Geomathematics: Theoretical Foundations, Applications and Future Developments

Foreword Preface 1. Complexity of the Geological Framework and Use of Mathematics 1.1 Use of Mathematics in Geology 1.2 Geological Data, Concepts and Maps 1.2.1 Map-Making 1.2.2 Geological Cross-Sections 1.2.3 Scientific Method in the Geosciences 1.2.4 Quality of Predictions 1.3 Use of Curves 1.3.1 Trend-Lines 1.3.2 Elementary Differential Calculus 1.3.3 Graphical Curve-Fitting 1.4 Use of Surfaces 1.4.1 Automated 3-Dimensional Map-Making: Central Baffin Example 1.4.2 Folds and Faults 1.5 Image Analysis 1.5.1 Geometrical Covariance, Intercept and Rose Diagram 1.5.2 Minkowski Operations: Bathurst Acidic Volcanics Example 1.5.3 Boundaries and Edge Effects 2. Probability and Statistics 2.1 History of Statistics 2.1.1 Emergence of Mathematical Statistics 2.1.2 Spatial Statistics 2.2 Probability Calculus and Discrete Frequency Distributions 2.2.1 Conditional Probability and Bayes' Theorem 2.2.2 Probability Generating Functions 2.2.3 Binomial and Poisson Distributions 2.2.4 Other Discrete Frequency Distributions 2.2.5 Oficina Formation Example 2.3 Continuous Frequency Distributions and Statistical Inference 2.3.1 Central-Limit Theorem 2.3.2 Frequency Distributions Derived from the Normal 2.3.3 Significance Tests and 95%-Confidence Intervals 2.3.4 Sum of Two Random Variables 2.4 Applications of Statistical Analysis 2.4.1 Statistical Inference: Grenville Potassium/Argon Ages Example 2.4.2 Q-Q Plots: Normal Distribution Example 2.5 Sampling 2.5.1 Pulacayo Mine Example 2.5.2 Virginia Mine Example 3. Maximum Likelihood, Lognormality and Compound Distributions 3.1 Maximum Likelihood Method with Applications to the Geologic Timescale 3.1.1 Weighting Function Defined for the Inconsistent Dates Model 3.1.2 Log-Likelihood and Weighting Functions 3.1.3 Caerfai-St David's Boundary Example 3.1.4 The Chronogram Interpreted as an Inverted Log-Likelihood Function 3.1.5 Computer Simulation Experiments 3.1.6 Mesozoic Timescale Example 3.2 Lognormality and Mixtures of Frequency Distributions 3.2.1 Estimation of Lognormal Parameters 3.2.2 Muskox Layered Intrusion Example 3.2.3 Three-Parameter Lognormal Distribution 3.2.4 Graphical Method of Reconstructing the Generating Process 3.3 Compound Random Variables 3.3.1 Compound Frequency Distributions and their Moments 3.3.2 Exploration Strategy Example 4. Correlation, Method of Least Squares, Linear Regression and the General Linear Model 4.1 Correlation and Functional Relationship 4.2 Linear Regression 4.2.1 Degree of Fit and 95% - Confidence Belts 4.2.2 Mineral Resource Estimation Example 4.2.3 Elementary Statistics of the Mosaic Model 4.3 General Model of Least Squares 4.3.1 Abitibi Copper Deposits Example 4.3.2 Forward Selection and Stepwise Regression Applied to Abitibi Copper 4.4 Abitibi Copper Hindsight Study 4.4.1 Incorporation of Recent Discoveries 4.4.2 Comparison of Weight Frequency Distributions of Copper Metal and Ore 4.4.3 Final Remarks on Application of the General Linear Model to Abitibi Copper 5. Prediction of Occurrence of Discrete Events 5.1 Weights-of-Evidence Modeling 5.1.1 Basic Concepts and Artificial Example 5.1.2 Meguma Terrane Gold Deposits Example 5.1.3 Flowing Wells in the Greater Toronto Area 5.1.4 Variance of the Contrast and Incorporation of Missing Data 5.2 Weighted Logistic Regression 5.2.1 Meguma Terrane Gold Deposits Example 5.2.2 Comparison of Logistic Model with General Linear Model 5.2.3 Gowganda Area Gold Occurrences Example 5.2.4 Results of the Gowganda Experiments 5.2.5 Training Cells and Control Areas 5.3 Modified Weights-of-Evidence 5.3.1 East Pacific Rise Seafloor Example 6. Autocorrelation and Geostatistics 6.1 Time Series Analysis 6.1.1 Spectral Analysis: Glacial Lake Barlow-Ojibway Example 6.1.2 Trend Elimination and Cross-Spectral Analysis 6.1.3 Stochastic Modeling 6.2 Spatial Series Analysis 6.2.1 Finite or infinite variance? 6.2.2 Correlograms and Semivariograms: Pulacayo Mine Example 6.2.3 Applications to Other Ore Deposits 6.2.4 Geometric Probability Modeling 6.2.5 Extension Variance 6.2.6 Short-Distance Nugget Effect Modeling 6.2.7 Spectral Analysis: Pulacayo Mine Example 6.2.8 KTB Copper Example 6.3 Autocorrelation of Discrete Data 6.3.1 KTB Geophysical Data Example 7. 2D and 3D Trend Analysis 7.1 2D and 3D Polynomial Trend Analysis 7.1.1 Top of Arbuckle Formation Example 7.1.2 Mount Albert Peridotite Example 7.1.3 Whalesback Copper Mine Example 7.2 Kriging and Polynomial Trend Surfaces 7.2.1 Top of Arbuckle Formation Example 7.2.2 Matinenda Formation Example 7.2.4 Sulphur in Coal: Lingan Mine Example 7.3 Logistic Trend Surface Analysis of Discrete Data 7.4 Harmonic Trend Surface Analysis 7.4.3 Whalesback Copper Deposit Exploration Example 7.4.4 East-Central Ontario Copper and Gold Occurrence Example 8. Statistical Analysis of Directional Features 8.1 Directed and Undirected Lines 8.1.1 Doubling the Angle 8.1.2 Bjorne Formation Paleodelta Example 8.1.3 Directed and Undirected Unit Vectors 8.2 Unit Vector Fields 8.2.1 San Stefano Quartzphyllites Example 8.2.2 Arnisdale Gneiss Example 8.2.3 TRANSALP Profile Example 8.2.4 Pustertal Tectonites Example 8.2.5 Tectonic Interpretation of Unit Vector Fields Fitted to Quartzphyllites in the Basement of the Italian Dolomites 8.2.6 Summary of Late Alpine Tectonics South of Periadriatic Lineament 8.2.7 Defereggen Schlinge Example 9. Automated Stratigraphic Correlation, Splining and Geological Timescales 9.1 Ranking and Scaling 9.1.1 Methods of Quantitative Stratigraphy 9.1.2 Artificial Example of Ranking 9.1.3 Scaling 9.1.4 Californian Eocene Nannofossils Example 9.2 Spline-Fitting 9.2.1 Smoothing Splines 9.2.2 Irregularly Spaced Data Points 9.2.3 Tojeira Sections Correlation Example 9.3 Large-Scale Applications of Ranking and Scaling 9.3.1 Sample Size Considerations 9.3.2 Cenozoic Microfossils Example 9.4 Automated Stratigraphic Correlation 9.4.1 NW Atlantic Margin and Grand Banks Foraminifera Examples 9.4.2 Central Texas Cambrian Riley Formation Example 9.4.3 Cretaceous Greenland-Norway Seaway Microfossils Example 9.5 Construction of Geologic Timescales 9.5.1 Timescale History 9.5.2 Differences between GTS2012 and GTS2004 9.5.3 Splining in GTS2012 9.5.4 Treatment of Outliers 9.5.5 Early Geomathematical Procedures 9.5.6 Re-Proportioning the Relative Geologic Time Scale 10. Fractals 10.1 Fractal Dimension Estimation 10.1.1 Earth's Topography and Rock Unit Thickness Data 10.1.2 Chemical Element Concentration Values: Mitchell-Sulphurets Example 10.1.3 Total Metal Content of Mineral Deposits: Abitibi Lode Gold Deposit Example 10.2 Fractal Modeling of Point Patterns 10.2.1 Cluster Density Determination of Gold Deposits in the Kirkland Lake Area on the Canadian Shield 10.2.2 Cluster Density Determination of Gold Deposits in the Larger Abitibi Area 10.2.3 Worldwide Permissive Tract Examples 10.3 Geochemical Anomalies versus Background 10.3.1 Concentration-Area (C-A) Method 10.3.2 Iskut River Area Stream Sediments Example 10.4 Cascade Models 10.4.1 The Model of de Wijs 10.4.2 The Model of Turcotte 10.4.3 Computer Simulation Experiments 11. Multifractals and Singularity Analysis 11.1 Self-Similarity 11.1.1 Witwatersrand Goldfields Example 11.1.2 Worldwide Uranium Resources 11.2 The Multifractal Spectrum 11.2.1 Method of Moments 11.2.2 Histogram Method 11.3 Multifractal Spatial Correlation 11.3.1 Pulacayo Mine Example 11.4 Multifractal Patterns of Line Segments and Points 11.4.1 Lac du Bonnet Batholith Fractures Example 11.4.2 Iskut River Map Gold Occurrences 11.5 Local Singularity Analysis 11.5.1 Gejiu Mineral District Example 11.5.2 Zhejiang Province Pb-Zn Example 11.6 Chen Algorithm 11.6.1 Pulacayo Mine Example 11.6.2 KTB Copper Example 12. Selected Topics for Further Research 12.1 Bias and Grouped Jackknife 12.1.1 Abitibi Volcanogenic Massive Sulphides Example 12.2 Compositional Data Analysis 12.2.1 Star Kimberlite Example 12.3 Non-Linear Process Modeling 12.3.1 The Lorentz Attractor 12.4 Three-parameter Model of de Wijs 12.4.1 Effective Number of Iterations 12.4.2 Au and As in South Saskatchewan Till Example 12.5 Other Modifications of the Model of de Wijs 12. 5.1 Random Cut Model 12.5.2 Accelerated Dispersion Model 12.6 Trends, Multifractals and White Noise 12.6.1 Computer Simulation Experiment 12.7 Universal Multifractals 12.7.1 Pulacayo Mine Example 12.8 Cell Composition Modeling 12.8.1 Permanent Frequency Distributions 12.8.2 The Probnormal Distribution 12.8.3 Bathurst Area Acidic Volcanics Example 12.8.4 Abitibi Acidic Volcanics Example 12.8.5 Asymmetrical Bivariate Binomial Distribution Index

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• Matemática aplicada
• Paleontologia
• Mineralogia e pedras preciosas
• Geologia e a litosfera