1. How Many Samples are Enough? Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. by K. Modis and S. Stavrou, Nat. Tech. Univ. of Athens, GREECE A. Christoforidis, HERACLES Cement Co APCOM 09 – 10-07-2009, Vancouver, CANADA

2. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 2 Contents Scope Scope About the Theory About the Theory Case Study: the Sesklo Clay Quarry Case Study: the Sesklo Clay Quarry Conclusions Conclusions

3. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 3 Scope

4. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 4 The Objective The objective of this work is to apply recently established theoretical results (Modis and Papaodysseus, 2006) in order to estimate the optimum sampling grid in a clay quarry in central Greece. The objective of this work is to apply recently established theoretical results (Modis and Papaodysseus, 2006) in order to estimate the optimum sampling grid in a clay quarry in central Greece. Scope

5. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 5 Where are we based on? The theoretical results are based on the sampling theorem of information which states that a band limited (i.e. slow varying) random waveform can be totally reconstructed by its samples if the sampling rate is greater than a critical value depending on its characteristics The theoretical results are based on the sampling theorem of information which states that a band limited (i.e. slow varying) random waveform can be totally reconstructed by its samples if the sampling rate is greater than a critical value depending on its characteristics Scope

6. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 6 Is it Reasonable? But, is it reasonable that there is a limit in sampling density, and more samples will add nothing to the quality of the approximation? But, is it reasonable that there is a limit in sampling density, and more samples will add nothing to the quality of the approximation? Scope

7. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 7 The Answer… The answer in fact is yes, under the condition that the frequency content of the sampled variable is limited. Band limitedness ensures slow variation. And it is easier to sample a slow varying phenomenon than a rapidly varying one. The answer in fact is yes, under the condition that the frequency content of the sampled variable is limited. Band limitedness ensures slow variation. And it is easier to sample a slow varying phenomenon than a rapidly varying one. There are two important facts about our approach … There are two important facts about our approach … Scope

8. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 8 Important Fact 1: It is the only way to estimate a theoretical limit to the sampling density, above which there is no significant improvement to the accuracy of estimation. It is the only way to estimate a theoretical limit to the sampling density, above which there is no significant improvement to the accuracy of estimation. Scope

9. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 9 Important Fact 2: In the case of an existing sampling grid, if the drill hole density is close to the ideal one, most interpolation algorithms converge to reality. In the case of an existing sampling grid, if the drill hole density is close to the ideal one, most interpolation algorithms converge to reality. Scope

10. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 10 About the Theory

11. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 11 About band-limitedness According to the aforementioned sampling theorem, only band-limited waveforms can be reconstructed by their samples. According to the aforementioned sampling theorem, only band-limited waveforms can be reconstructed by their samples. Modis and Papaodysseus (2006) have pointed out that earth- related phenomena represented by a variogram model with a sill (e.g. Spherical scheme) are approximately bandlimited. Modis and Papaodysseus (2006) have pointed out that earth- related phenomena represented by a variogram model with a sill (e.g. Spherical scheme) are approximately bandlimited. About the Theory

12. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 12 A Simple Example For example, the fourier transform of a linear variogram/ covariance model with a sill, For example, the fourier transform of a linear variogram/ covariance model with a sill, (b) 1 h γ(h) a-a 0 1 h R(h) a -a (a) 0 is approximately bandlimited About the Theory

13. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 13 A Common Example Or, the fourier transform of the spherical variogram/ covariance model with a sill, Or, the fourier transform of the spherical variogram/ covariance model with a sill, is also approximately bandlimited About the Theory

14. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 14 The Algorithm In that case, it is shown that the critical sampling interval equals half the range of influence of the underlying variogram model. In that case, it is shown that the critical sampling interval equals half the range of influence of the underlying variogram model. According to the above, the estimation of the critical sampling rate of a spatial phenomenon is done in two steps: According to the above, the estimation of the critical sampling rate of a spatial phenomenon is done in two steps: About the Theory

15. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 15 Step 1: During structural analysis, the variogram/ covariance model is estimated. If it is a model without a sill, the process stops here. During structural analysis, the variogram/ covariance model is estimated. If it is a model without a sill, the process stops here. About the Theory

16. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 16 Step 2: Step 2.At the estimation stage, the critical sampling interval is estimated by halving the range of influence of the underlying variogram model. Step 2.At the estimation stage, the critical sampling interval is estimated by halving the range of influence of the underlying variogram model. About the Theory

17. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 17 Case Study: the Sesklo Clay Quarry

18. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 18 The Sesklo Clay Quarry The Sesklo quarry is situated near the city of Volos in central Greece and exploited by the “Heracles” General Cement Company for more than 40 years. The Sesklo quarry is situated near the city of Volos in central Greece and exploited by the “Heracles” General Cement Company for more than 40 years. Its purpose is to supply a major Cement Plant. Its purpose is to supply a major Cement Plant. Case Study: the Sesklo Clay Quarry

19. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 19 Sampling The clay deposit has been surveyed by a drilling campaign, which includes 45 drill holes. The clay deposit has been surveyed by a drilling campaign, which includes 45 drill holes. Average distance between drill holes was 50 m Average distance between drill holes was 50 m Case Study: the Sesklo Clay Quarry

20. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 20 Physical Variable The 388 samples (3m long) were analyzed for Al 2 O 3 which, being the most important ingredient, was used for reserves characterization. The 388 samples (3m long) were analyzed for Al 2 O 3 which, being the most important ingredient, was used for reserves characterization. Case Study: the Sesklo Clay Quarry

21. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 21 Structural analysis An anisotropic spherical variogram with a sill 5.3 Al 2 O 3 (%) 2, a horizontal range 180 m and a vertical range 40 m was fitted to the data An anisotropic spherical variogram with a sill 5.3 Al 2 O 3 (%) 2, a horizontal range 180 m and a vertical range 40 m was fitted to the data Case Study: the Sesklo Clay Quarry

22. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 22 Critical Sampling Density Using our proposed formula of halving the variogram range to estimate the optimal sampling grid size: Using our proposed formula of halving the variogram range to estimate the optimal sampling grid size: D s = R/2 we get a value of 90 m. Since average distance between drill holes is 50 m, the orebody is over sampled. Since average distance between drill holes is 50 m, the orebody is over sampled. Case Study: the Sesklo Clay Quarry

23. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 23 Validate the Theory In order to examine the validity of the theory, clay reserves were estimated using three different drill hole patterns: In order to examine the validity of the theory, clay reserves were estimated using three different drill hole patterns:  The 50 x 50 m “dense grid”  The 100 x 100 m “medium grid”  The 150 x 150 m “sparse grid” Case Study: the Sesklo Clay Quarry

24. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 24 Reserves Estimation Two methods were used for the estimation of clay reserves: ordinary kriging and inverse distance squares (IDS). In all estimations, the unit block size was 20 x 50 x 5 m. Two methods were used for the estimation of clay reserves: ordinary kriging and inverse distance squares (IDS). In all estimations, the unit block size was 20 x 50 x 5 m. Case Study: the Sesklo Clay Quarry

25. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 25 Visual Example of plan maps estimated using IDS (left) and kriging (right) at +135, and a 50x50 m (a), 100x100 m (b) and 150x150 m (c) grid respectively Example of plan maps estimated using IDS (left) and kriging (right) at +135, and a 50x50 m (a), 100x100 m (b) and 150x150 m (c) grid respectively Case Study: the Sesklo Clay Quarry

26. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 26 Histograms Histogram of differences of estimates using dense vs. medium grids (left) and dense vs. sparse grids (right) for kriging models Histogram of differences of estimates using dense vs. medium grids (left) and dense vs. sparse grids (right) for kriging models Case Study: the Sesklo Clay Quarry

27. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 27 Correlation Correlation of estimates using dense vs. medium grids (left) and 5 dense vs. sparse grids (right) for kriging models Correlation of estimates using dense vs. medium grids (left) and 5 dense vs. sparse grids (right) for kriging models Case Study: the Sesklo Clay Quarry

28. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 28 Conclusions

29. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 29 Optimization The selection of an appropriate drill-hole network according to the proposed formula, can maximize information required for estimation ore reserves and also result in considerable savings in money and time. The selection of an appropriate drill-hole network according to the proposed formula, can maximize information required for estimation ore reserves and also result in considerable savings in money and time. Conclusions

30. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 30 Convergence of Algorithms If the density of the sampling grid is close to or greater than its critical value, the resource model can be produced using simpler interpolation algorithms, such as the IDS, and the accuracy would be similar to that derived by geostatistics. If the density of the sampling grid is close to or greater than its critical value, the resource model can be produced using simpler interpolation algorithms, such as the IDS, and the accuracy would be similar to that derived by geostatistics. Conclusions

31. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 31 The Sesklo Case In this study, a variogram model generated by the structural analysis of Al 2 O 3 spatial distribution was used to establish a critical sampling grid. In this study, a variogram model generated by the structural analysis of Al 2 O 3 spatial distribution was used to establish a critical sampling grid. From above analysis, it was shown that existing sampling grid was denser than required. From above analysis, it was shown that existing sampling grid was denser than required. The optimal sampling grid proposed here, will result to significant cost savings in the next drilling program which is now in plan by the company. The optimal sampling grid proposed here, will result to significant cost savings in the next drilling program which is now in plan by the company. Conclusions

32. Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. 32 Thank you!