Knowledge of the Earth's gravity field is an essential component for understanding the physical system of the Earth. Inside the masses, the field interacts with many other fields, according to complicated processes of physical and chemical nature; the study of these phenomena is the object of geophysics. Outside the masses, the gravity field smoothes out in agreement with the "harmonic" character of gravitation, while preserving, particularly close to the Earth's surface, the signature of the internal processes; the study of the gravity field on the boundary and in the external space is the object of physical geodesy. It is necessary to define a separation surface between the masses and the "free" space. This surface is the geoid, an equipotential surface of the gravity field in a stack of such surfaces, close to the surface of the sea. Determining the geoid, or some other surface closer to the Earth's surface, has become synonymous to modelling the gravity field in physical geodesy; this is the subject of this book. Nowadays, this knowledge has become a practical issue also for engineering and other applications, because the geoid is used as a reference surface (datum) of physical heights that is very important in order to relate such heights to purely geometric ones obtained, for example, from GNSS. The methods currently used to produce the geoid at the centimetre level require significant mathematical, stochastic and numerical analysis. The book is structured in such a way as to provide self consistently all the necessary theoretical concepts, from the most elementary ones, such as Newton's gravitation law, to the most complicated ones dealing with the stability of solutions of boundary value problems. It also provides a full description of the available numericaltechniques for precise geoid and quasi-geoid determination. In this way, the book can be used by both students at the undergraduate and graduate level, as well as by researchers engaged in studies in physical geodesy and in geophysics. The text is accompanied by a number of examples, from most elementary to more advanced, as well as by exercises that illustrate the main concepts and computational methods.

Résumé
This book will be based on the material of the lecture noties in several International Schools for the Determination and Use of the Geoid, organized by the International Geoid Serivice of the International Association of Geodesy. It consolidates, unifies, and streamlines this material in a unique way not covereed by the few other books that exist on this subjext. More specifically, the book presents (for the first time in a single volume) the theory and methodology of the most common technique used for precise determination of the geoid, including the computation of the marine geoid from satellite altimetry data. These are illustrated by specific examples and actual computations of local geoids. In addition, the book provides the fundamentals of estimating orthometric heights without spirit levelling, by properly combining a geoid with heights from GPS. Besides the geodectic and geophysical uses, this last application has made geoid computation methods very popular in recent years because the entire GPS and GIS user communities are interested in estimating geoid undulations in order to convert GPS heights to physically meaningful orthometric heights (elevations above mean sea level). The overall purpose of the book is, therefore, to provide the user community (academics, graduate students, geophysicists, engineers, oceanographers, GIS and GPS users, researchers) with a self-contained textbook, which will supply them with the complete roadmap of estimating geoid undulations, from the theoretical definitions and formulas to the available numerical methods and their implementation and the test in practice.

Contenu

Part I: 1. The forward modelling of the gravity field.- 2. Observable of physical geodesy and their analytical representation.- 3. Harmonic calculus and global gravity models.- 4. The local modellling of the gravity field: terrain effects.- 5. The local modelling of the gravity field by collocation.- Part II: 6. Global gravitational Models.- 7. Geoid determination by 3D least squares collaction.- 8. Mass reductions in geoid modelling.- 9. Marine gravity and geoid from satellite altimetry.- 10. Geoid determination by fast Fourier transform techniques.- 11.- Combination of heights.- Part III: 12. Hilbert spaces and deterministic collacation.- 13. On potential theory and HS of harmonic functions.- 14. A quick look to classical BVP solutions.- 15. The analysis of geodetic boundary value problems (BVP) in linear form.