Tonnage Uncertainty Assessment of Vein Type Deposits

Centre for Computational GeostatisticsSchool of Mining and Petroleum Engineering

Department of Civil & Environmental EngineeringUniversity of Alberta

Tonnage Uncertainty Assessment of Vein Type Deposits Using Distance Functions and Location-Dependent

VariogramsDavid F. Machuca-Mory, Michael J. Munroe and Clayton

V. Deutsch

APCOM 2009

1

Outline• Introduction• Distance Function Methodology• Locally Stationary Geostatistics • Example• Conclusions

(c) David F. Machuca-Mory, 2009

Introduction (1/2)• 3D modelling is required for delimiting

geologically and statistically homogeneous zones.

• Traditionally this is achieved by wireframe interpolation of interpreted geological sections:

– Highly dependent of a particular geological interpretation

– Can be highly demanding in professional effort

– Alternative scenarios may be difficult to produce

– No assessment of uncertainty provided

• Simulation techniques can be used for assessing the uncertainty of categorical variables

– They require heavy computational effort

– Results are not always geologically realistic

2(c) David F. Machuca-Mory, 2009

Introduction (1/2)

• Rapid geological modelling based on Radial Basis Functions (RBF)

– Fast for generating multiple alternative interpretation

– Locally varying orientations are possible

– No uncertainty assessment provided

• Proposed Approach:

– Distance functions are used for coding the sample distance to the contact.

– Locally stationary variogram models adapts to changes in the orientation, range and style of the spatial continuity of the vein/waste indicator.,

– The interpolation of the distance coding is done by locally stationary simple kriging with locally stationary variograms/correlograms


4

Outline• Introduction• Distance Function Methodology• Locally Stationary Geostatistics• Example• Conclusions


Distance Function (1/2)

• Beginning from the indicator coding of intervals:

• The anisotropic distance between samples and contacts:

• Is modified by

5

-1

-1

-2

-2-3-4-3

+2+3+4+5+6

+8+9

+7

+1

+1+2+3+4+5

+7+8+9

+10

+6

+10

+10.0

+10.0

+10.0

+10.0+10.0+10.0+10.0

+10.2+10.4+10.8+11.2+11.7

+12.8+13.5+14.1

+12.2

+10.1

+10.1+10.2+10.4+10.8+11.2

+12.2+12.8+13.5+14.1

+11.7

Distance Function (DF):Shortest Distance

Between Points withDifferent Vein Indicator

(VI)

+10.0

+10.0

1, if is located within the vein( )

0, otherwise u

u

VI αα

=

mod( ( ) ) / if ( ) 0

( )( ( ) ) if ( ) 1DF C VI

DFDF C VI

α αα

α α

ββ

+ == − + ⋅ =

u uu

u u

22 2( )u dx dy dzDF

hx hy hzα′ ′ ′ = + + ′ ′ ′


Distance Function (2/2)

• C is proportional to the width of the uncertainty bandwidth .

• β controls the position of the iso-zero surface

• β >1 dilates the iso-zero.• β <1 erodes the iso-zero.

6

mod( ( ) ) / if ( ) 0

( )( ( ) ) if ( ) 1DF C VI

DFDF C VI

α αα

α α

ββ

+ == − + ⋅ =

u uu

u u

C∆ − C∆ +Vein

ISO zero (Middle)Outer Limit (Maximum)Inner Limit (Minimum)

UncertaintyBandwidth

NonVein

Vein

Dilated (Increasing β )ISO Zero (β =1)Eroded (Decreasing β )

ββ

Position of ISO zero and Uncertainty bandwidth

NonVein


Selection of Distance Function Parameters

• Empirical selection, based on:

– Predetermined values

– Expert knowledge

• Partial Calibration

– C is chosen based on expert judgement.

– β is modified until p50 volume coincides with data ore/waste proportions or a deterministic model.

• Full Calibration, several C and β values are tried until:

Bias is minimum: Uncertainty is fair :

T*: DF model tonnage P*: DF model P intervalT : reference model tonnage P : Actual fraction 7

{ }{ }

*

1 0E T T

OE T

−=

T*

T Tru

e

O1 > 0

O1 < 0

O1 = 0

*

1

1

( )2 0

p

p

n

i ii

n

ii

P PO

P

=

=

−=

∑

∑

O2 = 0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Probability Interval -p

Act

ual F

ract

ion

O2 > 0

O2 < 0


Uncertainty Thresholds

• Simple Kriging is used for interpolating the DF values.

• The the inner and outer limits of the uncertainty bandwidth, DFmin and DFmax, respectively, are within the range:

with DS = drillhole spacing

• The p value of each cell is calculated by:

with DF* = interpolated distance value

8

[ ]min max1 1, ,2 2

C DSDF DF C DS ββ

⋅= − ⋅ ⋅

Vein minDF maxDF

NonVein

maxDF

minDF

Outside>1

Inside<1

NonVein

Vein

min

max min

*DF DFpDF DF

−=

−


9



The Assumption of Local-Stationarity

• Standard geostatistical techniques are constrained by the assumption of strict stationarity.

• The assumption of local stationarity is proposed:

• Under this assumption the distributions and their statistics are specific of each location.

• These are obtained by weighting the sample values inversely proportional to their distance to the prediction point o.

• The same set of weights modify all the required statistics.

• In estimation and simulation, these are updated at every prediction location.

10

{ } { }1 1Prob ( ) ,..., ( ) ; Prob ( ) ,..., ( ) ;

, and only if

u u o u h u h o

u u h =n K i n K jZ z Z z Z z Z z

D i jα α

α β

< < = + < + <

∀ + ∈ ,

( )nz u


Distance Weighting Function

• A Gaussian Kernel function is used for weighting samples at locations uα inversely proportional to their distance to anchor points o:

s is the bandwidth and ε controls the contribution of background samples.

• 2-point weights can be formed by the geometric average of 1-point weights:

11

( )

( )

2

2

2

21

( ; )exp

2( ; )

( ; )exp

2

u o

u ou o

GKn

ds

dn

s

α

αα

α

ε

ω

ε=

+ − =

+ −

∑

( , ; ) ( ; ) ( ; )α α α αω ω ω+ = ⋅ +u u h o u o u h o


Locally weighted Measures of Spatial Continuity(1/2)

• Location-dependent Indicator variogram

• Location-dependent Indicator covariances

• With:

12

( )

, ,1

( ; ) ( , ; ) ( ) ( ) ( ) ( )N

VI VI VIC VI VI F Fα α α αα

ω −=

′= + ⋅ ⋅ + − ⋅∑h

h +hh o u u h o u u h o o

( )

,1

( )

,1

( ; ) ( , ; ) ( ; ) ,

( ; ) ( , ; ) ( ; )

N

VI k k

N

VI k k

F s VI s

F s VI s

α α αα

α α αα

ω

ω

=

=

′= + ⋅

′= + ⋅ +

∑

∑

h

-h

h

+h

o u u h o u

o u u h o u h

( )

1

( , ; )( , ; )( , ; )

Nα α

α α

α αα

ωω

ω=

+′ + =+∑

hu u h ou u h o

u u h o

[ ]( )

2

1

1( ; ) ( , ; ) ( ) ( )2

N

VI VI VIα α α αα

γ ω=

′= + − +∑h

h o u u h o u u h


Locally weighted Measures of Spatial Continuity (2/2)

• Location-dependent indicator correlogram:

• With:

• Location-dependent correlograms are preferred because their robustness.

• Experimental local measures of spatial continuity are fitted semiautomatically.

• Geological knowledge or interpretation of the deposit’s geometry can be incorporated for conditioning the anisotropy orientation of the fitted models.

13

2 2, ,

( ; )( ; ) [ 1, 1]( ) ( )VI

VIVI VI

Cρ

σ σ−

= ∈ − +⋅h +h

h oh oo o

[ ][ ]

2

2

( ; ) ( ; ) 1 ( ; )

( ; ) ( ; ) 1 ( ; )h h h

h +h +h

o o o

o o ok k k

k k k

s F s F s

s F s F s

σ

σ− − −

+

= −

= −


Locally Stationary Simple Kriging

• Locally Stationary Simple Kriging (LSSK) is the same as traditional SK but the variogram model parameters are updated at each estimation location:

• The LSSK estimation variance is given by:

• And the LSSK estimates are obtained from:

14

( )( )

1( ) ( ; ) ( ; ) 1,..., ( )

nLSSK nβ α αβ

βλ ρ ρ α

=− = − =∑

oo u u o o u o o

( )2 ( )

1( ) (0; ) 1 ( ) ( ; )

nLSSK

LSSK C α αα

σ λ ρ=

= − −

∑

oo o o o u o

( ) ( )* ( ) ( )

1 1( ) ( )[ ( )] 1 ( ) ( )

n nLSSK LSSK

LSSKZ Z mα α αα α

λ λ= =

= + −

∑ ∑

o oo o u o o


15



Drillhole Data(Houlding, 2002)

• Drillhole fans separated by 40m• 2653 2m sample intervals coded by mineralization type.• Modelling restricted to the Massive Black Ore (MBO, red intervals in the figure).


Local variogram parameters (1/2)

• Anchor points in a 40m x 40m x 40m grid• Experimental local correlograms calculated using a GK

with 40m bandwidth.• Interpretation of the MBO structure bearing and dip was

used for guiding the fitting of .• Nugget effect was fixed to 0

17

1 ( ; )VIρ− h o

Local Azimuth Local Dip Local Plunge


Local variogram parameters (2/2)

Local range parallel to vein strike

18

Local range parallel to vein dip

Local range perpendicular to vein dip


Vein uncertainty model (2/2)

• Build by simple Kriging with location-dependent variogram models• Drillhole sample information is respected• Local correlograms allows the reproduction of local changes in the vein

geometry


Vein uncertainty model (1/2)

• Envelopes for vein probability >0.5

View towards North East View towards South West20(c) David F. Machuca-Mory, 2009

Uncertainty assessment

• A full uncertainty assessment in terms of accuracy and precision requires of reference models.

• In practice this may be demanding in time and resources.

• Partial calibration of the DF parameters leads to an unbiased distribution of uncertainty.

• The wide of this distribution is evaluated under expert judgement.


22

Outline• Introduction• Distance Function Methodology• Location-Dependent Correlograms• Example• Conclusions


Conclusions

• The distance function methodology allows producing uncertainty volumes for geological structures.

• Kriging the distance function values using locally changing variogram models allows adapting to local changes in the vein geometry.

• Partial calibration of the distance function parameters allows minimizing the bias of uncertainty volume

• Assessing the uncertainty width rigorously requires complete calibration.


Acknowledgements

• To the industry sponsors of the Centre for Computational Geostatistics for funding this research.

• To Angel E. Mondragon-Davila (MIC S.A.C., Peru) and Simon Mortimer (Atticus Associates, Peru) for their support in geological database management and 3D geological wireframe modelling.


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Tonnage Uncertainty Assessment of Vein Type Deposits

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