Journal of Geodesy

, Volume 92, Issue 6, pp 675–690 | Cite as

Solution to the spectral filter problem of residual terrain modelling (RTM)

  • Moritz RexerEmail author
  • Christian Hirt
  • Blažej Bucha
  • Simon Holmes
Original Article


In physical geodesy, the residual terrain modelling (RTM) technique is frequently used for high-frequency gravity forward modelling. In the RTM technique, a detailed elevation model is high-pass-filtered in the topography domain, which is not equivalent to filtering in the gravity domain. This in-equivalence, denoted as spectral filter problem of the RTM technique, gives rise to two imperfections (errors). The first imperfection is unwanted low-frequency (LF) gravity signals, and the second imperfection is missing high-frequency (HF) signals in the forward-modelled RTM gravity signal. This paper presents new solutions to the RTM spectral filter problem. Our solutions are based on explicit modelling of the two imperfections via corrections. The HF correction is computed using spectral domain gravity forward modelling that delivers the HF gravity signal generated by the long-wavelength RTM reference topography. The LF correction is obtained from pre-computed global RTM gravity grids that are low-pass-filtered using surface or solid spherical harmonics. A numerical case study reveals maximum absolute signal strengths of \(\sim 44\) mGal (0.5 mGal RMS) for the HF correction and \(\sim 33\) mGal (0.6 mGal RMS) for the LF correction w.r.t. a degree-2160 reference topography within the data coverage of the SRTM topography model (\(56^{\circ }\hbox {S} \le \phi \le 60^{\circ }\hbox {N}\)). Application of the LF and HF corrections to pre-computed global gravity models (here the GGMplus gravity maps) demonstrates the efficiency of the new corrections over topographically rugged terrain. Over Switzerland, consideration of the HF and LF corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 4.41 to 3.27 mGal, which translates into \(\sim 26\)% improvement. Over a second test area (Canada), our corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 5.65 to 5.30 mGal (\(\sim 6\)% improvement). Particularly over Switzerland, geophysical signals (associated, e.g. with valley fillings) were found to stand out more clearly in the RTM-reduced gravity measurements when the HF and LF correction are taken into account. In summary, the new RTM filter corrections can be easily computed and applied to improve the spectral filter characteristics of the popular RTM approach. Benefits are expected, e.g. in the context of the development of future ultra-high-resolution global gravity models, smoothing of observed gravity data in mountainous terrain and geophysical interpretations of RTM-reduced gravity measurements.


Residual terrain modelling Gravity forward modelling Spherical harmonics Newton’s integral 



This study has been designed and prepared in close collaboration with our dear colleague Simon Holmes, who passed away unexpectedly before the study could be finished. Our thoughts are with his family, friends and collaborators. Blažej Bucha was supported by the Project VEGA 1/0954/15. Christian Hirt acknowledges support by the German National Research Foundation (Grant No. Hi 1760/01).


  1. Balmino G, Lambeck K, Kaula W (1973) A spherical harmonic analysis of the Earth’s topography. J Geophys Res 78(2):478–521CrossRefGoogle Scholar
  2. Bucha B, Janák J (2014) A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders: efficient computation at irregular surfaces. Comput Geosci 66:219–227CrossRefGoogle Scholar
  3. Bucha B, Janák J, Papčo J, Beděk A (2016) High-resolution regional gravity field modelling in a mountainous area from terrestrial gravity data. Geophys J Int 207(2):949–966. CrossRefGoogle Scholar
  4. Claessens S, Hirt C (2013) Ellipsoidal topographic potential: new solutions for spectral forward gravity modelling of topography with respect to a reference ellipsoid. J Geophys Res 118(11):5991–6002. CrossRefGoogle Scholar
  5. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. OSU Report 355, Ohio State UniversityGoogle Scholar
  6. Forsberg R, Tscherning C (1981) The use of height data in gravity field approximation by collocation. J Geophys Res 86(B9):7843–7854CrossRefGoogle Scholar
  7. Fukushima T (2012) Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers. J Geod 86(4):271–285. CrossRefGoogle Scholar
  8. Grombein T, Seitz K, Heck B (2016) On high-frequency topography-implied gravity signals for a height system unification using GOCE-based global geopotential models. Surv Geophys. Google Scholar
  9. Hirt C (2010) Prediction of vertical deflections from high-degree spherical harmonic synthesis and residual terrain model data. J Geod 84(3):179–190CrossRefGoogle Scholar
  10. Hirt C (2013) RTM gravity forward-modeling using topography/bathymetry data to improve high-degree global geopotential models in the coastal zone. Mar Geodesy 36(2):1–20CrossRefGoogle Scholar
  11. Hirt C, Kuhn M (2014) A band-limited topographic mass distribution generates a full-spectrum gravity field: gravity forward modelling in the spectral and spatial domain revisited. J Geophys Res Solid Earth 119(4):3646–3661. CrossRefGoogle Scholar
  12. Hirt C, Featherstone W, Marti U (2010) Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data. J Geod 84(9):557–567CrossRefGoogle Scholar
  13. Hirt C, Gruber T, Featherstone W (2011) Evaluation of the first GOCE static gravity field models using terrestrial gravity, vertical deflections and EGM2008 quasigeoid heights. J Geod 85:723–740CrossRefGoogle Scholar
  14. Hirt C, Kuhn M, Featherstone W, Göttl F (2012) Topographic/isostatic evaluation of new-generation GOCE gravity field models. J Geophys Res Solid Earth 117(B05):407. Google Scholar
  15. Hirt C, Claessens S, Fecher T, Kuhn M, Pail R, Rexer M (2013) New ultra-high resolution picture of Earth’s gravity field. Geophys Res Lett. Google Scholar
  16. Hirt C, Kuhn M, Claessens S, Pail R, Seitz K, Gruber T (2014) Study of the Earth’s short-scale gravity field using the ERTM2160 gravity model. Comput Geosci 73:71–80CrossRefGoogle Scholar
  17. Hirt C, Reußner E, Rexer M, Kuhn M (2016) Topographic gravity modelling for global bouguer maps to degree 2160: validation of spectral and spatial domain forward modelling techniques at the 10 microgal-level. J Geophys Res Solid Earth 121(9):6846–6862. CrossRefGoogle Scholar
  18. Jarvis A, Reuter H, Nelson A, Guevara E (2008) Hole-filled SRTM for the globe v4.1.
  19. Kuhn M, Hirt C (2016) Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET). J Geod 90(9):883–902. CrossRefGoogle Scholar
  20. Kuhn M, Featherstone W, Kirby J (2009) Complete spherical Bouguer gravity anomalies over Australia. Aust J Earth Sci 56:213–223CrossRefGoogle Scholar
  21. Marti U (2004) High-precision combined geoid determination in Switzerland. In: Presented at gravity, geoid and space missions (GGSM) 2004 symposium, Porto, Portugal, 30 Aug–03 Sep 2004Google Scholar
  22. Mayer-Gürr T, Kurtenbach E, Eicker A (2010) ITG-Grace2010 gravity field model.
  23. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560. CrossRefGoogle Scholar
  24. Novak P (2010) Direct modelling of the gravitational field using harmonic series. Acta Geodyn Geomater 157(1):35–47Google Scholar
  25. NRC (2011) Canadian gravity data base. Natural Resources Canada, CanadaGoogle Scholar
  26. Pail R, Bruinsma S, Migliaccio F, Förste C, Goiginger H, Schuh WD, Höck E, Reguzzoni M, Brockmann JM, Abrikosov O, Veicherts M, Fecher T, Mayrhofer R, Krasbutter I, Sanso F, Tscherning CC (2011) First GOCE gravity field models derived by three different approaches. J Geod 85(11):819–843. (special issue: “GOCE: The Gravity and Steady-state Ocean Circulation Explorer”)CrossRefGoogle Scholar
  27. Pavlis N, Factor J, Holmes S (2007) Terrain-related gravimetric quantities computed for the next EGM. In: Kilioglu A, Forsberg R (eds) Proceedings of the 1st international symposium of the international gravity field service, vol 18, pp 318–323Google Scholar
  28. Pavlis N, Holmes S, Kenyon S, Factor J (2012) The developement and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res. Google Scholar
  29. Pavlis NK (2011) Spherical harmonic analysis applied to potential fields. In: Gupta HK (ed) Encyclopedia of Solid Earth Geophysics, 1st edn. Springer, Dordrecht, pp 1382–1391Google Scholar
  30. Pavlis N, Holmes S, Kenyon S, Factor J (2013) Correction to the development and evaluation of the Earth Gravitational Model 2008. J Geophys Res 118(5):2633–2633CrossRefGoogle Scholar
  31. Rexer M (2017) Spectral solutions to the topographic potential in the context of high-resolution global gravity field modelling. PhD thesis, Technische Universität MünchenGoogle Scholar
  32. Rexer M, Hirt C (2015a) Ultra-high-degree surface spherical harmonic analysis using the Gauss–Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of Earth, Mars and Moon. Surv Geophys 36(6):803–830.
  33. Rexer M, Hirt C (2015b) Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000. J Geod 89(9):887–909.
  34. Rexer M, Hirt C, Claessens S, Tenzer R (2016) Layer-based modelling of the Earth’s gravitational potential up to 10-km scale in spherical harmonics in spherical and ellipsoidal approximation. Surv Geophys 37(6):1035–1074. CrossRefGoogle Scholar
  35. Rummel R, Rapp R, Sünkel H, Tscherning C (1988) Comparisons of global topographic/isostatic models to the Earth’s observed gravity field. OSU report 388, Ohio State UniversityGoogle Scholar
  36. Scott D (1979) On optimal and data-based histograms. Biometrika 66:605–610CrossRefGoogle Scholar
  37. Sneeuw N (1994) Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective. Geophys J Int 118:707–716CrossRefGoogle Scholar
  38. Šprlák M, Gerlach C, Pettersen B (2012) Validation of GOCE global gravity field models using terrestrial gravity data in Norway. J Geod Sci 2(2):134–143Google Scholar
  39. Torge W (2003) Geodäsie. Walter de Gruyter, BerlinGoogle Scholar
  40. Tziavos I, Vergos G, Grigoriadis V, Tzanou E, Natsiopoulos D (2015) Validation of GOCE/GRACE satellite only and combined global geopotential models over Greece, in the frame of the GOCESeaComb Project. In: Rizos C, Willis P (eds) IAG 150 years, international association of geodesy symposia, vol 143. Springer, Basel. Google Scholar
  41. Vergos G, Grigoriadis V, Tziavos I, Kotsakis C (2014) Evaluation of GOCE/GRACE Global Geopotential Models over Greece with collocated GPS/Levelling observations and local gravity data. In: U Marti (ed) Gravity, geoid and height systems. International association of geodesy symposia, vol 141, pp 85–92.
  42. Vergos G, Erol B, Natsiopoulos D, Grigoriadis V, Isik M, Tziavos I (2017) Preliminary results of GOCE-based height system unification between Greece and Turkey over marine and land areas. Acta Geod Geophys. Google Scholar
  43. Willberg M, Gruber T, Vergos G (2017) Analysis of GOCE omission error and its contribution to vertical datum offsets in Greece and its Islands. In: 1st Joint Commission 2 and IGFS meeting (gravity, geoid and height system 2016-GGHS2016), International association of geodesy symposia, vol 146.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Moritz Rexer
    • 1
    Email author
  • Christian Hirt
    • 1
  • Blažej Bucha
    • 2
  • Simon Holmes
    • 3
  1. 1.Institute for Astronomical and Physical Geodesy, Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  2. 2.Department of Theoretical GeodesySlovak University of Technology in BratislavaBratislavaSlovakia
  3. 3.SGT Inc.GreenbeltUSA

Personalised recommendations