Geomathematics : theoretical foundations, applications and future developments
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Bibliographic Information
Geomathematics : theoretical foundations, applications and future developments
（Quantitative geology and geostatistics, 18）
Springer, 2014
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Includes bibliographical references and index
Description and Table of Contents
Description
This book provides a wealth of geomathematical case history studies performed by the author during his career at the Ministry of Natural Resources Canada, Geological Survey of Canada (NRCan-GSC). Several of the techniques newly developed by the author and colleagues that are described in this book have become widely adopted, not only for further research by geomathematical colleagues, but by government organizations and industry worldwide. These include Weights-of-Evidence modelling, mineral resource estimation technology, trend surface analysis, automatic stratigraphic correlation and nonlinear geochemical exploration methods. The author has developed maximum likelihood methodology and spline-fitting techniques for the construction of the international numerical geologic timescale. He has introduced the application of new theory of fractals and multi fractals in the geostatistical evaluation of regional mineral resources and ore reserves and to study the spatial distribution of metals in rocks. The book also contains sections deemed important by the author but that have not been widely adopted because they require further research. These include the geometry of preferred orientations of contours and edge effects on maps, time series analysis of Quaternary retreating ice sheet related sedimentary data, estimation of first and last appearances of fossil taxa from frequency distributions of their observed first and last occurrences, tectonic reactivation along pre-existing schistosity planes in fold belts, use of the grouped jackknife method for bias reduction in geometrical extrapolations and new applications of the theory of permanent, volume-independent frequency distributions.
Table of Contents
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
by "Nielsen BookData"