Tonnage Uncertainty Assessment of Vein Type Deposits

APCOM 2009

Tonnage Uncertainty Assessment of
Vein Type Deposits Using Distance
Functions and Location-Dependent
Variograms
David F. Machuca-Mory, Michael J. Munroe and Clayton
V. Deutsch

Centre for Computational Geostatistics
School of Mining and Petroleum Engineering
Department of Civil & Environmental Engineering
University of Alberta

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

(c) David F. Machuca-Mory, 2009 1

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


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


Outline
• Introduction
• Example
• Conclusions


Distance Function (1/2)
+10 +14.1
+9 +13.5
• Beginning from the indicator coding of +8 +12.8
intervals: +7 +12.2
+6 +11.7
+10.0
1, if uα is located within the vein +5 +11.2
VI (uα ) =  +4 +10.8
0, otherwise +3 +10.4
+2 +10.2
+1 +10.1
• The anisotropic distance between -1
+10.0
+10.0
samples and contacts: -2
Distance Function (DF): +10.0
-3 Shortest Distance +10.0
-4 Between Points with +10.0
2 2 2 -3
 dx ′   dy ′   dz ′  Different Vein Indicator +10.0
DF (uα ) =   + ′ + ′ -2 (VI) +10.0
 hx ′   hy   hz  -1 +10.0
+1 +10.1
+2 +10.2
+3 +10.4
• Is modified by +4 +10.8
+5 +11.2
( DF (uα ) + C ) / β if VI (uα ) =
0 +6 +11.7
DFmod (uα ) =  +7 +12.2
−( DF (uα ) + C ) ⋅ β if VI (uα ) = +8
1 +12.8
+9 +13.5
+10 +14.1

Distance Function (2/2)
Outer Limit (Maximum)
( DF (uα ) + C ) / β if VI (uα ) = Limit (Minimum)
0 ISO zero (Middle)
Inner
DFmod (uα ) = 
−( DF (uα ) + C ) ⋅ β if VI (uα ) =
1
Non
• C is proportional to the width of the uncertainty Vein

bandwidth . ∆C − ∆C +
Vein

Uncertainty
Bandwidth

Dilated (Increasing β )
• β controls the position of the iso-zero surface ISO Zero (β =1)
Non Eroded (Decreasing β )
• β >1 dilates the iso-zero. Vein

• β <1 erodes the iso-zero.
Vein β β

Position of ISO zero and
Uncertainty bandwidth


Selection of Distance Function Parameters

• Empirical selection, based on: O1 = 0

– Predetermined values O1 > 0

TTrue
– Expert knowledge

• Partial Calibration O1 < 0

T*
– 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:
1.0
O2 = 0
O2 > 0
Bias is minimum: Uncertainty is fair : 0.8
np

Actual Fraction
E {T * − T } ∑ (P i
*
− Pi ) 0.6

O1 = 0 O2 = i =1
0
0.4

E {T } np
O2 < 0
∑P
0.2

i 0.0
i =1 0.0 0.2 0.4 0.6 0.8 1.0
Probability Interval -p
T*: DF model tonnage P*: DF model P interval
T : reference model tonnage (c) David F. Machuca-Mory, 2009
P : Actual fraction 7

Uncertainty Thresholds

• Simple Kriging is used for interpolating the DF values. Non
Vein
• The the inner and outer limits of the uncertainty
bandwidth, DFmin and DFmax, respectively, are within
the range: Vein DFmin DFmax

 1 1 C ⋅ DS 
[ DFmin , DFmax ] =− C ⋅ DS ⋅ β ,
 2 2 β  

with DS = drillhole spacing
• The p value of each cell is calculated by:
Outside
DF * − DFmin >1 DFmax
p= Non
DFmax − DFmin Vein

with DF* = interpolated distance value
DFmin
Vein
Inside
<1


Outline
• Introduction
• Example
• Conclusions


The Assumption of Local-Stationarity

• Standard geostatistical techniques are constrained by the assumption of strict
stationarity.
• The assumption of local stationarity is proposed:

{
Prob {Z (uα ) < z1 ,..., Z (u n ) <= Prob Z (uα + h) < z1 ,..., Z (u n + h) < z K ; o j
z K ; oi } }
∀ uα , u β + h ∈ D, and only if i =j

• Under this assumption the distributions and their statistics are specific of each
location.
• These are obtained by weighting the sample values z (u n ) 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.


Distance Weighting Function

• A Gaussian Kernel function is used for weighting
samples at locations uα inversely proportional to
their distance to anchor points o:
 ( d (u ; o) )2 
α
ε + exp  − 
 2s 2 
ωGK (uα ; o) =  
n  ( d (u ; o) )2 
nε + ∑ exp  − α

α =1  2s 2 
 
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:

ω (uα , uα + h; o)
= ω (uα ; o) ⋅ ω (uα + h; o)


Locally weighted Measures of Spatial Continuity
(1/2)
• Location-dependent Indicator variogram
1 N (h )
∑ ω′(uα , uα + h; o) [VI (uα ) − VI (uα + h)]
2
γ=
VI (h; o)
2 α =1

• Location-dependent Indicator covariances

N (h )
CVI (h; o)
= ∑ ω′(uα , uα + h; o) ⋅VI (uα ) ⋅VI (uα + h) − FVI , −h (o) ⋅ FVI , +h (o)
α =1
• With:
N (h )
FVI ,-h ( sk ; o)
= ∑ ω′(uα , uα + h; o) ⋅VI (uα ; s ) ,
α =1
k

N (h )
FVI ,+h ( sk ; o)
= ∑ ω′(uα , uα + h; o) ⋅VI (uα + h; s )
α =1
k

ω (u , u + h; o)
ω ′(uα , uα + h; o) = α α
N (h )
∑ ω (uα , uα + h; o)
α =1 David F. Machuca-Mory, 2009
(c) 12

Locally weighted Measures of Spatial Continuity
(2/2)
• Location-dependent indicator correlogram:
CVI (h; o)
=ρVI (h; o) ∈ [−1, +1]
2 2
σ VI ,−h (o) ⋅ σ VI ,+h (o)

• With: = F−h ( sk ; o) [1 − F−h ( sk ; o)]
2
σ − h ( sk ; o )
= F+h ( sk ; o) [1 − F+h ( sk ; o) ]
2
σ + h ( sk ; o )

• 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.


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:
n (o )
∑ λβ( LSSK ) (o) ρ (u β − uα ; o) = (o − uα ; o)
ρ α = n(o)
1,...,
β =1

• The LSSK estimation variance is given by:
 n (o ) ( LSSK ) 
2
σ LSSK (o) =1 − ∑ λα
C (0; o)  (o) ρ (o − uα ; o) 
 α =1 

• And the LSSK estimates are obtained from:

n (o )  n (o ) ( LSSK ) 
= *
Z LSSK (o) ( LSSK )
λα ∑
(o)[ Z (uα )] + 1 − λα ∑(o)  m(o)
= 1= 1 α  α 


Outline
• Introduction
• Example
• Conclusions


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 1 − ρVI (h; o) .
• Nugget effect was fixed to 0

Local Azimuth Local Dip Local Plunge


Local variogram parameters (2/2)

Local range parallel to Local range perpendicular Local range parallel to
vein strike
to vein dip 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 West

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.


Outline
• Introduction
• 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.


Tonnage Uncertainty Assessment of Vein Type Deposits

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