Geomathematics: Theoretical Foundations, Applications and Future Developments

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