Journal of Geodesy

, Volume 90, Issue 9, pp 883–902 | Cite as

Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET)

  • Michael KuhnEmail author
  • Christian Hirt
Original Article


In gravity forward modelling, the concept of Rock-Equivalent Topography (RET) is often used to simplify the computation of gravity implied by rock, water, ice and other topographic masses. In the RET concept, topographic masses are compressed (approximated) into equivalent rock, allowing the use of a single constant mass–density value. Many studies acknowledge the approximate character of the RET, but few have attempted yet to quantify and analyse the approximation errors in detail for various gravity field functionals and heights of computation points. Here, we provide an in-depth examination of approximation errors associated with the RET compression for the topographic gravitational potential and its first- and second-order derivatives. Using the Earth2014 layered topography suite we apply Newtonian integration in the spatial domain in the variants (a) rigorous forward modelling of all mass bodies, (b) approximative modelling using RET. The differences among both variants, which reflect the RET approximation error, are formed and studied for an ensemble of 10 different gravity field functionals at three levels of altitude (on and 3 km above the Earth’s surface and at 250 km satellite height). The approximation errors are found to be largest at the Earth’s surface over RET compression areas (oceans, ice shields) and to increase for the first- and second-order derivatives. Relative errors, computed here as ratio between the range of differences between both variants relative to the range in signal, are at the level of 0.06–0.08 % for the potential, \(\sim \)3–7 % for the first-order derivatives at the Earth’s surface (\(\sim \)0.1 % at satellite altitude). For the second-order derivatives, relative errors are below 1 % at satellite altitude, at the 10–20 % level at 3 km and reach maximum values as large as \(\sim \)20 to 110  % near the surface. As such, the RET approximation errors may be acceptable for functionals computed far away from the Earth’s surface or studies focussing on the topographic potential only. However, for derivatives of the functionals computed near the Earth’s surface, the use of RET introduces very spurious errors, in some cases as large as the signal, rendering it useless for smoothing or reducing of field observation, thus rigorous mass modelling should be used for both spatial and spectral domain methods.


Rock-equivalent topography (RET) Topographic potential First and second-order derivatives Spherical approximation Discretized Newtonian integration 



We are grateful to Prof Mehdi Eshagh and two anonymous reviewers for their thorough reviews. We thank for access to the Pawsey Supercomputing Centre ( through the merit allocation scheme used for the calculations presented in this work. Christian Hirt thanks the German National Research Foundation for support via Grant Hi 1760.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Western Australian Geodesy Group and The Institute for Geoscience ResearchCurtin UniversityPerthAustralia
  2. 2.Institute for Astronomical and Physical Geodesy and Institute for Advanced StudyTechnische Universität MünchenMunichGermany

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