Handbook of Geomathematics. Zuhair Nashed, M. Zuhair Nashed, Thomas Sonar

Foreword Preface Contents About the Editors Contributors Part I General Issues, Historical Background,and Future Perspectives Geomathematics: Its Role, Its Aim, and Its Potential Contents 1 Introduction 2 Geomathematics as a Cultural Asset 3 Geomathematics as Task and Objective 4 Geomathematics as Interdisciplinary Science 5 Geomathematics as a Challenge 6 Geomathematics as Solution Potential 7 Geomathematics as Solution Method 8 Geomathematics: Three Exemplary ``Circuits'' 8.1 Circuit: Gravity Field from Deflections of the Vertical Mathematical Modeling of the Gravity Field (Classical Approach) Mathematical Analysis Development of a Mathematical Solution Method ``Back-Transfer'' to Application 8.2 Circuit: Oceanic Circulation from Ocean Topography Mathematical Modeling of Ocean Flow Mathematical Analysis Development of a Mathematical Solution Method ``Back-Transfer'' to Application 8.3 Circuit: Seismic Processing from Acoustic Wave Tomography Mathematical Modeling in Reservoir Detection Mathematical Analysis of Acoustic Wave Propagation Development of a Mathematical Solution Method ``Back-Transfer'' to Application 9 Final Remarks References Navigation on Sea: Topics in the History of Geomathematics 1 General Remarks on the History of Geomathematics 2 Introduction 3 The History of the Magnet 4 Early Modern England 5 The Gresham Circle 6 William Gilberts Dip Theory 7 The Briggsian Tables 8 The Computation of the Dip Table 9 Conclusion References Gauss' and Weber's ``Atlas of Geomagnetism'' (1840) Was not the first: the History of the Geomagnetic Atlases 1 Introduction 2 Samuel Dunn (1723–1794) 2.1 Biographical Notes 2.2 Dunn's Geomagnetic Maps 3 John Churchman (1753–1805) 3.1 Biographical Notes 3.2 The Magnetic Atlas (Philadelphia 1790) 3.3 The Magnetic Atlas (2. Edition, London 1794) 3.4 The Magnetic Atlas (3. Edition, New York, 1800; 4. Edition, London, 1804) 4 Christopher Hansteen (1784–1873) 4.1 Biographical Notes 4.2 The Magnetic Atlas (Christiania 1819) 4.3 Expedition to Russia (1828–1830) 4.4 Hansteen's Maps 5 Carl Friedrich Gauss (1777–1855) and Wilhelm Weber (1804–1891) 5.1 The Beginning 5.2 The ``Magnetic Association'' in Göttingen (1834–1843) 5.3 Gauss' ``General Theory of Geomagnetism'' (1839) 5.4 Gauss' and Weber's ``Atlas of Geomagnetism'' (Leipzig 1840) 5.5 The End of the ``Magnetic Association'' in Göttingen 6 Excursus: Berghaus' ``Physical Atlas'' 7 The Term ``Atlas'' References Part II Observational and Measurement Key Technologies Earth Observation Satellite Missions and Data Access 1 Introduction 2 End-to-End Earth Observation Satellite Systems 2.1 Space Segment 2.2 Ground Segment Flight Operations Segment Payload Data Segment 3 Overview on ESA Earth Observation Programs 3.1 Background 3.2 ERS-1 and ERS-2 Missions 3.3 Envisat Mission 3.4 Proba-1 3.5 Earth Explorers 3.6 GMES and Sentinels 3.7 Meteorological Programs 3.8 ESA Third Party Missions 4 Some Major Results of ESA Earth Observation Missions 4.1 Land 4.2 Ocean 4.3 Cryosphere 4.4 Atmosphere 4.5 ESA Programs for Data Exploitation 5 User Access to ESA Data 5.1 How to Access the EO Data at ESA 6 Conclusion References Satellite-to-Satellite Tracking (Low-Low/High-Low SST) 1 Introduction 2 Scientific Relevance 3 Conventions 3.1 Reference Systems 3.2 Basis Functions 4 Celestial Mechanics 5 Observation Models and Data Processing Strategies 5.1 Satellite-to-Satellite Tracking in the High–Low Mode Energy-Balance Approach Energy-Balance Approach Torus Approach Acceleration Approach Acceleration Approach Integral Equation Approach Integral Equation Approach Variational Equation Approach Variational Approach Colombo's Modification of Variational Equation Approach Colombo's Modification 5.2 Satellite-to-Satellite Tracking in the Low–Low Mode Potential-Difference Approach Potential-Difference Approach Line-of-Sight Gradiometry Line-of-Sight Gradiometry Approach Variational Equation Approach Variational Equation Approach 5.3 Integral Equations Approach Integral Equation Approach 6 Regional Gravity Field Models 6.1 Final Remarks 7 Missions and Outcomes 7.1 CHAMP Mission 7.2 GRACE Mission 8 Conclusions References GOCE: Gravitational Gradiometry in a Satellite 1 Introduction: GOCE and Earth Sciences 2 GOCE Gravitational Sensor System 3 Gravitational Gradiometry 4 GOCE Status 5 Conclusions: GOCE Science Applications References Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation 1 Sources of the Earth's Magnetic Field 1.1 Internal Field Sources: Core and Crust Core Field Crustal Field 1.2 Ionospheric, Magnetospheric, and Earth-Induced Field Contributions Ionospheric Contributions Magnetospheric Contributions Induction in the Solid Earth and the Oceans 2 Modern Geomagnetic Field Data 2.1 Definition of Magnetic Elements and Coordinates 2.2 Ground Data 2.3 Satellite Data 3 Making the Best of the Data to Investigate the Various Field Contributions: Geomagnetic Field Modeling References Part III Modeling of the System Earth (Geosphere, Cryosphere, Hydrosphere, Atmosphere, Biosphere, Anthroposphere) Classical Physical Geodesy 1 Introduction 1.1 Preliminary Remarks 1.2 What is Geodesy? 1.3 Reference Systems 2 Basic Principles 2.1 Gravitational Potential and Gravity Field 2.2 The Normal Field 2.3 The Geoid and Height Systems 2.4 Gravity Gradients and General Relativity Separability of Gravitation and Inertia Separability in First-Order Gradients Separability in Second-Order Gradients Satellite Orbits Applications Gravitation and Time 3 Key Issues of Theory: Harmonicity, Analytical Continuation, and Convergence Problems 3.1 Harmonic Functions and Spherical Harmonics 3.2 Convergence and Analytical Continuation 3.3 More About Convergence 3.4 Krarup's Density Theorem 4 Key Issues of Applications 4.1 Boundary-Value Problems of Physical Geodesy Nonlinear Inverse Problems of Functional Analysis The Standard Classical Model Linearization Solution by Spherical Harmonics: A Useful Formula for Spherical Harmonics Solution by Stokes-Type Integral Formulas Remarks 4.2 Collocation Principles Least-Squares Collocation Concluding Remarks 5 Future Directions 5.1 The Earth as a Nonrigid Body 5.2 The Smoothness of the Earth's Surface 5.3 Inverse Problems 6 Conclusion 6.1 The Geoid References Geodetic Boundary Value Problem 1 Introduction 2 The Linearized Scalar Molodensky Problem (LSMP) withOne Further Modification 3 The Simple Molodensky Problem (SiMP): Definition of the Relevant Spaces 4 Solution of the SiMP and of the LSMP 5 Conclusions References Time-Variable Gravity Field and Global Deformation of the Earth 1 Introduction 2 Mass and Mass Redistribution 3 Earth Model 4 Analysis of TVG and Deformation Pattern 5 Future Directions 6 Conclusions References Satellite Gravity Gradiometry (SGG): From Scalarto Tensorial Solution 1 Introduction 2 SGG in Potential Theoretic Perspective 3 Decomposition of Tensor Fields by Means of Tensor Spherical Harmonics 4 Formulation as Pseudodifferential Equation 4.1 SGG as Pseudodifferential Equation 4.2 Upward/Downward Continuation 4.3 Operator of the First-Order Radial Derivative 4.4 Pseudodifferential Operator for SST 4.5 Pseudodifferential Operator of the Second-Order Radial Derivative 4.6 Pseudodifferential Operator for Satellite Gravity Gradiometry 4.7 Survey on Pseudodifferential Operators Relevant in Satellite Technology 4.8 Classical Boundary Value Problems and Satellite Problems 4.9 A Short Introduction to the Regularization of Ill-Posed Problems 4.10 Regularization of the Exponentially Ill-Posed SGG-Problem 5 Future Directions 6 Conclusion References Spacetime Modeling of the Earth's Gravity Field by Ellipsoidal Harmonics 1 Introduction 2 Dirichlet Boundary-Value Problem on the Ellipsoid of Revolution 2.1 Formulation of the Dirichlet Boundary-Value Problem on an Ellipsoid of Revolution 2.2 Power-Series Representation of the Integral Kernel 2.3 The Approximation of O(e02) 2.4 The Ellipsoidal Poisson Kernel 2.5 Spatial Forms of Kernels Li(t, x) 2.6 Residuals Ri(t, x) 2.7 The Behavior at the Singularity 2.8 Conclusion 3 Stokes Boundary-Value Problem on the Ellipsoid of Revolution 3.1 Formulation of the Stokes Problem on an Ellipsoid of Revolution 3.2 The Zero-Degree Harmonic of T 3.3 Solution on the Reference Ellipsoid of Revolution 3.4 The Derivative of the Legendre Function of the Second Kind 3.5 The Uniqueness of the Solution 3.6 The Approximation up to O(e02) 3.7 The Ellipsoidal Stokes Function 3.8 Spatial Forms of Functions Ki(cosψ) 3.9 Conclusion 4 Vertical Deflections in Gravity Space 4.1 Representation of the Actual Gravity Vector as Well as the Reference Gravity Vector Both in a Global and a Local Frame of Reference 4.2 The Incremental Gravity Vector 5 Vertical Deflections and Gravity Disturbance in Geometry Space 5.1 Ellipsoidal Coordinates of Type Gauss Surface Normal 5.2 Jacobi Ellipsoidal Coordinates 6 Potential Theory of Horizontal and Vertical Components of the Gravity Field: Gravity Disturbance and Vertical Deflections 6.1 Ellipsoidal Reference Potential of Type Somigliana-Pizzetti 6.2 Ellipsoidal Reference Gravity Intensity of Type Somigliana-Pizzetti 6.3 Expansion of the Gravitational Potential in Ellipsoidal Harmonics 6.4 External Representation of the Incremental Potential Relative to the Somigliana-Pizzetti Potential Field of Reference 6.5 Time Evolution 7 Ellipsoidal Reference Potential of Type Somigliana-Pizzetti 7.1 Vertical Deflections and Gravity Disturbance in Vector-Valued Ellipsoidal Surface Harmonics 7.2 Vertical Deflections and Gravity Disturbance Explicitly in Terms of Ellipsoidal Surface Harmonics 8 Case Studies 9 Curvilinear Datum Transformations 10 Datum Transformations in Terms of Spherical Harmonic Coefficients 11 Datum Transformations of Ellipsoidal Harmonic Coefficients 12 Examples 13 Conclusions References Multiresolution Analysis of Hydrology and Satellite Gravitational Data 1 Introduction 2 Scientific Relevance of Multiresolution 2.1 Preliminaries 2.2 Multiresolution in Hilbert Spaces 2.3 Wavelets for the Time and Space Domain Legendre Wavelets Spherical Wavelets 3 Key Issues for the Comparison of GRACE and WGHM Data 3.1 Tensorial Time-Space Multiresolution 3.2 Correlation Analysis Between GRACE and WGHM 4 Fundamental Results 5 Future Directions 6 Conclusion References Time-Varying Mean Sea Level 1 Introduction 2 Theoretical Considerations 3 Scientific Relevance 4 Data and Methodology of Numerical Treatment 4.1 Fundamental Results 5 Future Directions 5.1 Accuracy of Sea-Level Observation 5.2 Observation at the Coast and in Open Ocean 5.3 Attribution of Sea-Level Rise 5.4 Reconstruction of Past Sea Level 5.5 Prediction of Sea-Level Evolution 5.6 Mathematical Representation of Time-Varying Sea Level 6 Conclusions References Self-Attraction and Loading of Oceanic Masses 1 Introduction 2 Theory 2.1 Derivation of Spherical Harmonic Equations 2.2 Reformulation with Green's Functions 2.3 Extension with the Sea-Level Equation 2.4 Parameterization and Simplification 3 Key Questions 3.1 Magnitude of SAL on Different Frequencies 3.2 Dynamical or Equilibrium Response? 3.3 How to Improve Computational Efficiency 4 Fundamental Results 4.1 Tidal Variations 4.2 Nontidal Variations 5 Future Directions 6 Conclusion References Unstructured Meshes in Large-Scale Ocean Modeling 1 Introduction 2 Dynamic Equations and Typical Approximations 3 Finite-Element and Finite-Volume Methods 3.1 FE Method 3.2 Finite Volumes 3.3 Discontinuous FE 3.4 Brief Summary 4 FE Consistency Requirements 4.1 Consistency Between Elevation and Vertical Velocity 4.2 Consistency of Tracer Spaces and Tracer Conservation 4.3 Energetic and Pressure Consistency 5 Nonconforming and Continuous Linear Representations in FEOM 5.1 Preliminary Remarks 5.2 Solving the Dynamical Part with NC Elements 5.3 Solving the Dynamical Part with the CL Approach 5.4 Vertical Velocity Pressure, and Tracers 6 C-Grid and FV Cell-Vertex Type of Discretization 6.1 C-Grid 6.2 P0-P1-Like Discretization 7 Conclusions References Numerical Methods in Support of Advanced TsunamiEarly Warning 1 Introduction 2 Tsunami Propagation and Inundation Modeling 2.1 Operational Tsunami Simulation Software 2.2 Unstructured Grid Tsunami Simulation 2.3 Adaptive Mesh Refinement for Tsunami Simulation 2.4 Advanced Topics 3 Forecasting Approaches 3.1 Existing Tsunami Early Warning Systems 3.2 Multi-sensor Selection Approach 3.3 Implementing the Matching Procedure 3.4 Experimental Confirmation 4 Future 5 Conclusion References Gravitational Viscoelastodynamics 1 Introduction 2 Basic Concepts 2.1 Kinematic Representations 2.2 Total, Initial, and Incremental Fields 2.3 Interface Conditions 3 Field Equations and Interface Conditions 3.1 Equations for the Total Fields 3.2 Equations for the Initial Fields 3.3 Equations for the Incremental Fields Local Form Constitutive Equation 3.4 Continuity and State Equations 4 Asymptotic Incremental Field Theories 4.1 Relaxation Functions 4.2 Asymptotic Relaxation Functions Large-s Asymptotes Small-s Asymptotes 4.3 Asymptotic Incremental Field Equations and Interface Conditions Small-t Asymptotes: Field Theory of GED Large-t Asymptotes: Field Theory of GVD 5 Approximate Incremental Field Theories 5.1 Local Incompressibility Equations for the Initial Fields Equations for the Incremental Fields: Local Form 5.2 Material Incompressibility Equations for the Initial Fields Equations for the Incremental Fields: Local Form 6 Summary References Elastic and Viscoelastic Response of the Lithosphereto Surface Loading 1 Loading Response and Their Observation 2 General Response of the Earth to Surface Loads 2.1 Rheological Stratification of the Solid Earth 2.2 Elastic Behavior 2.3 Anelasticity 2.4 Viscous Behavior 2.5 Viscoelasticity 3 Fundamental Results 3.1 Field Equations for a Spherical Planet 3.2 The Elastic Problem 3.3 The Viscoelastic Problem 4 Future Directions 5 Conclusion References Multiscale Model Reduction with Generalized Multiscale Finite Element Methods in Geomathematics 1 Introduction 2 Preliminaries 2.1 Some Selected Applications Single-Phase Flow Wave Equation Nonlinear Flows Multiphase Flow and Transport 2.2 Coarse Mesh Description 3 GMsFEM Framework 3.1 Local Basis Functions 3.2 Global Coupling Continuous Galerkin Coupling Discontinuous Galerkin Coupling 4 GMsFEM for Applications 4.1 Single-Phase Compressible Flow 4.2 Wave Equation 4.3 Nonlinear Flows 4.4 Multiphase Flow and Transport 5 Conclusions References Efficient Modeling of Flow and Transport in Porous Media Using Multi-physics and Multi-scale Approaches 1 Introduction 2 State of the Art 2.1 Definition of Scales 2.2 Upscaling and Multi-scale Methods Upscaling Methods Volume Averaging/Homogenization Methods Numerical Upscaling Multi-scale Methods Homogeneous Multi-scale Methods Heterogeneous Multi-scale Methods Variational Multi-scale Method Multi-scale Finite Volume Method Multi-scale Finite Element Method Multi-scale Methods and Domain Decomposition Vertical Equilibrium Methods Adaptive Upscaling Methods 2.3 Multi-physics Methods 3 Mathematical Models for Flow and Transport Processes in Porous Media 3.1 Preliminaries Basic Definitions Phases Components Fluid Parameters Compositions and Concentrations Density Viscosity Matrix Parameters Porosity Intrinsic Permeability Parameters Describing Fluid–Matrix Interaction Saturation Capillarity Relative Permeability Extended Darcy's Law Laws for Fluid-Phase Equilibria Dalton's Law Raoult's Law Henry's Law The Reynolds Transport Theorem 3.2 Multiphase Flow The Immiscible Case The Compositional Case 3.3 Decoupled Formulations The Immiscible Case Pressure Equation Global Pressure Formulation for Two-Phase Flow Phase Pressure Formulation for Two-Phase Flow Saturation Equation Global Pressure Formulation for Two-Phase Flow Phase Pressure Formulation for Two-Phase Flow The Compositional Case 3.4 Non-isothermal Flow 4 Numerical Solution Approaches 4.1 Solution of the Fully Coupled Equations 4.2 Solution of the Decoupled Equations The Immiscible Case The Compositional Case Flash Calculations 5 Application of Multi-physics and Multi-scale Methods 5.1 A Multi-physics Example Single-Phase Transport Model Coupling Practical Implementation Application Example 5.2 A Multi-scale Example The Multi-scale Approach Numerical Results 6 Conclusion References Convection Structures of Binary Fluid Mixtures in Porous Media 1 Introduction 2 Foundations 2.1 System and Basic Equations 2.2 Numerical Method 2.3 Conductive Ground State 3 Structures for Positive Separation Ratios 3.1 Bifurcation Behavior of Roll, Crossroll, and Square Convection 3.2 Structure of the Fields for Roll and Square Convection 3.3 Stability Results 4 Structures for Negative Separation Ratios 4.1 Bifurcation Behavior 4.2 Structure of the Fields for TW Convection 4.3 Lateral Currents 5 Conclusion References Numerical Dynamo Simulations: From Basic Concepts to Realistic Models 1 Introduction 2 Mathematical Formulation 2.1 Basic Equations 2.2 Boundary Conditions 2.3 Numerical Methods 3 Numerical Dynamo Solutions 3.1 Force Balances 3.2 Dynamo Regimes 3.3 Scaling Laws 3.4 Double Diffusive Approach 3.5 Dynamo Mechanism 3.6 Is There a Distinct Low Ekman Number Regime? The Influence of the Magnetic Fields on Rapidly Rotating Convection The Impact of Lorentz Forces and Buoyancy Boundary Conditions on the Flow Scales in Numerical Simulations Subcritical Dynamos and the Nature of the Dynamo Bifurcation Transitions in Low Ekman Number Rapidly Rotating Convection 4 Comparison with the Geomagnetic Field Dipole Properties and Magnetic Field Symmetries Persistent Features and Mantle Influence Inverse Magnetic Field Production and the Cause for Reversals Time Variability 5 Conclusion References Mathematical Properties Relevant to Geomagnetic Field Modeling 1 Introduction 2 Helmholtz's Theorem and Maxwell's Equations 3 Potential Fields 3.1 Magnetic Fields in a Source-Free Shell 3.2 Surface Spherical Harmonics 3.3 Magnetic Fields from a Spherical Sheet Current 4 Non-potential Fields 4.1 Helmholtz Representations and Vector Spherical Harmonics 4.2 Mie Representation 4.3 Relationship of B and J Mie Representations 4.4 Magnetic Fields in a Current-Carrying Shell 4.5 Thin-Shell Approximation 5 Spatial Power Spectra 6 Mathematical Uniqueness Issue 6.1 Uniqueness of Magnetic Fields in a Source-Free Shell 6.2 Uniqueness Issues Raised by Directional-Only Observations 6.3 Uniqueness Issues Raised by Intensity-Only Observations 6.4 Uniqueness of Magnetic Fields in a Shell Enclosinga Spherical Sheet Current 6.5 Uniqueness of Magnetic Fields in a Current-Carrying Shell 7 Concluding Comments: From Theory to Practice References Multiscale Modeling of the Geomagnetic Fieldand Ionospheric Currents 1 Introduction 2 Relevant Function Systems 2.1 Spherical Harmonics 2.2 Green's Function for the Beltrami Operator 2.3 Single Layer Kernel 3 Two Approaches to Multiscale Representations 3.1 Wavelets as Frequency Packages 3.2 Locally Supported Wavelets 4 Application to Geomagnetic Problems 4.1 Crustal Field Modeling and Separation of Sources Wavelets as Frequency Packages Locally Supported Wavelets 4.2 Reconstruction of Radial Current Systems Wavelets as Frequency Packages Locally Supported Wavelets 4.3 Multiscale Power Spectrum Wavelets as Frequency Packages Locally Supported Wavelets 5 Conclusion and Outlook References Toroidal-Poloidal Decompositions of Electromagnetic Green's Functions in Geomagnetic Induction 1 Introduction 2 The Electromagnetic Green's Functions in a Homogeneous Medium 3 Toroidal-Poloidal Decomposition Under Cartesian Geometry 4 Interaction with a Half-Space 5 The Quasi-static Limit 6 Toroidal-Poloidal Decomposition Under Spherical Geometry 7 Summary References Using B-Spline Expansions for Ionosphere Modeling 1 Introduction 2 Basic Parameters of the Ionosphere 2.1 Electron Density 2.2 Total Electron Content 3 Mathematical Approaches 3.1 Coordinate Systems 3.2 Basis Functions Polynomial B-Splines Trigonometric B-Splines 3.3 Series Expansions 1-D B-Spline Expansion 2-D Global B-Spline Expansion 3-D Global B-Spline Expansion 3-D Regional B-Spline Expansion 4-D Regional B-Spline Expansion 4 Multi-scale Representation 4.1 Two-Scale Relations and Decomposition Equation 4.2 1-D Multi-scale Representation 4.3 3-D Multi-scale Representation 5 Parameter Estimation 5.1 Linear Model 5.2 Combination of Observation Techniques 5.3 Prior Information 5.4 Variance Component Estimation 6 Input Data from Space-Geodetic Observation Techniques 6.1 Terrestrial GPS 6.2 Space-Based GPS 6.3 Doppler Orbitography and Radiopositioning Integratedby Satellite (DORIS) 6.4 Radar Altimetry (RA) 7 Example 8 Summary References The Forward and Adjoint Methods of Global Electromagnetic Induction for CHAMP Magnetic Data 1 Introduction 2 Basic Assumptions on EM Induction Modeling for CHAMP Magnetic Data 3 Forward Method of Global EM Induction 3.1 Formulation of EM Induction for a 3-D Inhomogeneous Earth 3.2 Special Case: EM Induction in an Axisymmetric Case 3.3 Gauss Representation of Magnetic Induction in the Atmosphere 4 Forward Method of EM Induction for the X Component of CHAMP Magnetic Data 4.1 Classical Formulation 4.2 Weak Formulation Ground Magnetic Data Satellite Magnetic Data 4.3 Frequency-Domain and Time-Domain Solutions 4.4 Vector Spherical Harmonic Parameterization over Colatitude 4.5 Finite-Element Approximation over the Radial Coordinate 4.6 Solid Vector Spherical Harmonic Parameterization of A0 5 Forward Method of EM Induction for the External Gauss Coefficients 5.1 Classical Formulation 5.2 Weak Formulation 6 Time-Domain, Spectral Finite-Element Solution 7 CHAMP Data Analysis 7.1 Selection and Processing of Vector Data 7.2 Two-Step, Track-by-Track Spherical Harmonic Analysis Change of the Interval of Orthogonality Extrapolation of Magnetic Data from Mid-latitudes to PolarRegions Selection Criteria for Extrapolation Examples of Spherical Harmonic Analysis of the CHAMP Magnetic Data 7.3 Power-Spectrum Analysis 8 Adjoint Sensitivity Method of EM Induction for the Z Component of CHAMP Magnetic Data 8.1 Forward Method 8.2 Misfit Function and Its Gradient in the Parameter Space 8.3 The Forward Sensitivity Equations 8.4 The Adjoint Sensitivity Equations 8.5 Boundary Condition for the Adjoint Potential 8.6 Adjoint Method 8.7 Reverse Time 8.8 Weak Formulation 9 Adjoint Sensitivity Method of EM Induction for the Internal Gauss Coefficients of CHAMP Magnetic Data 9.1 Forward Method 9.2 Misfit Function and Its Gradient in the Parameter Space 9.3 Adjoint Method 9.4 Weak Formulation 9.5 Summary 10 Sensitivity Analysis for CHAMP Magnetic Data 10.1 Brute-Force Sensitivities 10.2 Model Parameterization 10.3 Three-Layer, 1-D Conductivity Model Sensitivity Comparison Conjugate Gradient Inversion 10.4 Two-Layer, 2-D Conductivity Model Sensitivity Comparison Conjugate Gradient Inversion 11 Conclusions References Modern Techniques for Numerical Weather Prediction:A Picture Drawn from Kyrill 1 Introduction 1.1 What is a Weather Forecast? 2 Data Assimilation Methods: The Journey from 1d-Var to 4d-Var 2.1 Observational Nudging 2.2 Variational Analysis 3 Basic Equations 3.1 Vertical Coordinate System σ-Coordinates η-Coordinates 3.2 The Eulerian Formulation of the Continuous Equations 3.3 Physical Background Processes Clouds and Precipitation Clouds Precipitation 3.4 The Discretization 4 Ensemble Forecasts 5 Statistical Weather Forecast (MOS) 6 Applying the Techniques to Kyrill 6.1 Analysis of the Air Pressure and Temperature Fields 6.2 Analysis of Kyrills Surface Winds 6.3 Analysis of Kyrill's 850hPa Winds 6.4 Ensemble Forecasts 6.5 MOS Forecasts 6.6 Weather Radar 7 Conclusion References Radio Occultation via Satellites 1 Introduction 2 Physical Background of Radio Occultation 2.1 GPS Signals 2.2 The Radio Occultation Techniques Refraction Dry Density, Pressure, and Temperature Temperature Ambiguity of Humidity 3 Mathematical Modeling of Radio Occultation Data 3.1 Notation 3.2 Spherical Harmonics 3.3 Green's Function with Respect to the Beltrami Operator Δ* 3.4 Spherical Spline Functions 3.5 Results 4 Conclusion References Asymptotic Models for Atmospheric Flows 1 Introduction 2 Examples of Reduced Models, Scale Analysis, and Asymptotics 2.1 QG-Theory 2.2 Sound-Proof Models 3 Dimensional Considerations 3.1 Characteristic Scales and Dimensionless Parameters 3.2 Distinguished Limits 3.3 Remarks of Caution 4 Classical Single-Scale Models 4.1 Scalings of the Governing Equations Governing Equations Some Revealing Transformations The Exner Pressure A General Scale Transformation 4.2 Midlatitude Internal Gravity Wave Models 4.3 Balanced Models for Advection Time Scales Weak Temperature Gradient Models for the Mesoscales Synoptic Scales and the Quasi-geostrophic Approximation Ogura and Phillips' Anelastic Model for Weak Stratification 4.4 Scalings for Near-Equatorial Motions Equatorial Sub-Synoptic Flow Models Equatorial Synoptic and Planetary Models 4.5 The Hydrostatic Primitive Equations 5 Developments for Multiple Scales Regimes 5.1 Shaw and Shepherd's Parameterization Framework 5.2 Superparameterization 6 Conclusions References Stokes Problem, Layer Potentials and Regularizations,and Multiscale Applications 1 Introduction 2 Basic Equations 2.1 Conservation Laws 2.2 Constituting Equations 3 Stokes Boundary-Value Problems 3.1 Basic Nomenclature 3.2 Fundamental Solution, Stokeslet, and Stresslet 3.3 Limit and Jump Relations 3.4 Existence and Uniqueness of the Stokes Problems 3.5 The Stress Tensor of the Double-Layer Potential 4 Multiscale Regularizations of Layer Potentials 4.1 Scaling Functions and Wavelets 4.2 Scale Discretized Scaling Functions and Wavelets 4.3 Scale and Detail Spaces 4.4 Scaling Functions and Wavelets of the Second Kind 4.5 Spherical Multiresolution Analysis 4.6 A Case Study in Meteorology: The Storm Kyrill A Regular Surfaces B Kernel Functions C Scaling Functions and Wavelets C.1 Scaling Functions C.2 Wavelet Functions C.3 Scaling Functions of the Second Kind C.4 Wavelet Functions of the Second Kind References On High Reynolds Number Aerodynamics: Separated Flows 1 General Introduction and Motivation 2 Marginal Separation Theory 3 Cauchy Problems 3.1 Steady Problems 3.2 Ill-Posedness and Regularized Dynamics Abstract Cauchy Problems Operator Symbols and Regularization Explicit Versus Implicit Time Integration 3.3 Self-Similar Finite Time Blow-Up 4 Concluding Remarks References Turbulence Theory 1 Physical Description of the Tropospheric Turbulence 1.1 An Intuitive Approach of Turbulence 1.2 Troposphere The Lower Part of the Troposphere: The Boundary Layer Production of Turbulence in the Boundary Layer 1.3 Scales of Turbulence 1.4 Navier-Stokes Equations Conservation Equations Equation of State Conservation of Mass Conversation of Momentum Conservation of Internal Energy: First Law of Thermodynamics Potential Temperature and Virtual Potential Temperature Conservation of Specific Moisture Approximations Mean Turbulent Kinetic Energy 1.5 A Spectral Approach of Turbulence: The Concept of Eddies Energy Cascade Wavenumber Spectrum of Turbulence 2 Statistical Approach of Turbulence for Geodetic Applications 2.1 Introduction 2.2 Random Fields Stationary Increments and Local Homogeneity Stationary Increments Stationary Process Local Homogeneity and Isotropy Dealing with Anisotropy Locally Inhomogeneous Field with Smoothly Varying Mean Characteristics Kolmogorov Theory Difference Between Structure Function of Nth Order and of Nth Increments Phase Structure Function Links to Semivariogram Taylor's Frozen Hypothesis: The Temporal Structure Function Application of the Structure Function of the Second Increment: Allan Variance Structure Functions Limitation 2.3 Spectral Density Functions General Expression of the Spectral Density Atmospheric Transmission Spectral Density for Interferometric Measurements Spectral Density and Structure Function: A Comparison Power Law Dependencies and Statistical Processes 2.4 Conclusions 3 Covariance Models 3.1 Introduction 3.2 Covariance: Definition 3.3 Previous Work on Physical Correlations 3.4 Treuhaft and Lanyi model (1987) Simulation 3.5 Schön and Brunner model (2008a) Separation Distance Influence of the height at which the separation distance is computed Anisotropy Tropospheric Height Outer Scale Length Wind Taylor's Frozen Hypothesis and VLBI Structure Constant 3.6 Comparison Between the Schön and Brunner Model and the Treuhaft and Lanyi Model 4 Applications of the Covariance Matrix Models 4.1 Simulation of Slant Delay Cholesky Decomposition Eigenvalue Decomposition: Case Study 4.2 Improving the Stochastic Model of GPS or VLBI by Taking into Consideration Physical Correlations 4.3 Wavelet and Turbulence: Covariance Analysis 5 Conclusions References Forest Fire Spreading 1 Problem Statement and Status Quo 2 Physical Model, Development of the Equations, and Preliminary Tools 2.1 Physical Model Combustion Process Heat and Mass Transfer Mechanisms 2.2 Development of the Equations Comparison of Different Models 2.3 Preliminary Tools Nondimensionalization Numerical Prework 3 Collocational Solution Theory 3.1 Space Discretization 3.2 Time Discretization 3.3 Stabilization 4 Data Analysis 4.1 Weather and Fuel Data 4.2 Parameter Influence on the Model 4.3 Parameter Studies 5 Numerical Example 6 Perspective and Future Application Fields 6.1 Climate Change and Fire Risk 6.2 Climate Change Projections for Rhineland-Palatinate 6.3 Future Application Fields 6.4 Forest Management Options References Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation 1 Introduction 2 Mathematical Modeling 3 Numerical Method 3.1 Finite Volume Method Numerical Results for Shallow Water Flow 3.2 Positivity Preserving and Conservative Schemes Numerical Results for Positive Ordinary Differential Equations 3.3 Practical Applications 4 Conclusion References Model-Based Visualization of Instationary Geo-Data with Application to Volcano A