Introduction Geostatistic for Mineral Resources

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Introduction Geostatisticsfor

Mineral Deposit

Presented by

Bosta PratamaM AusIMM, M MGEI

Senior Consultant – Perth Western Australia

Agenda

• 08.00 – 08.30 : Introduction and Overview

• 08.30 – 10.00 : Sampling

• 10.00 – 10.15 : Break 1

• 10.15 – 11.45 : Geostatistics part 1

• 11.45 – 12.45 : Lunch Break

• 12.45 – 14.45 : Geostatistics part 2

• 14.45 – 15.00 : Break 2

• 15.00 – 16.00 : Estimations

• 16.00 – 17.00 : Discussion

OVERVIEW

Historical Perspective

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Geostatistics • Definition :

“ A branch of applied statistics which deals with spatially distributed data”

• What is :

A set of mathematical tools that can be use for :

DATA ANALYSIS SPATIAL MODELLING CHARACTERIZATION OF UNCERTAINTY RISK ANALYSIS

• Why is :

1. It bridges descriptive information and engineering analysis

2. Provides means for a sound scientific/engineering basis for remediation planning

3. Allows for the incorporation of qualitative and quantitative data

• QUALITATIVES :

1. Geology Maps

2. Structural information

3. Expert opinions

• QUANTITATIVES :

1. Sample

2. Indirect measurements

– Geostatistics must not be:

• Considered as a Mathematical tool which can do anything

• Used at all costs

• Used by the ill informed – Beware of Instant Experts

– Geostatistics consists of two words:

• Geo

• Statistics

– Remember that Geo comes before Statistics

• Understand your data

• Understand the geology and what controls what

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SAMPLING

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• In practice the squared difference between duplicate samples can never be reduced to zero.

• The squared difference is a measure of the dispersion or spread of sampling errors.

• Gy calls this the variance of the Fundamental Sampling Error (or FSE).

• Gy’s Sampling Theory allows us to calculate/ quantify the FSE.

Unless the size of the sample is equal to the size of the lot, we will incur a non-zero sampling variance.

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• The sampling nomograph is a graphical tool which enables visualisation of sampling protocols

• The nomograph is derived by taking the logarithms of both sides of Gy’s formula, giving us

Sampling Nomographs‘Safety Zone’‘Safety Zone’

crus

hing

grin

ding

pulv

eris

ing

B

σ 2(B)= σ 2(A)-7.98 x 10 -3

D

E

F

G

1/4“ sampling line

d=0.825cm

28 # sampling line

d=0.0595cm

200 # sampling line

d=0.0074cm

σ 2(D) =

σ 2(C)-5.8 x 10 -3

σ 2(G)= σ 2(E)-8.7 x 10 -3 C

A

Sampling lines derived from:

σ2=Kdα/M – 470xd1.5/M

Final Sample:σ2=22.48xd-3

σR=15%

10tonne roundUnknown Size

Comminution step

(ie. vertical line)

Sub-sampling or mass reduction

step

‘Safety Zone’‘Safety Zone’

crus

hing

grin

ding

pulv

eris

ing

B

σ 2(B)= σ 2(A)-7.98 x 10 -3

D

E

F

G

1/4“ sampling line

d=0.825cm

28 # sampling line

d=0.0595cm

200 # sampling line

d=0.0074cm

σ 2(D) =

σ 2(C)-5.8 x 10 -3

σ 2(G)= σ 2(E)-8.7 x 10 -3 C

A

Sampling lines derived from:

σ2=Kdα/M – 470xd1.5/M

Final Sample:σ2=22.48xd-3

σR=15%

10tonne roundUnknown Size

Comminution step

(ie. vertical line)

Sub-sampling or mass reduction

step

Sampling Nomographs

Comments

• Sampling theory is very powerful• But… the Bongarcon modification is

strongly advised for gold• If you are involved in setting up a

sampling programme or defining sampling protocols, application of Gy’s formula is strongly recommended.

What does Sampling Theory not apply to?

• The Sampling Theory does NOT directly assist us with questions regarding:– Drilling practice and sample recovery– Drill spacing and drill density– Grouping and segregation errors

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GEOSTATISTICS 1

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GEOSTATISTICS 2

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Established from the equation:

γ(h) = Σ(f(x) – f(x+h))2 / 2n

Where: f(x) is the value of the first samplef(x+h) is the value of the second sample of

distance h from f(x)n is the number of sample pairsγ(h) is the semi-variance

The semi-variogram can be plotted as a graph by plotting γ(h) against distance h

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ESTIMATIONS

– Numerous methods of resource estimation are available:• Geological Methods• Nearest Neighbour• Polygonal Methods• Triangular Methods• Random Stratified Grids• Inverse Distance Weighting• Trend Surface• Kriging

– All have good aspects and equally bad aspects

Linear Estimation– Basics:

• Method usually done as a check on most resource models• Area is divided into a series of polygons, centred upon an individual

point by the bisectors of lines drawn between sample points• Average grade assigned to polygon is that of the central sample

– Assumptions:• Similar to geological method

– Problems:• Each polygon of different area• Estimate based upon a single sample• Spurious high grade sample/sampling errors can have large impact• Shape of polygon dictated by data, not geology

Polygonal method

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– Basics:• Method became very popular with the introduction of the computer• Involves a large number of calculations• Deposit is divided into a series of blocks or panels and the value of

each one determined from the set of surrounding data values. The weight applied to each one is dependent upon distance from the block

• Samples closest to the block have the largest weights, the farthest samples the lowest weights

– Assumptions:• Data positions are well known• A mathematical function can be applied

– Problems:• How many samples do you use?• How do I select my samples?• What power do I use?

Inverse Distance method

• The Basic idea is to estimate the attribute value (say, porosity) at a location where we do not know the true value

where u refers to a location, Z*(u) is an estimate at location u, there are ndata values and λi refer to weights.

• What factors could be considered in assigning the weights? - closeness to the location being estimated - redundancy between the data values - anisotropic continuity (preferential direction) - magnitude of continuity / variability

Weighted Linear Estimator

1( ) ( )

n

i ii

Z Zλ∗

=

= ⋅∑u u

There are three equations to determine the three weights:

In matrix notation: (Recall that )

1 2 3

1 2 3

1 2 3

(1,1) (1,2) (1,3) (0,1)(2,1) (2, 2) (2,3) (0, 2)(3,1) (3, 2) (3,3) (0,3)

C C C CC C C CC C C C

λ λ λλ λ λλ λ λ

⋅ + ⋅ + ⋅ =⋅ + ⋅ + ⋅ =

⋅ + ⋅ + ⋅ =

( ) (0) ( )C C γ= −h h

1

2

3

(1 1) (1 2) (1 3) (0 1)(2 1) (2 2) (2 3) (0 2)(3 1) (3 2) (3 3) (0 3)

C C C CC C C CC C C C

λλλ

, , , , , , , = , , , , ,

Weighted Linear Estimator

Simple kriging with a zero nugget effect and an isotropic spherical variogram with three different ranges:

0.0000.0000.0001

0.001-0.0270.6485

0.0650.0120.781Range=10

λ3λ2λ1

Kriging

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Simple kriging with an isotropic spherical variogram with a range of 10 distance

units and three different nugget effects:

0.0000.0000.000100%

0.0530.1300.17275%

0.0640.2030.46825%

0.0650.0120.781Nugget=0%

λ3λ2λ1

Kriging Kriging

– Multiple Indicator Kriging (MIK)– Uniform Conditioning (UC)

Non Linear Estimation Recoverable Resources

‘Recoverable Resources’ is a term used in geostatistics to denote that the portion of in-situ resources that are recovered during mining.Recoverable Resources can be defined on a global or local basis.

Global: estimated for the whole field of interest.

e.g. estimation for the entire domain (or a large well-defined subset of the domain like an entire bench).

Local: recoverable resources on a panel/panel basis (see later).

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• The objective of looking at indicator variograms was to get an idea of the continuity of grade at different cut offs.

• Indicators are binary transforms of a variable into values of 1 or 0, depending on whether the variable is above or below a threshold or cutoff. Indicator variograms can be used as tools on capturing pattern of spatial continuity for that particular cutoff and since an indicator variable is either 0 or 1, indicator variograms do not suffer from the adverse effects of erratic outliers and usually behave fairly well (Isaaks and Srivastava, 1990).

Multiple Indicator Kriging

Steps:

1. Split distribution into classes (cut-offs);

2. Transform grades to 1’s and 0’s;

3. Krige indicators;

4. Estimate distribution within Panels;

5. Effect Change of Support; and

6. Calculate tonnage and grade for each cut-off.

Multiple Indicator Kriging

Kriging indicators with multiple cut-offs assumes that each cut-off is spatially independent from the next.

For example, Indicators at 0.6 are independent (spatially uncorrelated) to Indicators at 0.7!

The indicators are (generally) not independent Order relation problems (similar to initial lithology problem).

The ideal solution is:a) model a single variogram that is proportional or b) model variograms and cross variograms.

Multiple Indicator Kriging

Uniform Conditioning (UC) is a variation of Gaussian Disjunctive Kriging (DK).

UC aims at deriving the local conditional distributions of SMU’s.

Method considers the grade of the panel as known.

Assumes a diffusive model for grade distribution (and a few other assumptions).

Uniform Conditioning

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Steps:

1. Estimate panel (OK, MIK, IDW – OK usually);

2. (In Gaussian Space) Calculate (global) change of support coefficients for SMU and panel; and

3. Calculate Tonnage (proportion) and Metal using panel grade and change of support coefficients. Back calculate grades.

Uniform Conditioning

SIMULATIONS

Simulation ≠ Estimation. The simulation is usually made on the point data scale. Simulation of blocks is also possible.

Simulations reproduce sample histogram and variogram, with the assumption that these fully describes the sample population.

Conditional simulations also ‘honour the data’(when we do point simulation). Hence ‘conditional’

Grade profile

"Distance"

"Gra

de"

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Grade profile

"Distance"

"Gra

de"

Grade profile

"Distance"

"Gra

de"

Estimate: a path through each sample that Estimate: a path through each sample that minimisesminimisesthe distance (=error) to the distance (=error) to unsampledunsampled true valuestrue values

Grade profile

"Distance"

"Gra

de"

Less precisionLess precision butbutReproduction of Reproduction of variabiltyvariabilty

A: KrigingB: Non-Conditional SimulationC: Conditional Simulation

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Gaussian Related Algorithms

LU decomposition

Sequential Gaussian

Truncated Gaussian

Turning Band

Conditional Simulation

Indicator Based AlgorithmsAppropriate for categorical (discrete) and continuous variables

Sequential algorithm (SIS)

Suffers from the usual drawbacks: complex structural analysis, order-relationship problems

Conditional Simulation

Conditional Simulation – example

QUESTIONS ???

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TERIMA KASIH