Advanced Geostatistics, Simulations

and Environmental Applications

Roussos Dimitrakopoulos

Dept of Mining and Materials EngineeringMcGill 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

CONTENT – Part 1

Introduction

Stochastic Models and Simulation

An introduction to Monte Carlo simulation

Sequential simulation (Gaussian)

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

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

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

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 Kulatilakain “Real Options”

UNCERTAINTY IS NOT A “BAD THING”

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?

Options vs DCF view of ValueC

urr

en

t A

sset

Valu

e

FutureGold Price

$0

$-

$+ Real Options View:Current Value ofOption to Produce

Traditional DCF View(now or never)

No productionNPV = 0

ProductionNPV > 0

Contingent Decision Payoff Function

(future price known)

Accurate Uncertainty Assessment Needed

Unknown,trueanswer

Reserves

Accurateuncertaintyestimation

Single,oftenprecise,wronganswer

Reserves

Pro

bab

ility

1

“The goal of technical evaluation should be to strive for an accurate assessment of uncertainty, not a single precise answer”

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

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

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

Uncertainly in Orebody Modelling

All models have identical:data, stats, continuity, information

A laterite nickel deposit

Probability Maps and Site Rehabilitation

Nor

th

East

EC>0.6(dS/M) EC>0.8 EC>1.0

1.0

0.8

0.6

0.4

0.2

0.0

0

200

100

RedRed:: Over 80% chance to be above a contaminant concentration

PurplePurple: Over 80% chance below the contaminant concentration

BlueBlue: 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

Uncertainty at Various Selectivity

The higher the degree of selectivity the greater the uncertainty of the grade-tonnage information

0

50 000

100 000

150 000

200 000

0.9 1.0 1.1 1.2 1.3 1.4

Cut-off Grade (% Ni)

Con

tain

ed M

etal

(t)

OK Model 25x25x1

Min 25x25x1

Max 25x25x1

Min 15x15x1

Max 15x15x1

Min 10x10x5

Max 10x10x1

Min 5x5x1

Max 5x5x1

Flaws in Traditional Modelling have been Known

Nor

mal

ized

Oil

Rec

over

y

Injected Pore Volume

The expected oil production has little chance to be realized

Traditional

Stochastic Simulations

Most natural spatially distributed phenomena are too complex for simple modelling approaches

T im e

Intr

est R

ate

D is tan c e

Hei

ght

A data set

Interpretation: a bouncing ball.....

Interpretation: the stock market ...

The Need for Modelling