Beyond Ordinary Kriging Master7

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The Geostatistical Association of AustralasiaPO Box 1719 West Perth WA Australia

Beyond Ordinary Kriging:

Non-Linear Geostatistical Methods in Practice

Proceedings of a 1 day Symposium held at Rydges Hotel, Perth CBD on

Friday 30thOctober 1998

Major Sponsors

Mining & Resource

Technology


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MAJOR SPONSORS

Edith Cowan UniversityContact: Dr Lyn Bloom

[email protected]

Mining & Resource Technology

Contact: Bill Shaw

[email protected]

Resource Service Group

Contact: Dr Julian Barnes

[email protected]


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SATCHEL SPONSOR

Geoval Australias Geostatistical Experts.

Contact: John Vann

[email protected]

OTHER SPONSORS

Arne Berkmans, Mineral Resource Consulting

CSIRO Division of Mathematical and Information Sciences

Gemcom, Mining Software

Snowden & Associates, Mining Consultants


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Dedication

Thanks also to satchel sponsor, Geoval, and to those companies purchasing display

booth space for the symposium: Gemcom, Global Mining Services, Geoval and

Snowden Associates. Other sponsors were Arne Berckmans and the CSIRO Division

of Mathematical and Information Sciences.

Finally, thanks to authors, members of the GAA, and others, attending this first

historic symposium.

John Vann

GAA Executive Committee

Perth, October 1998.

2003 Addendum

This volume was re-edited to fit the standard GAA symposium volume format, for

publication as a compact disc. Proof-reading of the volume was undertaken by Stella

Searston, Roger Cooper and John Warner. John Vann designed the cover artwork,

which was then produced by John Warner. The CD design, layout and manufacture

were due to John Warner.

Stella Searston and John Warner,

May 2003.

Symposium on Beyond Ordinary Kriging 2


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Dedication

DEDICATION

This volume is dedicated to our colleagues:

Professor Michel

David

May 10, 2000

Professor Georges

Matheron

7 August, 2000



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Program

PROGRAM

8.30 to 9.00 am Registration, Level 1

Session 1.Chair: Dr John Henstridge

(Data Analysis Australia)


9.00 to 9.10 am Welcome and opening comments

John Vann (Geoval) Symposium Convenor

9:10 to 9:50 am Keynote AddressBeyond Ordinary Kriging a review of non-linear

estimation

John Vann, Daniel Guibal (Geoval)

9.50 to 10.20 am A practitioners implementation of Indicator Kriging

Ian Glacken, Paul Blackney (Snowden Associates)

10.20 to 10.50 am Coffee Break

Session 2

Chair: Mr Bill Shaw

(Mining and Resource Technology),


10.50 to 11.20 am The application of Indicator Kriging in the modelling of

geological data

Brett Gossage (Resource Service Group)

11.20 to 11.50 am Non-linear modelling of geological continuityDr John Henstridge (Data Analysis Australia)

11.50 to 12.20 pm Comparison of Median and full Indicator Kriging in the

analysis of gold mineralisation

Donna Hill, Dr Lyn Bloom, Dr Ute Mueller (Edith

Cowan University), Danny Kentwell (SRK Australia)

12.20 to 1.20 pm Lunch



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Program

Session 3

Chair: Dr Lyn Bloom

(Edith Cowan University) GAA Executive Committee

1.20 to 1.50 pm Local recoverable resource estimation: a case study in

Uniform Conditioning on the Wandoo project for

Boddington gold mine

Michael Humphreys (Geoval)

1.50 to 2.20 pm A case study using Indicator Kriging: the MountMorgan gold-copper deposit, Queensland

Ivor Jones (WMC Resources)

2.20 to 2.50 pm Practical application of Multiple Indicator Kriging to

recoverable resource estimation for the Halleys lateritic

nickel deposit

Ian Lipton, Richard Gaze, John Horton (Mining and

Resource Technology)

2.50 to 3.20 pm Coffee Break

Session 4

Chair: Daniel Guibal

(Geoval)

3.20 to 3.50 pm Median Indicator Kriging: a case study in iron ore

Alison Keogh, Craig Moulton (Hamersley Iron)

3.50 to 4.20 pm A proposed approach to change of support correction

for Multiple Indicator Kriging, based on p-field

simulation

Sia Khosrowshahi, Richard Gaze, Bill Shaw (Mining

and Resource Technology)



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Program

Discussion Panel

Chair: John Vann

(Geoval), GAA Executive Committee

Panel:

Dr Lyn Bloom (Edith Cowan University), GAA

Executive Committee

Richard Gaze (Mining and Resource Technology)

Brett Gossage (Resource Service Group)

Daniel Guibal (Geoval)

Vivienne Snowden (Snowden Associates)

4.20 to 5.10 General discussion on the floor and with the panel

5.10 to 6 pm Drinks on the Terrace



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Vann and Guibal Keynote presentation

CONTENTS

Foreword.......................................................................................................................1Dedication .....................................................................................................................1

Program ........................................................................................................................2

Beyond Ordinary Kriging An Overview of Non-linear Estimation ..................6

John Vann and Daniel Guibal Geoval.......................................................................6

A practitioners implementation of indicator kriging .............................................26

Ian Glacken and Paul Blackney Snowden Associates .............................................26

The Application of Indicator Kriging in the Modelling of Geological Data.........40

Brett Gossage Resource Service Group...................................................................40

Non-Linear Modelling of Geological Continuity ....................................................41

John Henstridge Data Analysis Australia................................................................41

Comparison of Median and Full Indicator Kriging in the Analysis of a GoldMineralisation ............................................................................................................50

Donna Hill, Ute Mueller, Lyn Bloom Edith Cowan University..............................50

Local recoverable estimation: A case study in uniform conditioning on the

Wandoo Project for Boddington Gold Mine ...........................................................63

Michael Humphreys Geoval ....................................................................................63

A case study using indicator kriging the Mount Morgan Gold-Copper

Deposit, Queensland ..................................................................................................76

Ivor Jones WMC Resources .....................................................................................76

Practical application of multiple indicator kriging and conditional simulation to

recoverable resource estimation for the Halleys lateritic nickel deposit.............88

Ian Lipton, Richard Gaze, John Horton and Sia Khosrowshahi Mining andResource Technology...............................................................................................88

Median Indicator Kriging - A Case Study in Iron Ore ........................................106

Alison Keoghand Craig MoultonHamersley Iron ................................................106

A proposed approach to change of support correction for multiple indicator

kriging, based onp-field simulation .......................................................................121

Sia Khosrowshahi, Richard Gaze and Bill Shaw Mining and Resource Technology

................................................................................................................................121

Author Contact Details............................................................................................132



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BEYOND ORDINARY KRIGING

AN OVERVIEW OF NON-LINEAR ESTIMATION

John Vann and Daniel Guibal

Geoval

Abstract

Many geostatistical variables have sample distributions that are highly

positively skewed. Because of this, significant deskewing of the histogram

and reduction of variance occurs when going from sample to blocksupport, where blocks are of larger volume than samples. When making

estimates in both mining and non-mining applications we often wish to

map the spatial distribution on the basis of block support rather than

sample support. The SMU or selective mining unit in mining geostatistics

refers to the minimum support upon which decisions (traditionally:

ore/waste allocation decisions) can be made. The SMU is usually

significantly smaller than the sampling grid dimensions, in particular at

exploration/feasibility stages. Linear estimation of such small blocks (for

example by inverse distance weighting IDW or ordinary Kriging

OK) results in very high estimation variances, i.e. the small block linear

estimates have very low precision. A potentially serious consequence ofthe small block linear estimation approach is that the grade-tonnage

curves are distorted i.e. prediction of the content of an attribute above

a cut-off based on these estimates is quite different to that based on true

block values. Assessment of project economics (or other critical decision

making) based on such distorted grade-tonnage curves will be riskier than

necessary. While estimation of very large blocks, say similar in

dimensions to the sampling grid, will result in lower estimation variance,

it also implies very low selectivity, which is often an unrealistic

assumption. This paper presents an overview of the geostatistical

approach to solving this problem: non-linear estimation. Linear

estimation is compared to non-linear estimation, the motivations of non-linear approaches are presented. A summary of the main geostatistical

non-linear estimators is included. In a non-linear estimation we estimate,

for each large block (by convention called a panel) the proportion of

SMU-sized parcels above a cut-off grade or attribute threshold. A series

of proportions above cut-off defines the SMU distribution. Use of such

non-linear estimates reduces distortion of grade-tonnage curves and

allows for better decision making. A partial bibliography of key

references on this subject is included.



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Key Words: geostatistics, non-linear estimation, mining, environmental

contamination, grade-tonnage curve, indicator, Gaussian transformation,

lognormal distribution



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Where are the weights, dare the distances from each sample location to the centroid

of the block to be estimated and is the power1. Once the power to be used is

specified, the ith sample is assigned a weight that depends solely upon its location

(distance to the centroid). Whether the sample at this location had an average or

extreme value does not have any impact whatsoever on the assignment of

di.

OK is a more sophisticated linear interpolator proposed by Matheron (1962, 1963a,

1963b). OKs advantage over IDW as a linear estimator is that it ensures minimum

estimation variance given:

(1)A specified model spatial variability (i.e. variogram or other characterisation ofspatial covariance/correlation), and

(2)A specified data/block configuration (in other words, the geometry of the

problem).

The second criterion involves knowing the block dimensions and geometry, the

location and support of the informing samples, and the search (or Kriging

neighbourhood) employed. Minimum estimation variance simply means that the

estimation error is minimised by OK. Given an appropriate variogram model, OK will

outperform IDW because the estimate will be smoothed in a manner conditioned by

the spatial variability of the data (known from the variogram).

Now, contrast linear regression with non-linear regression. There are many types of

non-linear relationships we can imagine betweenxandy, a simple example being:

y ax b= +2

This is a quadratic (or parabolic) regression, available in most modern spreadsheet

software, for example. Note that the relationship betweenxandyis now clearly non-

linear the nature of the relationship between x and y is clearly dependent upon the

particular x value considered. Non-linear geostatistical estimators therefore allocate

weights to samples that are functions of the grades themselves and not solely

dependent on the location of data.

Non-linear interpolators

Limitations of linear Interpolators

The fundamental limitations of linear estimation (of which OK provides the best

solution) are straightforward:

1The denominator of this fraction expresses the weight calculated as a proportion of the total weight allocated to all samples

found within the search



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1. We may be motivated to estimate the distribution rather than simply an expectedvalue at some location (or over some area/volume, if we are talking about block

estimation). Linear estimators cannot do this. The cases abound: recoverable ore

reserves in a mine, the proportion of an area exceeding some threshold of

contaminant content in an environmental mapping, etc.

2. We are dealing with a strongly skewed distribution, eg. a precious metal oruranium deposit, and simply estimating the mean by a linear estimator (for

example by OK) is risky, the presence of extreme values making any linear

estimate very unstable. We may require a knowledge of the distribution of grades

in order to get a better estimate of the mean. This usually involves making

assumptions about the distribution (for example, what is the shape of the tail of the

distribution?) even in situations where we are ostensibly distribution free (for

example using IK).

3. We may be studying a situation where the arithmetic mean (and therefore thelinear estimator used to obtain it) is an inappropriate measure of the average, for

example in situations of non-additivity like permeability for petroleum

applications or soil strength for geological engineering applications.

The specific problem of estimating recoverable resources was the origin of non-linear

estimation and has been the main application.

From a geostatistical viewpoint, non-linear interpolation is an attempt to estimate the

conditional expectation, and further the conditionaldistributionof grade at a location,as opposed to simply predicting the grade itself. In such a case we wish to estimate the

mean grade (expectation) at some location under the condition that we know certain

nearby sample values (conditional expectation).This conditional expectation, with a

few special exceptions (eg. under the Gaussian Model see later) is non-linear.

In summary, non-linear geostatistical estimators are those that use non-linear

functions of the data to obtain (or approximate) the conditional expectation.

Obtaining this conditional expectation is possible, in particular through the probability

distribution:

[ ]Pr ( )| ( )Z x Z xo i

This reads: the probability of the grade at location xogiven the

known sampling information at locations Z(xi) (i.e. Z(x1), Z(x2)

.Z(xN). This is the conditional distribution of grade at that

location. Once we know (or approximate) this distribution, we can

predict grade tonnage relationships (eg. how much of this block is

above a cut-offZC?).



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Available methods

There are many methods now available to make local (panel by panel) estimates of

such distributions, some of which are:

Disjunctive Kriging DK (Matheron, 1976, Armstrong and Matheron,1986a, 1986b);

Indicator Kriging IK (Journel, 1982, 1988) and variants (Multiple IndicatorKriging; Median Indicator Kriging, etc.);

Probability Kriging PK (Verly and Sullivan, 1985);

Lognormal Kriging LK (Dowd, 1982);

Multigaussian Kriging MK (Verly and Sullivan, 1985, Schofield, 1989a,1989b);

Uniform Conditioning UC (Rivoirard, 1994, Humphreys, 1998);

Residual Indicator Kriging RIK (Rivoirard, 1989).

In a non-linear estimation we estimate, for each large block (by convention called a

panel) the proportion of SMU-sized parcels above a cut-off grade or attribute

threshold. A series of proportions above cut-off defines the SMU distribution.

Note that there is a very long literature warning strongly against estimation of

small blocks by linear methods (Armstrong and Champigny, 1989; David, 1972;

David 1988; Journel, 1980, 1983, 1985; Journel and Huijbregts, 1978; Krige, 1994,

1996a, 1996b, 1997; Matheron, 1976, 1984; Ravenscroft and Armstrong, 1990;

Rivoirard, 1994; Royle, 1979). By small blocks, we mean blocks that are

considerably smaller than the average drilling grid (say appreciably less than half the

size, although in higher nugget situations, blocks with dimensions of half the drill

spacing may be very risky).

The authors strongly reiterate this warning here.The prevalence in Australia of

estimating blocks that are far too small is symptomatic of misunderstanding of basicgeostatistics. Even estimating such small blocks directly by a non-linear estimator

may be incorrect and risky. When using non-linear estimation for recoverable

resources estimation in a mine, the panels should generally have dimensions

approximately equal to the drill spacing, and only in rare circumstances (i.e. strong

continuity) can significantly smaller panels be specified.

Non-linear estimation provides the solution to the small block problem. We cannot

precisely estimate small (SMU-sized) blocks by direct linear estimation. However, we

canestimate the proportion of SMU-sized blocks above a specified cut-off, within a



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panel. Thus, the concept of change of support is critical in most practical applications

of non-linear estimation.

Support effect

Definition

"Support" is a term used in geostatistics to denote the volume upon which average

values may be computed or measured. Complete specification of support includes the

shape, size and orientation of the volume. If the support of a sample is very small in

relation to other supports considered, eg. drill hole sample upon which a gold assayhas been made, it is sometimes assumed to correspond to "point support".

Grades of mineralisation measured on a small support (eg. drill hole samples) can be

much richer or poorer than grades measured on larger supports, say selective mining

units (SMU) blocks. The grades on smaller supports are said to be more dispersed

than grades on larger supports. Dispersion is usually measured by variance.

Although the global mean grades measured (or estimated) on different supports (at

zero cut-off) are the same, the variance of the smaller supports is higher, i.e. very high

drill hole sample grades are possible, but large mining blocks have a smoother

distribution of grades (fewer very high and very low grades). "Support effect" is thisinfluence of the support on the distribution of grades.

The necessity for change of support

Change of support is vital for predicting recoverable reserves if we intend to

selectively mine a deposit. Before committing the capital required to mine such a

deposit, an economic decision must be made based only on the samples available

from exploration drilling. Because mining does notproceed with a selection unit of

comparable size to the samples, the difference in support between the samples and

the proposed SMU must be accounted for in any estimate to obtain achievable results.When there is a large nugget effect, or an important short-scale structure apparent

from the variography, then the impact of change of support will be pronounced.

The histogram of drill hole samples will usually have a much longer "tail" than the

histogram of mining blocks. Simplistic variance corrections, for example affine

corrections, do not reflect the fact that, in addition to variance reduction, change of



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support also involves symmetrisation of the histogram2.This is especially important

in cases where the histogram of samples is highly skewed.

Recoverable resources

Recoverable resources are the portion of in-situ resources that are recovered during

mining. The concept of recoverable resources involves both technical considerations,

such as cut-off grade, SMU definition, machinery selection etc., and also

economic/financial considerations such as site operating costs, commodity prices

outlook, etc. In this paper, only technical factors are considered. Recoverable

resources can be categorised as either global or local recoverable resources. Global

recoverable resources are estimated for the whole field of interest; eg. estimation of

recoverable resources for the entire orebody (or a large well-defined subset of the

orebody like an entire bench)3.Local recoverable resources are estimated for a local

subset of the orebody; eg. estimation of recoverable resources for a 25m x 50m x 5m

panel.

A summary of main non-linear methods

Indicators

The use of indicators is a strategy for performing structural analysis with a view to

characterising the spatial distribution of grades at different cut-offs. The transformed

distribution is binary, and so by definition does not contain extreme values.

Furthermore, the indicator variogram for a specified cut-off is physically

interpretable as characterising the spatial continuity of samples with grades exceeding

. Indicator transformations may thus be conceptually viewed as a digital contouring

of the data. They give very valuable information on the geometry of the

mineralisation.

zc

zc

A good survey of the indicator approach can be found in the papers of Andre Journel

(eg. 1983, 1987, 1989).

An indicator random variable is defined, at a locationI x zc( , ) x , for the cut-off as

the binary or step function that assumes the value 0 or 1 under the following

conditions:

zc

2This symmetry can be demonstrated via the central limit theorem of classical statistics, which states that the means of repeated

samplings of any distribution will have a distribution which is normal, regardless of the underlying distribution.. When we

consider block support, the aggregation of points to form blocks will thus deskew the histogram. In the ultimate case, we have a

single block, being the entire zone of stationarity and there is noskewness as such.

3Global recoverable resources can be very useful as checks on local recoverable estimation, a good first pass valuation or can be

used for checking the impact on grade-tonnage relationships of changing SMU, bench-height studies, etc. They are not

specifically discussed in this paper. The interested reader is referred to Vann and Sans (1995).



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I x z Z x z

I x z Z x z

c c

c c

( , ) ( )

( , ) ( )

=

= >

0

1

if

if

The indicators thus form a binomial distribution, and we know the mean and variance

of this distribution from classical statistics:

m p

p p

=

= 2 1( )

Where p is the proportion of 1s as defined above (for example, if

the cut-off, is equal to the median of the grade distribution, p

takes a value of0.5, and the maximum variance is defined as 0.25).

zc

After transforming the data, indicator variograms can be calculated easily by any

program written to calculate an experimental variogram. An indicator variogram is

simply the variogram of the indicator.

Indicator Kriging

Indicator Kriging is kriging of indicator transformed values using the appropriate

indicator variogram as the structural function. In general the kriging employed is

ordinary kriging. (OK). An IK estimate (i.e. kriging of a single indicator) must always

lie in the interval [0,1], and can be interpreted either as

1. probabilities (the probability that the grade is above the specified indicator) or2. as proportions (the proportion of the block above the specified cut-off on data

support).

In addition to its uses for indicator kriging (IK), multiple indicator kriging (MIK),

probability kriging (PK)and allied techniques, the indicator variogram can be useful

when making structural analysis to determine the average dimensions of mineralised

pods at different cut-offs. Indicators are also useful for charactering the spatial

variability of categorical variables (eg. presence or absence of a specific lithology,

alteration, vein type, soil type, etc.). Henstridge (1998) presents examples of such

applications for an iron deposit and Gossage (1998) give a more general overview of

such applications of indicator kriging.

Multiple Indicator Kriging

Multiple indicator kriging (MIK) involves kriging of indicators at several cut-offs (see

various publications by Andre Journel in the references to this paper as well as Hohn,

1988 and Cressie, 1993). MIK is an approach to recoverable resources estimation

which is robust to extreme values and is practical to implement. Theoretically, MIK

gives a worse approximation of the conditional expectation than disjunctive kriging



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(DK), which can be shown to approximate a full co-kriging of the indicators at all cut-

offs, but does not have the strict stationarity restriction of DK.

The major difficulties with MIK can be summarised as:

1. Order relation problems: i.e. because indicator variogram models may beinconsistent from one cut-off to another we may estimate more recovered metal

above a cut-off than for a lower cut-off , wherezc2 zc1 z zc1 c2< , which is clearlyimpossible in nature. While there is much emphasis on the triviality of order

relation problems and the ease of their correction in the literature, the authors have

observed quite severe difficulties in this regard with MIK. The theoretical solution

is to account for the cross-correlation of indicators at different cut-offs in the

estimation by co-kriging of indicators, but this is completely impractical from a

computational and time point of view. In fact, the motivation for developing

probability kriging (PK) was to approximate full indicator co-kriging (see below).

2. Change of support is not inherent in the method. In the authors experience, mostpractical applications of MIK involve using the affine correction, which assumes

that the shape of the distribution of SMUs is identical to that of samples, the sole

change in the distribution being variance reduction as predicted by Kriges

Relationship. There are clear warnings in the literature (by Journel, Isaaks and

Srivastava, Vann and Sans, and others) about the inherent deskewing of the

distribution when going from samples to blocks. The affine correction is not suited

to situations where there is a large decrease in variance (i.e. where the nugget is

high and/or there is a pronounced short-scale structure in the variogram of grades).

Other approaches can be utilised, e.g. lognormal corrections (very distributiondependent), or conditional simulation approaches (costly in time). A new proposal

for change of support in MIK is given by Khosrowshahi et al. (1998).

Median Indicator Kriging

Median indicator kriging is an approximation of MIK which assumes that the spatial

continuity of indicators at various cut-offs can be approximated by a single structural

function, that for zc = m~ , where ~m is the median of the grade distribution. The

indicator variogram at (or close to) the median is sometimes considered to be

representative of the indicator variograms at other cut-offs. This may or may not be

true, and needs to be checked. The clear advantage of median indicator kriging over

MIK is one of time (both variogram modelling and estimation). The critical risk is in

the adequacy of the implied approximation. If there are noticeable differences in the

shape of indicator variograms at various cut-offs, one should be cautious about using

median indicator kriging (Isaaks and Srivastava, 1989, pp 444). Hill et al (1998) and

Keogh and Moulton (1998) present applications of the method.

Probability Kriging



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Probability kriging (PK) was introduced by Sullivan (1984) and a case study is given

in Verly and Sullivan (1985). It represents an attempt to alleviate the order

relationship problems associated with MIK, by considering the data themselves

(actually their standardised rank transforms, distributed in [0,1]) in addition to the

indicator values. Thus a PK estimate is a co-kriging between the indicator and the

rank transform of the data U. When performed for ncut-offs, it requires the modellingof 2n+1 variograms: n indicator variograms, n cross-variograms between indicators

and U, and finally the variogram of U.The hybrid nature of this estimate as well as

the time-consuming complexity of the structural analysis makes it rather unpractical.

Indicator Co-kriging and Disjunctive Kriging

In general, any practical function of the data can be expressed as a linear combination

of indicators:

),()( nn

n zZIfZf =

Thus, estimating amounts to estimating the various indicators. The best linear

estimate of these indicators is their full co-kriging, which takes into account the

existing correlations between indicators at various cut-offs. Full indicator co-kriging

(also called Disjunctive Kriging, abbreviated to DK) theoretically ensures consistency

of the estimates (reducing order relationships to a minimum or eliminating them

altogether): this makes the technique very appealing, but there is a heavy price to pay:

if nindicators are used, n

)(Zf

2variograms and cross-variograms need to be modelled, and

this is unpractical as soon as n gets over 5 or 6, even with the use of modern automatic

variogram modelling software.

The various non-linear estimation methods can be considered as ways of simplifying

the full indicator cokriging. Roughly speaking, there are three possible paths to follow

(Rivoirard, 1994):

1. Ignore the correlations between indicators: this is the choice made by MIK alreadydiscussed. The authors consider this a fairly drastic choice.

2. Assume that there is intrinsic correlation, i.e.that all variograms and cross-variograms are multiples of one unique variogram. In that case, cokriging isstrictly equivalent to kriging; this is the hypothesis underlying median IK.

Needless to say, unfortunately, this very convenient assumption is rarely true in

practice (see median IK, above).

3. Express the indicators as linear combinations of uncorrelated functions(orthogonal functions), which can be calculated from the data. Cokriging of the

indicators is then equivalent to separate kriging of the orthogonal functions; this



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decomposition of the indicators is the basis of residual indicator kriging (RIK) and

of isofactorial disjunctive kriging.

Residual Indicator Kriging

In this particular model, within the envelope defined by a low cut-off, the higher

grades are randomly distributed. The proximity to the border of the envelope has no

direct incidence on the grade, and this corresponds to some types of vein

mineralisation, where there is little correlation between the geometry of the vein and

the grades. The validity of the model is tested by calculating the ratios

ij

i

h

h

( )

( )

(cross-variograms of indicators over variograms of indicator) for the cut-offszjhigherthan zi. If these ratios remain approximately constant, then the model is appropriate.

Note that an alternative decreasing model exists where one compares the cross-

variograms to the variogram associated with the highest cut - off (instead of the

lowest ).

The residuals are defined from the indicator functions by

1

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

=i

i

i

ii

T

zxZI

T

zxZIxH where [ ])),(( ii zxZIET = , i.e. the proportion of

grades higher than the cut-offzi.

i.e.

H x0 1( )=

1)

)( 11 z

xH),((

1

=T

xZI

......

1

1 )),((()(

=n

nn

T

zxZZIxH

)),(

n

n I

T

zx

The i( ) are uncorrelated and we have:

)()),((

0

xHT

zxZI i

j

j

i

i =

=

This means that the indicators can be factorised. In order to get a disjunctive estimate

of , it is enough to krige separately each of the residuals))(( xZf xi( ) . The Tare

simply estimated by the means of the indicators, .

i

)),(( izxZI



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In practice, the residuals are calculated at each data point, their variograms are then

evaluated and independent krigings are performed. Another check of the model

consists in directly looking at the cross variograms of the residuals: if they are flat,

indicating no spatial correlation, the model works. Thus, essentially this model

requires no more calculations than indicator kriging, while being more consistent

when it is valid.

The reader is referred to Rivoirard (1994, chapter 4) for a fuller explanation and a

case study (chapter 13) of this approach.

Residual indicators is one way to co-krige indicators by separately kriging

independent combinations of them and recombining these to form the co-kriged

estimate. Like MIK, this method involves working with many indicators and the same

number of variograms. Thus, it can be time consuming.

Isofactorial Disjunctive Kriging

There are several versions of isofactorial DK, by far the most common is Gaussian

DK.

Gaussian DK is based on an underlying diffusion model (where, in general, grade

tends to move from lower to higher values and vice versa in a relatively continuous

way).

The initial data are transformed into values with a Gaussian distribution, which can

easily be factorised into independent factors called Hermite polynomials (see

Rivoirard, 1994 for a full explanation and definition of Hermite polynomials anddisjunctive kriging). In fact, any function of a Gaussian variable, including indicators,

can be factorised into Hermite polynomials. These factors are then kriged separately

and recombined to form the DK estimate. The major advantage of DK is that you only

need to know the variogram of the Gaussian transformed values in order to perform

all the krigings required. The basic hypothesis made is that the bivariate distribution

of the transformed values is bigaussian, which is testable. Although order

relationships can occur, they are very small and quite rare in general. A very powerful

and consistent change of support model exists for DK: the discrete Gaussian model

(see Vann and Sans, 1995).

Gaussian disjunctive kriging has proved to be relatively sensitive to stationaritydecisions, (in most cases simple kriging is used in the estimation of the polynomials).

DK should thus only be applied tostrictlyhomogeneous zones.

Uniform Conditioning

Uniform conditioning (UC) is a variation of Gaussian DK more adapted to situations

where the stationarity is not very good.



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In order to ensure that the estimation is locally well constrained, a preliminary

ordinary kriging of relatively large panels is made, and the proportions of ore per

panel are conditional to that kriging value.

UC is a relatively robust technique. However, it does depend heavily upon the quality

of the kriging of the panels. As for DK, the discrete Gaussian model ensuresconsistent change of support. Humphreys (1998) gives a case study of application of

UC to a gold deposit.

Lognormal Kriging

Lognormal kriging (LK) is not linked to an indicator approach and belongs to the

conditional expectation estimates.

If the data are truly lognormal, then it is possible, by taking the log, and assuming that

the resulting values are multigaussian, to perform a lognormal kriging. The resultingestimate is the conditional expectation and is thus in theory the best possible estimate.

This type of estimation has been used very successfully in South Africa.

Unfortunately the lognormal hypothesis is very strict: any departure can result in

completely biased estimates.

Multigaussian kriging

A generalisation of the lognormal transformation is the Gaussian transformation

which applies to any reasonable initial distribution. Again, under the multigaussian

hypothesis, the resulting estimate represents the conditional expectation and is thusoptimal. This is a very powerful estimate much more largely applicable than

lognormal kriging, but requires very good stationarity to be used with confidence.

Compared to Gaussian DK, it is completely consistent, but based on stronger

multigaussian assumptions and its application to block estimation is more complex.

Conclusions and recommendations

1. As we approach the end of this century, and nearly 40 years since Matheronspioneering formulation of the Theory of Regionalised Variables, there are a

large number of operational non-linear estimators to choose from.Understanding the underlying assumptions and mathematics of these methods

is critical to making informed choices when selecting a technique.

2. We join the tedious chorus of geostatisticians over many years andrecommend that linear estimation of small blocks be consigned to the past,

unless it can be explicitly proved through very simple and long known kriging

tests that such estimation is adequate. It is our professional responsibility to

change to culture of providing what is asked for regardless of the

demonstrable and potentially serious financial risks of such approaches.



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26/140


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Glacken and Blackney A practitioners implementation of Indicator Kriging

APRACTITIONERS IMPLEMENTATION OF

INDICATOR KRIGING

Ian Glacken and Paul Blackney

Snowden Associates

Abstract

Indicator Kriging (IK) was introduced by Journel in 1983, and since that

time has grown to become one of the most widely-applied gradeestimation techniques in the minerals industry. Its appeal lies in the fact

that it makes no assumptions about the distribution underlying the sample

data, and indeed that it can handle moderate mixing of diverse sample

populations. However, despite the elegant and simple theoretical basis

for IK, there are many practical implementation issues which affect its

application and which require serious consideration. These include

aspects of order relations and their correction, the change of support,

issues associated with highly skewed data, and the treatment of the

extremes of the sample distribution when deriving estimates.

This paper discusses the theoretical and practical bases for theseconsiderations, and illustrates through examples and case studies how the

issues associated with the daily application of the IK algorithm are

addressed. Finally, some less commonly-used IK applications are

presented, and the limitations of IK are discussed, along with proposed

alternatives.

Key Words:geostatistics, indicator kriging, minerals industry, categorical

kriging, soft kriging.

Introduction

Indicator Kriging (IK) as a technique in resource estimation is over fifteen years old.Since its introduction in the geostatistical sphere by Journel in 1983, many authors

have worked on the IK algorithm or its derivatives. The original intention of Journel,

based on the work of Switzer (1977) and others, was the estimation of local

uncertainty by the process of derivation of a local cumulative distribution function

(cdf). The original appeal of IK was that it was non-parametric it did not rely upon

the assumption of a particular distribution model for its results. From slow

beginnings in the early eighties as a technique in mineral resource estimation, and in

many other natural resource mapping applications, IK has grown to become one of the



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Glacken and Blackney A practitioners implementation of Indicator Kriging

most widely-used algorithms, despite the relative difficulty in its application. It is the

prime non-linear geostatistical technique used today in the minerals industry.

This paper presents an overview of the theory of IK, followed by some discussion of

practical applications. A number of practical aspects concerning the implementation

of the IK algorithm and its variants are then discussed, including various ways toovercome some of the shortcomings of the technique. Finally, some of the less

common applications of the indicator approach are presented, and an approach which

is the successor to IK is proffered.

Overview of theory of Indicator Kriging

The concept of indicator coding of data is not new to science, but has only been

proposed in the estimation of spatial distributions since the work of Journel (1983).

The essence of the indicator approach is the binomial coding of data into either 1 or 0

depending upon its relationship to a cut-off value, zk. For a given value z(x),

=

1.

where is a parameter greater than or equal to one,zKis the grade of the maximum

cut off and is a constant such that (zK) = *(zK), the sample cumulative frequency.

Figure 9 shows the grade tonnage curves produced from the mIK and fIK estimates

using parameters = 1.5 and max = 0.995 together with the actual grade tonnagecurve. .



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Hill et al Comparison of Median and Full Indicator Kriging

Average Grade Tonnage Curves

0

2

4

6

8

10

12

14

16

18

0 10000 20000 30000 40000 50000 60000 70000

tonnage above cut off

averagegradeabovecutoff

blasthole

full IK

median IK

Figure 9: Average grade tonnage curves for mIK, fIK and blast hole

data.

Both grade tonnage curves produce tonnage estimates which underestimate the actual

value at the chosen cut off. For each cut off chosen the estimates derived from mIK

and fIK are almost identical, with fIK producing slightly higher values. However,

even though mIK and fIK underestimate the average grades at the lower cut offs, they

overestimate the average grades at the two highest cut offs.

Conclusions

This study reinforces the theory that little is lost by using the more time-efficient mIK

rather than the more involved fIK. This is true even here where we are dealing with a

highly skewed, sparse data set. However, as indicated earlier, care must be taken in

choosing the cut-off value for the common semivariogram to be used in the mIK

approach. Even though only one semivariogram is needed it may be worthwhile to

model the semivariogram at several cut offs close to the median in order to ensure a

sufficiently large range is used in the kriging procedure. For large data sets this may

not be important, but for sparse data sets such as the one we used, a judicious choice

for the cut off can help to minimise the number of locations at which only the global

cumulative distribution function is used.

Acknowledgements

The authors would like to thank WMC Resources for making available the raw data

(exploration and blast hole) from the Goodall mine. Thanks go also to D. J. Kentwell

(a former Edith Cowan University graduate student now at SRK Consulting) for



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Hill et al Comparison of Median and Full Indicator Kriging

allowing us to use his composited data for this study and for making available the

code for the bounding polygon used to delimit this irregularly shaped region.

References

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Kentwell, D. J. (1997). Fractal relationships and spatial distributions in ore body

modelling. Unpublished masters thesis, Edith Cowan University, Perth,

Western Australia.

Kentwell, D. J., Bloom, L. M., & Comber, G. A., (1997). Improvements in grade

tonnage curve prediction via sequential Gaussian fractal simulation.

Mathematical Geology. (to appear).

Quick, D., (1994). Exploration and geology of the Goodall gold mine. Proceedings

of the AUSIMM annual conference, Darwin August 1994. 75-82.



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Humphreys Case study in Uniform Conditioning, Wandoo project

LOCAL RECOVERABLE ESTIMATION:ACASE

STUDY IN UNIFORM CONDITIONING ON THE

WANDOO PROJECT FOR BODDINGTON GOLD

MINE

Michael Humphreys

Geoval

Abstract

A practical case study for estimation of the large, low grade Wandoo

deposit at Boddington Gold Mine is presented. This was broken down into

seven zones, primarily by geology. One of these zones was a higher grade

vein set that could be solid modelled and estimated separately. A

geological mineralisation envelope was applied to constrain the

estimation. This represented a broad, relatively continuous envelope in

keeping with the geology of the orebody and was loosely based on a 0.1

g/t Au cut-off. Data was composited to 9m to reflect the intended mining

bench height and the true variability expected from those benches. Waste

dykes were only excluded from this compositing if they were considered

large enough to be practically avoided when mining. Variograms of thecomposited data were not particularly clear for either Au or Cu, and a

Gaussian transform was applied to help determine a model. Gold and

copper do not exhibit the same anisotropy and there is significant

variability at less than, or equal to ,the average drill spacing. Tests were

conducted to determine the suitability of the Gaussian approach. This

approach best represents a diffusion model of spatial continuity. These

tests indicated that the Gaussian approach was suitable at Wandoo. A

global evaluation of resources was carried out using the Discrete

Gaussian Model. This was a first quick estimate of resources at cut-off

that was useful as an order of magnitude check on the final local

estimates. There was a requirement to represent selectivity in mining fora local resource estimate. It was unrealistic to try to achieve this by

directly estimating such a small block size taking into consideration the

average drillhole spacing. Therefore, the technique of Uniform

Conditioning was applied to calculate the expected proportions above a

cut-off. Previous experience has shown that this is a relatively robust

method.

Key Words: uniform conditioning, non-linear estimation, Discrete

Gaussian Model, mining selectivity.



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Introduction

Study data are from the Wandoo deposit at Boddington Gold Mine (BGM), located

approximately 130 km southeast of Perth, Western Australia. The numbers present

have been modified for confidentiality.

Mineralisation is hosted by intermediate volcanic and intrusive rocks of the Archaean

Saddleback Greenstone Belt. Unmineralised dolerite dykes transect the sequence.

The estimation area has been broken up into seven distinct geological zones. One of

these zones consists of two solid modelled, steeply-dipping actinolite veins a few

metres wide. These are generally associated with higher grades. The estimation was

confined to unoxidised host rocks.

Data

There are 2589 drillholes with over 118000 samples, mostly on 2m lengths, from

diamond (DDH) and reverse circulation (RC) drilling (Figure 1). These were coded

for rock type and zone. Exploration holes are variously spaced, but average a 25m x

25m spacing, with some zones drilled more sparsely. Inclinations vary from vertical

to sub-horizontal. Hole azimuths also vary widely. A 25m x 25m x 9m block model

was supplied, defining the geological zones, blocks to be estimated and rock type.

This block size was chosen from the data spacing and mining considerations. Smaller

blocks could not be used without possibly serious under-estimation of the variability(Vann and Guibal, 1998).

An outer boundary to possible mineralisation was created by BGM geologists at a 0.1

g/t Au cut-off. This boundary was relatively insensitive to increases in cut-off up to

approximately 0.5 g/t Au. An advantage of defining the outer boundary at a

geological cut-off was that it allowed the application of different mining cut-offs

within this boundary. Conversely, estimating with a high mining cut-off initially

would probably require re-estimation if lower cut-offs were subsequently

contemplated.

It is very important in estimation to work with equal support (volume) samples. Thisis why the data were composited to equal lengths. A bench height of 9m was

envisaged, therefore the samples were composited to that length along drillholes

within the geological envelope (excluding defined, barren dolerite samples) to best

represent real 9m bench variability. For this estimation, some dolerite was considered

as unavoidably mined, and included. Dykes that were large enough to be easily

excluded when mining, were excluded from compositing and estimation. Not to do so

may bias the estimation.



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In gold, the effect of outlying values is usually significant and some approach must be

taken to account for these. There is no single accepted method for determining upper

cuts with theoretical justification available, and no strong argument to choose one

method over another. A final cut of 40 g/t Au was employed.

Figure 1 Drillhole Location Map

Two tests for sensitivity of grade variability to cut-off were carried out (Figure 2) in

each zone. These showed that (i) removing approximately the highest 10 values,

and (ii) employing an upper cut of around 40 g/t Au, both reduced the outlier effect

significantly.



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RC and DDH composites were compared in an area that was adequately covered by

both datasets. Tests showed little difference in the statistical characteristics. This,

along with the greater volume of RC drilling, led to the decision to keep both sets of

data for the estimation. Combining data types should not be an automatic decision,

but one consciously made with supporting results.

Mean Grade and CV by Number of High Grades Removed for

Zone 1

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100

Number of Highest Grades Removed

Grade(g/t),CV

Mean

"CV"

Mean Declustered Grade and CV by Upper Cut for Zone 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.82

10 30 50 70 90 110 130

Upper Cut (g/t)

Grade(g/t),CV

Mean

"CV"

Figure 2 Sensitivity tests to cut-off grade

The spatial distribution of data is not uniform due to an irregular drilling grid, varied

length and inclination of holes. There fore we used a Declustering (weighting)

procedure so that statistics were not biased by preferential spatial position (eg many

close-spaced holes in a high grade area). This does not decrease the number of

composites used, but simply weights the histogram to produce an unbiased mean and

variance. A simple Declustering usees the number of points in a block but we



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required the more accurate method using kriging weights which is far more time

consuming. Table 1 shows the weighted means and variances compared to the original

statistics. The weighted variances and means are generally lower than the unweighted

statistics.

Au Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7

Number 8460 4240 3339 1178 2078 776 129

Mean 1.14 0.64 0.88 0.48 1.21 1.20 10.17

Variance 2.82 0.79 4.06 0.54 4.14 11.22 168.7

Weighted

Mean

1.07 0.64 0.66 0.43 0.91 0.87 8.78

Weighted

Variance

2.15 1.30 2.02 0.32 2.21 3.35 134.9

Cu Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7

Mean 1416.2 1650.9 1057.9 646.5 503.1 1556.9 2234.6

Variance 1546842 1572393 1256807 272742 318646 1233720 3177538

Weighted

Mean

1472.1 1599.7 939.9 645.1 511.0 1409.3 1905.5

Weighted

Variance

1536447 1592774 921811 297998 265296 1100511 2766932

Variograms were then calculated for Au and Cu in each of the zones on the 9m

composites. These were not particularly clear and a Gaussian transform was

employed to help define the underlying structure. Calculation of the Gaussian

transform utilised the Declustering weights previously discussed. Variograms of the

Gaussian transformed data presented clearer structures and were more easily

modelled. Models were fitted in consultation with BGM geologists taking into

account the known geological and mineralisation trends (see Figures 3 and 4).

Models for the Gaussian variables were then transformed back to models on the raw

data which will now reflect the declustered 3D spatial variability.

The Gaussian transform is very powerful, and is part of the Discrete Gaussian method

(diffusion methods). It is the only method having a built-in change of support to

reflect a deskewing of the histogram for different volumes (eg samples versus blocks).

Three tests were conducted to see whether the Discrete Gaussian model was

applicable at Wandoo, as follows:



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The Gaussian transform is very powerful and is part of the Discrete Gaussian Method

(diffusion methods). It is the only method having a built-in change of support to

reflect a deskewing of the histogram for different volumes (eg samples versus blocks).

Three tests were conducted to see whether the Discrete Gaussian Model was

applicable at Wandoo as follows:

1. checking indicator residuals (see Rivoirard, 1994, for example). If these showsome spatial correlation (as seen in cross variograms) then a Gaussian approach is

justified.

c:\msoffice\winword\template\normal.dot

Figure 3 Zone 1 Gaussian Au Variograms



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Figure 4 Zone 1 Gaussian Cu Variograms

2. Using the Deutsch and Lewis normalised indicator approach. If the Gaussianreconstructed indicator variograms are the same as those calculated immediatelyfrom the data, then the Gaussian approach is justified (see Figure 5).

3. Checking ratios of indicator variograms. If the ratio of cross variogram tovariogram increases with distance, then a diffusion or Discrete Gaussian Model is

applicable (see Figure 6).

All these conditions were satisfied (see Figures 5 and 6).



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Theoretical indicator variogram for 50th

percentile

0.00

0.10

0.20

0.30

0.40

0.50

0 50 100 150

Distance, m

Variogram Major

Semi-major

Minor

Figure 5a Gaussian reconstructed indicator variograms (test 2)

Experimental variogram for 50th percentile

0.00

0.10

0.20

0.30

0.40

0.50

2 60 120 180

Distance, m

Variogram Major

Semi-major

Minor

Figure 5b Indicator variograms (test 2)

70Symposium on Beyond Ordinary Kriging


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Figure 6 Indicator variogram ratio test for Zone 3 (test 3)

Knowing that the Discrete Gaussian Model was applicable a global estimate wasmade (see also Vann and Sans, 1995 or Guibal, 1987). By modelling the composite

histogram and knowing the Gaussian transform function, the histogram of any size

block that we want to consider for estimation can be obtained. This gives a prediction

of the global tonnes and grade above a cut-off and is a good first pass approximation

to a local result.

The last step before local estimation is to test the estimation/neighbourhood

parameters. This is important and is all too rarely performed. A consequence of not

testing and using too small a neighbourhood would be a biased, poor quality and

badly representative estimate. Many different configurations were tested and results

compared for estimation variance, bias (slope of the regression of true value with the

estimated value) and weight of the mean (a measure of the need for closer spaced

and/or more data in the neighbourhood). For further discussion see Armstrong and

Champigny (1989), Krige (1994, 1996a, 1996b and 1997), Ravenscroft and

Armstrong (1990) and Royle (1979).

Examining the kriging weights can also help determine if large negative weights or

other possible problems exists. Table 2 shows some results. It is desirable that the



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weight of the mean is below 10%, the slope of the regression is close to 1.0 (above 0.9

is preferable) and that estimation variance is minimised.

Table 2 Kriging Neighbourhood Test Results for Zone 1

Sample grid 25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

No. of informingcomposites

3 x 3 x 5 3 x 3 x 7 3 x 3 x 9 3 x 3 x11

3 x 5 x 5 3 x 5 x 7 3 x 5 x 9 3 x 5 x11

Estimated block size 25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

25 x 25x 9

Ordinary kriging result

Estimation variance 0.0971 0.0911 0.0886 0.0874 0.0937 0.0893 0.0874 0.0866

Slope of the regressionZ/ZE

0.9029 0.9382 0.9580 0.9701 0.9524 0.9745 0.9862 0.9931

Simple kriging result

Weight assigned to themean

0.1179 0.0773 0.0539 0.0393 0.0647 0.0361 0.0201 0.0104

Sample grid 25 x 25 x9

25 x 25 x9

25 x 25 x9

25 x 25 x9

25 x 25 x9

25 x 25 x9

25 x 25 x9

No. of informing composites 5 x 5 x 5 5 x 5 x 7 5 x 5 x 9 5 x 5 x11

5 x 7 x

11

5 x 7 x

13

9 x 9 x

13

Estimated block size 25 x 25 x6

25 x 25 x

6

25 x 25 x

6

25 x 25 x

6

25 x 25 x

6

25 x 25 x

6

25 x 25 x

6

Ordinary kriging result

Estimation variance 0.0920 0.0884 0.0869 0.0862 0.0862 0.0859 0.0858

Slope of the regressionZ/ZE 0.9877 0.9989 1.0042 1.0071 1.0090 1.0093 1.0024Simple kriging result

Weight assigned to the mean 0.0190 0.0018 -0.0072 -0.0125 -0.0188 -0.0203 -0.0122

The Gaussian model, the variograms and the results of neighbourhood testing provide

the parameters necessary for the kriging estimation and the non-linear local estimation

by Uniform Conditioning.

Ordinary kriging was performed but this could not be used to give a resource

reflecting the real mining selectivity. Kriging of smaller blocks would seriouslyunderstate the true variability. Therefore, Uniform Conditioning was applied to

obtain a more realistic resource estimate corresponding to the intended mining

selectivity. A selective mining unit (SMU) of 8.3 x 8.3 x 9m was used as the

minimum basis for determining ore or waste parcels.

Uniform Conditioning (Rivoirard, 1994) takes the locally estimated ordinary kriging

result and applies a change of support to calculate the expected histogram of grades

for that 25 x 25 x 9m block based on an SMU of 8.3 x 8.3 x 9m. Results are then

reported for each large block as the proportion of the block above cut-off and the



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grade above cut-off from that expected SMU histogram, knowing the estimated grade

of the entire block. Histograms comparing composites, kriged 25 x 25 x 9m blocks

and SMU results are given in Figure 7. These show the expected deskewing effect of

larger block sizes. The Discrete Gaussian Method is one of the few approaches that

takes this important deskewing into account. Affine corrections do not for example.

Figure 7 Comparing Histograms for Different Supports

he global results obtained previously can be used as an order of magnitude check forT

the Uniform Conditioning local results. In this study, agreement between global and

local results was good. If alternative cut-offs are required then it is simply a matter of



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running only the Uniform Conditioning step with the new cut-offs no other work

need be re-done.

Acknowledgements

The help, assistance and permission of the Boddington joint venture comprising

Normandy Mining, Acacia Resources and Newcrest Mining is gratefully

acknowledged. Thanks to David Burton and Simon Williams who provided assistance

on the project concerned. Thanks also to Olivier Bertoli, Henri Sanguinetti and Daniel

Guibal of Geoval who provided much valuable assistance and input.

References

Armstrong, M. And Champigny, N., 1989. A study on kriging small blocks. CIM

Bulletin. Vol. 82, No. 923, pp. 128-133.

Deutsch, C.V. and Lewis, R.W., Advances in the Practical Implementation of

Indicator Geostatistics: Appendix A: A Test for the Validity of Parametric

Methods. 23rdAPCOM Proceedings

Guibal, D., 1987. Recoverable Reserves Estimation at an Australian gold project.

Geostatistical Case Studies, G. Matheron and M. Armstrong (eds.), pp149-

168. Kluwer Academic Publishers.

Krige, D.G., 1994. An analysis of some essential basic tenets of geostatistics not

always practised in ore valuations. Proceedings of the Regional APCOM,

Slovenia.

Krige, D.G., 1996a. A basic perspective on the roles of classical statistics, data

search routines, conditional biases and information smoothing effects in ore

block valuations.Proceedings of the Regional APCOM, Slovenia.

Krige, D.G., 1996b. A practical analysis of the effects of spatial structure and data

available and accessed, on conditional biases in ordinary kriging. In:Geostatistics Wollongong 96. Proceedings of the International

Geostatistical Congress, Wollongong, NSW, Australia, September, 1996, pp.

799-810.

Krige, D.G., 1997. Block kriging and the fallacy of endeavouring to reduce or

eliminate smoothing.Proceedings of the Regional APCOM, Moscow.



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Ravenscroft, P.J., and Armstrong, M., 1990. Kriging of block models the

dangers re-emphasised.Proceedings of APCOM XXII, Berlin, September 17-

21, 1990, pp. 577-587.

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

Clarendon Press (Oxford), 181pp.

Royle, A.G., 1979. Estimating small blocks of ore, how to do it with confidence.

World Mining, April 1979.

Vann J. and Guibal D., 1998. Beyond Ordinary Kriging An overview of non-

linear estimation. Beyond Ordinary Kriging Seminar, Geostatistical

Association of Australasia, Perth, WA(this volume).

Vann J. and Sans H., 1995. Global resource estimation and change of support at the

Enterprise Gold Mine, Pine Creek, Northern Territory Application of the

geostatistical Discrete Gaussian model. Proceedings of the APCOM XXV1995 Conference (Brisbane), Aus. I.M.M., PP. 171-180.



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Jones Case study Indicator Kriging and the Mount Morgan goldcopper deposit

ACASE STUDY USING INDICATOR KRIGING

THE MOUNT MORGAN GOLD-COPPER

DEPOSIT,QUEENSLAND

Ivor Jones

WMC Resources

Abstract

In August 1882, the Morgan brothers recognised a mineral deposit, now

known as the Mount Morgan Gold-Copper Deposit. The final productionfigures for the mine were 250 tonnes of gold and 360,000 tonnes of copper

from 50 million tonnes of ore, making the average grades 4.99g/t gold and

0.72% copper.

A three dimensional grade model was made of the pre-mined gold

distribution within the Main Pipe mineralisation between the Slide Fault

(to the west) and the Andesite Dyke (to the east), and bound to the south

by the South Dyke.

Indicator kriging provided a method for estimating the grade in the

strongly skewed gold distribution, without the problems of smearing of the

high grades as seen in linear techniques. The application of indicatorkriging using grade thresholds based on the declustered sample decile

values was shown to be a poor application of indicator kriging, but was

greatly improved by modifying grade thresholds above the median so that

the amount of contained metal was evenly distributed between these

classes.

The pre-mined resource estimate for this portion of the Main Pipe

mineralisation using a 2g/t lower selection limit was 3,526,800 tonnes

with an average grade of 11.98g/t, equivalent to 42.25 tonnes gold.

Key Words: geostatistics, non-linear grade estimation, indicator kriging.

Introduction

The Mount Morgan Mine in central Queensland was for many years described as the

greatest gold mine on earth with grades as high as 2000 ounces per tonne. The aim of

the study was to prepare a detailed three dimensional computer model of the grade

distribution within a select area of the Mount Morgan deposit (pre-mining), and to

compare some different estimation techniques. Indicator kriging was used to estimate

the grades in this paper.



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The study area is a section of the Main Pipe mineralisation between the Andesite

Dyke and the Slide Fault, and north of the South Dyke (Figure 1). The northern,

lower and upper boundaries are those of the known data.

Andesite Dyke

Slide

Fault

South Dyke

Figure 1 Location of the Study Area in relation to the major

geological features within the Mt Morgan deposit.

The Main Pipe mineralisation was a concentrically zoned sub-horizontal pipe-like

orebody (Jones and Golding, 1994), with the highest grades in the centre of the pipe.

At the western end of the study area, there is a narrow vertical high grade gold shoot.

The aim of the study was to prepare a detailed three dimensional computer model of

the grade distribution within a select area of the Mount Morgan deposit (pre-mining).

The model itself was used to investigate the three dimensional grade distribution of

the deposit relating it to the recorded geology using computer visualisation.

In order to do this, a three dimensional computer model of the distribution of metalwithin the study area was prepared, and indicator kriging was used for grade

estimation. A comparative study using two different methods of determining grade

thresholds in the study was also performed.

Indicator kriging, as proposed by Journel (1982), has been a well accepted technique

by the geostatistical community for dealing with skewed distributions and extreme

values. It is a non-parametric estimation procedure, is not based on the data fitting a

particular statistical distribution, is resistant to the influence of outliers, and is based

on the knowledge that different parts of a mineralisation can have different spatial



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characteristics. High grade mineralisation may be limited in spatial extent, and in

strongly skewed distributions, the contained metal can contribute a significant

proportion of the ore reserves (Journel and Arik, 1988), although not all high grade

occurrences may have been sampled. Alternatively, low grade mineralisation may be

pervasive, and spatially extensive. It was therefore a suitable procedure for gold grade

estimation in the Mount Morgan Deposit.

Data analysis

Underground mining at Mount Morgan was primarily by square set stoping, and large

open stopes or chambers. A face sample was taken for each square set and assayed

for gold, copper and in select areas silica, the assays being recorded on level plans.

The square sets, were approximately 5 feet by 6 feet square by 7 feet 9 inches high

(Patterson and Thomas, 1910). The assays for each square set from every fourth level

were digitised as a point representing the centre of the square set location. The square

set stope data was the basis of this study.

Moving window statistics

Moving Window Statistics were calculated for the study area as well as the rest of the

Main Pipe mineralisation. They were used as a tool for examination of the data in a

spatial context, and as a tool for determining if high grade zones could be separated

from the remainder of the deposit for modelling purposes.

The contoured plans of the moving window statistics (Figure 1) and visual

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