Geomathematics

ForewordPrefaceChapter 1. Mathematical Models in Geology 1.1 Introduction 1.2 Scientific Methods in Geology 1.3 Use of Geomathematics 1.4 Mathematical Techniques in Use 1.5 Use of Empirical FunctionsChapter 2. Review of Calculus 2.1 Introduction 2.2 Classical Geometry 2.3 Variables and Functions 2.4 Transformations 2.5 Other Concepts of Plane Analytic Geometry 2.6 Differential Calculus 2.7 Infinite Series 2.8 Propagation of Error 2.9 Application of Taylor Series 2.10 Integral Calculus 2.11 Differential Equations 2.12 Complex NumbersChapter 3. Elementary Matrices 3.1 Introduction 3.2 Basic Definitions 3.3 Determinants 3.4 The Inverse Matrix 3.5 Linear Equations 3.6 Method of Pivotal Condensation 3.7 Homogeneous Equations; Singularity 3.8 Partitioned Matrices 3.9 Canonical Form, Eigenvalues and Eigenvectors 3.10 Powers of a Matrix 3.11 Spectral Components of a Matrix 3.12 Miscellaneous ConceptsChapter 4. Geometry 4.1 Introduction 4.2 Geometrical Interpretation of Vectors and Matrices 4.3 Lines and Planes in Three-Dimensional Space 4.4 Quadric Surfaces 4.5 Centers and Axes of Quadrics 4.6 Method of Least Squares 4.7 Major AxisChapter 5. Factor Analysis 5.1 Introduction 5.2 Practical Example: Mount Albert Peridotite Intrusion 5.3 Some Thermodynamical Considerations 5.4 Statistical Relationship Between the Mineralogical Variables 5.5 Factor Analysis 5.6 Application of Factor AnalysisChapter 6. Probability and Statistics 6.1 Introduction 6.2 Set Theory 6.3 Calculus of Probabilities 6.4 The Binomial Distribution 6.5 Discrete and Continuous Frequency Distributions 6.6 The Normal Distribution 6.7 Significance Test and Confidence Interval 6.8 Method of Moments 6.9 Properties of Expectation and Variance 6.10 Continuous Frequency Distributions Derived from the Normal Distribution 6.11 Age Determinations for Structural Provinces in the Canadian Shield 6.12 A Simple Mathematical Model to Estimate Time Span 6.13 Statistical Inference of Age DeterminationsChapter 7. Frequency Distributions and Functions of Independent Random Variables 7.1 Introduction 7.2 Functions of Random Variables 7.3 Normal Probability Paper 7.4 Other Methods of Frequency-Distribution Analysis 7.5 A Computer Simulation Experiment 7.6 Probability Generating Functions 7.7 Examples of Discrete Random Variables 7.8 Compound Random Variables 7.9 A Problem of Exploration Strategy 7.10 Desirable Properties of Estimators 7.11 Maximum Likelihood and Other Methods of EstimationChapter 8. Statistical Dependence; Multiple Regression 8.1 Introduction 8.2 Statistical Dependence 8.3 Covariance, Correlation Coefficient and Conditional Probability 8.4 The Bivariate Normal Distribution 8.5 Bivariate Regression Analysis 8.6 Multiple Regression 8.7 Geometric Interpretation of Least Squares 8.8 Regression Analysis and Component Analysis 8.9 Methods of Sequential Regression Analysis 8.10 Dummy VariablesChapter 9. Trend Analysis 9.1 Introduction 9.2 Empirical Trend Functions 9.3 Applied Trend Analysis 9.4 Mineralogical Variations in the Mount Albert Intrusion 9.5 Specific Gravity in Mount Albert Intrusion 9.6 Analysis of Covariance 9.7 Spatial Variation of Copper in the Whalesback Deposit, Newfoundland 9.8 Trend Surface Analysis, 425-ft Level 9.9 Three-Dimensional Trend Analysis 9.10 Three-Dimensional Analysis of Surface-Hole DataChapter 10. Stationary Random Variables and Kriging 10.1 Introduction 10.2 Space Series 10.3 The Model of De Wijs 10.4 Correlograms and Variograms 10.5 Main Properties of the Autocorrelation Function 10.6 Theoretical Autocorrelation Functions 10.7 Transformations of Stationary Series 10.8 Two-Dimensional Autocorrelation Functions 10.9 Theoretical Spatial Autocorrelation Functions 10.10 Variance of Average Concentrations for Volumes of Rock 10.11 Matheron's Geostatistical Theory 10.12 Kriging 10.13 Least-Squares Model with Deterministic and Stochastic ComponentsChapter 11. Harmonic Analysis and Power Spectra 11.1 Introduction 11.2 One-Dimensional Fourier Series 11.3 Spectral Analysis 11.4 Stochastic Model for Varve Deposition in Glacial Lake Barlow-Ojibway 11.5 Spectral Analysis of Minerals in Thin Section 11.6 Two-Dimensional Harmonic Analysis 11.7 Harmonic Analysis of Copper and Gold Occurrences in East-Central Ontario 11.8 FilteringChapter 12. Powers of Matrices and Markov Chains 12.1 Introduction 12.2 Sylvester's Theorem 12.3 Applications of Sylvester's Theorem 12.4 An Iterative Method for Extracting Eigenvectors 12.5 Iterative Procedures, Review of Powering Methods 12.6 Computer Algorithms for Extracting Eigenvalues 12.7 Markov Chains; Explicit Form of Transition Matrix 12.8 Embedded Markov Chains 12.9 Upper Pennsylvanian Cyclothems in KansasChapter 13. Multivariate Stochastic Process-Models with Applications to the Petrology of Basalts 13.1 Introduction 13.2 Petrological Models 13.3 Projection of a p-Dimensional Situation Onto the Harker Diagram 13.4 The Yellowknife Volcanic Belt 13.5 Calculation of Trend Lines in the Triangular Diagram 13.6 Best-Fitting Trends in a Nine-Component System 13.7 Other Petrological Models of Basalt Magma; Comparison of Principal Components to Indices 13.8 The Kolmogorov Differential Equations 13.9 The Transition Matrix of Multivariate Series 13.10 Properties of the First Spectral Component 13.11 Extrapolation by Kolmogorov Differential EquationsChapter 14. Calculation of Preferred Orientations from Vectorial Data 14.1 Introduction 14.2 Methods of Treating Angular Data 14.3 Methods of Calculating the Mean for Vectorial Data in the Plane 14.4 Comparison of Means 14.5 Systematic Variation of Locality Means 14.6 Unit Vectors in Space 14.7 Comparison of Vector Means 14.8 Geological Introduction to Unit Vector Fields 14.9 Method of Fitting Unit Vector Fields 14.10 The Pusteria TectonitesChapter 15. Spatial Variability of Multivariate Systems 15.1 Introduction 15.2 Problems Related to Complexity of the Geological Framework 15.3 Methods of Multivariate Analysis 15.4 Discriminant Analysis 15.5 Calculation of a Probability Index 15.6 Generalization to Multivariate Situations 15.7 Large Copper Deposits in the Abitibi Area 15.8 Two-Stage Least-Squares ModelsBibliographySelected ReadingReferences CitedIndex