1. Advanced Geostatistics, Simulations and Environmental Applications Roussos Dimitrakopoulos Dept of Mining and Materials Engineering McGill University, Canada Email: roussos.dimitrakopoulos@mcgill.ca URL: http://www.iamg.org/ International Association for Mathematical Geosciences Distinguished Lecturer - 2009 & 2010 Seminar - May 28, 2009, U of Athens, Greece

2. CONTENT – Part 1 Introduction Stochastic Models and Simulation An introduction to Monte Carlo simulation Sequential simulation (Gaussian)

3. CONTENT – Part 2 Quantification of Mine Spoil Variability and Rehabilitation Decision Making Classification and Remediation of Mercury Contaminated Soils The examples are all available in text - see materials provided

4. GEOSTATISTICAL TEXTBOOKS An Introduction to Applied Geostatistics Isaaks, EH and RM Srivastava, Oxford University Press, 1989 Geostatistics for the Next Century Dimitrakopoulos, R, Kluwer, 1994 Fundamentals of Geostatistics in Five Lessons Journel, AG Short Course in Geology, v. 8, AGU, 1989 GSLIB Geostatistical Software Library and User's Guide Deutsch, CV and Journel, AG, 2nd Edition, Oxford, 1997 Geostatistics for Natural Resources Evaluation Goovaerts, P, Oxford, 1997 Geostatistics – Modelling Spatial Uncertainty Chiles, J-P and Delfiner, P, Wiley, 1999

5. Geostatistical Glossary and Multilingual Dictionary, Olea, R, Oxford, 1991 GEOSTATISTICAL TEXTBOOKS Applied Geostatistics with SGeMS: A User’s Guide Remy, N, Boucher, A, and Wu, J, Oxford, 2009 Geostatistics for Environmental Scientists Webster, R and Oliver, MA, Wiley, 2007

6. Many managers believe that uncertainty is a problem and should be avoided….. … you can take advantage of uncertainty. Your strategic investments will be sheltered from its adverse effects while remaining exposed to its upside potential. Uncertainty will create opportunities and value. Once your way of thinking explicitly includes uncertainty, the whole decision-making framework changes. Martha Amram and Nalin Kulatilaka in “Real Options” UNCERTAINTY IS NOT A “BAD THING”

7. Real Option Theory Derived from the Nobel Prize-winning work of Black, Merton and Scholes –What is the value of a contract that gives you the right, but not the obligation, to purchase a share of ‘GoldMin’ for $30 six months from now? –Separates risk from expected return on investment and includes timing Applications to real (non-financial) assets -What is the value of starting a project that gives you the right, but not the obligation, to commence production for a cost of $7M six months from now? -What is the value of delaying production to get additional information to reduce uncertainty? –What is the value of building in flexibility to manage uncertainty?

8. Options vs DCF view of Value Current Asset Value Future Gold Price $0 $- $+ Real Options View: Current Value of Option to Produce Traditional DCF View (now or never) No production NPV = 0 Production NPV > 0 Contingent Decision Payoff Function (future price known)

9. Accurate Uncertainty Assessment Needed Unknown, true answer Reserves Accurate uncertainty estimation Single, often precise, wrong answer Reserves Probability 1 “The goal of technical evaluation should be to strive for an accurate assessment of uncertainty, not a single precise answer”

10. Information about the deposit or contaminated site or aquifer Actual but unknown mineral deposit or contaminated site or aquifer … Probable models of the deposit or contaminated site or aquifer or … Describing the Uncertainty about Spatially Distributed Phenomena

11. Probable models of..... Process/Model Parameters of interest Transfer Function Response 1 Response Parameter Response Distribution Response 2 Response m Diagrammatic Representation of the Proposed Simulation Framework

12. Two Important Points Transfer functions are generally non-linear. As a consequence, (i) an average type “block” model may not provide an average map of the space of response uncertainty; and (ii) a criterion for generating possible models may be defined: the simulation technique selected for modelling must be evaluated in for its mapping of the response uncertainty

13. Uncertainly in Orebody Modelling All models have identical: data, stats, continuity, information A laterite nickel deposit

14. Probability Maps and Site Rehabilitation North East EC>0.6(dS/M)EC>0.8EC>1.0 1.0 0.8 0.6 0.4 0.2 0.0 0 200 100

15. Red: Red: Over 80% chance to be above a contaminant concentration Purple Purple: Over 80% chance below the contaminant concentration Blue Blue: Additional drilling would probably be needed Map of the probability of being above a contaminant concentration provides a possible basis for deciding on additional sampling, evaluation and rehabilitation decisions.... Additional Sampling

16. Uncertainty at Various Selectivity The higher the degree of selectivity the greater the uncertainty of the grade-tonnage information

17. Flaws in Traditional Modelling have been Known Normalized Oil Recovery Injected Pore Volume The expected oil production has little chance to be realized Traditional Stochastic Simulations

18. Most natural spatially distributed phenomena are too complex for simple modelling approaches A data set Interpretation: a bouncing ball..... Interpretation: the stock market... The Need for Modelling