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Modelling the geometry of a vein is a crucial step in resources estimation. The resulting models are used as mineralization domain boundaries and have a direct impact on the tonnage of estimated resources. Deterministic models are often built using time consuming wireframing techniques usually based on hand interpretation of the drillhole intercepts. Another approach consists in coding the drillhole samples by a function of their distance to the veins contacts. The coding is subsequently used for modelling the vein contacts away from drillholes. This is a more efficient approach and is able to provide a measure of tonnage uncertainty. The use of location-dependent correlograms improves the modelling by incorporating local changes in the anisotropy of the vein structure. This combined approach results in more realistic vein models, particularly when the geometry of the vein has been are altered by folding, shearing and other structural processes. The approach is illustrated on a realistic case study.
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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
3(c) David F. Machuca-Mory, 2009
4
Outline• Introduction• Distance Function Methodology• Locally Stationary Geostatistics• Example• Conclusions
(c) David F. Machuca-Mory, 2009
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′ ′ ′ = + + ′ ′ ′
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
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
−=
−
(c) David F. Machuca-Mory, 2009
9
Outline• Introduction• Distance Function Methodology• Locally Stationary Geostatistics • Example• Conclusions
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
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− − −
+
= −
= −
(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
15
Outline• Introduction• Distance Function Methodology• Locally Stationary Geostatistics • Example• Conclusions
(c) David F. Machuca-Mory, 2009
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).
16(c) David F. Machuca-Mory, 2009
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
(c) David F. Machuca-Mory, 2009
Local variogram parameters (2/2)
Local range parallel to vein strike
18
Local range parallel to vein dip
Local range perpendicular to vein dip
(c) David F. Machuca-Mory, 2009
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
19(c) David F. Machuca-Mory, 2009
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.
21(c) David F. Machuca-Mory, 2009
22
Outline• Introduction• Distance Function Methodology• Location-Dependent Correlograms• Example• Conclusions
(c) David F. Machuca-Mory, 2009
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.
23(c) David F. Machuca-Mory, 2009
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.
24(c) David F. Machuca-Mory, 2009