Chapter

# The gradiometric-geodynamic boundary value problem

DOI: 10.1007/3-540-26932-0_61 In book: Gravity, Geoid and Space Missions, pp.352-357

**ABSTRACT**

The role of gravity gradients is investigated in the framework of geodetic-geodynamic boundary value problems. The time variation of the Eötvös tensor can be separated into three parts. The first part is a surface movement term, the second is the time variation of the gravity field in the original point, and the third is a coupling term. The first and third terms can be computed by a third order derivative tensor of the gravity potential. These terms can be formulated in spherical and planar approximation.

Gravity gradients have the advantage over gravity measurements that certain gravity gradient combinations are insensitive to site movements, thereby allowing the time variation of the gravity field to be determined without repeated height measurements. The solution of the corresponding gradiometric-geodynamic boundary value problem is shown with repeated gravity gradient measurements of the torsion balance.

Several test computations with a simple mass density model were performed in order to analyze the time variation of gravity gradients and give estimates of their respective magnitudes.

Gravity gradients have the advantage over gravity measurements that certain gravity gradient combinations are insensitive to site movements, thereby allowing the time variation of the gravity field to be determined without repeated height measurements. The solution of the corresponding gradiometric-geodynamic boundary value problem is shown with repeated gravity gradient measurements of the torsion balance.

Several test computations with a simple mass density model were performed in order to analyze the time variation of gravity gradients and give estimates of their respective magnitudes.

- Acta Geodaetica et Geophysica Hungarica 06/2007; 42(2). DOI:10.1556/AGeod.42.2007.2.2 · 0.35 Impact Factor
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**ABSTRACT:**Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth’s surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5):282–289, 1997) for degree 360 models, produce cm- to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultra-high resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth’s surface.Journal of Geodesy 09/2012; 86(9). DOI:10.1007/s00190-012-0550-y · 2.70 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**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.Journal of Geodesy 02/2015; 89(2):141-157. DOI:10.1007/s00190-014-0767-z · 2.70 Impact Factor

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