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Physical heights is one of the most important topics in physical geodesy. Their original concept, introduced in the 19-th century, defined physical heights as lengths of plumblines of the Earth's gravity field between the geoid and points of interest. There are orthometric heights of surface points, that have been traditionally estimated by spirit levelling and measured gravity; however, the knowledge of the density distribution of topographic masses (masses between the geoid and Earth's surface) is required that significantly affects their determinability. This was also the main reason why a new type of physical heights was proposed in the mid of the 20-th century. Normal heights approximate orthometric heights in a sense that the Earth's gravity field is replaced by the normal gravity field, an analytic model based on the theory of an equipotential ellipsoid. This height system has been introduced since that time in different countries in Europe and beyond. Contrary to the classical height system based on orthometric heights, its counterpart based on normal heights may have slightly different definitions. Moreover, normal heights are often defined as heights of points above the quasigeoid. This contribution reviews alternative definitions of normal heights and respective height systems. It is argued that both orthometric and normal heights refer to the geoid. In the case physical heights are estimated by satellite positioning, normal heights must be computed through the height anomaly estimated at each point of interest, whether it is below, at or above the Earth's surface. On the contrary, orthometric heights of all points along the same plumbline, be it below, at or above the Earth's surface, are estimated by introducing one value of the geoid height. Normal heights of surface points can be estimated by spirit levelling easier than orthometric heights as no topographic mass density hypothesis is required; however, one has to keep in mind the gravity field approximation used both for their definition and realization. [ABSTRACT FROM AUTHOR]

The gravitational field of the Earth reflects Earth's surface mass redistribution and its inner structure and dynamics. Satellite gravimetry techniques have been used to observe the Earth's external gravitational field and its temporal variations on a global scale. The global gravitational models from satellite gravimetry, typically in terms of spherical harmonic coefficients, are crucial in geodetic, geodynamic, geophysical, hydrological, glaciological, oceanographic, and many other geoscience applications. In this paper, we provide a comprehensive overview of theoretical definitions describing relationships between the spherical harmonic coefficients and different satellite gravimetry observables such as orbital perturbations in terms of satellite positions, velocities, and accelerations; satellite-to-satellite range rates; and gravitational gradients. Products and applications of the Earth's static global gravitational models are presented and discussed in the context of determination of the gravimetric geoid and physical heights, gravimetric and isostatic crustal thickness, bathymetric depths, glacier bedrock relief, sediment thickness, geostrophic and eddy currents, Earth's inertia tensor and dipole, precession and nutation parameters of the Earth's rotation, and prediction of the satellite orbital geometry. Furthermore, applications and advances of the Earth's time-variable gravitational models for monitoring large earthquakes, hydrological mass transport, Earth's rotation parameters, and vertical crustal motions (due to the glacial isostatic adjustment and other phenomena) are presented. Finally, future trends and prospects in the satellite gravimetry are discussed. • A comprehensive mathematical foundations of satellite gravimetry is presented. • A complete progress for applications of satellite gravimetry products is shown. • An overview for future satellite gravimetry missions and trend is given. [ABSTRACT FROM AUTHOR]