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
Mining & Resource Technology
Contact: Bill Shaw
Resource Service Group
Contact: Dr Julian Barnes
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SATCHEL SPONSOR
Geoval Australias Geostatistical Experts.
Contact: John Vann
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.
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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)
GAA Executive Committee
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),
GAA Executive Committee
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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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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Armstrong, M. and Champigny, N., 1989. A study on kriging small blocks. CIM
Bulletin. Vol. 82, No. 923, pp.128-133.
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Baafi, E., and Schofield, N.A., 1997. Geostatistics Wollongong 96. Proceedings of
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David, M., 1972. Grade tonnage curve: use and misuse in ore reserve estimation.Trans. IMM, Sect. A., Vol. 81, pp.129-132.
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Developments in Geomathematics 6. Elsevier (Amsterdam), 216pp.
Dowd, P.A., 1982. Lognormal kriging the general case. Mathematical Geology,
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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
Deutsch, C. V., & Journel, A. G. (1992).GSLIB: Geostatistical software library andusers guide. New York: Oxford University Press.
Goovaerts, P. (1997). Geostatistics for natural resources evaluation. New York:
Oxford University Press.
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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Humphreys Case study in Uniform Conditioning, Wandoo project
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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Humphreys Case study in Uniform Conditioning, Wandoo project
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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Humphreys Case study in Uniform Conditioning, Wandoo project
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)
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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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Jones Case study Indicator Kriging and the Mount Morgan goldcopper deposit
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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Jones Case study Indicator Kriging and the Mount Morgan goldcopper deposit
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