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.
Author(s): Willi Freeden
Series: Geosystems Mathematics
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 484
City: Cham
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
