Encyclopedia of Mathematical Geosciences (Encyclopedia of Earth Sciences Series)

Foreword References Preface Acknowledgments Contents About the Editors Editorial Board Contributors A Accuracy and Precision Definitions The Mathematical Geosciences Context Resolution of the Measuring Device: Significant Figures Sensed Data Integrating Sensed and Directly Measured Data Using Integrated Data Types for Estimation Summary Cross-References Bibliography Additive Logistic Normal Distribution Synonyms Definition Sample Space of Compositional Data and Its Geometry The Logistic Normal Model Key Properties Applications to Geosciences Summary Cross-References Bibliography Additive Logistic Skew-Normal Distribution Synonyms Definition Sample Space of Compositional Data and Its Geometry The Multivariate Skew-Normal Model The Logistic Skew-Normal Model Definition Key Properties Application to Geochemical Data Summary Cross-References Bibliography Agterberg, Frits Biography Bibliography Aitchison, John Biography Bibliography Algebraic Reconstruction Techniques Definition Tomographic Projection Model ART Algorithm Convergence of ART Algorithm Characteristics of the Solution Advantages and Disadvantages of ART Simultaneous Iterative Reconstructive Technique (SIRT) Simultaneous Algebraic Reconstructive Technique (SART) Modeling the Forward Projection Process Summary and Conclusions Bibliography Allometric Power Laws Definition Introduction: Allometric Power Laws and Self-Similarity Examples for Allometric Power Laws in Geosciences Allometric Relations with Modified Power Laws Allometric Power Laws for Size Distribution Concluding Remarks: The Power of Allometric Power Laws Cross-References References Argand Diagram Synonyms Definition Argand Diagram Historical Origin of the Argand Diagram The Argand Diagram: A Geometric Plot Some Applications of the Argand Diagram in Geosciences Summary Bibliography Artificial Intelligence in the Earth Sciences Definition Introduction What Is Artificial Intelligence? Development and Activity Domains Expert Systems Natural Language Processing Machine Learning Computer Vision Planning Systems Robotics Applications in the Earth Sciences Planning Systems Conclusion and Prospectus Bibliography Artificial Neural Network Definition Introduction ANN Models Components of ANNs Organization Learning Applications Conclusions Cross-References Bibliography Autocorrelation Synonyms Definition Discussion Time Series Extensions Summary Cross-References Bibliography Automatic and Programmed Gain Control Definition Introduction: Stationary and Nonstationary Processes, Gain Control Programmed Gain Control (``True Amplitude Recovery´´) of Seismic Data Automatic Gain Control (AGC) A Few Concluding Remarks on the Use of PGC and AGC Cross-References References B Backprojection Tomography Definition Introduction Parallel-Beam Image Reconstruction Radon Transform Fourier Slice Theorem Filtered Backprojection Algorithm Fan-Beam Image Reconstruction Reconstruction with Rebinning Reconstruction without Rebinning Filtered Backprojection fan-Beam Algorithm Cone-Beam Image Reconstruction Feldkamp´s Algorithm Summary and Conclusions References Bayes´s Theorem Synonyms Definition Application to Mineral Exploration or a ``Mineral Adventure´´ Application to Geological Systems Modeling or ``Like Oil and Water´´ Derivation Bayesian Statistics Conclusions Cross-References Bibliography Bayesian Inversion in Geoscience Definition Introduction Methods Bayesian Linear Inversion Monte Carlo Methods Example Conclusions Bibliography Bayesian Maximum Entropy Definition Introduction BME Methodology The Knowledge Bases (KB) The Chronotopologic Domain The Spatiotemporal Random Field Model The BME Workflow BME Chronotopologic Estimation and Uncertainty Assessment BME Features and Applications BME Versatility The Emergence of Stochastic Logic in Geostatistics BME in Software Applications of BME in the Real World Conclusions Acknowledgments Bibliography Best Linear Unbiased Estimation Definition Introduction Basic Ideas Application: Kriging Inverse Problems Basic Form Unknown Coefficients in the Mean The Kalman Filter The Model Kalman Filter Algorithm Prediction Updating Kalman Filter with Unknown Inputs Successive Linearization Variable Transformation Algorithms for Large Problems The Ensemble Kalman Filter Relation to Bayesian Methods Summary and Conclusions References Big Data Definition Introduction Infrastructure and Characteristics of Big Data Open Data to Change the Mode of Geoscience A Smart Data Ecosystem for Geoscience: A Vision for the Future Technological Approaches Toward a Smart Data Ecosystem Machine-Readable FAIR Metrics of Geoscience Data Semantic Representation of Scientific Workflows for Provenance Documentation Context-Aware Data and Resource Recommendation for Geoscience Data Science Approaches for Geoscience in the Era of Big Data Summary and Conclusions Cross-References Bibliography Binary Mathematical Morphology Definition Basic MM Operators on Binary Images Filtering on Binary Images Other Operators on Binary Images Summary Cross-References Bibliography Binary Partition Tree Definition Illustration Construction and Processing the BPT BPT in Edge-Weighted Graphs Summary Cross-References Bibliography Bonham-Carter, Graeme Biography Bibliography Bootstrap Definition Introduction Traditional Bootstrap Resampling Spatial Bootstrap Resampling Discussion Summary and Conclusions Cross-References Bibliography Burrough, Peter Alan Biography Bibliography C Cartogram Synonyms Definition Area Cartogram Rectangular Cartograms Mosaic Cartograms Circular Cartograms Noncontiguous Cartograms Contiguous Cartograms Distance Cartogram Summary Cross-References Bibliography Chaos in Geosciences Definition Introduction Model of de Wijs Gold in South Saskatchewan till Example The Lorenz Attractor Summary and Conclusions Cross-References Bibliography Chayes, Felix Biography Bibliography Cheng, Qiuming Biography Bibliography Circular Error Probability Definition Introduction Doubling the Angles Bjorne Formation Paleodelta Example Directed and Undirected Unit Vectors TRANSALP Profile Example Summary and Conclusions Cross-References Bibliography Cloud Computing and Cloud Service Synonyms Definitions Background What Is Cloud Computing? Using Cloud in Geosciences Geoscientists on Cloud Geoscientific Use Cases Cloud-Based Data Storage Cloud-Based Data Analysis Challenges and Opportunities Turning Geoscientific Applications to Cloud Native Managing Cloud Expenses Data Security and Privacy Governance Private Cloud Summary Bibliography Cluster Analysis and Classification Definition Historical Material Essential Concepts and Applications Data Problems in the Context of Cluster Analysis Conclusions Cross-References Bibliography Compositional Data Synonyms Definitions Introduction Principles The Sample Space of Compositional Data The Aitchison Geometry for CoDa Linear Functionals on Compositions The Principle of Working in Coordinates Non-Orthogonal Log-Ratio Representations Orthogonal Log-Ratio Representations Properties of an Orthonormal Representation Sequential Binary Partitions Zeros Exploratory CoDa Tools: Variation Matrix, Biplot, and CoDa-Dendrogram Varanasi Dataset Center and Variability Compositional Biplot Compositional Dendrogram Software for CoDa Conclusions Books on Compositional Data Analysis Cross-References Bibliography Computational Geoscience Definition Censored Data Process Discovery Multivariate Data Analysis Techniques Fractal Methods Visualizing the Coordinates and Elemental Associations of a Multivariate Metric Geospatial Coherence Recognizing Rare Events - Under-Sampled Processes in Principal Component Analysis Process Validation - Testing Models Derived from Process Discovery Example of Process Discovery and Process Prediction of Mineral Resource Potential Using Geochemical Survey Data Process Discovery AOV Geospatial Coherence in Process Discovery Process Validation Summary Cross-References Bibliography Computer-Aided or Computer-Assisted Instruction Definition Introduction Types of CAI Programs Features of CAI Types of Student Interactions in CAI How Is CAI Implemented? Advantages of CAI Limitations of CAI Application of CAI in Geospatial Analysis Conclusion Cross-References Bibliography Concentration-Area Plot Definition Introduction Methodology: Concentration-Area Fractal Model Case Study: Sweden Summary of the Other Models Developed Based on the C-A Fractal Model Summary and Conclusions Cross-References Bibliography Constrained Optimization Definition Introduction Constrained Optimization Application in Geospatial Science Case Study Summary and Conclusions Cross-References Bibliography Convex Analysis Definition Introduction Convex Sets Convex Functions Operations that preserve Convexity Duality Example - Least Squares Solution Summary and Conclusions Cross-References Bibliography Copula in Earth Sciences Synonyms Definition Introduction Measures of Spatial Variability Some Fields of Application Basic Concepts Some Features of Copulas Copula Families Copulas for Describing Spatial Variability Estimation Aspects Copula Modeling Validation of the Copula Models Vine Copula Conclusions Cross-References Bibliography Correlation and Scaling Definition Correlation Between Sections in Stratigraphy Examples of Ranking, Scaling, and Correlation Summary and Conclusions Cross-References Bibliography Correlation Coefficient Definition Introduction Scatter Graph Random Variable Population and Sample Basic Statistics Correlation Coefficient Hypothesis Testing Rank Correlation Summary Bibliography Coupled Modeling Definition Introduction Hydromechanical Coupling Solid Deformation Model and Its Coupled Form Fluid Flow Model and Its Coupled Form Thermomechanical Coupling Thermo-Hydromechanical Coupling Chemo-Hydromechanical Coupling Solving Coupled Models Coupling Approaches Summary Cross-References Bibliography Cracknell, Arthur Phillip Biography Acknowledgments Cressie, Noel A.C. Biography Cross-References References Crystallographic Preferred Orientation Synonyms Definition Historical Notes Introduction Definitions Estimation of an Orientation Density Function Estimation with Integral Orientation Measurements Estimation with Individual Orientation Measurements Grain Modeling with Individual Orientation Measurements Orientation Statistics Estimation of Macroscopic Directional Properties Applications of Texture Analysis Typical Applications in General Texture of a Hematite Specimen from the Pau Branco Deposit, Quadriltero Ferrífero (Iron Quadrangle), Brazil Conclusions Bibliography Cumulative Probability Plot Definition Illustration Relation to Probability Density and Probability Mass Functions Some Important Properties Uses in Statistics Summary Cross-References Bibliography D Dangermond, Jack Biography Data Acquisition Definition Terminology Introduction Remote Sensing Data Acquisition Principles Types Sensors Resolution Conclusions Bibliography Data Dictionary Definition Introduction Types of Data Dictionary Active Data Dictionary Passive Data Dictionary Components of a Data Dictionary Significance of Data Dictionary Data Dictionary in NLP Data Dictionary in Geoscience Summary Cross-References References Data Life Cycle Definition Introduction Conceptualization Data Collection Data Processing Data Archiving Data Analysis Data Sharing Data Product Summary Cross-References Bibliography Data Mining Synonyms Definition What Geoscience Data Can Be Mined? What Methods Are Used in Data Mining? Summary and Conclusions Cross-References Bibliography Data Visualization Definition Overview Geovisualization and Geovisual Analytics Challenges, Opportunities, and Future Scope Bibliography Database Management System Definition Terminology Introduction Metadata Management Quality Security Integration Governance Storage Data Compression Archive Formats Storage Devices Backup Types Cloud Storage Big Data Management Remote Sensing Data Management Conclusions Bibliography David, Michel Biography Bibliography Davis, John Clements Biography Bibliography De Wijs Model Definition Introduction De Wijs Data Selected Case Study of De Wijs Model Summary and Conclusions Bibliography Decision Tree Definition Terminology Introduction Stability in a Decision Tree Practical Implications of Decision Tree Analysis The Eastern Himalayas Example The Western Himalayan Example Conclusions Cross-References References Deep CNN-Based AI for Geosciences Definition Introduction Convolutional Layer Pooling Layer Fully Connected Layers Training a Deep CNN Model Transfer Learning Some Commonly Used CNN Architecture AlexNet (Year) VGGNet Inception Net/GoogLeNet ResNet (2015) Inception ResNet MobileNet Applications in Geosciences Classification Segmentation Detection Other Summary Cross-References Bibliography Deep Learning in Geoscience Synonyms Definition Introduction Convolutional Neural Networks (CNNs) Recurrent Neural Networks (RNNs) Autoencoders (AEs) Graph Neural Networks (GNNs) Summary Acknowledgments Bibliography Dendrogram Definition A Clustering Point of View Applications in Geosciences Summary Cross-References Bibliography Differential Equations Definition Introduction Classification History Solution Methods Notation Conclusion Cross-References Bibliography Digital Elevation Model Definition and Scope Introduction Extracting Elevation/3D Coordinates from Stereo Imagery: A Generic Photogrammetric Data Adjustment Approach Satellite Photogrammetry for Line-scanner Imagery Rigorous Sensor Model (RSM) Coordinate Systems Estimation of Image Coordinates from Ground Coordinates Estimation of Ground Coordinates from Image Coordinates Rational Polynomial Model (RPM) (Grodecki 2001; Tao and Ha 2001; Sanjay et al. 2008) Some Applications of Photogrammetric Adjustment Are: Reconstruction of Attitude Behavior Reconstruction of Orbital Parameters Corrections to Viewing Geometry with RPCs 3D Point Coordinates Computation DEM with RSM DEM with RPM Photogrammetric Adjustment of Aerial Stereo Photo Blocks DEMs from Unmanned Aerial Vehicles (UAV) Using Structure from Motion (SfM) Approach Image Matching Hierarchical Image Matching (HIM) Image Condensation Interest Point Generation Local Mapping Image Matching Scale Invariant Feature Transform (SIFT) Scale-Space Extrema Detection Keypoint Localization Orientation Assignment Keypoint Descriptor DEM Interpolation Weighted Average Polynomial Fitting Finite Element Method Kriging Gerchberg Algorithm DEM Formats Summary Bibliography Digital Filters Definition Introduction Single-Channel Convolutional Filtering in Time Domain, the z-Transform Filtering in Fourier Domain Transfer Function of Digital Filters Single-Channel Optimal Filters Appendix: Recursive Filtering (Shanks, 1967) Concluding Remarks on the Use of Digital Filtering Cross-References References Digital Geological Mapping Definition The History of Digital Geological Mapping Digital Geological Mapping System The Process of Digital Geological Mapping Summary Cross-References Bibliography Digital Twins of the Earth Synonyms Definition Essential Concepts and Components DT Versus Digital Model Some Significant Research and Innovation Applications Current Investigations, Controversies, and Gaps in Current Knowledge Summary or Conclusions Cross-References Bibliography Dimensionless Measures Synonyms Definition Introduction Dimensionless Measures in Geoscience Example 1: Pixel and Pixel Value Example 2: Scale Example 3: Contrast Ratio (CR) Example 4: Spectral Index or Band Ratio Example 5: Error Matrix and Measures Through It Summary and Conclusion Cross-References Bibliography Discrete Prolate Spheroidal Sequence Definition Overview Methodology Formulation of the Concentration Problem Properties of the Slepian Eigenvectors and Eigenvalues Applications in Spectral Estimation Periodogram Estimator Welch´s Method Multitaper Method for Spectral Density Estimation Selection of the MTM Parameters DPSS in the Geosciences Software Implementations Summary or Conclusions Cross-References Bibliography Discriminant Analysis Definition Introduction Linear Discriminant Analysis Stage I: Calculation of Between-Class Variance Stage II: Calculation of Within-Class Variance Stage III: Construct Lower Dimensional Space Example Evaluation Metrics Conclusions Cross-References Bibliography Discriminant Function Analysis Definition Introduction Discrimination, Classification, and Identification Fisher (Two-Group) Linear Discriminant Analysis Multiple Group Linear Discriminant Analysis (Canonical Variates Analysis) Performance Testing and Statistical Significance Conclusion Bibliography Doveton, John Holroyd Biography Bibliography E Earth Surface Processes Definition Earth Surface Processes (ESP) and Landforms Mathematical Concepts to Study Various ESP and Landforms Differential Equation Hillslope Evolution Bedrock Channel Evolution Exponential Function Power-Law Function Fractals in ESP Matrices Statistical Approaches in ESP Numerical Methods in ESP Machine Learning Application in ESP Conclusions Bibliography Earth System Science Definition Goal of Earth System Science History of Earth System Science Research Tasks of Earth System Science New Concepts in Earth System Science Information Processing and Expression in Earth System Science Next Generation of Earth System Science Conclusions Bibliography Earthquake Complexity Definition Introduction The Implausibility of Earthquake Prediction Pervasive Patterns Consistent with Complexity The 1906 San Francisco Earthquake: Unequivocally Unpredictable Burridge-Knopoff Model (1967) as a Paradigm for Complexity Summary and Conclusions Bibliography Eigenvalues and Eigenvectors Definition Overview Applications with Focus on Covariance Matrix Summary Cross-References Bibliography Electrofacies Definition Facies and Electrofacies Supervised and Unsupervised Models Parametric Methods Nonparametric and Neural Network Methods Summary Cross-References Bibliography Ensemble Kalman Filtering Definition The Kalman Filter The Extended Kalman Filter The Ensemble Kalman Filter The Ensemble Kalman Filter with Augmented State The Normal-Score Ensemble Kalman Filter Kalman Filter and Kriging Summary and Conclusions Cross-References References Entropy Definition Historical Development and Properties Historical Development Basic Properties of (Z) Conditional Entropy and Mutual Information Differential Entropy and Multivariate Case Applications Characterizing Spatial Uncertainty Maximum Entropy Principle for Sampling Design Summary and Final Remarks Bibliography Euler Poles of Tectonic Plates Definition Introduction Modeling the Geometry of Tectonic Plates and the Kinematics of Their Motion Geometric Modeling of Tectonic Plates Rotations in 3 Kinematic Modeling of Plate Motion-Finite Rotations Kinematic Modeling of Plate Motion-lnfinitesimal Rotations Rotating Several Polygons Path of Migrating Euler Pole Conclusions Bibliography Expectation-Maximization Algorithm Definition Overview Applications with Focus on Remotely Sensed Images Summary Cross-References References Exploration Geochemistry Synonyms Definition Introduction Historical Developments Univariate and Bivariate Methods Multivariate Methods Spatial Methods Directions Summary Cross-References Bibliography Exploratory Data Analysis Definition Introduction Graphical Analysis Robust Classification of Univariate Data Data Standardization and Transformation Exploratory Analysis of Multivariate Relationships Concluding Remarks Cross-References Bibliography F FAIR Data Principles Synonyms Definition Introduction Findable F ``Principle F1: (meta)data are assigned a globally unique and persistent identifier´´ ``Principle F2: data are described with rich metadata´´ ``Principle F3: (meta)data clearly and explicitly include the identifier of the data it describes´´ ``Principle F4: (meta)data are registered or indexed in a searchable resource´´ Accessible A ``Principle A1: (meta)data are retrievable by their identifier using a standardized communications protocol´´ ``Principle A1.1: the protocol is open, free, and universally implementable´´ ``Principle A1.2: the protocol allows for an authentication and authorization procedure, where necessary´´ ``Principle A2: (meta)data are accessible, even when the data are no longer available´´ Interoperable I ``Principle I1: (meta)data use a formal, accessible, shared, and broadly applicable language for knowledge representation´´ ``Principle I2: (meta)data use vocabularies that follow FAIR principles´´ ``Principle I3: (meta)data include qualified references to other (meta)data´´ Reusable R ``Principle R1: (meta)data are richly described with a plurality of accurate and relevant attributes´´ ``Principle R1.1: (meta)data are released with a clear and accessible data usage license´´ ``Principle R1.2: (meta)data are associated with detailed provenance´´ ``Principle R1.3: (meta)data meet domain-relevant community standards´´ Summary and Conclusions Acknowledgments Bibliography Fast Fourier Transform Definition Introduction Application to Drill-Hole Data from the Whalesback Copper Deposit Summary and Conclusions Cross-References Bibliography Fast Wavelet Transform Definition Introduction Wavelet Transform CWT Versus DWT Fast Wavelet Transform (FWT) Summary and Conclusion Cross-References Bibliography Favorability Analysis Definition Introduction Basic Concepts Classification of Spatial Variables Quantification of Geoscience Maps Control Area and Statistical Analogy Characteristic Analysis Favorability Analysis Concluding Remarks Bibliography Fisher, Ronald A. Biography Bibliography Flow in Porous Media Synonyms Definition Principles Quantifying Flow in Porous Media Open Questions and Unresolved Issues Conclusion Bibliography Forward and Inverse Stratigraphic Models Synonyms Definition Introduction Forward Stratigraphic Modeling Inverse Stratigraphic Modeling Forward Stratigraphic Models Geometrical Models Topography-Control (Diffusion, or Potential-Gradient) Models Hydraulic Process-Response Models Comparison Between Various Equations for Bed-Load Transport Mixed Algorithmic, Heuristic, and Fuzzy-Logic Models Inverse Stratigraphic Models Current Investigations, Controversies, and Gaps in Current Knowledge Limitations to Present Models Summary Bibliography Fractal Geometry and the Downscaling of Sun-Induced Chlorophyll Fluorescence Imagery Synonyms Definition Introduction Fractal Geometry and the Power Law (PL) Distribution of Sun-Induced Chlorophyll Fluorescence (SIF)-Emitting Objects Case Study: ``Fractal Geometry for SIF-Downscaling´´ Summary Cross-References Bibliography Fractal Geometry in Geosciences Definition Introduction Fractal Dimension Estimation Separation of Geochemical Anomalies from Background by Fractal Methods Fractal Perimeter-Area Relation The Multifractal Spectrum Singularity Analysis Model of de Wijs Multifractal Spatial Correlation Fractal/Multifractal Modeling of Point Patterns Lognormal and Pareto Size-Frequency Distributions for Mineral Resources Summary Cross-References References Fractal Landscape Definition Introduction A Review of Fractals: A Unifying Basis to Describe Nature´s Complexity Self-similar Fractals Self-affine Fractals Multifractals Scaling of Earth´s Landscape Proposed Dynamical Models for Fractal Landscapes Conclusion Bibliography Frequency Distribution Definition Different Forms Statistics Cautions and Warnings Summary and Conclusions Cross-References Bibliography Frequency-Wavenumber Analysis Definition Introduction Sampling of Rocks: Pulacayo Zinc Deposit Example Short-Distance Nugget Effect Modeling Universal Multifractals Summary and Conclusions Cross-References Bibliography Full Normal Plot Definition The Method Application Summary Cross-References References Fuzzy C-Means Clustering Definition Overview Applications with a Focus on Image Processing Future Scope Cross-References References Fuzzy Inference Systems for Mineral Exploration Definition FIS for Mineral Exploration Architecture of a FIS Types of FISs Example of a FIS Designed for Gold Mineralization Selected Case Studies Summary and Conclusions Cross-References Bibliography Fuzzy Set Theory in Geosciences Definition Introduction of Fuzzy Set Theory Stages of Fuzzy Logic Modeling Further Definition Using Various Case Studies Geothermal Resource Exploration Exploration Geology Using Remote Sensing Data Application of Iterative Fuzzy Edge Detection Algorithm to Blurred Satellite Images Summary or Conclusions Cross-References Bibliography G Genetic Algorithms Definition Introduction GAs for Clustering Example Conclusion Cross-References Bibliography Geocomputing Synonyms Definition The Why and How Selected Applications and Challenges Conclusions Cross-References Bibliography Geographical Information Science Definition Introduction Spatial Data Concepts Spatial Data Sources Geographical Coordinate System Map Projections Spatial Data Models Spatial Database Creation Spatial Relations Spatial Data Analysis Vector Data Analysis Raster Data Analysis Spatial Data Interpolation Digital Terrain Mapping Network Analysis Statistical Analysis Visualization of Data Analysis Spatial Decision Support Systems Spatial Data Accuracy Assessment Applications Research Challenges in Geographical Information Science Research Areas Conclusions Cross-References Bibliography Geographically Weighted Regression Definition Introduction Model Formulation Model Calibration Model Output Case Study Extensions to GWR Conclusions Cross-References Bibliography Geohydrology Synonyms Definition Introduction Open-Pit Mine Dewatering Dewatering Techniques Passive Dewatering Active (Advance) Dewatering Groundwater Flow Model Study Schedule Conclusions Bibliography Geoinformatics Definition Introduction Quantitative Geosciences: Evolving Computational Systems Quantitative Geosciences: Early Computer Engagement Quantitative Geosciences: The Twenty-First Century Current Investigation Examples Predicting Earthquakes Mapping and Analyzing Global Geospatial Data Geoinformatics: Discussion Topics Summary Cross-References Bibliography Geologic Time Scale Estimation Definition Introduction Methods and Ages Conventions and Standards The McKerrow Method Smoothing Splines Smoothing Spline Software Devonian Example Paleozoic ``Super-Spline´´ Cretaceous and Jurassic Periods Cubic Spline Summary Cross-References References Geomathematics Definition Introduction Relation of Geomathematics to Mathematical Statistics Involvement by Mathematical Statisticians Relation of Geomathematics to the Geosciences Geostatistics Mineral Resource Estimation Compositional Data Analysis Directional Features Mathematical Morphology Quantitative Stratigraphy Geologic Time Scale Estimation Fractals and Multifractals Lognormal and Pareto Frequency Distributions Summary Cross-References References Geomatics Definition Geomatic Components Land Surveying Survey Measurements Geodesy Photogrammetry Remote Sensing Global Navigation Satellite Systems (GNSS) Geographic Information Systems (GIS) Cartography Innovations in Geomatics Geomatic Applications Summary Cross-References Bibliography Geomechanics Definition Introduction Subsurface Stresses and Pore Pressure Subsurface Stresses Pore Pressure Vertical Stress Research Path of Geomechanics Conclusion Bibliography Geoscience Signal Extraction Definition Introduction Pulacayo Mine and Main KTB Borehole Examples Spectral Analysis: Glacial Lake Barlow-Ojibway Example Unit Vector Fields Summary and Conclusions Cross-References Bibliography Geosciences Digital Ecosystems Synonyms Definition Essential Concepts and Components Ecosystem Principles, Patterns, and Governance Some Significant Research and Innovation Aspects Evolutionary Aspect Metasystemic Aspect Summary and Conclusions Cross-References Bibliography Geostatistical Seismic Inversion Definition Geophysical Inversion Trace-by-Trace Geostatistical Seismic Inversion Global Geostatistical Seismic Inversion Summary Cross-References Bibliography Geostatistics Definition Introduction Statistical Formulation Stationarity Properties of the Covariance and Variogram Functions Nonstationary Random Functions and the Generalized Covariance Function Coregionalization Estimating Covariance and Variogram Functions Data Configurations Modeling Experimental Covariances and Semivariograms Kriging Linear Kriging Cokriging Kriging of ZV(x0) and Functions of Z(x0) Non-Parametric Estimation: Multiple Indicator Kriging Projection onto Larger Spaces Disjunctive Kriging Bivariate Distributions and Isofactorial Models The DK Equations Disjunctive Kriging in the Bivariate Gaussian Model Change of Support Simulation Simulation of a Second-Order Stationary Process Sequential Gaussian Simulation Sequential Indicator Simulation Simulation by Matrix Decomposition Conditional Simulation Using Successive Residuals Truncated Gaussian Simulation and Plurigaussian Simulation The Gibbs Sampler Multi-Point Simulation (SNESIM) Other Simulation Methods Concluding Remarks Cross-References References GeoSyntax Background Essential Concepts of the GeoSyntax Approach Current Status Extension to Two- and Three-Dimensional Sedimentary Bodies Summary Bibliography Geotechnics Definition Introduction Research Path of Geotechnics Conclusion Bibliography Geothermal Energy Definition Subsurface Temperature Modeling Application of Geostatistics Application of Machine Learning Remote Sensing Applications Outline of Remote Sensing Surface Temperature Estimation Detection and Mapping of Hydrothermally Altered Minerals Lineament-Based Fracture Characterization Continuous Mapping of Topographic Change by D-InSAR Fluid Flow Simulation Summary Cross-References Bibliography Global and Regional Climatic Modeling Definition Introduction Climate: Basic Approaches to Analysis Modeling: Tools and Approaches Global Versus Local Summary and Conclusions Cross-References Acknowledgments Bibliography Goodchild, Michael F. Biography Cross-References Referenc Foreword References Preface Acknowledgments Contents About the Editors Editorial Board Contributors A Accuracy and Precision Definitions The Mathematical Geosciences Context Resolution of the Measuring Device: Significant Figures Sensed Data Integrating Sensed and Directly Measured Data Using Integrated Data Types for Estimation Summary Cross-References Bibliography Additive Logistic Normal Distribution Synonyms Definition Sample Space of Compositional Data and Its Geometry The Logistic Normal Model Key Properties Applications to Geosciences Summary Cross-References Bibliography Additive Logistic Skew-Normal Distribution Synonyms Definition Sample Space of Compositional Data and Its Geometry The Multivariate Skew-Normal Model The Logistic Skew-Normal Model Definition Key Properties Application to Geochemical Data Summary Cross-References Bibliography Agterberg, Frits Biography Bibliography Aitchison, John Biography Bibliography Algebraic Reconstruction Techniques Definition Tomographic Projection Model ART Algorithm Convergence of ART Algorithm Characteristics of the Solution Advantages and Disadvantages of ART Simultaneous Iterative Reconstructive Technique (SIRT) Simultaneous Algebraic Reconstructive Technique (SART) Modeling the Forward Projection Process Summary and Conclusions Bibliography Allometric Power Laws Definition Introduction: Allometric Power Laws and Self-Similarity Examples for Allometric Power Laws in Geosciences Allometric Relations with Modified Power Laws Allometric Power Laws for Size Distribution Concluding Remarks: The Power of Allometric Power Laws Cross-References References Argand Diagram Synonyms Definition Argand Diagram Historical Origin of the Argand Diagram The Argand Diagram: A Geometric Plot Some Applications of the Argand Diagram in Geosciences Summary Bibliography Artificial Intelligence in the Earth Sciences Definition Introduction What Is Artificial Intelligence? Development and Activity Domains Expert Systems Natural Language Processing Machine Learning Computer Vision Planning Systems Robotics Applications in the Earth Sciences Planning Systems Conclusion and Prospectus Bibliography Artificial Neural Network Definition Introduction ANN Models Components of ANNs Organization Learning Applications Conclusions Cross-References Bibliography Autocorrelation Synonyms Definition Discussion Time Series Extensions Summary Cross-References Bibliography Automatic and Programmed Gain Control Definition Introduction: Stationary and Nonstationary Processes, Gain Control Programmed Gain Control (``True Amplitude Recovery ́ ́) of Seismic Data Automatic Gain Control (AGC) A Few Concluding Remarks on the Use of PGC and AGC Cross-References References B Backprojection Tomography Definition Introduction Parallel-Beam Image Reconstruction Radon Transform Fourier Slice Theorem Filtered Backprojection Algorithm Fan-Beam Image Reconstruction Reconstruction with Rebinning Reconstruction without Rebinning Filtered Backprojection fan-Beam Algorithm Cone-Beam Image Reconstruction Feldkamp ́s Algorithm Summary and Conclusions References Bayes ́s Theorem Synonyms Definition Application to Mineral Exploration or a ``Mineral Adventure ́ ́ Application to Geological Systems Modeling or ``Like Oil and Water ́ ́ Derivation Bayesian Statistics Conclusions Cross-References Bibliography Bayesian Inversion in Geoscience Definition Introduction Methods Bayesian Linear Inversion Monte Carlo Methods Example Conclusions Bibliography Bayesian Maximum Entropy Definition Introduction BME Methodology The Knowledge Bases (KB) The Chronotopologic Domain The Spatiotemporal Random Field Model The BME Workflow BME Chronotopologic Estimation and Uncertainty Assessment BME Features and Applications BME Versatility The Emergence of Stochastic Logic in Geostatistics BME in Software Applications of BME in the Real World Conclusions Acknowledgments Bibliography Best Linear Unbiased Estimation Definition Introduction Basic Ideas Application: Kriging Inverse Problems Basic Form Unknown Coefficients in the Mean The Kalman Filter The Model Kalman Filter Algorithm Prediction Updating Kalman Filter with Unknown Inputs Successive Linearization Variable Transformation Algorithms for Large Problems The Ensemble Kalman Filter Relation to Bayesian Methods Summary and Conclusions References Big Data Definition Introduction Infrastructure and Characteristics of Big Data Open Data to Change the Mode of Geoscience A Smart Data Ecosystem for Geoscience: A Vision for the Future Technological Approaches Toward a Smart Data Ecosystem Machine-Readable FAIR Metrics of Geoscience Data Semantic Representation of Scientific Workflows for Provenance Documentation Context-Aware Data and Resource Recommendation for Geoscience Data Science Approaches for Geoscience in the Era of Big Data Summary and Conclusions Cross-References Bibliography Binary Mathematical Morphology Definition Basic MM Operators on Binary Images Filtering on Binary Images Other Operators on Binary Images Summary Cross-References Bibliography Binary Partition Tree Definition Illustration Construction and Processing the BPT BPT in Edge-Weighted Graphs Summary Cross-References Bibliography Bonham-Carter, Graeme Biography Bibliography Bootstrap Definition Introduction Traditional Bootstrap Resampling Spatial Bootstrap Resampling Discussion Summary and Conclusions Cross-References Bibliography Burrough, Peter Alan Biography Bibliography C Cartogram Synonyms Definition Area Cartogram Rectangular Cartograms Mosaic Cartograms Circular Cartograms Noncontiguous Cartograms Contiguous Cartograms Distance Cartogram Summary Cross-References Bibliography Chaos in Geosciences Definition Introduction Model of de Wijs Gold in South Saskatchewan till Example The Lorenz Attractor Summary and Conclusions Cross-References Bibliography Chayes, Felix Biography Bibliography Cheng, Qiuming Biography Bibliography Circular Error Probability Definition Introduction Doubling the Angles Bjorne Formation Paleodelta Example Directed and Undirected Unit Vectors TRANSALP Profile Example Summary and Conclusions Cross-References Bibliography Cloud Computing and Cloud Service Synonyms Definitions Background What Is Cloud Computing? Using Cloud in Geosciences Geoscientists on Cloud Geoscientific Use Cases Cloud-Based Data Storage Cloud-Based Data Analysis Challenges and Opportunities Turning Geoscientific Applications to Cloud Native Managing Cloud Expenses Data Security and Privacy Governance Private Cloud Summary Bibliography Cluster Analysis and Classification Definition Historical Material Essential Concepts and Applications Data Problems in the Context of Cluster Analysis Conclusions Cross-References Bibliography Compositional Data Synonyms Definitions Introduction Principles The Sample Space of Compositional Data The Aitchison Geometry for CoDa Linear Functionals on Compositions The Principle of Working in Coordinates Non-Orthogonal Log-Ratio Representations Orthogonal Log-Ratio Representations Properties of an Orthonormal Representation Sequential Binary Partitions Zeros Exploratory CoDa Tools: Variation Matrix, Biplot, and CoDa-Dendrogram Varanasi Dataset Center and Variability Compositional Biplot Compositional Dendrogram Software for CoDa Conclusions Books on Compositional Data Analysis Cross-References Bibliography Computational Geoscience Definition Censored Data Process Discovery Multivariate Data Analysis Techniques Fractal Methods Visualizing the Coordinates and Elemental Associations of a Multivariate Metric Geospatial Coherence Recognizing Rare Events - Under-Sampled Processes in Principal Component Analysis Process Validation - Testing Models Derived from Process Discovery Example of Process Discovery and Process Prediction of Mineral Resource Potential Using Geochemical Survey Data Process Discovery AOV Geospatial Coherence in Process Discovery Process Validation Summary Cross-References Bibliography Computer-Aided or Computer-Assisted Instruction Definition Introduction Types of CAI Programs Features of CAI Types of Student Interactions in CAI How Is CAI Implemented? Advantages of CAI Limitations of CAI Application of CAI in Geospatial Analysis Conclusion Cross-References Bibliography Concentration-Area Plot Definition Introduction Methodology: Concentration-Area Fractal Model Case Study: Sweden Summary of the Other Models Developed Based on the C-A Fractal Model Summary and Conclusions Cross-References Bibliography Constrained Optimization Definition Introduction Constrained Optimization Application in Geospatial Science Case Study Summary and Conclusions Cross-References Bibliography Convex Analysis Definition Introduction Convex Sets Convex Functions Operations that preserve Convexity Duality Example - Least Squares Solution Summary and Conclusions Cross-References Bibliography Copula in Earth Sciences Synonyms Definition Introduction Measures of Spatial Variability Some Fields of Application Basic Concepts Some Features of Copulas Copula Families Copulas for Describing Spatial Variability Estimation Aspects Copula Modeling Validation of the Copula Models Vine Copula Conclusions Cross-References Bibliography Correlation and Scaling Definition Correlation Between Sections in Stratigraphy Examples of Ranking, Scaling, and Correlation Summary and Conclusions Cross-References Bibliography Correlation Coefficient Definition Introduction Scatter Graph Random Variable Population and Sample Basic Statistics Correlation Coefficient Hypothesis Testing Rank Correlation Summary Bibliography Coupled Modeling Definition Introduction Hydromechanical Coupling Solid Deformation Model and Its Coupled Form Fluid Flow Model and Its Coupled Form Thermomechanical Coupling Thermo-Hydromechanical Coupling Chemo-Hydromechanical Coupling Solving Coupled Models Coupling Approaches Summary Cross-References Bibliography Cracknell, Arthur Phillip Biography Acknowledgments Cressie, Noel A.C. Biography Cross-References References Crystallographic Preferred Orientation Synonyms Definition Historical Notes Introduction Definitions Estimation of an Orientation Density Function Estimation with Integral Orientation Measurements Estimation with Individual Orientation Measurements Grain Modeling with Individual Orientation Measurements Orientation Statistics Estimation of Macroscopic Directional Properties Applications of Texture Analysis Typical Applications in General Texture of a Hematite Specimen from the Pau Branco Deposit, Quadriltero Ferrífero (Iron Quadrangle), Brazil Conclusions Bibliography Cumulative Probability Plot Definition Illustration Relation to Probability Density and Probability Mass Functions Some Important Properties Uses in Statistics Summary Cross-References Bibliography D Dangermond, Jack Biography Data Acquisition Definition Terminology Introduction Remote Sensing Data Acquisition Principles Types Sensors Resolution Conclusions Bibliography Data Dictionary Definition Introduction Types of Data Dictionary Active Data Dictionary Passive Data Dictionary Components of a Data Dictionary Significance of Data Dictionary Data Dictionary in NLP Data Dictionary in Geoscience Summary Cross-References References Data Life Cycle Definition Introduction Conceptualization Data Collection Data Processing Data Archiving Data Analysis Data Sharing Data Product Summary Cross-References Bibliography Data Mining Synonyms Definition What Geoscience Data Can Be Mined? What Methods Are Used in Data Mining? Summary and Conclusions Cross-References Bibliography Data Visualization Definition Overview Geovisualization and Geovisual Analytics Challenges, Opportunities, and Future Scope Bibliography Database Management System Definition Terminology Introduction Metadata Management Quality Security Integration Governance Storage Data Compression Archive Formats Storage Devices Backup Types Cloud Storage Big Data Management Remote Sensing Data Management Conclusions Bibliography David, Michel Biography Bibliography Davis, John Clements Biography Bibliography De Wijs Model Definition Introduction De Wijs Data Selected Case Study of De Wijs Model Summary and Conclusions Bibliography Decision Tree Definition Terminology Introduction Stability in a Decision Tree Practical Implications of Decision Tree Analysis The Eastern Himalayas Example The Western Himalayan Example Conclusions Cross-References References Deep CNN-Based AI for Geosciences Definition Introduction Convolutional Layer Pooling Layer Fully Connected Layers Training a Deep CNN Model Transfer Learning Some Commonly Used CNN Architecture AlexNet (Year) VGGNet Inception Net/GoogLeNet ResNet (2015) Inception ResNet MobileNet Applications in Geosciences Classification Segmentation Detection Other Summary Cross-References Bibliography Deep Learning in Geoscience Synonyms Definition Introduction Convolutional Neural Networks (CNNs) Recurrent Neural Networks (RNNs) Autoencoders (AEs) Graph Neural Networks (GNNs) Summary Acknowledgments Bibliography Dendrogram Definition A Clustering Point of View Applications in Geosciences Summary Cross-References Bibliography Differential Equations Definition Introduction Classification History Solution Methods Notation Conclusion Cross-References Bibliography Digital Elevation Model Definition and Scope Introduction Extracting Elevation/3D Coordinates from Stereo Imagery: A Generic Photogrammetric Data Adjustment Approach Satellite Photogrammetry for Line-scanner Imagery Rigorous Sensor Model (RSM) Coordinate Systems Estimation of Image Coordinates from Ground Coordinates Estimation of Ground Coordinates from Image Coordinates Rational Polynomial Model (RPM) (Grodecki 2001; Tao and Ha 2001; Sanjay et al. 2008) Some Applications of Photogrammetric Adjustment Are: Reconstruction of Attitude Behavior Reconstruction of Orbital Parameters Corrections to Viewing Geometry with RPCs 3D Point Coordinates Computation DEM with RSM DEM with RPM Photogrammetric Adjustment of Aerial Stereo Photo Blocks DEMs from Unmanned Aerial Vehicles (UAV) Using Structure from Motion (SfM) Approach Image Matching Hierarchical Image Matching (HIM) Image Condensation Interest Point Generation Local Mapping Image Matching Scale Invariant Feature Transform (SIFT) Scale-Space Extrema Detection Keypoint Localization Orientation Assignment Keypoint Descriptor DEM Interpolation Weighted Average Polynomial Fitting Finite Element Method Kriging Gerchberg Algorithm DEM Formats Summary Bibliography Digital Filters Definition Introduction Single-Channel Convolutional Filtering in Time Domain, the z-Transform Filtering in Fourier Domain Transfer Function of Digital Filters Single-Channel Optimal Filters Appendix: Recursive Filtering (Shanks, 1967) Concluding Remarks on the Use of Digital Filtering Cross-References References Digital Geological Mapping Definition The History of Digital Geological Mapping Digital Geological Mapping System The Process of Digital Geological Mapping Summary Cross-References Bibliography Digital Twins of the Earth Synonyms Definition Essential Concepts and Components DT Versus Digital Model Some Significant Research and Innovation Applications Current Investigations, Controversies, and Gaps in Current Knowledge Summary or Conclusions Cross-References Bibliography Dimensionless Measures Synonyms Definition Introduction Dimensionless Measures in Geoscience Example 1: Pixel and Pixel Value Example 2: Scale Example 3: Contrast Ratio (CR) Example 4: Spectral Index or Band Ratio Example 5: Error Matrix and Measures Through It Summary and Conclusion Cross-References Bibliography Discrete Prolate Spheroidal Sequence Definition Overview Methodology Formulation of the Concentration Problem Properties of the Slepian Eigenvectors and Eigenvalues Applications in Spectral Estimation Periodogram Estimator Welch ́s Method Multitaper Method for Spectral Density Estimation Selection of the MTM Parameters DPSS in the Geosciences Software Implementations Summary or Conclusions Cross-References Bibliography Discriminant Analysis Definition Introduction Linear Discriminant Analysis Stage I: Calculation of Between-Class Variance Stage II: Calculation of Within-Class Variance Stage III: Construct Lower Dimensional Space Example Evaluation Metrics Conclusions Cross-References Bibliography Discriminant Function Analysis Definition Introduction Discrimination, Classification, and Identification Fisher (Two-Group) Linear Discriminant Analysis Multiple Group Linear Discriminant Analysis (Canonical Variates Analysis) Performance Testing and Statistical Significance Conclusion Bibliography Doveton, John Holroyd Biography Bibliography E Earth Surface Processes Definition Earth Surface Processes (ESP) and Landforms Mathematical Concepts to Study Various ESP and Landforms Differential Equation Hillslope Evolution Bedrock Channel Evolution Exponential Function Power-Law Function Fractals in ESP Matrices Statistical Approaches in ESP Numerical Methods in ESP Machine Learning Application in ESP Conclusions Bibliography Earth System Science Definition Goal of Earth System Science History of Earth System Science Research Tasks of Earth System Science New Concepts in Earth System Science Information Processing and Expression in Earth System Science Next Generation of Earth System Science Conclusions Bibliography Earthquake Complexity Definition Introduction The Implausibility of Earthquake Prediction Pervasive Patterns Consistent with Complexity The 1906 San Francisco Earthquake: Unequivocally Unpredictable Burridge-Knopoff Model (1967) as a Paradigm for Complexity Summary and Conclusions Bibliography Eigenvalues and Eigenvectors Definition Overview Applications with Focus on Covariance Matrix Summary Cross-References Bibliography Electrofacies Definition Facies and Electrofacies Supervised and Unsupervised Models Parametric Methods Nonparametric and Neural Network Methods Summary Cross-References Bibliography Ensemble Kalman Filtering Definition The Kalman Filter The Extended Kalman Filter The Ensemble Kalman Filter The Ensemble Kalman Filter with Augmented State The Normal-Score Ensemble Kalman Filter Kalman Filter and Kriging Summary and Conclusions Cross-References References Entropy Definition Historical Development and Properties Historical Development Basic Properties of (Z) Conditional Entropy and Mutual Information Differential Entropy and Multivariate Case Applications Characterizing Spatial Uncertainty Maximum Entropy Principle for Sampling Design Summary and Final Remarks Bibliography Euler Poles of Tectonic Plates Definition Introduction Modeling the Geometry of Tectonic Plates and the Kinematics of Their Motion Geometric Modeling of Tectonic Plates Rotations in 3 Kinematic Modeling of Plate Motion-Finite Rotations Kinematic Modeling of Plate Motion-lnfinitesimal Rotations Rotating Several Polygons Path of Migrating Euler Pole Conclusions Bibliography Expectation-Maximization Algorithm Definition Overview Applications with Focus on Remotely Sensed Images Summary Cross-References References Exploration Geochemistry Synonyms Definition Introduction Historical Developments Univariate and Bivariate Methods Multivariate Methods Spatial Methods Directions Summary Cross-References Bibliography Exploratory Data Analysis Definition Introduction Graphical Analysis Robust Classification of Univariate Data Data Standardization and Transformation Exploratory Analysis of Multivariate Relationships Concluding Remarks Cross-References Bibliography F FAIR Data Principles Synonyms Definition Introduction Findable F ``Principle F1: (meta)data are assigned a globally unique and persistent identifier ́ ́ ``Principle F2: data are described with rich metadata ́ ́ ``Principle F3: (meta)data clearly and explicitly include the identifier of the data it describes ́ ́ ``Principle F4: (meta)data are registered or indexed in a searchable resource ́ ́ Accessible A ``Principle A1: (meta)data are retrievable by their identifier using a standardized communications protocol ́ ́ ``Principle A1.1: the protocol is open, free, and universally implementable ́ ́ ``Principle A1.2: the protocol allows for an authentication and authorization procedure, where necessary ́ ́ ``Principle A2: (meta)data are accessible, even when the data are no longer available ́ ́ Interoperable I ``Principle I1: (meta)data use a formal, accessible, shared, and broadly applicable language for knowledge representation ́ ́ ``Principle I2: (meta)data use vocabularies that follow FAIR principles ́ ́ ``Principle I3: (meta)data include qualified references to other (meta)data ́ ́ Reusable R ``Principle R1: (meta)data are richly described with a plurality of accurate and relevant attributes ́ ́ ``Principle R1.1: (meta)data are released with a clear and accessible data usage license ́ ́ ``Principle R1.2: (meta)data are associated with detailed provenance ́ ́ ``Principle R1.3: (meta)data meet domain-relevant community standards ́ ́ Summary and Conclusions Acknowledgments Bibliography Fast Fourier Transform Definition Introduction Application to Drill-Hole Data from the Whalesback Copper Deposit Summary and Conclusions Cross-References Bibliography Fast Wavelet Transform Definition Introduction Wavelet Transform CWT Versus DWT Fast Wavelet Transform (FWT) Summary and Conclusion Cross-References Bibliography Favorability Analysis Definition Introduction Basic Concepts Classification of Spatial Variables Quantification of Geoscience Maps Control Area and Statistical Analogy Characteristic Analysis Favorability Analysis Concluding Remarks Bibliography Fisher, Ronald A. Biography Bibliography Flow in Porous Media Synonyms Definition Principles Quantifying Flow in Porous Media Open Questions and Unresolved Issues Conclusion Bibliography Forward and Inverse Stratigraphic Models Synonyms Definition Introduction Forward Stratigraphic Modeling Inverse Stratigraphic Modeling Forward Stratigraphic Models Geometrical Models Topography-Control (Diffusion, or Potential-Gradient) Models Hydraulic Process-Response Models Comparison Between Various Equations for Bed-Load Transport Mixed Algorithmic, Heuristic, and Fuzzy-Logic Models Inverse Stratigraphic Models Current Investigations, Controversies, and Gaps in Current Knowledge Limitations to Present Models Summary Bibliography Fractal Geometry and the Downscaling of Sun-Induced Chlorophyll Fluorescence Imagery Synonyms Definition Introduction Fractal Geometry and the Power Law (PL) Distribution of Sun-Induced Chlorophyll Fluorescence (SIF)-Emitting Objects Case Study: ``Fractal Geometry for SIF-Downscaling ́ ́ Summary Cross-References Bibliography Fractal Geometry in Geosciences Definition Introduction Fractal Dimension Estimation Separation of Geochemical Anomalies from Background by Fractal Methods Fractal Perimeter-Area Relation The Multifractal Spectrum Singularity Analysis Model of de Wijs Multifractal Spatial Correlation Fractal/Multifractal Modeling of Point Patterns Lognormal and Pareto Size-Frequency Distributions for Mineral Resources Summary Cross-References References Fractal Landscape Definition Introduction A Review of Fractals: A Unifying Basis to Describe Nature ́s Complexity Self-similar Fractals Self-affine Fractals Multifractals Scaling of Earth ́s Landscape Proposed Dynamical Models for Fractal Landscapes Conclusion Bibliography Frequency Distribution Definition Different Forms Statistics Cautions and Warnings Summary and Conclusions Cross-References Bibliography Frequency-Wavenumber Analysis Definition Introduction Sampling of Rocks: Pulacayo Zinc Deposit Example Short-Distance Nugget Effect Modeling Universal Multifractals Summary and Conclusions Cross-References Bibliography Full Normal Plot Definition The Method Application Summary Cross-References References Fuzzy C-Means Clustering Definition Overview Applications with a Focus on Image Processing Future Scope Cross-References References Fuzzy Inference Systems for Mineral Exploration Definition FIS for Mineral Exploration Architecture of a FIS Types of FISs Example of a FIS Designed for Gold Mineralization Selected Case Studies Summary and Conclusions Cross-References Bibliography Fuzzy Set Theory in Geosciences Definition Introduction of Fuzzy Set Theory Stages of Fuzzy Logic Modeling Further Definition Using Various Case Studies Geothermal Resource Exploration Exploration Geology Using Remote Sensing Data Application of Iterative Fuzzy Edge Detection Algorithm to Blurred Satellite Images Summary or Conclusions Cross-References Bibliography G Genetic Algorithms Definition Introduction GAs for Clustering Example Conclusion Cross-References Bibliography Geocomputing Synonyms Definition The Why and How Selected Applications and Challenges Conclusions Cross-References Bibliography Geographical Information Science Definition Introduction Spatial Data Concepts Spatial Data Sources Geographical Coordinate System Map Projections Spatial Data Models Spatial Database Creation Spatial Relations Spatial Data Analysis Vector Data Analysis Raster Data Analysis Spatial Data Interpolation Digital Terrain Mapping Network Analysis Statistical Analysis Visualization of Data Analysis Spatial Decision Support Systems Spatial Data Accuracy Assessment Applications Research Challenges in Geographical Information Science Research Areas Conclusions Cross-References Bibliography Geographically Weighted Regression Definition Introduction Model Formulation Model Calibration Model Output Case Study Extensions to GWR Conclusions Cross-References Bibliography Geohydrology Synonyms Definition Introduction Open-Pit Mine Dewatering Dewatering Techniques Passive Dewatering Active (Advance) Dewatering Groundwater Flow Model Study Schedule Conclusions Bibliography Geoinformatics Definition Introduction Quantitative Geosciences: Evolving Computational Systems Quantitative Geosciences: Early Computer Engagement Quantitative Geosciences: The Twenty-First Century Current Investigation Examples Predicting Earthquakes Mapping and Analyzing Global Geospatial Data Geoinformatics: Discussion Topics Summary Cross-References Bibliography Geologic Time Scale Estimation Definition Introduction Methods and Ages Conventions and Standards The McKerrow Method Smoothing Splines Smoothing Spline Software Devonian Example Paleozoic ``Super-Spline ́ ́ Cretaceous and Jurassic Periods Cubic Spline Summary Cross-References References Geomathematics Definition Introduction Relation of Geomathematics to Mathematical Statistics Involvement by Mathematical Statisticians Relation of Geomathematics to the Geosciences Geostatistics Mineral Resource Estimation Compositional Data Analysis Directional Features Mathematical Morphology Quantitative Stratigraphy Geologic Time Scale Estimation Fractals and Multifractals Lognormal and Pareto Frequency Distributions Summary Cross-References References Geomatics Definition Geomatic Components Land Surveying Survey Measurements Geodesy Photogrammetry Remote Sensing Global Navigation Satellite Systems (GNSS) Geographic Information Systems (GIS) Cartography Innovations in Geomatics Geomatic Applications Summary Cross-References Bibliography Geomechanics Definition Introduction Subsurface Stresses and Pore Pressure Subsurface Stresses Pore Pressure Vertical Stress Research Path of Geomechanics Conclusion Bibliography Geoscience Signal Extraction Definition Introduction Pulacayo Mine and Main KTB Borehole Examples Spectral Analysis: Glacial Lake Barlow-Ojibway Example Unit Vector Fields Summary and Conclusions Cross-References Bibliography Geosciences Digital Ecosystems Synonyms Definition Essential Concepts and Components Ecosystem Principles, Patterns, and Governance Some Significant Research and Innovation Aspects Evolutionary Aspect Metasystemic Aspect Summary and Conclusions Cross-References Bibliography Geostatistical Seismic Inversion Definition Geophysical Inversion Trace-by-Trace Geostatistical Seismic Inversion Global Geostatistical Seismic Inversion Summary Cross-References Bibliography Geostatistics Definition Introduction Statistical Formulation Stationarity Properties of the Covariance and Variogram Functions Nonstationary Random Functions and the Generalized Covariance Function Coregionalization Estimating Covariance and Variogram Functions Data Configurations Modeling Experimental Covariances and Semivariograms Kriging Linear Kriging Cokriging Kriging of ZV(x0) and Functions of Z(x0) Non-Parametric Estimation: Multiple Indicator Kriging Projection onto Larger Spaces Disjunctive Kriging Bivariate Distributions and Isofactorial Models The DK Equations Disjunctive Kriging in the Bivariate Gaussian Model Change of Support Simulation Simulation of a Second-Order Stationary Process Sequential Gaussian Simulation Sequential Indicator Simulation Simulation by Matrix Decomposition Conditional Simulation Using Successive Residuals Truncated Gaussian Simulation and Plurigaussian Simulation The Gibbs Sampler Multi-Point Simulation (SNESIM) Other Simulation Methods Concluding Remarks Cross-References References GeoSyntax Background Essential Concepts of the GeoSyntax Approach Current Status Extension to Two- and Three-Dimensional Sedimentary Bodies Summary Bibliography Geotechnics Definition Introduction Research Path of Geotechnics Conclusion Bibliography Geothermal Energy Definition Subsurface Temperature Modeling Application of Geostatistics Application of Machine Learning Remote Sensing Applications Outline of Remote Sensing Surface Temperature Estimation Detection and Mapping of Hydrothermally Altered Minerals Lineament-Based Fracture Characterization Continuous Mapping of Topographic Change by D-InSAR Fluid Flow Simulation Summary Cross-References Bibliography Global and Regional Climatic Modeling Definition Introduction Climate: Basic Approaches to Analysis Modeling: Tools and Approaches Global Versus Local Summary and Conclusions Cross-References Acknowledgments Bibliography Goodchild, M