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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 …

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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- 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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