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