Journal of Geodesy

, Volume 93, Issue 10, pp 2089–2108 | Cite as

Residual terrain modelling (RTM) in terms of the cap-modified spectral technique: RTM from a new perspective

  • Blažej BuchaEmail author
  • Christian Hirt
  • Meng Yang
  • Michael Kuhn
  • Moritz Rexer
Original Article


We present two novel approaches to residual terrain modelling (RTM), one of which is practical and the other rather theoretical. The first provides a solution to the harmonic correction issue and the high-frequency error of the spectral filter problem. As its key feature, cap-modified spectral gravity forward modelling is applied to deliver near-zone gravity effects induced by the reference (smooth) topography. Thanks to its spectral nature, the sought gravity can be evaluated at the problematic points inside the smooth topography by regularized downward continuation (solution for the harmonic correction problem), and, at the same time, it is band-limited in spherical harmonics (solution for the high-frequency error). A validation over two mountainous areas, Switzerland and Slovakia, reveals that this technique is at least comparable with two other common RTM variants (RMS agreement up to 0.1 mGal). Finally, we formulate the theoretical RTM concept, showing that the harmonic correction issue and the spectral filter problem are caused by filtering of the mass model in the topography domain. When properly filtering gravity effects in the gravity domain, that is, avoiding the concept of the reference topography, both problems disappear. Limitations of both RTM approaches include the possible divergence effect of spherical harmonic series on the Earth’s surface and a conceptual inconsistency between two involved types of spherical harmonic coefficients. Applications of this study could be found in the development of combined gravity models (the fill-in procedure) or in the spectral enhancement of spherical harmonic gravity models.


Gravity field modelling Residual terrain modelling Topography Spherical harmonics Divergence effect Runge–Krarup theorem 



Swisstopo (Dr. Urs Marti) and Zahorec et al. (2017) are kindly acknowledged for providing ground-truth gravity data over Switzerland and Slovakia, respectively. Some of the computations were performed at the HPC centres at the Slovak University of Technology in Bratislava, the Slovak Academy of Sciences and the University of Žilina, which are parts of the Slovak Infrastructure of High Performance Computing (SIVVP Project, ITMS code 26230120002, funded by the European region development funds, ERDF). BB was supported by the Project VEGA 1/0750/18. CH received funding from the German National Research foundation via Grant Hi 1760/1. The maps were produced using the Generic Mapping Tools (Wessel and Smith 1998).

Author Contributions

BB, CH, MY, MK and MR designed the study; BB conducted all the numerical experiments and drafted the manuscript; MY performed independent check of the results from spatial-domain gravity forward modelling; all authors discussed and commented on the manuscript.

Compliance with ethical standards

Data availability

The Swiss gravity data are the property of Swisstopo. The ownership of the Slovak gravity database is distributed among Geocomplex Inc., Comenius University in Bratislava and Slovak University of Technology in Bratislava. None of the two gravity databases can be made available to third parties. The EIGEN-6C4 gravity model is available at The original MERIT topography is due to Yamazaki et al. (2017) and is available at MATLAB-based software packages for (i) ultra-high-degree spherical harmonic synthesis (GrafLab and isGrafLab), (ii) ultra-high-degree surface spherical harmonic analysis and (iii) the evaluation of Molodensky’s truncation coefficients are freely available at Attached are also the truncation coefficients used in the numerical experiments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Blažej Bucha
    • 1
    Email author
  • Christian Hirt
    • 2
    • 3
  • Meng Yang
    • 2
  • Michael Kuhn
    • 4
  • Moritz Rexer
    • 2
    • 3
  1. 1.Department of Theoretical GeodesySlovak University of Technology in BratislavaBratislavaSlovak Republic
  2. 2.Institute for Astronomical and Physical GeodesyTechnische Universität MünchenMunichGermany
  3. 3.Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  4. 4.School of Earth and Planetary Sciences, Western Australian Geodesy GroupCurtin UniversityPerthAustralia

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