Journal of Geodesy

, Volume 91, Issue 2, pp 167–194 | Cite as

Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components

  • Michal ŠprlákEmail author
  • Pavel Novák
Original Article


New spherical integral formulas between components of the second- and third-order gravitational tensors are formulated in this article. First, we review the nomenclature and basic properties of the second- and third-order gravitational tensors. Initial points of mathematical derivations, i.e., the second- and third-order differential operators defined in the spherical local North-oriented reference frame and the analytical solutions of the gradiometric boundary-value problem, are also summarized. Secondly, we apply the third-order differential operators to the analytical solutions of the gradiometric boundary-value problem which gives 30 new integral formulas transforming (1) vertical-vertical, (2) vertical-horizontal and (3) horizontal-horizontal second-order gravitational tensor components onto their third-order counterparts. Using spherical polar coordinates related sub-integral kernels can efficiently be decomposed into azimuthal and isotropic parts. Both spectral and closed forms of the isotropic kernels are provided and their limits are investigated. Thirdly, numerical experiments are performed to test the consistency of the new integral transforms and to investigate properties of the sub-integral kernels. The new mathematical apparatus is valid for any harmonic potential field and may be exploited, e.g., when gravitational/magnetic second- and third-order tensor components become available in the future. The new integral formulas also extend the well-known Meissl diagram and enrich the theoretical apparatus of geodesy.


Boundary-value problem Differential operators Gravitational tensor Gravitational gradient Integral transform Meissl diagram 



The authors were supported by the project No. GA15-08045S of the Czech Science Foundation. Thoughtful and constructive comments of the three anonymous reviewers are gratefully acknowledged. Thanks are also extended to the editor-in-chief Prof. Jürgen Kusche and the responsible editor Prof. Wolfgang Keller for handling our manuscript.

Supplementary material

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Authors and Affiliations

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

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