Multigaussian Kriging Min-max Autocorrelation factors

Total and soluble copper grade estimation using minimum/maximum autocorrelation

factors and multigaussian krigingAlejandro 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

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

Objective

• To jointly estimate total and soluble copper grades– Inequality relationship should be reproduced as well as possible

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


• 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


• 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


• 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


• 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

Proposed approach

• The proposed approach is similar to MAF co-simulation, except that simulation step is replaced by multigaussiankriging in order to obtain estimated values of total and soluble copper grades

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

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.

Units Exotic

– Green oxides: chrysocolla, malachite.

– Mixed: trazes chrysocolla, malachite and copper wad.

– Black oxides: copper wad, limonitepitch and pseudomalachite

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

Samples scatter plot by unit

Green oxides

Black oxides

Mixed

All

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

2*,Z

Raw Variables Cut and Cus

Gaussian Variables F1, F2: uncorrelated

Kriging F1 F2

N( )

2*,Z

Gaussian local distribution

Local data distributiion( local average)

1

Normal score transformation

Normal score back transformation

1MAF

MAF

Numerical integrationgaussian simulation

2*,Z



Kriging F1 F2

N( )

2*,Z



1



1MAF

MAF


2*,Z



Kriging F1 F2

N( )

2*,Z



1



1MAF

MAF


2*,Z



Kriging F1 F2

N( )

2*,Z



1



1MAF

MAF


2*,Z



Kriging F1 F2

N( )

2*,Z



1



1MAF

MAF


Application

• Comparison with ordinary kriging and cokriging─ Local estimates

DataOrdinary

Kriging

Ordinary

Cokriging

Multigaussian

kriging + MAF

Variable Mean value Correlation Mean value Correlation Mean value Correlation Mean value Correlation

Total copper

grade0.381

0.939

0.386

0.85

0.348

0.904

0.383

0.966Soluble

copper grade0.173 0.167 0.149 0.170

Application

─ Dependence between total and soluble copper grades

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 heterotopicsampling

Acknowledgements

• GeoInnova

• ALGES Laboratory at University of Chile

• Codelco Chile– Ricardo Boric

– Enrique Chacón