1. Total and soluble copper grade estimation
using 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. 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. Objective
• To jointly estimate total and soluble copper grades
– Inequality relationship should be reproduced as well as possible
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. 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. 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. 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. 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. 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. 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. 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. Units Exotic
– Green oxides: chrysocolla,
malachite.
– Mixed: trazes chrysocolla,
malachite and copper wad.
– Black oxides: copper wad, limonite
pitch and pseudomalachite
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
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. Raw Variables
Gaussian Variables F1, F2: uncorrelated
Cut 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. Raw Variables
Gaussian Variables F1, F2: uncorrelated
Cut 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. Raw Variables
Gaussian Variables F1, F2: uncorrelated
Cut 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. Raw Variables
Gaussian Variables F1, F2: uncorrelated
Cut 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. Raw Variables
Gaussian Variables F1, F2: uncorrelated
Cut 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. 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 Correlation
Total 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 0.170
copper grade
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
