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- 1. 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
- 2. Outline • Introduction • Distance Function Methodology • Locally Stationary Geostatistics • Example • Conclusions (c) David F. Machuca-Mory, 2009 1
- 3. 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 (c) David F. Machuca-Mory, 2009 2
- 4. 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 (c) David F. Machuca-Mory, 2009 3
- 5. Outline • Introduction • Distance Function Methodology • Locally Stationary Geostatistics • Example • Conclusions (c) David F. Machuca-Mory, 2009 4
- 6. 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 (c) David F. Machuca-Mory, 2009 5
- 7. 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 (c) David F. Machuca-Mory, 2009 6
- 8. 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
- 9. 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 (c) David F. Machuca-Mory, 2009 8
- 10. Outline • Introduction • Distance Function Methodology • Locally Stationary Geostatistics • Example • Conclusions (c) David F. Machuca-Mory, 2009 9
- 11. 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. (c) David F. Machuca-Mory, 2009 10
- 12. 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) (c) David F. Machuca-Mory, 2009 11
- 13. 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
- 14. 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. (c) David F. Machuca-Mory, 2009 13
- 15. 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 (c) David F. Machuca-Mory, 2009 14
- 16. Outline • Introduction • Distance Function Methodology • Locally Stationary Geostatistics • Example • Conclusions (c) David F. Machuca-Mory, 2009 15
- 17. 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). (c) David F. Machuca-Mory, 2009 16
- 18. 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 (c) David F. Machuca-Mory, 2009 17
- 19. Local variogram parameters (2/2) Local range parallel to Local range perpendicular Local range parallel to vein strike to vein dip vein dip (c) David F. Machuca-Mory, 2009 18
- 20. 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 (c) David F. Machuca-Mory, 2009 19
- 21. Vein uncertainty model (1/2) • Envelopes for vein probability >0.5 View towards North East View towards South West (c) David F. Machuca-Mory, 2009 20
- 22. 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. (c) David F. Machuca-Mory, 2009 21
- 23. Outline • Introduction • Distance Function Methodology • Location-Dependent Correlograms • Example • Conclusions (c) David F. Machuca-Mory, 2009 22
- 24. 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. (c) David F. Machuca-Mory, 2009 23
- 25. 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. (c) David F. Machuca-Mory, 2009 24

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