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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]

Values of scalar, vector and second-order tensor parameters of the Earth's gravitational field have been collected by various sensors in geodesy and geophysics. Such observables have been widely exploited in different parametrization methods for the gravitational field modelling. Moreover, theoretical aspects of these quantities have extensively been studied and well understood. On the other hand, new sensors for observing gravitational curvatures, i.e., components of the third-order gravitational tensor, are currently under development. As the gravitational curvatures represent new types of observables, their exploitation for modelling of the Earth's gravitational field is a subject of this study. Firstly, the gravitational curvature tensor is decomposed into six parts which are expanded in terms of third-order tensor spherical harmonics. Secondly, gravitational curvature boundary-value problems defined for four combinations of the gravitational curvatures are formulated and solved in spectral and spatial domains. Thirdly, properties of the corresponding sub-integral kernels are investigated. The presented mathematical formulations reveal some important properties of the gravitational curvatures and extend the so-called Meissl scheme, i.e., an important theoretical framework that relates various parameters of the Earth's gravitational field. [ABSTRACT FROM AUTHOR]

Integral transforms of the disturbing gravitational potential derived from satellite altimetry onto satellite gradiometric data are formulated, investigated and applied in this article. First, corresponding differential operators, that relate the disturbing gravitational potential to the six components of the disturbing gradiometric tensor in the spherical local north-oriented frame, are applied to the spherical Abel-Poisson integral equation. This yields six new integral equations for which respective kernel functions are given in both spectral and spatial forms. Second, truncation error formulas for each of the integral transforms are provided in the spectral form. Also expressions for the corresponding truncation error coefficients are derived. Third, practical estimators for evaluation of the disturbing gravitational gradients are formulated, and their correctness and expected accuracy are investigated. Finally, the practical estimators are applied for validation of a sample of the gradiometric data provided by the GOCE satellite mission. Obtained results demonstrate applicability of the new apparatus as an alternative validation method of the satellite gravitational gradients. [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]