Surveys in Geophysics

, Volume 37, Issue 6, pp 1035–1074 | Cite as

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

  • Moritz RexerEmail author
  • Christian Hirt
  • Sten Claessens
  • Robert Tenzer


Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.


Gravity forward modelling Ellipsoidal topographic potential Harmonic combination method Spherical harmonics Spherical approximation Ellipsoidal approximation Layer concept Earth2014 



This study received support from the German National Research Foundation (DFG), project \(\hbox {n}^{\circ }\) Hi 1760/1. It was also supported by the Technische Universität München - Institute for Advanced Study, funded by the German Excellence Initiative (and the European Union Seventh Framework Programme under grant agreement \(\hbox {n}^{\circ }\, 291763\)). We also acknowledge the Czech Ministry of Education, Youth and Sport for a financial support by the National Program of Sustainability, Project No.: LO1506. We thank Simon Holmes and NGA for the distribution of the land/ocean masks used in the creation of EGM2008. We further thank Simon Holmes for generously sharing the software facilitating Jekeli’s transform with us. We thank Torsten Mayer-Gürr and Jan-Martin Brockmann for providing the satellite gravity normal equations for this study. We further acknowledge the work of all other producers of data used in this work. Finally, we thank two anonymous reviewers for their detailed reviews.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Moritz Rexer
    • 1
    Email author
  • Christian Hirt
    • 1
  • Sten Claessens
    • 2
  • Robert Tenzer
    • 3
    • 4
  1. 1.Institute of Astronomical and Physical Geodesy, Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  2. 2.Department of Spatial Sciences, The Institute for Geoscience Research, Western Australian Geodesy GroupCurtin UniversityPerthAustralia
  3. 3.The Key Laboratory of Geospace Environment and GeodesyWuhan UniversityWuhanChina
  4. 4.New Technologies for the Information Society (NTIS)University of West BohemiaPlzeňCzech Republic

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