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.
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Notes
SHTOOLs is an open-source software package for spherical harmonic transforms which has originally been developed by Mark Wieczorek (https://shtools.oca.eu/shtools/). The ultra-high degree extension is available here: https://www.researchgate.net/publication/291102839_ultra_high_degree_extension_v1_SHTOOLS.
References
Balmino G, Lambeck K, Kaula W (1973) A spherical harmonic analysis of the Earth’s topography. J Geophys Res 78(2):478–521
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 66:219–227
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. https://doi.org/10.1093/gji/ggw311
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. https://doi.org/10.1002/2013JB010457
Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. OSU Report 355, Ohio State University
Forsberg R, Tscherning C (1981) The use of height data in gravity field approximation by collocation. J Geophys Res 86(B9):7843–7854
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. https://doi.org/10.1007/s00190-011-0519-2
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. https://doi.org/10.1007/s10712-016-9400-4
Hirt C (2010) Prediction of vertical deflections from high-degree spherical harmonic synthesis and residual terrain model data. J Geod 84(3):179–190
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):1–20
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. https://doi.org/10.1002/2013JB010900
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):557–567
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 85:723–740
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. https://doi.org/10.1029/2011JB008878
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. https://doi.org/10.1002/grl.50838
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 73:71–80
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. https://doi.org/10.1002/2016JB013249
Jarvis A, Reuter H, Nelson A, Guevara E (2008) Hole-filled SRTM for the globe v4.1. http://srtm.csi.cgiar.org
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. https://doi.org/10.1007/s00190-016-0917-6
Kuhn M, Featherstone W, Kirby J (2009) Complete spherical Bouguer gravity anomalies over Australia. Aust J Earth Sci 56:213–223
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 2004
Mayer-Gürr T, Kurtenbach E, Eicker A (2010) ITG-Grace2010 gravity field model. www.igg.uni-bonn.de/apmg/index.php?id=itg-grace2010
Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560. https://doi.org/10.1007/s00190-006-0094-0
Novak P (2010) Direct modelling of the gravitational field using harmonic series. Acta Geodyn Geomater 157(1):35–47
NRC (2011) Canadian gravity data base. Natural Resources Canada, Canada
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. https://doi.org/10.1007/s00190-011-0467-x (special issue: “GOCE: The Gravity and Steady-state Ocean Circulation Explorer”)
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 318–323
Pavlis N, Holmes S, Kenyon S, Factor J (2012) The developement and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res. https://doi.org/10.1029/2011JB008916
Pavlis NK (2011) Spherical harmonic analysis applied to potential fields. In: Gupta HK (ed) Encyclopedia of Solid Earth Geophysics, 1st edn. Springer, Dordrecht, pp 1382–1391
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):2633–2633
Rexer M (2017) Spectral solutions to the topographic potential in the context of high-resolution global gravity field modelling. PhD thesis, Technische Universität München
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. https://doi.org/10.1007/s10712-015-9345-z
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. https://doi.org/10.1007/s00190-015-0822-4
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. https://doi.org/10.1007/s10712-016-9382-2
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 University
Scott D (1979) On optimal and data-based histograms. Biometrika 66:605–610
Sneeuw N (1994) Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective. Geophys J Int 118:707–716
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):134–143
Torge W (2003) Geodäsie. Walter de Gruyter, Berlin
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. https://doi.org/10.1007/1345_2015_60
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. https://doi.org/10.1007/978-3-319-10837-7_11
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. https://doi.org/10.1007/s40328-017-0204-x
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. https://doi.org/10.1007/1345_2017_3
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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Rexer, M., Hirt, C., Bucha, B. et al. Solution to the spectral filter problem of residual terrain modelling (RTM). J Geod 92, 675–690 (2018). https://doi.org/10.1007/s00190-017-1086-y
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DOI: https://doi.org/10.1007/s00190-017-1086-y