Journal of Geodesy

, Volume 91, Issue 3, pp 233–251 | Cite as

Gravity field error analysis for pendulum formations by a semi-analytical approach

  • Huishu Li
  • Tilo Reubelt
  • Markus Antoni
  • Nico Sneeuw
Original Article


Many geoscience disciplines push for ever higher requirements on accuracy, homogeneity and time- and space-resolution of the Earth’s gravity field. Apart from better instruments or new observables, alternative satellite formations could improve the signal and error structure compared to Grace. One possibility to increase the sensitivity and isotropy by adding cross-track information is a pair of satellites flying in a pendulum formation. This formation contains two satellites which have different ascending nodes and arguments of latitude, but have the same orbital height and inclination. In this study, the semi-analytical approach for efficient pre-mission error assessment is presented, and the transfer coefficients of range, range-rate and range-acceleration gravitational perturbations are derived analytically for the pendulum formation considering a set of opening angles. The new challenge is the time variations of the opening angle and the range, leading to temporally variable transfer coefficients. This is solved by Fourier expansion of the sine/cosine of the opening angle and the central angle. The transfer coefficients are further applied to assess the error patterns which are caused by different orbital parameters. The simulation results indicate that a significant improvement in accuracy and isotropy is obtained for small and medium initial opening angles of single polar pendulums, compared to Grace. The optimal initial opening angles are \(45^\circ \) and \(15^\circ \) for accuracy and isotropy, respectively. For a Bender configuration, which is constituted by a polar Grace and an inclined pendulum in this paper, the behaviour of results is dependent on the inclination (prograde vs. retrograde) and on the relative baseline orientation (left or right leading). The simulation for a sun-synchronous orbit shows better results for the left leading case.


Pendulum formation Error assessment Semi-analytical approach Opening angle Grace 



We are grateful to the four anonymous reviewers and the editor whose comments and suggestions have significantly improved the manuscript. We like to thank Prof. Min Zhong, Prof. Zebing Zhou and Prof. Hou-Tse Hsu for their helpful suggestions. This work was started during the stay of the first author in GIS, thanks to the scholarship provided by the China Scholarship Council. This work was supported by the State Key Program of National Natural Science Foundation of China (Grants 11235004) and Strategic Leading Science and Technology project, B category (Grants Y604111011).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Geophysics, School of PhysicsHuazhong University of Science and TechnologyWuhanChina
  2. 2.Institute of GeodesyUniversity of StuttgartStuttgartGermany
  3. 3.Institute of Geodesy and GeophysicsChinese Academy of SciencesWuhanChina

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