As presented to the CIM Thunder Bay Branch, by Alan Aubut, February 9, 2011 (with some minor additions)

Presentation is divided into 4 parts:

1. Introduction to Geostatistics

2. The Resource Estimation Process

3. Resource Classification

4. Conclusions

5. Bibliography

- Geostatistics is the analysis and prediction of spatial or temporal phenomena, such as metal grades.

- It is used to analyze and predict values of a variable distributed in space that are implicitly assumed to be correlated with each other.

- Predictions at unsampled locations are made using "kriging" or they can be simulated using "conditional simulations.”

- Traditional estimation methods (ID, Polygonal, Etc.) are biased:

- Weights assigned to each sample are strictly arbitrary.

- Geostatistics determines weights based on the correlation of the samples themselves (determined using the Variogram).

- Kriging strives to reduce the error.

- BLUE : Best Linear Unbiased Estimator

Stationarity: the mean and variance of values do not depend on location.

The Mean is assumed to be constant and independent of location.

The variogram is assumed to exist and the increments depend only on the difference between any two points (sample locations).

You must choose a “Region of Stationarity”- samples that are related to one another (a single population)

The foundation of Geostatistics.

It is a two-point statistic that is the mean squared difference for samples at a specific separation (h), or “lag”.

Function of distance (h) and direction.

Quantifies the variability between two random variables separated by a distance vector h (the lag).

It is used to model the way pairs of samples in space are correlated.

𝛾 𝐡 =1

2𝑁(𝐡) [𝑧 𝐮 − 𝑧(𝐮+ 𝐡)]2

𝑁(𝐡)

Each point is the average of all pairs for a particular “lag”.

N

Plan View

xh=5’

x+hx h=10’ x+h

Ni Grade at point (x)

Ni

Grad

e a

t p

oin

t (x

+h

)

1:1 li

ne

++++

+

++

+

+

+

+

+

+

++

++

Variogram=moment of inertia=i=1

n

2

1 (xi - (xi +h))2

H-scattergram for different lags”.

Ni Grade at point (x)

Ni

Grad

e a

t p

oin

t (x

+h

)

1:1 li

ne

++++

+

++

+

+

+

+

+

+

++

++

Variogram=moment of inertia=i=1

n

2

1 (xi - (xi +h))2

Distance between Samples

Sill = Variance of the sample values

Range = Distance where maximumvariability between samplesis reached (ie the sill).

Nugget = analytical error + natural variability

Spatial Variance

Ni Grade at point (x)

Ni

Gra

de a

t p

oin

t (x

+h

)

1:1 li

ne

++++

+

++

+

+

+

+

+

+

++

++

Variogram=moment of inertia=i=1

n

2

1 (xi - (xi +h))2

Variogram Model

A geostatistical estimation technique that uses a linear combination of surrounding sampled values to make a prediction.

Kriging allows you to derive weights from the variogram that result in optimal and unbiased estimates.

Weights are applied to each surrounding sampled data.

Kriging attempts to: (a) minimize the error variance; and (b) systematically set the mean of the prediction errors to zero, so that there are no over- or under-estimates.

Ordinary Kriging - the sum of the weights add up to one.

Simple Kriging – the mean and the variance of the system are known

Indicator Kriging – the probability that a value is 0 or 1.

Multiple Indicator Kriging (MIK) – uses Indicator Kriging to estimate the probability that a value is above a cut-off, or “indicator”. The combined probabilities can then be combined to get a point estimate. Used for highly skewed distributions.

Universal Kriging – estimating data that has a regional trend.

Resource Estimation- least biased method.

Data Interpolation– can be used with many data set that are spatially of temporally distributed

- Must honour “Stationarity”

Conditional Simulation– best way to analyze “Risk”.

Used extensively in the Petroleum industry

Can be and should be used more in the Minerals industry

Kriging (and all other weighted averaging methods) over-smooths the values

Smoothing results in the blocks having a variance lower than the true variance

Conditional Simulation builds many equally probable densely informed point realizations.

Each realization reproduces the input sample histogram and the variogram

Each realization is different because of the uncertainty away from known points (thus simulations and not estimates)

1. Turning Bands- Simple method that does not require data to be on a regular

grid- Requires two passes; non-conditional simulation and then

conditioning- Requires Gaussian transformation of the data

2. Sequential Gaussian- Requires a Gaussian transformation of the data- Data needs to be on a regular grid- Simulation is done an a random path through the grid utilizing

both hard and soft data at the same time- Very fast computationally

3. Sequential Indicator- Does not require a Gaussian transformation- Best for categorical variables (i.e. Rock Type)

Reproduces the Variability of the input data

HISTOGRAM

Reproduces the Spatial Continuity of the input data

VARIOGRAM

Measures the likelihood of the desired outcome (risk)

Recognizes that many equally-likely models of reality exist

Mean = 0

Variance = 1

Flowchart of the various stages of construction of a simulation matching the histogram, the variogram and the data.

Summary of the conditioning process for the turning bands method (after David, 1977, p. 327). From Olea, 1999, p. 162.

NN Kriged

Sim10 Sim14

Variable grade dispersion

Smoothed estimates

Simulations preserves natural variability

Average of all the Simulations

One Simulation Realization

Conditional Simulations allow the spatial grade variability to be properly characterized (assuming sample data histogram, variogram and domain size, shape and location are representative).

The results from Conditional Simulations can be used to determine the range of possible resources and reserves as defined by various levels of mining selectivity.

Conditional Simulations allow practitioners to quantify the uncertainty associated with any given reserve estimate and to perform better sensitivity analyses to estimate the risk of not achieving the project’s financial targets.

Conditional Simulation provides an additional tool in mineral assessments to enable key decision makers to better understand the risk associated with projects.

1. Assemble drill hole sample data

2. Validate

3. Create Mineral Envelope

4. Capture and composite samples

5. Create unfold Strings

6. Unfold data

7. Variography

8. Create NN and OK Models

9. Apply Smoothing Correction

10. Report Resources

Validate Drill hole Data

Use Histograms

Use Scatter plots

Evaluate sections and plans

Check for Co-ordinate system errors

Check for Azimuth problems

Check for Elevation Problems

Check for sample Overlaps

Median(50%)

Mean(average)

stdv

Element of Interest (%)

Dispersion - The degree of scatter of data, usually about an average value

Skewness - the degree of asymmetry of a distribution around its mean

mode

Normal Distribution – Symmetric– Unimodal

Skewed Distribution

– Asymmetric– can be positively skewed or negatively skewed

arithmeticmean

Au (oz/t)

1m samples

5m samples

10m samples

Kurtosis - the degree of “peakedness” of a distribution around its mode

Bi-modal Distribution – There are two distinct populations.

- Likely due to presence of multiple domains.- Cannot assume “Stationarity”- If possible need to separate the two populations.

Ni (%)

Ni = a * Cu + b

b

a

outlier

Geostatistics is the study of spatial data.

Errors in the data result in errors in the analysis.

A single aberrant datum can have a significant effect on the variogram.

Histograms and correlation plots can be used to systematically evaluate the data prior to doing variography.

What was sample recovery and did it vary?

Are the samples consistently sampled and analyzed over time and space?

Are the sample sizes (especially diameter) consist over time and space?

Model the mineral system (define the Stationarity System)

Do not model economic cut-offs

Snap interpretation lines to boreholes

Interpretation must be in 3 Dimensions

Need to be “snug” to mineralisation

Respect different mineralisation styles

Separate into different domains if need be

Must be Samples that are related to one another

Each domain is modelled separately

37000 Section 37250 Section

Use mineral domain wire frame to capture samples

Assess samples for missing assays and Density or Specific Gravity (SG)

Use polynomial regression to calculate Density/SG

Composite to uniform sample length based on mean sample length and intended block size.

All calculations from this point onward treat all data as point data.

Each point represents the same volume, or “support”, otherwise a bias is introduced.

You need to “Unfold” if:

1. There is a change in dip

2. There is a change in strike and dip

3. There are embayment structures

4. There are pinch and swell structures

5. There is folding of the mineralisation

• Needed because variogram is best determined with a flat data set.

Use of Unfolding simplifies Domaining

“pinch and swell” structurevariable dip

Used to deal with various geological problems

Geological Distance

Cartesian Distance

12

3

• This unfolding procedure resulted in the creation of a new

set of co-ordinates (UCSA [across dip direction], UCSB [down

dip direction], and UCSC [along strike direction]) for each

sample and each discretization point of the deposit

A – AcrossB – parallel to unfold stringsC – Orthogonal to B

The variogram

1. Independent of location and depends only on distance between sample pairs

2. Measure of similarity of samples as a function of distance

3. Foundation of geostatistics

4. Kriging relies on the variogram

• The variogram determines what weight will be applied to each sample.

• All weights for each estimate must sum to 1.

The variogram “Search Cone” and its components.

Lag – distance between samples

Distance

Variogram Model

“Experimental Variogram”

C2

a2

C1

C0

a1

Also known as “Structural Analysis”

Involves constructing a model of the experimental semi-variogram.

Requires knowledge of the phenomenon being modelled (i.e. grade)

Also requires good “craft” based on experience and/or advice of those with experience.

The purpose of the variogram is to condense the main structural features of the phenomenon being studied into an operational form.

Four primary models used and about a dozen available

Each is defined by a mathematical formula.

Spherical Model Exponential Model

Gaussian ModelPower Model

a – the “Range”h – the “Lag”c – the “sill” of the structure

Variations between samples are due to many causes over a range of scales including:

Sample support due to variability in measurements.

Petrographic level due to variations between minerals.

The mineralised horizon due to the variation between mineralised material and waste.

All come into play simultaneously

Result in needing to “nest” models to represent all of these structures – additively combine multiple models.

Isotropic – no difference with direction

Anisotropy – differs with direction

Geometric Anisotropy – all rise to the same sill but each primary axis has a different range.

Typical in mineralised systems.

Zonal Anisotropy – there are at least two sills, depending on direction.

Usually associated with multiple bedded mineral horizons

The Dimensions are derived from the Variogram Ranges.

The data used to calculate the semi-variogram must be of the same support (sample dimensions)

Core samples should not be mixed with samples of another type (i.e. RC)

Hole sizes should be similar.

Sample lengths should be similar.

If not, composite to a uniform sample length

Use ranges determined from modelling to establish how far we need to look for samples before we estimate block.

Try and eliminate bias through clustering by applying sample selection restrictions.

Typically use an “Octant Search”

Search ellipsoid is divided into octants.

Minimum and maximum number of samples per octant.

Minimum number of octants

To ensure all blocks are estimated use 3 nested searches

Parameters are less restrictive than for the previous search

- The size of the blocks in the model will best match mining selectivity, sampling density and anticipated grade control method (dynamic process if drilling density is changed or a different mining method is considered)

- SMU (Smallest Mining Unit)

- Selectivity will be a function of the equipment/mining method that possibly will be used

- Allows inclusion of internal waste and some external waste

- Resultant model will be “diluted”

- Closer to mining reality

Block Model Definition

Model first using Nearest Neighbour estimation

Provides declusterised global statistics

Global Mean should be the same no matter what method is used thus use NN Mean as check (Stationarity).

Model using Kriging

Used because it reduces error to a minimum

Least biased

Was the grade distributed in the NN model in a reasonable fashion?

Is the mean of the OK Model essentially the same as the mean for the NN Model?

Was the grade distributed in the OK model in a reasonable fashion?

Do both models honour the drill holes in a reasonable fashion, allowing for “change of support” smoothing?

NN Model OK Model

Kriging tendencies:

Underestimates values above the mean

Overestimates values below the mean

This distortion is called Conditional Bias

Alternative name is “Smoothing”

For skewed distributions it results in not enough high grade samples (the upper tail is fore-shortened)

Variance is lower than expected.

Evaluate using following relationship:

Sample Variance in deposit (Vd) = Variance between blocks (VB) + Sample variance in Block (Vb)

Zv*

Zv

Regression line

(no bias)

m*

m

waste

processed

Ore

dumped

Zc

Est

imate

d G

rad

e

Actual Grade

Two methods commonly used to correct for Smoothing

1. Affine

A direct variance correction

All secondary modes are reproduced

Can result in negative values

2. Log Normal Shortcut

For highly skewed (~ Log Normal) distributions

Both restore variance to expected levels

Classification of Resources is wholly dependant of Confidence in the estimate

Two primary factors to consider

1. During which search was the block estimated?

Most restrictive criteria result in highest confidence in resulting estimate.

2. What confidence do we have in the Variogram which was used to determine our primary search criteria (the ranges) and from which the Kriging Weights were derived?

A Mineral Resource is a concentration or occurrence of diamonds, natural solid inorganic material, or natural solid fossilized organic material including base and precious metals, coal, and industrial minerals in or on the Earth’s crust in such form and quantity and of such a grade or quality that it has reasonable prospects for economic extraction. The location, quantity, grade, geological characteristics and continuity of a Mineral Resource are known, estimated or interpreted from specific geological evidence and knowledge.

Some practitioners interpret “reasonable prospects for economic extraction” to mean that mineability must be demonstrated.

a mining method must be selected

a “transfer function” needs to be applied.

• Use “Whittle” to demonstrate mining using open pit method

• Use “Floating Stope” to demonstrate mining by underground methods.

Reasonable – sensible

Prospects - Something expected; a possibility.

Application of a Transfer Function to select what will be called Resources is too restrictive.

1. No feasibility study has been made to determine the economic viability of a mining method.

2. The total resources initially in-place is constant.

Not the case if filtered using a transfer function.

A

B

CD A – mineral domainB – Inferred ResourcesC – Indicated ResourcesD – Measured ResourcesB+C+D = “the Resource”

Exploration Results

Mineral Resources Mineral Reserves

Inferred Possible

Indicated Plausible Probable

Measured Presumed Proven

Consideration of only

mining and economic

factors (the "Transfer

Function")

Consideration of

mining, metallurgical,

economic, marketing,

legal, environmental,

social and

governmental factors

(the "Modifying

Factors)

Increasing Probability of economic viability

Increasing level of

geological

knowledge and

confidence

http://www.unece.org/energy/se/pdfs/UNFC/UNFCemr.pdf

Geostatistics uses the data from the mineral domain to determine the weights to use when estimating.

The Variogram

Estimation is done using Kriging

Reduces error to a minimum

If done properly it is the least biased estimation method.

Resource Estimation requires an integrated approach with emphasis on reducing bias at all stages.

A smoothing correction needs to be applied to compensate for too much averaging (high grade will be under reported).

Classification of resources is dependant on confidence in the estimate which in turn is dependant on confidence in the Variogram.

Currently there is a tendency to under-report resources by application of a transfer function.

In part due to current classification scheme

UN sponsored classification scheme that is more robust has been developed and should be adopted.

Some good references:

Sinclair, A.J. and Blackwell, G.H. (2006) Applied Mineral Inventory Estimation; Cambridge University Press, New York, 381p.

Leuangthong, O., Khan, K.D., and Deutsch, C.V. (2008) Solved Problems in Geostatistics; John Wiley and Sons, New Jersey, 207.

Isaaks, E.H. and Srivastava, R.M. (1989) Applied Geostatistics; Oxford University Press, New York, 561p.

Deutsch, C.V. and Journel, A.G. (1998) GSLIB Geostatistical Software Library and User's Guide; Oxford University Press, New York, 369p.

Journel, A.G. and Huijbreghts, C.J. (1991) Mining Geostatistics; The Blackburn Press, New Jersey, 600p.

Goovaerts, P. (1997) Geostatistics for Natural Resources Evaluation; Oxford University Press, New York, 483p.