Geomathematics: Theoretical Foundations, Applications and Future Developments (eBook)

ForewordPreface1.  Complexity of the Geological Framework and Use of Mathematics1.1  Use of Mathematics in Geology1.2  Geological Data, Concepts and Maps1.2.1  Map-Making 1.2.2  Geological Cross-Sections1.2.3  Scientific Method in the Geosciences1.2.4  Quality of Predictions1.3  Use of Curves1.3.1  Trend-Lines1.3.2  Elementary Differential Calculus1.3.3  Graphical Curve-Fitting1.4  Use of Surfaces1.4.1  Automated 3-Dimensional Map-Making: Central Baffin Example1.4.2  Folds and Faults1.5  Image Analysis1.5.1  Geometrical Covariance, Intercept and Rose Diagram1.5.2  Minkowski Operations: Bathurst Acidic Volcanics Example1.5.3  Boundaries and Edge Effects 2.  Probability and Statistics2.1  History of Statistics2.1.1  Emergence of Mathematical Statistics2.1.2  Spatial Statistics2.2  Probability Calculus and Discrete Frequency Distributions2.2.1 Conditional Probability and Bayes’ Theorem2.2.2  Probability Generating Functions2.2.3  Binomial and Poisson Distributions2.2.4  Other Discrete Frequency Distributions2.2.5  Oficina Formation Example2.3  Continuous Frequency Distributions and Statistical Inference2.3.1  Central-Limit Theorem 2.3.2  Frequency Distributions Derived from the Normal2.3.3  Significance Tests and 95%-Confidence Intervals2.3.4  Sum of Two Random Variables2.4  Applications of Statistical Analysis2.4.1  Statistical Inference: Grenville Potassium/Argon Ages Example2.4.2  Q-Q Plots: Normal Distribution Example2.5  Sampling2.5.1  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 Timescale3.1.1  Weighting Function Defined for the Inconsistent Dates Model3.1.2  Log-Likelihood and Weighting Functions3.1.3  Caerfai-St David’s Boundary Example 3.1.4  The Chronogram Interpreted as an Inverted Log-Likelihood Function3.1.5  Computer Simulation Experiments 3.1.6  Mesozoic Timescale Example 3.2  Lognormality and Mixtures of Frequency Distributions3.2.1  Estimation of Lognormal Parameters3.2.2  Muskox Layered Intrusion Example 3.2.3  Three-Parameter Lognormal Distribution3.2.4  Graphical Method of Reconstructing the Generating Process3.3  Compound Random Variables3.3.1  Compound Frequency Distributions and their Moments3.3.2  Exploration Strategy Example 4.  Correlation, Method of Least Squares, Linear Regression and the General Linear Model4.1  Correlation and Functional Relationship4.2  Linear Regression4.2.1  Degree of Fit and 95% - Confidence Belts 4.2.2  Mineral Resource Estimation Example4.2.3  Elementary Statistics of the Mosaic Model4.3  General Model of Least Squares4.3.1  Abitibi Copper Deposits Example4.3.2  Forward Selection and Stepwise Regression Applied to Abitibi Copper4.4  Abitibi Copper Hindsight Study4.4.1  Incorporation of Recent Discoveries4.4.2  Comparison of Weight Frequency Distributions of Copper Metal and Ore4.4.3 Final Remarks on Application of the General Linear Model to Abitibi Copper 5.  Prediction of Occurrence of Discrete Events5.1  Weights-of-Evidence Modeling5.1.1  Basic Concepts and Artificial Example5.1.2  Meguma Terrane Gold Deposits Example5.1.3  Flowing Wells in the Greater Toronto Area5.1.4  Variance of the Contrast and Incorporation of Missing Data5.2  Weighted Logistic Regression5.2.1   Meguma Terrane Gold Deposits Example5.2.2   Comparison of Logistic Model with General Linear Model5.2.3  Gowganda Area Gold Occurrences Example5.2.4  Results of the Gowganda Experiments5.2.5  Training Cells and Control Areas5.3  Modified Weights-of-Evidence5.3.1  East Pacific Rise Seafloor Example 6.  Autocorrelation and Geostatistics6.1  Time Series Analysis6.1.1  Spectral Analysis: Glacial Lake Barlow-Ojibway Example6.1.2  Trend Elimination and Cross-Spectral Analysis6.1.3  Stochastic Modeling 6.2  Spatial Series Analysis6.2.1  Finite or infinite variance?6.2.2  Correlograms and Semivariograms: Pulacayo Mine Example6.2.3  Applications to Other Ore Deposits6.2.4  Geometric Probability Modeling6.2.5  Extension Variance 6.2.6  Short-Distance Nugget Effect Modeling6.2.7  Spectral Analysis: Pulacayo Mine Example6.2.8  KTB Copper Example6.3  Autocorrelation of Discrete Data6.3.1  KTB Geophysical Data Example 7.  2D and 3D Trend Analysis7.1  2D and 3D Polynomial Trend Analysis7.1.1  Top of Arbuckle Formation Example7.1.2  Mount Albert Peridotite Example7.1.3  Whalesback Copper Mine Example7.2  Kriging and Polynomial Trend Surfaces7.2.1  Top of Arbuckle Formation Example7.2.2  Matinenda Formation Example7.2.4  Sulphur in Coal: Lingan Mine Example7.3  Logistic Trend Surface Analysis of Discrete Data7.4  Harmonic Trend Surface Analysis7.4.3  Whalesback Copper Deposit Exploration Example7.4.4  East-Central Ontario Copper and Gold Occurrence Example 8.  Statistical Analysis of Directional Features8.1  Directed and Undirected Lines8.1.1  Doubling the Angle8.1.2  Bjorne Formation Paleodelta Example8.1.3  Directed and Undirected Unit Vectors8.2  Unit Vector Fields8.2.1  San Stefano Quartzphyllites Example8.2.2  Arnisdale Gneiss Example8.2.3  TRANSALP Profile  Example8.2.4  Pustertal Tectonites Example8.2.5  Tectonic Interpretation of Unit Vector Fields Fitted to Quartzphyllites in the Basement of the Italian Dolomites8.2.6  Summary of Late Alpine Tectonics South of Periadriatic Lineament8.2.7  Defereggen Schlinge Example 9.  Automated Stratigraphic Correlation, Splining and Geological Timescales9.1  Ranking and Scaling9.1.1  Methods of Quantitative Stratigraphy9.1.2  Artificial Example of Ranking9.1.3  Scaling9.1.4  Californian Eocene Nannofossils Example9.2  Spline-Fitting9.2.1  Smoothing Splines9.2.2  Irregularly Spaced Data Points9.2.3  Tojeira Sections Correlation Example9.3  Large-Scale Applications of Ranking and Scaling9.3.1  Sample Size Considerations9.3.2  Cenozoic Microfossils Example9.4  Automated Stratigraphic Correlation9.4.1  NW Atlantic Margin and Grand Banks Foraminifera Examples9.4.2  Central Texas Cambrian Riley Formation Example9.4.3  Cretaceous Greenland-Norway Seaway Microfossils Example9.5   Construction of Geologic Timescales9.5.1  Timescale History9.5.2  Differences between GTS2012 and GTS20049.5.3  Splining in GTS20129.5.4  Treatment of Outliers9.5.5  Early Geomathematical Procedures9.5.6  Re-Proportioning the Relative Geologic Time Scale 10.  Fractals10.1  Fractal Dimension Estimation10.1.1  Earth’s Topography and Rock Unit Thickness Data10.1.2  Chemical Element Concentration Values: Mitchell-Sulphurets Example10.1.3  Total Metal Content of Mineral Deposits: Abitibi Lode Gold Deposit Example10.2  Fractal Modeling of Point Patterns10.2.1 Cluster Density Determination of Gold Deposits in the Kirkland Lake Area on the Canadian Shield10.2.2 Cluster Density Determination of Gold Deposits in the Larger Abitibi Area10.2.3  Worldwide Permissive Tract Examples10.3  Geochemical Anomalies versus Background10.3.1  Concentration-Area (C-A) Method10.3.2  Iskut River Area Stream Sediments Example10.4  Cascade Models10.4.1  The Model of de Wijs10.4.2  The Model of Turcotte10.4.3 Computer Simulation Experiments 11.  Multifractals and Singularity Analysis11.1  Self-Similarity11.1.1  Witwatersrand Goldfields Example11.1.2  Worldwide Uranium Resources11.2  The Multifractal Spectrum11.2.1   Method of Moments11.2.2   Histogram Method11.3  Multifractal Spatial Correlation11.3.1   Pulacayo Mine Example11.4   Multifractal Patterns of Line Segments and Points11.4.1   Lac du Bonnet Batholith Fractures Example11.4.2   Iskut River Map Gold Occurrences11.5  Local Singularity Analysis11.5.1  Gejiu Mineral District Example11.5.2  Zhejiang Province Pb-Zn Example11.6  Chen Algorithm11.6.1  Pulacayo Mine Example11.6.2  KTB Copper Example 12.  Selected Topics for Further Research12.1  Bias and Grouped Jackknife                        12.1.1  Abitibi Volcanogenic Massive Sulphides Example12.2  Compositional Data Analysis12.2.1  Star Kimberlite Example12.3  Non-Linear Process Modeling12.3.1  The Lorentz Attractor12.4  Three-parameter Model of de Wijs12.4.1  Effective Number of Iterations12.4.2  Au and As in South Saskatchewan Till Example12.5  Other Modifications of the Model of de Wijs12. 5.1  Random Cut Model12.5.2  Accelerated Dispersion Model12.6   Trends, Multifractals and White Noise12.6.1  Computer Simulation Experiment12.7  Universal Multifractals12.7.1  Pulacayo Mine Example12.8  Cell Composition Modeling12.8.1  Permanent Frequency Distributions12.8.2  The Probnormal Distribution12.8.3  Bathurst Area Acidic Volcanics Example12.8.4  Abitibi Acidic Volcanics Example12.8.5  Asymmetrical Bivariate Binomial DistributionIndex