Geostatistic for mineral resources

1. 1
Introduction Geostatistics
for
Mineral Deposit
Presented by
Bosta Pratama
M 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

2. 2
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

3. 3
SAMPLING

4. 4

5. 5

6. 6

7. 7
• 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.

8. 8
• 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
‘SafetyZone’‘SafetyZone’
crushing
grinding
pulverising
B
σ 2
(B)=
σ 2
(A)-7.98
x
10 -3
D
E
F
G
1/4“ sam
pling
line
d=0.825cm
28
#
sam
pling
line
d=0.0595cm
200
#
sam
pling
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 round
Unknown Size
Comminution
step
(ie. vertical line)
Sub-sampling
or mass reduction
step
‘SafetyZone’‘SafetyZone’
crushing
grinding
pulverising
B
σ 2
(B)=
σ 2
(A)-7.98
x
10 -3
D
E
F
G
1/4“ sam
pling
line
d=0.825cm
28
#
sam
pling
line
d=0.0595cm
200
#
sam
pling
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 round
Unknown 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

9. 9
GEOSTATISTICS 1

10. 10

11. 11

12. 12

13. 13

14. 14
GEOSTATISTICS 2

15. 15

16. 16

17. 17

18. 18

19. 19
Established from the equation:
γ(h) = Σ(f(x) – f(x+h))2 / 2n
Where: f(x) is the value of the first sample
f(x+h) is the value of the second sample of
distance h from f(x)
n is the number of sample pairs
γ(h) is the semi-variance
The semi-variogram can be plotted as a graph by plotting
γ(h) against distance h

20. 20
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

21. 21
– 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 n
data 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 i
i
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 C
C C C C
C C C C
λ λ λ
λ λ λ
λ 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 C
C C C C
C C C C
λ
λ
λ
 
 
 
 
 
 
 
 
, , , ,   
   , , , = ,
   
   , , , ,   
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

22. 22
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).

23. 23
• 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

24. 24
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"
"Grade"

25. 25
Grade profile
"Distance"
"Grade"
Grade profile
"Distance"
"Grade"
Estimate: a path through each sample thatEstimate: a path through each sample that minimisesminimises
the distance (=error) tothe distance (=error) to unsampledunsampled true valuestrue values
Grade profile
"Distance"
"Grade"
Less precisionLess precision butbut
Reproduction ofReproduction of variabiltyvariabilty
A: Kriging
B: Non-Conditional Simulation
C: Conditional Simulation

26. 26
Gaussian Related Algorithms
LU decomposition
Sequential Gaussian
Truncated Gaussian
Turning Band
Conditional Simulation
Indicator Based Algorithms
Appropriate 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 ???

27. 27
TERIMA KASIH