Journal of Geodesy

, Volume 89, Issue 9, pp 887–909 | Cite as

Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000

  • Moritz RexerEmail author
  • Christian Hirt
Original Article


Classical degree variance models (such as Kaula’s rule or the Tscherning-Rapp model) often rely on low-resolution gravity data and so are subject to extrapolation when used to describe the decay of the gravity field at short spatial scales. This paper presents a new degree variance model based on the recently published GGMplus near-global land areas 220 m resolution gravity maps (Geophys Res Lett 40(16):4279–4283, 2013). We investigate and use a 2D-DFT (discrete Fourier transform) approach to transform GGMplus gravity grids into degree variances. The method is described in detail and its approximation errors are studied using closed-loop experiments. Focus is placed on tiling, azimuth averaging, and windowing effects in the 2D-DFT method and on analytical fitting of degree variances. Approximation errors of the 2D-DFT procedure on the (spherical harmonic) degree variance are found to be at the 10–20 % level. The importance of the reference surface (sphere, ellipsoid or topography) of the gravity data for correct interpretation of degree variance spectra is highlighted. The effect of the underlying mass arrangement (spherical or ellipsoidal approximation) on the degree variances is found to be crucial at short spatial scales. A rule-of-thumb for transformation of spectra between spherical and ellipsoidal approximation is derived. Application of the 2D-DFT on GGMplus gravity maps yields a new degree variance model to degree 90,000. The model is supported by GRACE, GOCE, EGM2008 and forward-modelled gravity at 3 billion land points over all land areas within the SRTM data coverage and provides gravity signal variances at the surface of the topography. The model yields omission errors of \(\sim \)9 mGal for gravity (\(\sim \)1.5 cm for geoid effects) at scales of 10 km, \(\sim \)4 mGal (\(\sim \)1 mm) at 2-km scales, and \(\sim \)2 mGal (\(\sim \)0.2 mm) at 1-km scales.


Degree variance Omission error Discrete Fourier transform Ultra-high resolution gravity GGMplus Spherical approximation Ellipsoidal approximation 



With the support of the Technische Universität München—Institute for Advanced Study, funded by the German Excellence Initiative. We thank the Australian Research Council for funding via Grant DP120102441. We thank Roland Pail for sharing his knowledge on Fourier transforms, Reiner Rummel for directing us to Parseval’s theorem, and Sten Claessens for the discussions related to degree variances. We are grateful for very constructive and thorough reviews received from four anonymous reviewers, improving the clarity of presentation and stimulating future research.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for 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

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