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This paper represents a milestone in the UNB effort to formulate an accurate and self-consistent theory for regional geoid determination. To get the geoid to a sub-centimetre accuracy, we had to formulate the theory in a spherical rather than linear approximation, advance the modelling of the effect of topographic mass density, formulate the solid spherical Bouguer anomaly, develop the probabilistic downward continuation approach, incorporate improved satellite determined global gravitational models and introduce a whole host of smaller improvements. Having adopted Auvergne, an area in France as our testing ground, where the mean standard deviation of observed gravity values is 0.5 mGal, according to the Institute Geographique Nationale (Duquenne in Proceedings of the 1st international symposium of the international gravity field service "gravity field of the earth", International gravity field service meeting, Istanbul, Turkey, 2006), we obtained the standard deviation of the gravity anomalies continued downward to the geoid, as estimated by minimizing the L2 norm of their residuals, to be in average 3 times larger than those on the surface with large spikes underneath the highest topographic points. The standard deviations of resulting geoidal heights range from a few millimetres to just over 6 cm for the highest topographic points in the Alpine region (just short of 2000 m). The mean standard deviations of the geoidal heights for the whole region are only 0.6 cm, which should be considered quite reasonable even if one acknowledges that the area of Auvergne is mostly flat. As one should expect, the main contributing factors to these uncertainties are the Poisson probabilistic downward continuation process, with the maximum standard deviation just short of 6 cm (the average value of 2.5 mm) and the topographic density uncertainties, with the maximum value of 5.6 cm (the average value of 3.0 mm). The comparison of our geoidal heights with the testing geoidal heights, obtained for a set of 75 control points (regularly spaced throughout the region), shows the mean shift of 13 cm which is believed to reflect the displacement of the French vertical datum from the geoid due to sea surface topography. The mean root square error of the misfit is 3.3 cm. This misfit, when we consider the estimated accuracy of our geoid, indicates that the mean standard deviation of the "test geoid" is about 3 cm, which makes it about 5 times less accurate than the Stokes–Helmert computed geoid. [ABSTRACT FROM AUTHOR]

New spherical integral formulas between components of the second- and third-order gravitational tensors are formulated in this article. First, we review the nomenclature and basic properties of the second- and third-order gravitational tensors. Initial points of mathematical derivations, i.e., the second- and third-order differential operators defined in the spherical local North-oriented reference frame and the analytical solutions of the gradiometric boundary-value problem, are also summarized. Secondly, we apply the third-order differential operators to the analytical solutions of the gradiometric boundary-value problem which gives 30 new integral formulas transforming (1) vertical-vertical, (2) vertical-horizontal and (3) horizontal-horizontal second-order gravitational tensor components onto their third-order counterparts. Using spherical polar coordinates related sub-integral kernels can efficiently be decomposed into azimuthal and isotropic parts. Both spectral and closed forms of the isotropic kernels are provided and their limits are investigated. Thirdly, numerical experiments are performed to test the consistency of the new integral transforms and to investigate properties of the sub-integral kernels. The new mathematical apparatus is valid for any harmonic potential field and may be exploited, e.g., when gravitational/magnetic second- and third-order tensor components become available in the future. The new integral formulas also extend the well-known Meissl diagram and enrich the theoretical apparatus of geodesy. [ABSTRACT FROM AUTHOR]

The geoid-to-quasigeoid separation is often computed only approximately as a function of the simple planar Bouguer gravity anomaly and the height of the computation point while disregarding the contributions of terrain geometry and anomalous topographic density as well as the sub-geoid masses. In this study we demonstrate that these contributions are significant and, therefore, should be taken into consideration when investigating the relation between the normal and orthometric heights particularly in the mountainous, polar and geologically complex regions. These contributions are evaluated by applying the spectral expressions for gravimetric forward modelling and using the EIGEN-6C4 gravity model, the Earth2014 datasets of terrain, ice thickness and inland bathymetry and the CRUST1.0 sediment and (consolidated) crustal density data. Since the global crustal density models currently available (e.g. CRUST1.0) have a limited accuracy and resolution, the comparison of individual density contributions is-for consistency-realized with a limited spectral resolution up to a spherical harmonic degree 360 (or 180). The results reveal that the topographic contribution globally varies between $$-$$ 0.33 and 0.57 m, with maxima in Himalaya and Tibet. The contribution of ice considerably modifies the geoid-to-quasigeoid separation over large parts of Antarctica and Greenland, where it reaches $$\sim $$ 0.2 m. The contributions of sediments and bedrock are less pronounced, with the values typically varying only within a few centimetres. These results, however, have still possibly large uncertainties due to the lack of information on the actual sediment and bedrock density. The contribution of lakes is mostly negligible; its maxima over the Laurentian Great Lakes and the Baikal Lake reach only several millimetres. The contribution of the sub-geoid masses is significant. It is everywhere negative and reaches extreme values of $$-$$ 4.43 m. According to our estimates, the geoid-to-quasigeoid separation globally varies within $$-$$ 4.19 and 0.26 m while the corresponding values computed according to a classical definition are only negative and reach extreme values of $$-$$ 3.5 m. A comparison of these results reveals that inaccuracies caused by disregarding the terrain geometry and mass density heterogeneities distributed within the topography and below the geoid surface can reach $$\pm $$ 2 m or more in the mountainous regions. [ABSTRACT FROM AUTHOR]

A new mathematical model for evaluation of the third-order (disturbing) gravitational tensor is formulated in this article. Firstly, we construct corresponding differential operators for the components of the third-order (disturbing) gravitational tensor in a spherical local north-oriented frame. We show that the differential operators may efficiently be decomposed into an azimuthal and an isotropic part. The differential operators are even more simplified for a certain class of isotropic kernels. Secondly, the differential operators are applied to the well-known integrals of Newton, Abel-Poisson, Pizzetti and Hotine. In this way, 40 new integral formulas are derived. The new integral formulas allow for evaluation of the components of the third-order (disturbing) gravitational tensor from density distribution, disturbing gravitational potential, gravity anomalies and gravity disturbances. Thirdly, we investigate the behaviour of the corresponding integral kernels in the spatial domain. The new mathematical formulas extend the theoretical apparatus of geodesy, i.e. the well-known Meissl scheme, and reveal important properties of the third-order gravitational tensor. They may be exploited in geophysical studies, continuation of gravitational field quantities and analysing the gradiometric-geodynamic boundary value problem. [ABSTRACT FROM AUTHOR]