Journal of Geodesy

, Volume 93, Issue 1, pp 65–83 | Cite as

Cap integration in spectral gravity forward modelling: near- and far-zone gravity effects via Molodensky’s truncation coefficients

  • Blažej BuchaEmail author
  • Christian Hirt
  • Michael Kuhn
Original Article


Spectral gravity forward modelling is a technique that converts a band-limited topography into its implied gravitational field. This conversion implicitly relies on global integration of topographic masses. In this paper, a modification of the spectral technique is presented that provides gravity effects induced only by the masses located inside or outside a spherical cap centred at the evaluation point. This is achieved by altitude-dependent Molodensky’s truncation coefficients, for which we provide infinite series expansions and recurrence relations with a fixed number of terms. Both representations are generalized for an arbitrary integer power of the topography and arbitrary radial derivative. Because of the altitude-dependency of the truncation coefficients, a straightforward synthesis of the near- and far-zone gravity effects at dense grids on irregular surfaces (e.g. the Earth’s topography) is computationally extremely demanding. However, we show that this task can be efficiently performed using an analytical continuation based on the gradient approach, provided that formulae for radial derivatives of the truncation coefficients are available. To demonstrate the new cap-modified spectral technique, we forward model the Earth’s degree-360 topography, obtaining near- and far-zone effects on gravity disturbances expanded up to degree 3600. The computation is carried out on the Earth’s surface and the results are validated against an independent spatial-domain Newtonian integration (\(1\,\upmu \mathrm {Gal}\) RMS agreement). The new technique is expected to assist in mitigating the spectral filter problem of residual terrain modelling and in the efficient construction of full-scale global gravity maps of highest spatial resolution.


Spectral gravity forward modelling Spherical harmonics Molodensky’s truncation coefficients Topographic potential Gradient approach Residual terrain modelling 



Blažej Bucha was supported by the ProjectVEGA 1/0954/15 and acknowledges the computational resources made available by the HPC centres at the Slovak University of Technology in Bratislava and at the Slovak Academy of Sciences, which are parts of the Slovak Infrastructure of High Performance Computing (SIVVP project, ITMS code 26230120002, funded by the European region development funds, ERDF). Christian Hirt would like to thank the German National Research Foundation (DFG) for providing funding under Grant Agreement Hi 1760/01. The spatial-domain Newtonian integration was performed using the supercomputing resources kindly provided by Western Australia’s Pawsey Supercomputing Center. The maps were produced using the Generic Mapping Tools (Wessel and Smith 1998).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Theoretical GeodesySlovak University of Technology in BratislavaBratislavaSlovak Republic
  2. 2.Institute for Astronomical and Physical Geodesy, Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  3. 3.Western Australian Geodesy Group, School of Earth and Planetary SciencesCurtin UniversityPerthAustralia

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