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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]

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]

Due to the ESA's satellite mission GOCE launched in March 2009, gravitational gradients sampled along the orbital trajectory approximately 250 km above the Earth's surface have become available. Since 2010, gravitational gradients have routinely been applied in geodesy for the derivation of global Earth's gravitational models provided in terms of fully normalized coefficients in a spherical harmonic series representation of the Earth's gravitational potential. However, in geophysics, gravitational gradients observed by spaceborne instruments have still been applied relatively seldom. This contribution describes their possible geophysical applications in structural studies where gravitational gradients observed at satellite altitudes are compared with those derived by a spectral forward modeling technique using available global models of selected Earth's mass components as input data. In particular, GOCE gravitational gradients are interpreted in terms of a superposition principle of gravitation as combined gravitational effects generated by a homogeneous reference ellipsoid of revolution, mean topographic and ice mass density distributions, depth-dependent mass density contrasts within bathymetry and lateral mass density anomalies with sediments and crustal layers. Respective gravitational effects are one by one removed from gravitational gradients observed at approximately 250 km elevation above ground. Removing respective gravitational gradients from observed gravitational gradients gradually reveals problematic geographic areas with model deficiencies. For the full interpretation of observed gravitational gradients, deficiencies of CRUST2.0 must be corrected and effects of deeper laying mass anomalies not included in the study considered. These findings are confirmed by parameters describing spectral properties of the gravitational gradients. The methodology can be applied for validating Earth's gravitational models and for constraining crustal models in the development phase. [ABSTRACT FROM AUTHOR]

We derive expressions for computing the gravitational field (potential and its radial derivative) generated by an arbitrary homogeneous or laterally varying density contrast layer with a variable depth and thickness based on methods for a spherical harmonic analysis and synthesis of gravity field. The newly derived expressions are utilised in the gravimetric forward modelling of major known density structures within the Earth's crust (excluding the ocean density contrast) beneath the geoid surface. The gravitational field quantities due to the sediments and crust components density contrasts, shown in numerical examples, are computed using the 2 × 2 arc-deg discrete data from the global crustal model CRUST2.0. These density contrasts are defined relative to the adopted value of the reference crustal density of 2670 kgm. All computations are realised globally on a 1 × 1 arc-deg geographical grid at the Earth's surface. The maxima of the gravitational signal due to the sediments density contrast are mainly along continental shelf regions with the largest sedimentary deposits. The corresponding maxima due to the consolidated crust components density contrast are over areas of the largest continental crustal thickness with variable geological structure. [ABSTRACT FROM AUTHOR]

Spherical harmonic transformation is of practical interest in geodesy for transformation of globally distributed quantities such as gravity between space and frequency domains. The increasing spatial resolution of the latest and forthcoming gravitational models pose true computational challenges for classical algorithms since serious numerical instabilities arise during the computation of the respective base functions of the spherical harmonic expansion. A possible solution is the evaluation of the associated Legendre functions in the Fourier domain where numerical instabilities can be circumvented by an independent frequency-wise scaling of numerical coefficients into a numerically suitable double precision range. It is then rather straightforward to commit global fast data transformation into the Fourier domain and to evaluate subsequently spherical harmonic coefficients. For the inverse, the computation of respective Fourier coefficients from a given spherical harmonic model is performed as an inverse Fast Fourier Transform into globally distributed data points. The two-step formulation turns out to be stable even for very high resolutions as well as efficient when using state-of-the-art shared memory/multi-core architectures. In principle, any functional of the geopotential can be computed in this way. To give an example for the overall performance of the algorithm, we transformed an equiangular 1 arcmin grid of terrain elevation data corresponding to spherical harmonic degree and order 10800. [ABSTRACT FROM AUTHOR]

In the last few decades, satellites have acquired various potential data sets hundreds of kilometers above the Earth's surface. Conventionally, these global magnetic and gravitational data sets are approximated by using spherical harmonics that allow straightforward work with both fields outside the Earth's mass. In this article, we present an alternative approach for working with potential data in mass-free space given over a regular coordinate grid on a spherical surface. The algorithm is based on an iterative scheme and the Poisson integral equation for the sphere. With help from the Fourier transform, global potential (magnetic or gravitational) data can efficiently be continued from a mean orbital sphere down to a reference surface without using the spherical harmonics. This is illustrated both with simulated magnetic field data and with real data from the satellite gradiometry mission GOCE. In the case of simulated magnetic data and the downward continuation for 450 km, we have achieved a root mean square at the level of 0.05 nT, while it was <1 E (eotvos) for real GOCE data continued for 250 km. The crucial point is to apply the algorithm twice as a large part of noise can be removed from the input data. [ABSTRACT FROM AUTHOR]