On Combining the Directional Solutions of the Gravitational Curvature Boundary-Value Problem
In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundary-value problem can formally be formulated for the vertical-vertical-vertical,
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In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundaryvalue problem can formally be formulated for the vertical-vertical-vertical, vertical-verticalhorizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal components of the third-order gravitational tensor. Each integral equation provides an independent set of spherical harmonic coefficients because each component of the third-order gravitational tensor is sensitive to gravitational changes in the different directions. In this contribution, estimations of spherical harmonic coefficients of the gravitational potential are carried out by combining four solutions of the gravitational curvature boundary-value problem using three methods, namely an arithmetic mean, a weighted mean and a conditional adjustment model. Since the third-order gradients of the gravitational potential are not yet observed by satellite sensors, we synthesise them at the satellite altitude of 250 km from a global gravitational model up to the degree 360 while adding a Gaussian noise with the standard deviation of 6.3 1019 m1 s2 . Results of the numerical analysis reveal that the arithmetic mean model provides the best solution in terms of the RMS fit between predicted and reference values. We explain this result by the facts that the conditions only create additional stochastic bindings between estimated parameters and that more complex numerical schemes for the error propagation are unnecessary in the presence of only a random noise. Keywords
Conditional adjustment Gravitational curvature Spherical harmonics
1 M. Pitoˇnák () P. Novák NTIS – New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Plzeˇn, Czech Republic e-mail: [email protected] M. Šprlák School of Engineering and Built Environment, University of Newcastle, Callaghan, NSW, Australia R. Tenzer Department of Land Surveying and Geo-informatics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
Introduction
Solutions to a spherical boundary-value problem lead to spherical harmonic series or surface integrals with Green’s kernel functions (e.g., Jekeli 2009). When solving this problem for higher-order gradients of the gravitational potential as boundary conditions, more than one solution is obtained. The solutions to the gravimetric, gradiometric and gravitational curvature boundary-value problems (Martinec 2003; Šprlák and Novák 2016) lead to two, three and four formulas, respectively. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.
International Association of Geodesy Symposia, https://doi.org/10.1007/1345_2019_68, © Springer Nature Switzerland AG 2019
ˇ et al. M. Pitonák
Fig. 1 Differences between vertical-vertical-vertical (VVV), verticalvertical-horizontal (VVH), vertical-horizontal-horizontal (VHH
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