Lithos 112S (2009) 73–82

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Uncertainty-based grade modelling of kimberlite: A case study of the Jay kimberlitepipe, EKATI Diamond Mine, Canada

Sara Harrison a,⁎, Oy Leuangthong b, Barbara Crawford a, Peter Oshust a

a BHP Billiton Diamonds Inc. 1102, 4920-52nd Street Yellowknife, NT, Canada X1A 3T1b University of Alberta, Civil & Environmental Engineering Dept., 3-133 Markin/CNRL NREF Bldg, Edmonton, AB, Canada, T6G 2W2

⁎ Corresponding author. BHP Billiton Diamonds Inc. 1knife, NT, Canada X1A 3T1. Tel.: +1 250 869 5738; fax:

E-mail address: sara.sa.harrison@BHPBilliton.com (S

0024-4937/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.lithos.2009.04.047

a b s t r a c t

a r t i c l e i n f o

Article history:Received 15 September 2008Accepted 26 April 2009Available online 16 June 2009

Keywords:KimberliteResource estimationSequential Gaussian SimulationClassificationUncertainty

Understanding uncertainty in resource models is a significant requirement of mineral resourceevaluation. Geostatistical simulation is one method that can be used to quantify uncertainty and SequentialGaussian Simulation (SGS) is one of the easiest techniques to understand and implement. Using SGS providesboth a spatial model of a given variable and the ranges around that variable at any number of scales.The Jay kimberlite pipe is located in the southeastern quadrant of the EKATI property. Drilling to date hasidentified three kimberlitic domains characterized by varying lithological properties. These domains are notseparated by hard contacts, but rather by boundaries that are transitional. Within these domains, verticaltrends exist; in particular, diamond grade increases with depth. For these reasons, Jay required an in-depthinvestigation of the best uncertainty-based grade modelling method to use.Grade was modelled by organic SGS and by using the stepwise conditional transform (SCT) to incorporate atrend into the simulation routine. Although the SGS results were valid, they did not fully reproduce the trendand therefore, the results did not fully match the geological interpretation of the deposit. The SCT resultsreproduced the trend, however, did not correspond to the variability of the data and therefore under-represented the actual uncertainty in the model. This was confirmed through detailed uncertainty calculationand probabilistic resource classification. Therefore, the SGS model was chosen as the preferred uncertainty-based grade model for the Jay pipe.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Understanding uncertainty in resource models is a significantrequirement of mineral resource evaluation. Uncertainty analysisallows the modeller to explicitly quantify the risk around their es-timate, as well as to provide high and low estimates for mine design,assess the project up-side or down-side, model financial forecasts, andreconcile the deposit after mining. There are many sources ofuncertainty in a mineral resource, such as tonnage, diamond value,or grade. Geostatistical simulation is one method that can be used toquantify uncertainty around spatial variables such as grade andSequential Gaussian Simulation (SGS) is one of the easiest techniquesto understand and implement. Using SGS provides both a spatialmodel of a given variable and the ranges around that variable at anynumber of scales.

The Jay kimberlite pipe is located in the southeastern quadrant ofthe EKATI property (Fig. 1) in a 30 m deep depression in Lac duSauvage, approximately 2 km from the lake shore. It is within theEKATI Resource Development Plan and is currently at a concept study

102, 4920-52nd Street Yellow-+1 867 669 9293.. Harrison).

l rights reserved.

level. As a relatively large, yet lower value pipe, understanding the riskassociated with its value per tonne, and therefore diamond grade, isimperative.

Drilling to date has identified three kimberlite domains charac-terized by varying lithological characteristics, diamond grade, dry bulkdensity, andmoisture content. These domains are not separated by hardcontacts, but rather by boundaries that are transitional. Within thesedomains, vertical trends exist; in particular, diamond grade increaseswith depth. These conditions impacted grade modelling and thereforespecial attention was paid to ensure the results were the best possible.

Understanding the geology of Jay was paramount to choosing thebest modelling method and understanding the results. This contri-bution will introduce the geology of the Jay kimberlite pipe anddiscuss how the geological interpretation affected the grade model-ling technique chosen. Two modelling methods will be compared anddiscussed with a focus on uncertainty analysis.

2. Exploration history and data

The Jay kimberlite pipewasfirst identified as a conductive feature onan airborne electromagnetic survey in 1992. A core hole collared in 1993confirmed the anomaly as kimberlite and following this, Jay wasdelineated by eleven additional core holes, with the deepest kimberlite

Fig. 1. A map of the EKATI Claim Block and the kimberlites in the current Resource Development Plan. Grey indicates bodies of water.

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intersection at 20 m elevation (roughly 380 m below the kimberlitesurface). Five 31 cm reverse circulation (RC) holes drilled in 1996provided initial grade data from samples collected at 30 m intervals. Afurther twelve 45 cmRC holes drilled in 2006, with samples collected at15 m intervals, provided a detailed assessment of grade and filled in aroughly 50 m grid across the surface area of the pipe.

Core and RC holes have been macroscopically logged in detail withmicroscopic inspection where required. For RC holes, logging sampleswere collected every 2 m to provide material for description. Due to thedisaggregated and non-continuous nature of the material collected,small-scale properties are difficult to ascertain from RC drill holes.Detailed macroscopic and microscopic descriptions have been under-taken on drill cores, with RC drilling proving the spatial continuity of thelarger-scale properties. Two spatially representative core holes weresampled for thin sectionanalyses to confirmpetrographic classifications.

To date, 223 valid grade samples totalling over 1100 dry metrictonnes have been collected, with the deepest sample endingapproximately 350 m below the surface of the kimberlite at 50 melevation. All samples were processed at +1mm cut-off. They includeenough stones (average 154 stones) and are materially large enough(average 5 tonnes per sample) to be deemed statistically represen-tative of the Jay kimberlite.

Downhole caliper surveys were undertaken for all RC drillholes toprovide a measure of sample volume. Recovered carats and samplevolumes were used to calculate carats per cubic meter (cpm3), themodelled grade variable. Collected “slough” carats (carats collected frommass caving events during drilling) were applied to all up-drillhole

samples weighted on the number of carats collected and the “sloughpercentage” of a given sample. Slough percentage is defined as [(calipervolume−theoretical volume)/theoretical volume]×100%, where the-oretical volume is that estimated based on the nominal hole diameter.

All sampleswere assigned a single spatial coordinate at themidpointof the sample and allocated to a lithological domain based on the samplemidpoint. A dry bulk densitymodel was created and used to convert thecpm3 model to carats per dry metric tonne (cpt). This contribution willfocus on the techniques and uncertainty surrounding the cpm3 model.Further information on the methods used for RC drilling and gradecalculation are outlined in Dyck et al., 2004.

3. Geology

The EKATI property is situated in the center of the Slave StructuralProvince of the Canadian Shield. The general regional setting andgeneral bedrock geology of the EKATI property is well-documented; itis comprised of supercrustal metasediments, intrusive granitoids, andmafic dyke swarms (Nowicki et al., 2004; Kjarsgaard et al., 1994;LeCheminant and van Breemen, 1994; Wilkinson et al., 2001). Adescription of the host-rocks surrounding Jay and the kimberlite pipe'smorphology and internal geological domains are provided below.

3.1. Host-rock geology

Regional geophysics and drilling indicate that the Jay kimberlite pipeis situated fully within granitoid host-rocks, ranging from granite to

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granodiorite in composition. To the west of Jay, a poorly defined contactbetween the granitic intrusion and metasediments exists; mineralalignment in the granitoids is relatively weak, but increases near thiscontact. The metasediments are described as moderately to stronglyfoliated very fine- to medium-grained schistose rocks. An east–westtrending diabase dyke occurs to the north of Jay, and based on regionalgeophysical data, a bodyof tonalite is interpreted tobepresent to the east.

East–west and north–south trending regional structures have beeninterpreted from geophysics to be present to the north andwest of Jay,respectively. Both structures were confirmed by drilling and arerecognised by an increase in jointing and fractured core.

3.2. Kimberlite morphology

The Jay pipe is one of the largest known kimberlite bodies on theEKATI property, interpreted to be 11 ha near surface and roughly circularin shape. The morphology is similar to most other EKATI kimberlites inthat it is a steep-sided diatreme that tapers with depth (Fig. 2). Drillingsuggests that thepipe tapers fairlyconsistentlyat approximately−80° tosurface, except on the north side of the pipe, where the taper is −75°.The north side is likely controlled by both the east–west structure andthe diabase dyke.

The kimberlite/host-rock contact is sharp with little to nobrecciation. A brecciated contact zone was encountered in one deepcore hole at 85 m elevation. The internal granite (i.e. xenolithsoccurring with the kimberlite) in this zone is weakly to moderatelydesilicified and abundant silica veining is present. The presence of thisbreccia zone suggests that the pipe morphology and contacts may bemore irregular at depth.

3.3. Internal domains

From detailed geological logging of core holes and RC samples,three internal geological domains have been interpreted: resedi-

Fig. 2. A cut-away three-dimensional isometric view of the Jay kimberlite, illustratingthe pipe morphology; the relationships between the resedimented volcaniclastickimberlite (RVK), transitional kimberlite (TransK), and primary volcaniclastic kimber-lite (PVK) domains; and the locations of both the core and RC drilling completed to date.

mented volcaniclastic kimberlite (RVK); transitional kimberlite(TransK); and primary volcaniclastic kimberlite (PVK). Boundariesbetween these domains are transitional in nature and domain clas-sification is based on the appearance or disappearance of keykimberlitic textures and types. Fig. 2 illustrates the spatial relation-ship of these domains.

3.3.1. Resedimented volcaniclastic kimberliteThe upper-most domain of the Jay pipe comprises 150 to 175 m of

variable, yet generally olivine-poor resedimented volcaniclastickimberlite (RVK). Olivine size ranges from fine- to coarse-grained.The kimberlite matrix consists of blackmud to silt and is very clay-richmaking the overall rock unconsolidated to weak. Small-scale chaoticbedding is present; these beds are defined by silty to sandylaminations as well as variations in olivine abundance and size.

Variable amounts and sizes of black, pale grey, blue–grey, blue–green, brown, and tan coloured mudstone and siltstone xenolithscharacterize the RVK unit (Fig. 3a). Along with cobble-sized, fine- tomedium-grained granite xenoliths, these multi-coloured mud andsiltstone lithics are sometimes present in concentrations over 50%.

3.3.2. Transitional kimberliteA 25 to 85m thick alternating sequence of variably altered RVK and

primary volcaniclastic kimberlite (PVK) occurs below the RVKdomain. This mixed domain has been termed transitional kimberlite(TransK; Fig. 3b). The boundary from RVK toTransK is transitional andis marked by the appearance of short intervals (less than 50 cm) offresh to highly altered PVK.

Overall, the RVK intervals are similar to the upper domain; they arelocally variable but generally olivine-poor, fine- to coarse-grained andrich in matrix mud and lithic clasts.

The PVK intervals are dark blue–grey and competent when fresh,but show varying degrees of alteration with intense serpentinizationin places. Mudstone clasts are present but nomud occurs in thematrix(confirmed by microscopic inspection, Fig. 3c). Cored and uncoredjuvenile clasts are present and are mostly classified as monticellitekimberlite.

Variations in alteration are relatively sharp, with transitions fromfresh to intensely altered material in less that 50 cm. Intensely alteredmaterial is generally pale green in colour, with completely pseudo-morphed olivine (ranging from pale to very dark green) and intenselyalteredmatrix. From thin section analysis of intensely altered samples,olivine is seen to be completely pseudomorphed by serpentine (andless common carbonate), and in places, by clay. Grain boundaries arestill visible along with juvenile clasts allowing the textural classi-fication of this material as PVK. Less altered material has a relativelyfresh matrix, but olivine has been almost entirely pseuodomorphed,and in many cases removed resulting in a vuggy texture.

3.3.3. Primary volcaniclastic kimberliteBelow the TransK domain, the pipe is infilled with olivine-rich

PVK to the extent of drilling (approximately 20 m elevation). Theupper contact is transitional and is marked by the disappearance ofRVK and highly altered PVK. The PVK is grey–blue to green in colourand is highly competent with pervasive mild alteration. Olivine ismedium- to coarse-grained and sub-rounded to irregular in shape.Indicator minerals are abundant, dominated by peridotitic garnetsand lesser chrome diopside.

From thin section analysis, mud is absent from the interclast matrix,olivine clasts are broken, and in some sections, fine ash particles arereadily discerned (Fig. 3d). Both cored and uncored juvenile clasts arecommon, with the groundmass dominated by monticellite.

Small, irregular-shaped mud clasts are present, but decrease in abun-dance with depth. Pervasively altered granitic xenoliths are also present,ranging from 2 mm to 4 cm in size. These xenoliths also have strongalteration halos giving the kimberlite an overall patchy appearance.

Fig. 3. Photographs and photomicrographs of the Jay kimberlite domains. (a) An example of lithic olivine-rich RVK of the RVK domain. This sample has abundant olivine macrocrysts(OLVm) and shale xenoliths (SHX). Also shown is an example of a silty lamination (SLT LAM). (b) A typical section of the TransK domain in drill core showing the variable nature ofthis domain. This interval transitions quickly between very altered primary volcaniclastic kimberlite (vaPVK), altered PVK (aPVK), mud-rich RVK (mRVK), lithic olivine-poor RVK(lRVK), back into aPVK, and fresh PVK (PVK). (c) A thin section of PVK from the TransK domain, showing OLVm texture, mud clasts (MUD), and both cored and uncored juvenile clasts(JCu and JCc). (d) A thin section of the PVK domain illustrating the abundant olivine phenocrysts (OLVp), olivine macrocrysts (OLVm) and a mudstone clast (MUD).

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4. Grade modelling

The results of uncertainty-based modelling include both a spatialmodel of the variable, in this case grade, and a quantification of uncer-tainty ranges around the model at a given scale. The spatial modeltypically takes the form of a regular three-dimensional grid for whichvariables of interest are estimated (i.e. a block model), while theuncertainty estimate consists of grade ranges (either in the units ofinterest or as a percentage of themean) at different confidence intervalsand scales. The current uncertainty-based modelling method used atEKATI is Sequential Gaussian Simulation (SGS; Isaaks, 1990). SGS is acombination of simple kriging andMonte Carlo simulation; the former isused to determine the local distribution of uncertaintywhile the latter isused to randomly drawa simulated value from this distribution.Multipleequally probable realizations are generated by drawing different randomnumbers. From these realizations, range analyses canbe carried out froma block-by-block scale to a global level (Goovaerts, 1997).

While this technique ensures reproduction of spatial data, it isfounded on a very strong assumption of stationarity — in particular,the mean is assumed to be the same at every location in the field(Journel, 1994). Domains can generally be used to constrain fieldswith a commonmean; yet, when no hard boundary between varyingmeans exists, such as when there is a continuous trend in data, thestationarity assumption can not be maintained. Geological bodies,including kimberlites, often exhibit trends, which may require aspecial modelling technique. In this study the results of two SGSmethods will be compared — one without specifically dealing withthe trend in data, and another incorporating the modelled trend intosimulation using the stepwise conditional transform (Leuangthong,2003). The results of thesemethods will be discussed, including theirbenefits and drawbacks.

The steps involved in completing any uncertainty-based modellingusing SGS are as follows:

1. Exploratory data analyses (EDA) investigating data integrity, samplestatistics, the nature of internal domains, boundaries, trends,distributions;

2. Declustering data to ensure that there is no spatial bias;3. Data transformation to Gaussian space, either using a normal scores

transformor the stepwise conditional transform (Leuangthong, 2003);4. Variography to characterize the spatial variability of the deposit;5. Simulation using the SGS algorithmto generatemultiple realizations;6. Validation and back-transformation to original units; and7. Post-processing, including calculating the uncertainty around the

estimate and classifying the resource.

In this study, one-hundred realizations were simulated within themodelled pipe shell at a grid size of 5×5×5m. The SGS algorithmworksbest when grid spacing is as small as possible. This grid size isapproximately one-tenth thatof the sample spacing and is small enoughto accommodate SGS, yet is large enough for timely processing.

4.1. Exploratory data analyses

As the Jay kimberlite contains three distinct geological domains, itis important to understand the relationship of these domains to grade.Classical methods, including histograms, scatter-plots, and contactanalyses, were used to describe the sample populations and to assessthe appropriate method for grade modelling. As well, data integrity,potential bias and sample support were assessed to ensure that alldata is representative of the deposit and useful during simulation.

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Jay was sampled during two drill campaigns nearly a decade apartand although the drilling methods and relative sample tonnage arethe same, general changes in drilling, sampling, and sample proces-sing may add bias to the data set. The results of both campaigns werecompared using classical statistics and plots and appear to havesimilar distributions and means; however, given the relatively fewsamples from the 1996 campaign, validating this further was difficultand both datasets were assumed to have similar support.

When all data are plotted together as a histogram (Fig. 4a), no cleardomains are evident. However, geologically, some difference in gradebetween domains is expected and these differences become apparentwhen looking at each domain separately (Fig. 4b through d). In par-ticular, the RVK domain is negatively skewed, while the PVK domain isslightly positively skewed and the TransK domain, although it has veryfew samples, is approaching normal. All domains have differentstatistical means and ranges, which indicates that these populationsare dissimilar. Further analyses, including hypothesis testing andquantile–quantile plots, were conducted to ensure the distributions areindeed different.

There are no significant outliers present in any of the domains. Thenumber of samples available for both the TransK and PVK domains isrelatively low; in particular, there may be too few samples to reliablyspatially model the TransK domain alone.

Although these analyses prove that the domains are different intermsof their grade characteristic, theydonot illustrate thenature of thecontacts between these domains. A moving-window technique wasused to analyse these geological contacts (Fig. 5). As all RC holes are

Fig. 4. Histograms, cumulative distribution functions, and summary statistics for: (a) all data

verticalwith negligible deviation, for each sample the distance along thedrillhole from its midpoint to the given contact was calculated. A 30 mdistance-from-contact window spacing was used to calculate theaverage grade at varying distances from the contact. The results indicatethat there is no significant change in grade at either contact and bothwere determined to be transitional in terms of grade.

Due to the near-horizontal domain boundaries at Jay, vertical ornear-vertical trends are the most likely trends to occur. From theabove contact analyses there is a clear trend of increasing grade withdepth through the RVK and TransK domains. An elevation versusgrade scatter-plot illustrates that this trend continues into the PVKdomain to approximately 100 m elevation where it flattens (Fig. 6a).This trend is important to consider not only due to its magnitude(more than a factor of two over the given depth), but due to the few ofsamples available to reproduce it during modelling.

These analyses show that no hard grade boundaries are presentwithin the kimberlite; therefore, grade will be modelled treating theentire pipe as a single domain, paying attention to the trend ofincreasing grade with depth.

4.2. Trend modelling

The observed trend must be represented spatially to be used in thegrade modelling process. The simulated grade model will becompared against this trend model, and should inherently reproduceit. If required, the trendmodel will be used in themodelling process tocontrol simulation.

plotted together; (b) the RVK domain; (c) the TransK domain; and (d) the PVK domain.

Fig. 5. Contact analysis illustrating the transitional nature between the RVK and TransKdomains. Sample grades are plotted against the distance along the drillhole from itsmidpoint to the RVK/TransK contact. These grades were averaged in 30m intervals fromthe contact or “windows” to assess the significance of the contact to grade, and plottedagainst distance to graphically show the trend in data compared to the contact. Thistrend is smooth as it transitions between the RVK and TransK domains and therefore,the contact is deemed transitional in nature.

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Trend models should be smooth, representing the large scalefeatures of the data rather than small scale fluctuations. It is easy toover-fit the trendwhichmay lead to anover-constrained simulation (i.e.no variability between simulations). Trendmodels can take a number offorms— from a 1-dimensional linear equation to a 3-dimensional blockmodel. Avariety ofmethods are also available tomodel trends, includinghand mapping, best-fit line calculations, moving window averages,inverse distance, and kriging (Leuangthong and Deutsch, 2004;McLennan, 2007).

For the Jay pipe, a one-dimensional linear trend was modelled as asecond-order polynomial (Fig. 6a) and the equation of the line wasused to create a gridded model for use in simulation. This trend fitswith the geological interpretation of Jay. Grade increases with depththrough the RVK and TransK domains, and becomes constant oncetransitioned into the PVK domain. This change in grade through theupper portion could be explained by decreased dilution with depth(Scott Smith and Smith, this issue).

Rather than using the equation of the line to calculate trend values,they were extracted from the gridded model at the sample locations.This ensures exact reproduction of the trend values after trend-controlled simulation.

4.3. Declustering

As geological sampling programs are not entirely random, it isimportant to remove potential bias or “decluster” the data. Decluster-ing is a well-known technique that adjusts the summary statistics tobe representative of the entire area of interest and thereby removespotential bias. It assigns a weight to each data point based on its

closeness to surrounding data. One method, termed cell declustering,divides the area of interest into regular cells at a given grid spacing(Pyrcz and Deutsch, 2003). The weight for each cell, and all datapoints inside that cell, is calculated as 1/(# of data points in acell×total # of occupied cells). To determine the correct cell size,weights are calculated at multiple cell spacings and cell size is plottedagainst the mean of the data with the weights applied (thedeclustered mean). The grid size that either minimizes or maximisesthe mean is the final grid size used. Often, the cell size that is roughlyequal to the average sample spacing is an appropriate choice.

The Jay data was declustered using a grid size of 75 m in the X–Yand 22.5 m in the Z (1/3 the X–Y grid size). The declustered meanwasmaximised to 4.08 cpm3.

4.4. Simulation without the trend model

The most common modelling method used to create block modelsis kriging. The aim of the kriging algorithm is to provide the best localestimate of a variable without considering the global implications ofthe estimates together (i.e. each point is estimated individually andnothing links them together). The result is a smooth model with thewrong variability between predicted locations (Goovaerts, 1997;Deutsch and Journel, 1998). In contrast, simulation focuses onreproducing the global features (such as the histogram and trends)and the inherent variability of the data, thus allowing the assessmentand quantification of both local and global uncertainty.

Sequential Gaussian Simulation (SGS) is onemethod of simulation.SGS uses simple kriging and a random residual selected with MonteCarlo simulation to calculate the local value. Themean of these residualsmust be zero; however to determine the shape, data is transformed toGaussian space (and therefore has a standarddistribution and avarianceof 1). Each subsequent local estimate uses the previously calculatedresults as data so that the variance between all of the simulated values isreproduced. Each location is visited in a random order, and once all arecomplete, another equally probably realization is created startingwith adifferent random number seed (Deutsch and Journel, 1998).

To transform the data to Gaussian space, the normal scorestransformwas used with declustering weights applied. This transformis done by matching each sample's frequency on the cumulativedistribution function (CDF) to the corresponding frequency on thenormal CDF and translating this normal frequency to a Gaussian value(between −1 and 1).

An omni-directional spherical variogram with a nugget of 0.45(as the variogram corresponds to the normal scores data, the sill isequal to 1) and range of 185 mwas modelled from the Gaussian data.One-hundred SGS realizations were created and back-transformed toreal space using the reverse of the normal scores transform. Sensi-tivity analyses were run testing the input parameters of the SGSalgorithm (e.g. minimum and maximum number of original data,maximum number of simulated data, search region and distance).The aim is for the output histogram and block variogram of thesimulations to approach the declustered histogram and variogram(the global features of the data).

The results achieved are considered valid from both visual andstatistical investigation (Fig. 6a and b). The realizations are variable, yetreproduce the data points, the declustered histogram, and the Gaussianvariogram.However, the localmeanof thedatawas notmaintained nearsurface and after 100 m elevation — the results deviated towards thepopulation mean. These results did not fully reproduce the trend andgeological characteristics of the deposit, and subsequently the trendwasincorporated into simulation to compare the results.

4.5. Simulation incorporating the trend

A common approach to working with trends is to subtract the trendvalue from the original variable to calculate a residual, which is then

Fig. 6. Data and results of the simulationwithout the trendmodel. (a) A scatter-plot of sample data against elevation overlain by the linear trendmodel and realizations 1 through 10of the simulation results. (b) Vertical cross-section of one realization of the resulting blockmodel. Grade increases fromwhite to black blocks. Note the presence of lower grade blocksat depth demonstrating how the simulations tend back towards the mean.

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simulated (Deutsch, 2002). However, this method may lead to negativevalues as the relationship between the trend and residual is ignored,particularly when the data minimum approaches zero (Leuangthongand Deutsch, 2004). For Jay, residuals were initially used to incorporatethe trend into simulation; however negative results were returned andtherefore, the stepwise conditional transform (SCT) was used.

SCT is a multivariate Gaussian transform whereby the kth variableis transformed conditional to the probability classes of the first k−1variables (Rosenblatt, 1952; Leuangthong, 2003). In the case of twovariables (such as the trend and the original grade in this study), thesecondary variable is transformed to normal score based on binning ofthe primary variable. Independently transforming each class of dataresults in transformed variables that are Gaussian and have therelationship between the variables effectively removed.

To perform SCT there must be sufficient data. With two variables,between 102 and 202 data are required to perform SCT into 10 to 20classes with 10 to 20 data each (Leuangthong and Deutsch, 2003). TheJay sample population contains 223 data points and therefore, it isvalid to use SCT.

Grade data were transformed conditional to the trend using SCTwith declustering weights applied. Ten classes with a minimum of tendata each were used; sensitivities were performed and theseparameters resulted in the most normal population and removedthe most correlation between the transformed and normalised trendvariables.

An omni-directional exponential variogram with a nugget of 0.50and range of 165 m was modelled for the transformed data. Severalsensitivities were performed testing the number of data, searchparameters, and input variogram.

The model was then back-transformed to original units using thetransformation table produced during SCT. The resulting one-hundred

equally probably realizations reproduced the trend (Fig. 7a and b);however, the variability of these simulations was much less thanexpected and therefore, uncertainty surrounding both models wasquantified to compare the results.

4.6. Uncertainty quantification

Uncertainty is often of interest as it relates to multiple locationssimultaneously and can be quantified on a variety of scales (e.g. overregions that represent production periods, mining units, and/or overthe entire pipe) — it is important to understand the scale of thecalculation and the results. At EKATI, results are reported on twoscales: a global scale and a mining bench or level scale. To calculategrade uncertainty ranges at a given scale, the mean grade of eachrealization at that scale is calculated, realizations are ranked andassigned a probability, and uncertainty ranges are calculated at givenconfidence intervals.

Uncertainty ranges for both SGS models were calculated at a 15 mbench scale (Fig. 8a and b). The calculated ranges should compare tothose calculated for other kimberlite pipes on the EKATI property withsimilar amounts of data. However, the ranges calculated for the SCTmodel are significantly smaller — the 80% confidence interval of theSCT model is roughly one third of that of the regular SGS model. Basedon these results it appears that the trend model has over-constrainedsimulation.

4.7. Probabilistic resource classification

Historically, EKATI's kimberlite resources have been classified usingeither a probabilistic or a drill-hole spacingmethod. Probabilistic-basedclassification is more robust than that based on drill-spacing as it takes

Fig. 7. Results of simulation incorporating the trend model. (a) Moving window averages of realizations 1 through 10 of the simulation results; sample data; and the linear trendmodel plotted against elevation. The overall trend of the results follows the modelled trend at depth. (b) Vertical cross-section of one realization of the resulting block model. Gradeincreases from white to black blocks. Grade is higher at depth, following the interpreted geology and trend of the deposit.

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into account not only the confidence in the resource based on sampling,but also the variability in the sample grades. However, probabilisticclassification can not be used in isolation; it must be compared withdrill-spacing methods and take into account other variables.

As grade is generally the geostatistically calculated variable withthe highest uncertainty, it is used to classify the resource usingprobabilities. Uncertainty ranges are calculated at given probabilityintervals over set production periods based on a viable miningmethod. The tollgates used at EKATI are given in Table 1 and are basedon those proposed by Yeates and Hodson, 2006. Sensitivities are runby reasonably varying the production rates and tollgates to ensurethat the classification does not drastically change. Once probabilisticclassification is complete, the results are compared to the drillholespacingmethod to ensure reasonable consistency. The results are thendowngraded to reflect uncertainty around other non-geostatisticallymodelled properties such as diamond value, metallurgical character-istics, or geotechnical properties. Results are also clipped to the extentof reasonable mining.

To further compare the results of bothmodels, eachwas classified ona block-by-block basis using a conceptual open-pit mining rate. Usingthe tollgates in Table 1, the majority of the SGS model was classified asIndicated, with Inferred material near surface (due to increasedvariability and lower grades) and near the depth-extent of sampling(where sampling was sparser), andMeasuredmaterial around samples.The sensitivity of these results was tested by varying the productionperiods and tollgates and found to be stable. These results are consistentwith the drillhole spacingmethod, as themajority of the resourcewouldbe classified as indicated based on the average 50 m sample spacing.Importantly, using the probabilistic classification also captures the

increased variability near the surface of Jay, a feature that is not reflectedin classification based on drill-hole spacing.

In contrast, when the SCT model was classified using the samemethod, much of the model was classified as measured with someindicated. As this is inconsistent with the intuitive results and drillholespacing method, the trend model has very likely over-constrainedthe simulationprocess and thismodel should not be used for uncertaintyquantification and resource classification.

Jay has been modelled and classified to the lowest intersection ofgeological continuity as confirmed by a core intersection. Based on thedecreased level of confidence in other variables, such as diamondvalue, metallurgical studies, and geotechnical studies, all measuredblocks were down-graded to indicated. Classified material below thelowest grade sample but still within the known geological continuitywere downgraded to inferred. As a mineral resource must havereasonable prospects of economic extraction, mine planning softwarewas used to design a conceptual maximumopen-pit and this was usedto further limit the extent of the resource.

5. Discussion and conclusion

The uncertainty-based diamond grade models presented in thisstudy are consistent with standard geostatistical modelling practices.The presence of a trend required further consideration during simula-tion to ensure that it was properly reproduced. The stepwise conditionaltransformation is a relatively easy method to incorporate a trend intosimulation. Trends are common in kimberlite deposits due to emplace-ment processes such as airfall, dilution, sedimentation, and magmaticfractionation. Added to this, grade sampling is expensive given the large

Fig. 8. Grade from 100 simulations averaged by bench plotted with 80% confidenceinterval grade ranges (P10 and P90) for (a) the SGS model; and (b) the SCT model. Notethe very narrow ranges around the SCT model and the decreasing grade at depth in theSGS model.

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sample size required and therefore, sampling is generally sparsecompared to otherdeposits. These conditionsmay require incorporationof trends into the SGS routine.

The two methods of simulation compared in this study both havetheir benefits and drawbacks. The regular SGS method did not entirelyreproduce the modelled trend while the SCT model artificially reducedvariability. The SCT method is also much more complex to apply andadds anadditional level of subjectivity. Interestingly, in areaswhere dataare available, the trend is easily reproduced by both approaches; thissuggests that there are sufficient data to honour a trend in grade even inthe absence of an explicit trendmodel. It is only in the higher and lowerelevations of the model, where there is little data and high samplevariability that the SGS model deviates from the modeled trend andapproach the global mean. This is an expected result since there are noexplicit controls in SGS to reproduce a trend. Therefore, in areaswhere atrend is easily observed from the data, both simulation approaches yield

Table 1Resource classification tollgates used at EKATI (from Yeates and Hodson, 2006).

Inferred Indicated Measured

Production period Annual Annual QuarterlyAcceptable range ±30% ±30% ±15%Confidence 80% 95% 95%

similar averages in the simulated values. In the absence of data, thedifference in the resulting average grade profile is attributable to thesimulation methodology.

When comparing the models based on the resulting uncertaintyranges, clearly, the SCT model yields a much reduced uncertaintymodel. To accept this model places much faith in a trend model that isbased on relatively sparse data. For this reason, the SCT approach isconsidered to be too constraining an uncertainty model compared tothat of SGS. As we have seen above, the SGSmodel not only adequatelyreproduces the trend in the data, but also generates reasonableuncertainty ranges about this trend. The additional fact that an explicittrend model is not required to achieve these results makes the SGSapproach the preferred modeling tool in this case.

It is important to note that this assessment only takes into accountthe uncertainty around the grade model itself, and does not take intoaccount a variety of additional factors including sampling error andparameter uncertainty such as global statistics, the representativehistogram, and the model variogram. Despite this, the uncertaintyassessment discussed in this paper is compliant with industry practicegiven that there are no standard procedures for taking into accountthese other sources of uncertainty. They remain longstanding challengesin the area of resource modelling.

Along with grade, other areas such as pipe morphology (i.e.volume and tonnage) and geological interpretation, add mining riskand should be assessed in the resource model. The pipe modelpresented here was deterministically modelled using three-dimen-sional software, honouring all drillholes; however, there is inherentuncertainty between these pierce points. Other variables such asdiamond value and bulk density add to the risk associated with anykimberlitic resource evaluation. It is important to have a full under-standing of the uncertainty of each of these variables to understandthe full risk associated with the deposit.

Acknowledgments

The authors would like to thank BHP Billiton for permission topublish this study. Petrographic study of the Jay kimberlites wascompleted by Mineral Services, with special thanks to Casey Hetman.Guidance and suggestions from Darren Dyck, Jon Carlson and TomNowicki are much appreciated. Lastly, the authors would like to thankthe many EKATI geologists who logged the Jay kimberlite.

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