Surveys in Geophysics

, Volume 37, Issue 3, pp 681–700 | Cite as

Spherical Harmonic Analysis of Gravitational Curvatures and Its Implications for Future Satellite Missions

  • Michal Šprlák
  • Pavel Novák
  • Martin Pitoňák


In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth’s gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical–vertical–vertical, vertical–vertical–horizontal, vertical–horizontal–horizontal and horizontal–horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of \({\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}\) Hz\(^{-1/2}\).


Differential accelerometry Earth’s gravitational field Gravitational curvature Gravitational gradient Spherical harmonic analysis 



The authors were supported by the Project No. GA15-08045S of the Czech Science Foundation. Thoughtful and constructive comments of two anonymous reviewers are gratefully acknowledged. Thanks are also extended to the editor-in-chief Dr. Michael J. Rycroft for handling our manuscript.


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.NTIS – New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic

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