Multigaussian Kriging Min-max Autocorrelation factors

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The estimation version of Cosimulation using MAF

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Multigaussian Kriging Min-max Autocorrelation factors

  1. 1. Total and soluble copper grade estimationusing minimum/maximum autocorrelation factors and multigaussian kriging Alejandro Cáceres, Rodrigo Riquelme, Xavier Emery, Jaime Díaz, Gonzalo Fuster Geoinnova Consultores Ltda Department of Mining Engineering, University of Chile Advanced Mining Technology Centre, University of Chile Codelco Chile, División MMH
  2. 2. Introduction• Joint estimation of coregionalised variables – grades of elements of interest, by-products and contaminants – abundances of mineral species – total and recoverable copper grades• Multivariate estimation methods must account for the dependence relationships between variables
  3. 3. Objective• To jointly estimate total and soluble copper grades – Inequality relationship should be reproduced as well as possible
  4. 4. Current approaches for modelling total and soluble copper grades• Separate kriging and cokriging – Provide unbiased and accurate estimates – Cokriging accounts for the spatial correlation between the variables – Do not reproduce the inequality relationship estimated grades must be post-processed
  5. 5. Current approaches for modelling total and soluble copper grades• Gaussian co-simulation – Transform each grade variable into Gaussian – Calculate direct and cross variograms and fit a linear model of coregionalisation – Co-simulate the Gaussian variables, conditionally to the data – Back-transform the simulated variables into grades Again, this approach does not reproduce the inequality relationship simulated grades must be post-processed
  6. 6. Current approaches for modelling total and soluble copper grades• Co-simulation via a change of variables – Consider the total copper grade and the solubility ratio – Consider the soluble and insoluble copper grades variables are no longer linked by an inequality constraint
  7. 7. Current approaches for modelling total and soluble copper grades• Co-simulation via orthogonalisation – Transform original grades into spatially uncorrelated variables (factors) that may ideally be seen as independent. – Main orthogonalisation approaches include principal component analysis (PCA), minimum/maximum autocorrelation factors (MAF), and stepwise conditional transformation
  8. 8. Current approaches for modelling total and soluble copper grades• Example: co-simulation via MAF orthogonalisation – Transform original grades into Gaussian variables – Transform Gaussian variables into factors, using MAF – Perform variogram analysis of each factor – Simulate the factors – Back-transform simulated factors into Gaussian variables – Back-transform Gaussian variables into grades – Post-process realisations in order to correct for inconsistencies
  9. 9. Proposed approach• The proposed approach is similar to MAF co-simulation, except that simulation step is replaced by multigaussian kriging in order to obtain estimated values of total and soluble copper grades
  10. 10. Proposed approach• Algorithm – Transform total and soluble copper grades into Gaussian variables – Transform Gaussian variables into uncorrelated factors, using MAF – Perform variogram analysis of each factor – Perform multigaussian kriging of each factor. At each target location, one obtains the conditional distribution of each factors, which can be sampled via Monte Carlo simulation
  11. 11. Proposed approach– Back-transform simulated factors into a Gaussian variables, then into total and soluble copper grades– From the distributions of simulated grades, compute the mean values as the estimates at the target locations.
  12. 12. Units Exotic– Green oxides: chrysocolla, malachite.– Mixed: trazes chrysocolla, malachite and copper wad.– Black oxides: copper wad, limonite pitch and pseudomalachite
  13. 13. Application• 1289 DDH samples(1.5 m) , with information of total and soluble copper grades, from oxides unit of Mina Ministro Hales (MMH)• Isotopic data set
  14. 14. Samples scatter plot by unit All Black oxides Mixed Green oxides
  15. 15. Application• Steps ─ Gaussian transformation of copper grades ─ Orthogonalisation with minimum/maximum autocorrelation factors. A lag distance of 50 m is considered to construct factors ─ Variogram analysis of the factors. Variogram model contain nugget effect, anisotropic spherical and exponential structures ─ Multigaussian kriging (point support) ─ Back-transformation to Gaussian, then to grades ─ Calculation of expected grade values
  16. 16. Raw Variables Gaussian Variables F1, F2: uncorrelatedCut and Cus Kriging F1 F2 MAF N( Z * , 2 ) Normal score transformation Local data distributiion ( local average) Normal score Gaussian local back transformation distribution Z *, 2 1 Numerical integration 1 MAF gaussian simulation
  17. 17. Raw Variables Gaussian Variables F1, F2: uncorrelatedCut and Cus Kriging F1 F2 MAF N( Z * , 2 ) Normal score transformation Local data distributiion ( local average) Normal score Gaussian local back transformation distribution Z *, 2 1 Numerical integration 1 MAF gaussian simulation
  18. 18. Raw Variables Gaussian Variables F1, F2: uncorrelatedCut and Cus Kriging F1 F2 MAF N( Z * , 2 ) Normal score transformation Local data distributiion ( local average) Normal score Gaussian local back transformation distribution Z *, 2 1 Numerical integration 1 MAF gaussian simulation
  19. 19. Raw Variables Gaussian Variables F1, F2: uncorrelatedCut and Cus Kriging F1 F2 MAF N( Z * , 2 ) Normal score transformation Local data distributiion ( local average) Normal score Gaussian local back transformation distribution Z *, 2 1 Numerical integration 1 MAF gaussian simulation
  20. 20. Raw Variables Gaussian Variables F1, F2: uncorrelatedCut and Cus Kriging F1 F2 MAF N( Z * , 2 ) Normal score transformation Local data distributiion ( local average) Normal score Gaussian local back transformation distribution Z *, 2 1 Numerical integration 1 MAF gaussian simulation
  21. 21. Application • Comparison with ordinary kriging and cokriging ─ Local estimates Ordinary Ordinary Multigaussian Data Kriging Cokriging kriging + MAF Variable Mean value Correlation Mean value Correlation Mean value Correlation Mean value CorrelationTotal copper 0.381 0.386 0.348 0.383 grade 0.939 0.85 0.904 0.966 Soluble 0.173 0.167 0.149 0.170copper grade
  22. 22. Application─ Dependence between total and soluble copper grades
  23. 23. Conclusions• Proposed approach combines multigaussian kriging in order to model local uncertainty, and MAF transformation in order to model dependence relationship between grade variables.  It better reproduces the inequality constraint and linear correlation between total and soluble copper grades than traditional approaches.  Applications possible in polymetalic deposit or geometallurgical modelling  It is faster than simulation  MAF transformation loses information in the case of a heterotopic sampling
  24. 24. Acknowledgements• GeoInnova• ALGES Laboratory at University of Chile• Codelco Chile – Ricardo Boric – Enrique Chacón

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