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Journal of Geodesy

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

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

  • Moritz Rexer
  • Christian Hirt
  • Blažej Bucha
  • Simon Holmes
Original Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

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).

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Copyright information

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

Authors and Affiliations

  • Moritz Rexer
    • 1
  • 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

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