APPLIED MINING GEOSTATISTICSAPPLIED MINING GEOSTATISTICSMODULE I

Workshop

Newmont Gold Corporation

Boca Ratón, Florida, 24-28 September 2007

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INTRODUCTIONINTRODUCTIONGoals of this workshop:Goals of this workshop:–– Discuss and understand when and where Discuss and understand when and where

geostatisticsgeostatistics can add value to your product.can add value to your product.–– Understand data issues when applying Understand data issues when applying

geostatisticsgeostatistics..–– Introduce or deepen (as the case may be) the Introduce or deepen (as the case may be) the

ability to use GSLib or other programs to ability to use GSLib or other programs to develop resource models.develop resource models.

–– Step through major aspects of resource Step through major aspects of resource modeling.modeling.

–– Understand the limitations of the techniques Understand the limitations of the techniques discussed and the resulting models.discussed and the resulting models.

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INTRODUCTION INTRODUCTION (cont)(cont)

Understand the objectives of a Resource Understand the objectives of a Resource Model. Model. Different types of resource models for Different types of resource models for different objectives. Longdifferent objectives. Long--, medium, medium--, and , and shortshort--term models. term models. Conversion of Resources into Reserves.Conversion of Resources into Reserves.Conditional Simulation (CS) studies, the Conditional Simulation (CS) studies, the concept.concept.Discussion of basic geostatistical concepts.Discussion of basic geostatistical concepts.Emphasis on practical aspects and Emphasis on practical aspects and applications.applications.

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INTRODUCTION INTRODUCTION (cont)(cont)

Sampling and data quality issues.Sampling and data quality issues.Dilution.Dilution.Variability vs. smoothing.Variability vs. smoothing.Uncertainty and Risk.Uncertainty and Risk.How do all this fit in resource and reserve How do all this fit in resource and reserve modeling work, and in mine development modeling work, and in mine development and mine planning tasks?and mine planning tasks?Some geostatistical knowledge assumed!Some geostatistical knowledge assumed!

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INTRODUCTION INTRODUCTION (cont)(cont)

Brief Table of Contents:Brief Table of Contents:–– Random Variables and Random Functions.Random Variables and Random Functions.–– Statistical Analysis (EDA), including clustering, Statistical Analysis (EDA), including clustering,

dealing with outlier data, and geologic and dealing with outlier data, and geologic and estimation domain definition.estimation domain definition.

–– Variography.Variography.–– Recoverable ResourcesRecoverable Resources–– Various forms of Kriging.Various forms of Kriging.–– And minor discussions, time permitting, on And minor discussions, time permitting, on

model validation and reconciliation, conversion model validation and reconciliation, conversion of resource into reserves, and geostatistical of resource into reserves, and geostatistical Conditional Simulations.Conditional Simulations.

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A LITTLE HISTORYA LITTLE HISTORYFormal modern Probability Theory dates Formal modern Probability Theory dates back to the 1600back to the 1600’’s (s (BlaiseBlaise Pascal and Pascal and Fermat).Fermat).The foundation for The foundation for geostatisticsgeostatistics was laid was laid out in the early to mid 1900out in the early to mid 1900’’s by s by KolmogorovKolmogorov, , MaternMatern, Weiner, and , Weiner, and GandinGandin..Geostatistics started in the late 1950Geostatistics started in the late 1950’’s and s and 19601960’’s with Krige and s with Krige and SichelSichel in South in South Africa; and formalized by Matheron in Africa; and formalized by Matheron in France France ““Theory of Regionalized VariablesTheory of Regionalized Variables””..

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A LITTLE HISTORY A LITTLE HISTORY (cont)(cont)

Two of Matheron students (Journel and Two of Matheron students (Journel and David) moved to North America and David) moved to North America and established geostatistical centers.established geostatistical centers.Popular first in mining and meteorology, Popular first in mining and meteorology, geostatisticsgeostatistics in now used in many fields: in now used in many fields: forestry, petroleum industry, forestry, petroleum industry, environmental assessment and environmental assessment and remediation, fisheries, image processing, remediation, fisheries, image processing, alternative energy sources, etc.alternative energy sources, etc.Current main centers for Current main centers for geostatisticsgeostatistics are are FountainbleauFountainbleau, Stanford University, , Stanford University, University of Alberta, and others. University of Alberta, and others.

88Concept and slide courtesy of DAT Mine Consulting.

MODELING AND PLANNING WORK IS AN MODELING AND PLANNING WORK IS AN ITERATIVE PROCESSITERATIVE PROCESS

SCHEDULING

ANALYSISPIT LIMIT

MODELRESOURCE

AND COSTINGEQUIPMENT SIZING

THROUGHPUTMINE SIZE/

ANALYSISCUT-OFF GRADE

SEQUENCINGPHASING AND

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MODELING AND PLANNING WORK IS MODELING AND PLANNING WORK IS AN ITERATIVE PROCESSAN ITERATIVE PROCESS

Long Term Plan Short Term Plan

OperationsProduction Drilling Haulage and Transport

Finances

Marketing

Geotech Geology

Surveying

Laboratory. Metallurgy

Concept and slide courtesy of DAT Mine Consulting.

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CLASSIC STATISTICS OR CLASSIC STATISTICS OR GEOSTATISTICS?GEOSTATISTICS?

Most of classical statistics is based on the Most of classical statistics is based on the Central Limit Theorem (Gaussian theory). Central Limit Theorem (Gaussian theory). Limited applications to spatially correlated Limited applications to spatially correlated variables.variables.Geostatistics also has been developed in Geostatistics also has been developed in large measure based on Gaussian theory, large measure based on Gaussian theory, but it has been adapted to deal with but it has been adapted to deal with spatially correlated variables.spatially correlated variables.Geostatistics was born out of practical Geostatistics was born out of practical needs, and is likely to remain an applied needs, and is likely to remain an applied science, a specialized branch of classical science, a specialized branch of classical statistics.statistics.

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ADVANTAGES OF USING ADVANTAGES OF USING GEOSTATISTICSGEOSTATISTICS

Uses the spatial continuity of the data, Uses the spatial continuity of the data, which is a reflection of the physics of the which is a reflection of the physics of the underlying processes.underlying processes.The modeling of the spatial continuity The modeling of the spatial continuity accounts for quantitative data (assays, accounts for quantitative data (assays, core samples, fuzzy seismic), and also for core samples, fuzzy seismic), and also for the qualitative information the qualitative information (geologist/expert information). (geologist/expert information). The physical/geological interpretation and The physical/geological interpretation and modeling of the variables allows to go modeling of the variables allows to go beyond actual data.beyond actual data.

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ADVANTAGES OF USING ADVANTAGES OF USING GEOSTATISTICS GEOSTATISTICS (Cont)(Cont)

The physical/geological interpretation and The physical/geological interpretation and modeling of the variables allows to go modeling of the variables allows to go beyond actual data.beyond actual data.Provides a framework for Provides a framework for ““transportingtransporting””geologic/physical interpretations further geologic/physical interpretations further downstream.downstream.Mathematical models (and the Mathematical models (and the incorporated geology) are to be used in incorporated geology) are to be used in the process design. There should be the process design. There should be consistency between the two.consistency between the two.Provides the tool for assessing modeling Provides the tool for assessing modeling uncertainty.uncertainty.

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DISADVANTAGES OF DISADVANTAGES OF USING GEOSTATISTICSUSING GEOSTATISTICS

Does not replace or create actual data!Does not replace or create actual data!Does not interpret or model by itself. Only Does not interpret or model by itself. Only provides tools for interpretation and provides tools for interpretation and modeling. modeling. Does not replace expert decisions: Does not replace expert decisions: modeling cannot be done by a computer.modeling cannot be done by a computer.Forces the geologists, Forces the geologists, geostatisticiansgeostatisticians, and , and mining engineers to work harder! Most mining engineers to work harder! Most importantly, forces the experts to make importantly, forces the experts to make decisions consistent with the data decisions consistent with the data we we are dataare data--driven! driven!

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SOME BASIC CONCEPTS SOME BASIC CONCEPTS 1.1. Modeling Scale:Modeling Scale: mining does not occur at core mining does not occur at core

(drill hole data) scale, but at a much larger (drill hole data) scale, but at a much larger scale. Keep in mind also the time factor in the scale. Keep in mind also the time factor in the definition of the scale of interest. definition of the scale of interest.

2.2. Numerical Modeling: numbers replace Numerical Modeling: numbers replace subjective concepts to some extent. There is subjective concepts to some extent. There is only only oneone true value that we will never know; so true value that we will never know; so our probabilistic models are our best guesses.our probabilistic models are our best guesses.

3.3. Stationarity, how it Stationarity, how it relatsrelats to domaining.to domaining.4.4. Uncertainty: it is not a property of the Uncertainty: it is not a property of the

mineralization. It stems from our incomplete mineralization. It stems from our incomplete knowledge (and sometimes incorrect use of knowledge (and sometimes incorrect use of numerical models.numerical models.

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SOME BASIC CONCEPTS SOME BASIC CONCEPTS (Cont(Cont))

5.5. Smoothing: most of the methods we use Smoothing: most of the methods we use sacrifice representing the true variability of the sacrifice representing the true variability of the information for a better guess of the average information for a better guess of the average value. Not suited for modeling extreme of data value. Not suited for modeling extreme of data distributions.distributions.

6.6. Dilution, its relation to smoothing, and how to Dilution, its relation to smoothing, and how to control/model it.control/model it.

7.7. Where does conditional bias fit into all of this?Where does conditional bias fit into all of this?8.8. Impact of the mistakes we make: differences Impact of the mistakes we make: differences

between uncertainty and risk. The concept of between uncertainty and risk. The concept of Loss and its use in modeling.Loss and its use in modeling.

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BREAK BEFORE WE GO INTO BREAK BEFORE WE GO INTO DETAILS?DETAILS?

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REGIONALIZED VARIABLES REGIONALIZED VARIABLES AND RANDOM FUNCTIONSAND RANDOM FUNCTIONS

Consider an attribute (mineral Consider an attribute (mineral grades, seam thicknesses, pH, or grades, seam thicknesses, pH, or just about anything!) varying in the just about anything!) varying in the space space A: A: z(z(xx), ), xx εε A.A.The value The value z(z(xx) is interpreted as a ) is interpreted as a realization of realization of Z(Z(xx). ). z(z(xx) is a regionalized variable, while ) is a regionalized variable, while Z(Z(xx) is the random variable. ) is the random variable.

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REGIONALIZED VARIABLES REGIONALIZED VARIABLES AND RANDOM FUNCTIONS AND RANDOM FUNCTIONS (Cont)(Cont)

The approach is then to randomize the variables The approach is then to randomize the variables {{z(z(xx), ), xx εε A},A}, interpreting them as realizations of interpreting them as realizations of the set the set {{Z(Z(xx), ), xx εε A}.A}.The random variables The random variables {{Z(Z(xx), ), xx εε A} A} have allhave all the the same probability distribution function: same probability distribution function:

FF((xx1,...,,...,xxN; ; zz1,...,,...,zN) = ) = Prob{Prob{ZZ((xx1) ) ≤≤ zz1,..., ,..., ZZ((xxN) ) ≤≤ zN }, also written as}, also written as

Prob{Z(xProb{Z(x) ) ≤≤ z} = z} = F(zF(z),), independent of independent of xx..

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REGIONALIZED VARIABLES REGIONALIZED VARIABLES AND RANDOM FUNCTIONS AND RANDOM FUNCTIONS (Cont)(Cont)

The univariate cdf of the RV Z(x) is used The univariate cdf of the RV Z(x) is used to characterize uncertainty about the to characterize uncertainty about the value z(x); the multivariate cdf is used to value z(x); the multivariate cdf is used to characterize joint uncertainty about the N characterize joint uncertainty about the N values z(values z(xx11) , ... , z() , ... , z(xxNN).).The bivariate (N=2) cdf of any two RVs The bivariate (N=2) cdf of any two RVs Z(Z(xx11), Z(), Z(xx22), is particularly important ), is particularly important since conventional geostatistical since conventional geostatistical procedures are restricted to univariate procedures are restricted to univariate (F(x;z)) and bivariate distributions:(F(x;z)) and bivariate distributions:

F(F(xx11, , xx22;z;z11,z,z22)=Prob{Z()=Prob{Z(xx11) ) ≤≤ zz11,Z(,Z(xx22) ) ≤≤ zz22}}

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REGIONALIZED VARIABLES REGIONALIZED VARIABLES AND RANDOM FUNCTIONSAND RANDOM FUNCTIONS (cont)(cont)

The description just made is the definition The description just made is the definition of of stationarity. stationarity. It is an expertIt is an expert’’s decision, who s decision, who decidesdecideswhat area what area AA is statistically homogeneous is statistically homogeneous (stationary)(stationary). . Stationarity is necessary because it is the Stationarity is necessary because it is the only way we can pool data values coming only way we can pool data values coming from different locations (and/or times) from different locations (and/or times) into a single population.into a single population.The RV The RV Z(Z(xx) ) are not independent from are not independent from each other.each other. Their setTheir set is calledis called the the Random FunctionRandom Function {{Z(Z(xx), ), xx εε A}.A}.

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REGIONALIZED VARIABLES REGIONALIZED VARIABLES AND RANDOM FUNCTIONSAND RANDOM FUNCTIONS (cont)(cont)

It is important to always keep in mind that It is important to always keep in mind that the only real attributes are the the only real attributes are the regionalized variables, regionalized variables, z(z(xx),), i.e., the i.e., the actual samples. The rest is only a actual samples. The rest is only a construction necessary for geostatistical construction necessary for geostatistical modeling.modeling.Matheron defined stationarity based on Matheron defined stationarity based on the existence of moments of the Random the existence of moments of the Random Function Function {{Z(Z(xx), ), xx εε A}. A}. The concepts may The concepts may be more obscure, but familiar at the same be more obscure, but familiar at the same time. time.

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CONVENTIONAL DEFINITION CONVENTIONAL DEFINITION OF STATIONARITYOF STATIONARITY

Strict Stationarity: all the Strict Stationarity: all the multivariate statistical moments of multivariate statistical moments of the RF defined are the same. This is the RF defined are the same. This is also called also called ““Invariance under Invariance under TranslationTranslation””, meaning every RV at , meaning every RV at every location every location x x has the same has the same distribution function as the others.distribution function as the others.

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CONVENTIONAL DEFINITION CONVENTIONAL DEFINITION OF STATIONARITY OF STATIONARITY (cont) (cont)

Stationarity of Order 2: only the first two Stationarity of Order 2: only the first two moments (Expected value and Covariance) of the moments (Expected value and Covariance) of the RF are invariant. If the covariance exists RF are invariant. If the covariance exists (stationarity of Order 2), then the variance and (stationarity of Order 2), then the variance and

the covariance will necessarily exist.the covariance will necessarily exist.

{ } ( ) ( )E Z m zdF z zf z dz+∞ +∞

−∞ −∞

= = =∫ ∫

{ , } {[ ][ ]} { }

( )( ) ( , )

X Y X Y

X Y XY

Cov X Y E X m Y m E XY m m

E dx x m y m f x y dy+∞ +∞

−∞ −∞

= − − = −

= − −∫ ∫

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CONVENTIONAL DEFINITION CONVENTIONAL DEFINITION OF STATIONARITY OF STATIONARITY (cont)(cont)

““IntrinsicIntrinsic”” stationarity is when the stationarity is when the expected value of the RF does not expected value of the RF does not depend on the location (as before), depend on the location (as before), and the variance of all increments and the variance of all increments hhis finite and does not depend on its is finite and does not depend on its location location xx::

( ) ( ) ( ) ( ) ( ){ }22 Var Y Y E Y Yγ = − + = − + h u u h u u h

2727

CONVENTIONAL DEFINITION CONVENTIONAL DEFINITION OF STATIONARITY OF STATIONARITY (cont)(cont)

QuasiQuasi--stationarity is when the stationarity is when the moments of the RF are inferred only moments of the RF are inferred only for some local areas within the for some local areas within the overall volume. This is in practice overall volume. This is in practice what is used most in what is used most in geostatisticsgeostatistics. . This is also called This is also called ““locallocal”” stationarity, stationarity, and commonly found when we have and commonly found when we have large volumes (estimation domains) large volumes (estimation domains) and many data to work with.and many data to work with.

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STATIONARITY: HYPOTHESES STATIONARITY: HYPOTHESES OR DECISION?OR DECISION?

The most important decision is to define The most important decision is to define what part of the physical space what part of the physical space AA is is statistically statistically ““homogeneoushomogeneous””..Can this decision be tested Can this decision be tested a prioria priori??Stationarity implies that all data within Stationarity implies that all data within AAcan be pooled together. Statistical can be pooled together. Statistical moments can be obtained, and all other moments can be obtained, and all other population properties are appropriate population properties are appropriate descriptors of the Random Functiondescriptors of the Random Function. .

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STATIONARITY: HYPOTHESES STATIONARITY: HYPOTHESES OR DECISION? OR DECISION? (Cont.)(Cont.)

Any statistical test would be performed on Any statistical test would be performed on the model, the model, notnot on the real space on the real space AA..The decision of The decision of ““homogeneityhomogeneity”” can only be can only be assessed assessed a posterioria posteriori..In addition, in most cases the decision is In addition, in most cases the decision is affected by practical reasoning: lack of affected by practical reasoning: lack of data, knowledge about the physics of the data, knowledge about the physics of the phenomenon, consequences or impact of phenomenon, consequences or impact of the variables being modeled, etc.the variables being modeled, etc.

3131

STATISTICAL ANALYSESSTATISTICAL ANALYSESProvides a first look at the data. Also called Exploratory Provides a first look at the data. Also called Exploratory Data Analysis (EDA).Data Analysis (EDA). Should include:Should include:–– Histograms;Histograms;–– ScattergramsScattergrams and correlation plots;and correlation plots;–– QQ--Q and PQ and P--P plots;P plots;–– Contour Maps;Contour Maps;–– Moving Averages;Moving Averages;–– Declustering techniques;Declustering techniques;–– Etc.Etc.

The objective is to identify spatial and global features that The objective is to identify spatial and global features that would allow for a more informed stationarity decision, such would allow for a more informed stationarity decision, such as:as:–– Trends;Trends;–– Sampling errors;Sampling errors;–– Presence of multiple populations;Presence of multiple populations;–– Impact of outliers;Impact of outliers;–– Etc.Etc.