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Physical geodesy applies potential theory to study the Earth’s gravitational field in space outside and up to a few km inside the Earth’s mass. Among various tools offered by this theory, boundary-value problems are particularly popular for the transformation or continuation of gravitational field parameters across space. Traditional problems, formulated and solved as early as in the nineteenth century, have been gradually supplemented with new problems, as new observational methods and data are available. In most cases, the emphasis is on formulating a functional relationship involving two functions in 3-D space; the values of one function are searched but unobservable; the values of the other function are observable but with errors. Such mathematical models (observation equations) are referred to as deterministic. Since observed data burdened with observational errors are used for their solutions, the relevant stochastic models must be formulated to provide uncertainties of the estimated parameters against which their quality can be evaluated. This article discusses the boundary-value problems of potential theory formulated for gravitational data currently or in the foreseeable future used by physical geodesy. Their solutions in the form of integral formulas and integral equations are reviewed, practical estimators applicable to numerical solutions of the deterministic models are formulated, and their related stochastic models are introduced. Deterministic and stochastic models represent a complete solution to problems in physical geodesy providing estimates of unknown parameters and their error variances (mean squared errors). On the other hand, analyses of error covariances can reveal problems related to the observed data and/or the design of the mathematical models. Numerical experiments demonstrate the applicability of stochastic models in practice. [ABSTRACT FROM AUTHOR]

The regional gravimetric geoid solved using boundary-value problems of the potential theory is usually determined in two computational steps: (1) downward continuing ground gravity data onto the geoid using inverse Poisson's integral equation in a mass-free space and (2) evaluating geoidal heights by applying Stokes integral to downward continued gravity. In this contribution, the two integration steps are combined in one step and the so-called one-step integration method in spherical approximation is implemented to compute the regional gravimetric geoid model. Advantages of using the one-step integration method instead of the two integration steps include less computational cost, more stable numerical computation and better utilization of input ground gravity data (reduced in each integration step to avoid edge effects). A discrete form of the one-step integral equation is used to convert mean values of ground gravity anomalies into mean values of geoidal heights. To evaluate mean values of the integral kernel in the vicinity of the computation point, a fast and numerically accurate analytical formula is proposed using planar approximation. The proposed formula is tested to determine the regional gravimetric geoid of the Auvergne test area, France. Results show a good agreement of the estimated geoid with geoidal heights estimated at GNSS-levelling reference points, with the standard deviation for the difference of 3.3 cm. Considering the uncertainty of geoidal heights derived at the GNSS/levelling reference points, one can conclude the geoid models computed by the one-step and two-step integration methods have negligible differences. Thus, the one-step method can be recommended for regional geoid modelling with its methodological and numerical advantages. [ABSTRACT FROM AUTHOR]

The static Earth’s gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth’s surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet’s boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green’s kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth. [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]

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]

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme. [ABSTRACT FROM AUTHOR]