1. Kriging for Estimation of Mineral Resources GISELA/EPIKH School Exequiel Sepúlveda Department of Mining Engineering, University of Chile, Chile ALGES Laboratory, Advanced Mining Technology Centre (AMTC), University of Chile, Chile

2. Kriging Most used method in estimation of mineral resources in mining industry Broadly known algorithm and present in many mining softwares Few implementations are up to date with modern computation techniques Does not consider multicore era Sample data + Geology Exploratory data analysis  estimation units VariogramsBlock estimated grades

3. Kriging Algorithm Few parameters are required Search parameters Variographic model Locations to be estimated Three main steps to estimate a location Sampling data search Covariances computing Linear systems solving

4. Techniques used in Kriging k-nearest neighbor Superblocks indexing Complexity: Linear System Solving LU Decomposition Complexity:

5. KNN in Kriging Multiple indexing techniques for spatial data KDTree (multidimension) Octree (3D) More efficient Complexity:

6. KNN in Kriging

7. Solvers in Kriging Kriging systems are symmetric Ordinary Kriging  symmetric Simple Kriging  symmetric definite positive Symmetric system solving is faster than LU decomposition Ordinary Kriging  Simple Kriging 

8. Solvers in Kriging

9. Kriging Tradeoff High demanding parameters Speed

10. Kriging Challenges Calculation techniques updates High demanding parameters with best performance Parallelism for computing optimization Scaling is needed to estimate large deposits

11. Case Study Copper deposit +230,000 drill hole samples (composites) 5,000,000 blocks to be estimated 2x2x2 block discretization Variographic model Nugget effect 2 anisotropic structures GSLIB comparison Most used open source tool for estimation of mineral grades in mining industry Fortran programming language Sequential and non-scalable algorithm Very fast

12. Kriging Upgrading Applying newer calculation techniques: Octree indexing Symmetric solvers Sequential mode No use of multicore Speed-up 2.6x

13. Kriging Upgrading Multicore era is here Can it be faster than 2.6x speed-up? Solution: Mutiprocess strategy

14. Kriging Parallelization Kriging Algorithm is easily parallelizable Estimating any location is not dependant of estimating another one Build groups of “to-be-estimated” locations, then process them in parallel Next sample 6 groups are defined Each location within group is estimated in parallel

15. Kriging Parallelization 12 34 5 6

16. Speed depends on how many estimations are computed using N processes What would happen considering P processors? Is it P times faster?

17. Amdahl’s Law

18. Kriging Parallelization Upgraded algorithm Parallelized Faster [SpeedUp 11x] 8 cores system

19. Kriging Parallelization Today we don’t have hardware limitations Can it be faster than a 11x speed-up? Solution: Distributed strategy 3 nodes system

20. Distributed Kriging Upgraded algorithm ParallelizedDistributed Fastest [SpeedUP ≈ 20x] 4 nodes with 8 cores system

21. Challenges of Kriging on Gird Maximum Speedup Output data ≈ 500 MB Synchronization of results

22. Conclusions Upgraded, parallelized and distributed Kriging allows fastest model estimation High demanding parameters are allowed with faster computation Stronger hardware capabilities are available Fully scalable Is there good performance on Grid?

23. Kriging for Estimation of Mineral Resources GISELA/EPIKH School Exequiel Sepúlveda Department of Mining Engineering, University of Chile, Chile ALGES Laboratory, Advanced Mining Technology Centre (AMTC), University of Chile, Chile