Journal of Geodesy

, Volume 88, Issue 2, pp 179–197 | Cite as

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

  • Michal ŠprlákEmail author
  • Josef Sebera
  • Miloš Val’ko
  • Pavel Novák
Original Article


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.


GOCE Gravitational gradient Integral equation  Meissl scheme Satellite gradiometry 



This work is supported in the framework of ESA’s project GOCE-GDC AO/1-6367/10/NL/AF. M. Šprlák and J. Sebera were supported by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic. P. Novák was supported by the project 209/12/1929 of the Czech Science Foundation. Constructive comments of the three anonymous reviewers are gratefully acknowledged. Thanks are extended to the editor-in-chief Prof. Roland Klees and the responsible editor Prof. Wolfgang Keller for handling our manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michal Šprlák
    • 1
    Email author
  • Josef Sebera
    • 1
  • Miloš Val’ko
    • 1
  • Pavel Novák
    • 1
  1. 1.New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic

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