1 Introduction .................................................. 1
2 Review of the Main Uncertainties and Risks in Geology ......... 3
2.1 Sources of Uncertainties in Geology and Problems of
Their Definition .......................................... 3
2.2 The Main Risks in Geology ................................ 11
3 Probabilistic Approaches for Handling Uncertainties .......... 13
3.1 Uncertainties of the Input Data .......................... 13
3.2 Evaluation of Uncertainties by Probability-Based
Methods .................................................. 15
3.3 Uncertainties Due to Spatial Variability ................. 19
3.4 Comparison of the Frequentist and the Bayesian
Approaches ............................................... 21
3.4.1 The Relative Frequency Approach to Probability ..... 22
3.4.2 The Subjective/Bayesian Approach to Probability .... 23
3.4.3 Conditional Probability and Independence ........... 24
3.4.4 Bayes' Theorem ..................................... 24
3.4.5 Solving Statistical Problems ....................... 26
3.5 Dempster-Shafer Theory of Evidence ....................... 28
3.5.1 Frame of Discernment ............................... 28
3.5.2 Basic Probability Assignment ....................... 28
3.5.3 Belief and Plausibility Functions .................. 29
3.5.4 Dempster's Combination Rule ........................ 32
3.6 Monte Carlo Simulation ................................... 32
3.7 Bootstrap Methods ........................................ 35
3.8 Entropy and Uncertainty .................................. 37
4 Review of Uncertainty Oriented Mathematical Methods .......... 41
4.1 Interval Arithmetic ...................................... 41
4.2 Fuzzy Sets ............................................... 46
4.2.1 Definition and Representations of Fuzzy Sets ....... 46
4.2.2 Basic Connectives for Fuzzy Sets ................... 50
4.2.3 Possibility and Necessity .......................... 51
4.2.4 Linguistic Variables ............................... 52
4.2.5 Fuzzy Relations .................................... 54
4.2.6 The Extension Principle ............................ 59
4.3 Fuzzy Numbers ............................................ 62
4.3.1 Fuzzy Quantities, Intervals and Numbers ............ 62
4.3.2 The Calculus of Fuzzy Quantities with
Noninteractive Variables ........................... 64
4.3.3 Extension of Usual Operations ...................... 65
4.3.4 Practical Calculations with Fuzzy Numbers and
Intervals .......................................... 67
4.4 Triangular and Trapezoidal Fuzzy Numbers ................. 70
4.4.1 Sum of Two Trapezoidal Fuzzy Numbers ............... 72
4.4.2 Product of a Trapezoidal Fuzzy Number and a Real
Number ............................................. 72
4.4.3 Difference of Two Trapezoidal Fuzzy Numbers ........ 73
4.4.4 Product of Two Trapezoidal Fuzzy Numbers ........... 73
4.4.5 Quotient of Two Trapezoidal Fuzzy Numbers .......... 76
4.5 Comparison of Fuzzy Quantities ........................... 77
4.5.1 Some Defuzzification Methods ....................... 78
4.5.2 Ranking Fuzzy Intervals via Defuzzification ........ 80
4.5.3 Comparing a Fuzzy Quantity and a Crisp Number ...... 81
4.5.4 Fuzzy Ordering of Fuzzy Intervals .................. 81
4.6 Fuzzy Measures ........................................... 82
4.7 Probability Versus Membership Degree ..................... 85
4.8 Statistics of Fuzzy Data ................................. 88
4.8.1 Linguistic Probabilities ........................... 89
4.8.2 Univariate Statistics for Fuzzy Data ............... 90
4.8.3 Multivariate Statistics for Fuzzy Data ............. 91
4.9 A Hybrid Approach for Handling Probabilistic and Fuzzy
Parameters ............................................... 93
4.10 Rough Sets ............................................... 96
4.11 Methodological Problems of Risk Analysis ................. 99
4.11.1 Problems of the Application of Traditional and
Uncertainty Oriented Methods to Risk Analysis ..... 99
4.11.2 Risk Analysis with Fuzzy Logic ................... 101
4.11.3 An Illustrative Example .......................... 103
5 Test Calculations and Other Applications .................... 111
5.1 Estimation of Resources of Solid Mineral Deposits ....... 111
5.1.1 Basic Concepts of Resource Estimation ............. 112
5.1.2 Limitations of the Traditional Resource
Estimation Methods ................................ 116
5.1.3 How to Apply the New Mathematical Methods to
Resource Estimation? .............................. 119
5.2 Quantitative Mineralogical Phase Analysis of Rocks ...... 129
5.2.1 Phase Analysis by X-ray Diffractometry ............ 130
5.2.2 Phase Analysis by Thermoanalytic Methods
(derivatography) .................................. 141
5.3 Application to Paleontological Biometry ................. 148
5.3.1 General Considerations ............................ 148
5.3.2 Results of Biometric Measurements ................. 150
5.3.3 Application of the Fuzzy Sets to
the Investigation of Uncertainties and
Transitions ....................................... 156
5.4 Safety Assessment of Radioactive Waste Disposal ......... 161
5.4.1 General Considerations ............................ 161
5.4.2 Safety Assessment of the Püspökszilágy
Repository, Hungary ............................... 166
5.5 Transitions, Sources of Geological Uncertainty .......... 181
5.6 Application of the Bootstrap Method ..................... 184
5.7 The Hybrid Approach in Risk Assessment .................. 187
5.8 Other Geological Applications ........................... 192
6 Conclusions ................................................. 199
References .................................................... 201
Glossary ...................................................... 211
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