Decorrelative Mollifier Gravimetry: Basics, Ideas, Concepts, and Examples

Description

This monograph presents the geoscientific context arising in decorrelative gravitational exploration to determine the mass density distribution inside the Earth. First, an insight into the current state of research is given by reducing gravimetry to mathematically accessible, and thus calculable, decorrelated models. In this way, the various unresolved questions and problems of gravimetry are made available to a broad scientific audience and the exploration industry. New theoretical developments will be given, and innovative ways of modeling geologic layers and faults by mollifier regularization techniques are shown. This book is dedicated to surface as well as volume geology with potential data primarily of terrestrial origin. For deep geology, the geomathematical decorrelation methods are to be designed in such a way that depth information (e.g., in boreholes) may be canonically entered.  Bridging several different geo-disciplines, this book leads in a cycle from the potential measurements made by geoengineers, to the cleansing of data by geophysicists and geoengineers, to the subsequent theory and model formation, computer-based implementation, and numerical calculation and simulations made by geomathematicians, to interpretation by geologists, and, if necessary, back. It therefore spans the spectrum from geoengineering, especially geodesy, via geophysics to geomathematics and geology, and back. Using the German Saarland area for methodological tests, important new fields of application are opened, particularly for regions with mining-related cavities or dense development in today's geo-exploration.  Preface Structure of the Book Acknowledgments Contents About the Author 1 Introductory Remarks 1.1 Cycle of Measurement and Modeling 1.2 Potential Methods: Historical Stages 1.3 Potential Methods: Geomathematical Aspects 1.4 Dirac and Newton Mollifiers 1.5 Exploratory Obligations Involving Gravity Part I Gravitation and Gravimetry 2 Gravitation 2.1 Gravity and Gravitation 2.2 Newton's Law and Subsequent Concepts 2.3 External and Internal Earth's Gravitational Field 2.4 Key Observational Quantities 3 Gravimetry 3.1 Absolute Gravimetry 3.2 Relative Gravimetry 3.3 Gravity Reduction Part II Potential Theory 4 Classical Context 4.1 Notation and Nomenclature 4.2 Key Results of Classical Potential Theory 4.3 Newton Potentials 5 Newton–Haar Mollifier Theory and Applications 5.1 Newton–Haar Potentials 5.2 Haar Mollifier Scaling and Wavelet Functions 5.3 Haar Wavelet-Based Density Decorrelation from Density Data 5.4 Haar Wavelet-Based Density Data Compression 5.5 Multi-Scale Signal-to-Noise Ratio 5.6 Haar Wavelet-Based Density Decorrelation from Potential Data 6 Disturbing Potential 6.1 Molodensky's Problem 6.2 Hörmander's Linearization 6.3 Standard Geodetic Conventions Part III Surface Decorrelation 7 Space versus Frequency Surface Modeling 7.1 Spherical Harmonics 7.2 Kernel Functions 7.3 Splines 7.4 Wavelets 7.5 Options in Surface Modeling 8 Surface Applications 8.1 Decorrelation of the Global Earth's GravitationalModel (EGM) 8.2 Mollifier Potential from Gravity Disturbances and Anomalies 8.3 Decorrelation of Gravity Disturbances for Galapagos 8.4 Decorrelation of Gravity Anomalies for Galapagos 8.5 Mollifier Potential from Deflections of the Vertical 8.6 Decorrelation of Deflections of the Vertical for Hawaii 8.7 Decorrelation of Deflections of the Vertical for Iceland Part IV Inverse Potential Theory 9 Gravimetry as an Ill-Posed Inverse Problem 9.1 Ill-Posedness and Regularization Methods 9.2 Direct and Inverse Gravimetry 9.3 Heuristic Perception of the Ill-Posedness 9.4 Space- and Frequency-Based Inversion Part V Volume Decorrelation 10 Volume Methodology 10.1 Multi-Scale Dirac Mollifier Context 10.2 Multi-Scale Newton Mollifier Context 10.3 Mollifier Spline Inversion 10.4 Mollifier Wavelet Inversion 11 Volume Applications 11.1 Test Area Saarland/Palatinate: Initial Situation 11.2 Surface Decorrelation and Geological Surface Interpretation 11.3 Surface Comparison of Gravimetry and Magnetometry 11.4 Depth Modeling and Interpretation 11.5 Mollifier Spline Inversion 11.6 Mollifier Wavelet Inversion Part VI Decorrelative Potential Methods 12 Decorrelative Monopole Potential-Based Gravimetry 12.1 Key Aspects 12.2 Innovative Ingredients 12.3 Perspectives 13 Decorrelative Dipole Potential-Based Magnetometry 13.1 Essential Constituents of the Earth's Magnetic Field 13.2 Dipole Potentials 13.3 Susceptibility and Permeability 13.4 Inverse Mollifier Magnetometry 14 Decorrelative Acoustic Potential-Based Exploration 14.1 Background 14.2 Acoustic Wave Equation 14.3 Helmholtz Equation-Based Mollifier Context 15 Decorrelative Elastic Potential-Based Exploration 15.1 From Euler to Cauchy-Navier Equation 15.2 Cauchy-Navier Mollifier Method and Decorrelation 15.3 Decorrelation by Cauchy-Navier Wavelets Part VII Outlook 16 Concluding Remarks 16.1 Gravimetric and Magnetometric Mollifier Exploration 16.2 Tomographic and Scattering Mollifier Exploration Part VIII Appendix 17 Supporting Material 17.1 List of Symbols 17.2 List of Acronyms References Index