for all experimental lithotype variograms is produed. The outome of the iterative proess is shown in Fgures 23, 24 and 25, where the indiator variograms (lines) for the unonditional simulation are ompared to the experimental indiator variograms (dotted line) for eah lithotype. The two underlying Gaussian models, G1 (horizontal axis) and G2 (vertial axis), are defined in Figure 22, where one threshold (t 1 ) is required for G1 between lithotype L1 and the other four lithoptyes ombined, and three thresholds (t 2, t 3 and t 4 ) are required for G2 between lithotypes L2, L3, L4 and L5. Figure 22 Underlying Gaussians and assoiated thresholds for PGS simulation. 57

68 Table 6 desribes the Gaussian model parameters used in the PGS proess. Eah Gaussian is defined using the spherial model defined in (26). The fit of the experimental variogram models is satisfatory with the exeption of L4 (green) in Figure 25. This is the least adequate variogram model of all lithotypes, with the short lag fit being overestimated and the longer range lags being underestimated, however the iterative proess demands that a balane be reahed in all variograms, and the underlying model parameters (Table 6) allowed a good fit for the remaining lithotype variograms. Table 6 Parameters for underlying Gaussian models. Gaussian model Model Sill Range Anisotropi Sale Fators G1 Spherial (7.8,110,20) G2 Spherial (7.8,50,16) 58

69 Figure 23 Experimental (dashed) and model (solid) horizontal variograms for lithotypes for L1 (top), L2 (middle) and L3 (bottom). 59

70 Figure 24 Experimental (dashed) and model (solid) horizontal variograms for lithotypes L4 (top) and L5 (bottom) 60

71 Figure 25. Experimental (dashed) and model (solid) vertial variograms for lithotypes 1, 2and 3 (top) and lithotypes 4 and 5 (bottom). 61

72 5.6 Conditional plurigaussian simulation One the roktype rule and the variogram models have been defined, simulations onditioned on drill hole data an now be performed. The moving neighbourhood searh ellipsoid is defined at this stage. This defines the region used for seletion of neighbourhood samples used for the kriging step. The Gibbs Sampler has been employed to alloate normal sore values for the lithotypes at data points. A turning bands algorithm with 100 bands is employed to initially determine an unonditional simulation from the underlying ovariane funtion and then to a onditional simulation is determined from a kriging of the error between the onditioning data at known loations and the unonditional simulation. These simulated Gaussian values are finally bak-transformed to lithotypes. A utaway 3D grid representation is shown in Figure 26, followed by horizontal slies at 10m depth intervals through the study region, beginning from z 110m (top left) through to z 30m (bottom right). These slies will be used as a basis for the physial omparison with both the IK and SIS models. L1 L2 L3 L4 L5 Figure 26 3D utaway representation of the onditional PGS over the study region. 62

73 Figure 27. Horizontal slies at 10m intervals of PGS output in the study region from z 110m (top left) to z 30m (bottom right) The above figure reflets a number of features seen in the original data as displayed in Figure 12. Lithotype 4 (green) predominates in the north (high y values) and in the south lithotype 3 (blue) is more evident at high z values transitioning to lithotype 4 at lower z levels in this region. Lithotype 2 (orange) has the least global proportion (2.93%) in the raw data and importantly it has not been removed from the simulation through under-estimation. 63

74 Figure 28 displays the variations inherent in the simulation proess. The six diagrams show a horizontal slie at z 110m from the output of six different simulation runs of the plurigaussian algorithm. There is a onsisteny in both the proportions of lithotypes present and their relative positions within the slie. F Figure 28. Six different realisations of PGS at depth z 110m 64

75 Chapter 6: Comparison of output of models In order to assess the differenes between the three models in this study, three omparisons will be made: visual inspetion, regional rok type proportions and a alulation of the numbers of lithotype ontats violating the roktype rule (a refletion of lithotype ontats in the experimental data). 6.1 Visual omparison between simulations and estimation. L1 L2 L3 L4 L5 Figure 29 Exavated 3D projetions of IK (top left), SIS (top right) and PGS (bottom) The three models shown in Figure 29 display regionally similar lithotype distributions, with PGS having less of the lithotype lustering effet evident in IK yet also a less fine 65

76 mosai effet evident in SIS. This suggests a more balaned and regionally sensitive output on a global sale for PGS although it is lear that IK and PGS have partiularly similar spatial distributions of L5 (pink) and L3 (blue). Whilst L2 ours in the same thin mid-level band in all models, the PGS model plaes L2 within one distint upper region of the study area whereas SIS and IK models spread very small pokets of L2 through the entire the study region. Figures 30 and 31 learly show the unrealisti lustering of lithotypes from the IK model as well as random alloations of L1 over the study region. The SIS slies show a gradual hange of harater as z values inrease whilst the PGS outputs are more definite in displaying the regional lithotype proportions seen in Figure 12 (lower panel). All models display varying degrees of so-alled inadmissible lithotype ontats (ontrary to the ontats appearing in the sample data and defined in the rok type rule), generally when lithotype 2 (orange) ontats either lithotype 4 (green) or 5 (pink). 66

77 Figure 30. IK (left), SIS (middle) and PGS (right) horizontal slies at z 110m (top), z 100m (middle) and z 90m (bottom). 67

78 Figure 31. IK (left), SIS (middle) and PGS (right) horizontal slies at z 70m (middle) and z 60m (bottom). z 80m (top), 68

79 6.2 Global and regional rok type proportions Table 7 displays global proportions for the drill-hole data set and the three models onsidered. The IK model has, as expeted, reprodued the global lithotype proportions very losely. Both SIS and PGS have muh higher proportions of the boundary lithotype L1 and ompensatory lower proportions of the other lithotypes. The variation within eah of the lithotype proportions for SIS is far greater than for PGS, shown both in summary in Table 7 and over the 100 simulations in Figure 32. Table 7 Summary lithotype perentage statistis for Data set, IK, SIS (100 realisations) and PGS (100 realisations). Data Set IK SIS PGS min mean max st. dev min mean max st. dev L L L L L

80 Figure 32 Lithotype proportions for 100 simulations of SIS (top) and PGS (bottom) The use of the Soares lassifiation algorithm has enfored the proportions from the data set to be reprodued in the IK model (Table 7). For SIS the relative magnitude of proportions is the same as for IK and the data set. Figure 32 indiates that for SIS (generally) the proportion of L1 (red) is lower than L3 (blue), however for PGS it appears to be different: there appears to be a greater proportion of loations where L1 is alloated rather than L3. The onern that the ompensatory balaning required due to the higher proportion of the boundary ategory L1 has lowered the validity of the PGS output is unfounded. Figure 33 shows the relative proportions of eah of L2, L3, L4 and L5 within the SIS and PGS models when removing the L1 proportion from the 70

81 alulation. The effet of the variation in the proportions of the boundary lithotype L1 has had little effet with respet to the other lithotypes within eah model. Relative proportions are not only very similar in eah of the simulations they are also similar to the drill-hole data set and IK proportions, shown in Table 8. Whilst the PGS model has the underlying goal of mathing global experimental lithotype proportions, the filling in of the boundary lithotype has had little effet on the onsisteny of the PGS model in terms of validity of lithotype proportions modelled in the body of the study region. Figure 33 Relative proportions of L2, L3, L4 and L5 with L1 removed for SIS (top) and PGS (bottom). 71

82 In Figure 33 variation in SIS output for the four lithotypes is evident, ontrasted against the onsistent nature of the lithotype outputs of PGS. Table 8 onfirms that SIS has between 3 and 4 times the variation of eah lithotype proportion than PGS however mean proportions of L2, L3 and L4 for SIS are loser to the data set proportions than for PGS. In all 100 PGS simulations the proportions of L4 and L2 are below that of the data set whilst the L3 proportion is above the data set. Table 8 Summary data of ategory perentages from SIS (top) and PGS (middle) and IK and Data Set (bottom) in the absene of L1. SIS L2 L3 L4 L5 min max mean st dev PGS L2 L3 L4 L5 min max mean st dev L2 L3 L4 L5 Data Set IK The 3D PGS exavation box perspetive shown in Figure 29 (lower) indiates that the L1 output is onentrated, as expeted, at the top and bottom boundaries of the study region. This is onfirmed by the histograms of lithotype proportions for PGS, IK and SIS shown in Figures 34 and 35 taken at 10m depth intervals. These histograms are ompared to histograms of lithotype proportions of 10m wide setions (in the vertial diretion) of the data set (in red), the width of the setions allowing meaningful numbers of data points to be inluded in eah histogram. These data set histograms give a regional lithotype proportion and so allow omparison to the model histograms. The relative proportions of all models orrelate with the data set, with the lithotype of greatest frequeny mathing in eah ase, with the only exeption being the PGS model at z 60m in Figure 35, where the proportion of L1 is greater than L3, possibly a 72

83 result of the ompletion of the VPCs with L1 in PGS. The histograms also indiate that SIS has less regional variation in terms of the proportions of the five lithotypes. This is of onern when it is lear that suh variation does exist in the experimental data (Figure 12). Figure 34 Lithotype proportions at z 110m (top), z 100m (middle) and z 90m (bottom) for (from left to right) IK, SIS, PGS and the data set. 73

84 Figure 35 Lithotype proportions at z 80m (top), z 70m (middle) and z 60m (bottom) for (from left to right) IK, SIS, PGS and the data set. 74

85 As well as omparison of lithotype proportions over horizontal slies of the study region, it is useful to onsider individual grid node lithotype proportions for PGS and SIS over the 100 simulations of eah model. For four grid nodes in the study area frequenies of lithotype alloations over the 100 simulations for eah model have been alulated and Figure 36 displays the output histograms. PGS usually has a better defined output mode than SIS SIS PGS SIS PGS Loation (606m,3250m,-160m) Loation (606m,2750m,-120m) SIS PGS SIS PGS Loation (622m,2950m, -140m) Loation (622m,2550m,-180m) Figure 36 Lithotype proportions at four grid loations in the study region for 100 simulations of PGS (red) and SIS (blue). 75