Advertisement

Cap integration in spectral gravity forward modelling: near- and far-zone gravity effects via Molodensky’s truncation coefficients

  • Blažej Bucha
  • Christian Hirt
  • Michael Kuhn
Original Article
  • 130 Downloads

Abstract

Spectral gravity forward modelling is a technique that converts a band-limited topography into its implied gravitational field. This conversion implicitly relies on global integration of topographic masses. In this paper, a modification of the spectral technique is presented that provides gravity effects induced only by the masses located inside or outside a spherical cap centred at the evaluation point. This is achieved by altitude-dependent Molodensky’s truncation coefficients, for which we provide infinite series expansions and recurrence relations with a fixed number of terms. Both representations are generalized for an arbitrary integer power of the topography and arbitrary radial derivative. Because of the altitude-dependency of the truncation coefficients, a straightforward synthesis of the near- and far-zone gravity effects at dense grids on irregular surfaces (e.g. the Earth’s topography) is computationally extremely demanding. However, we show that this task can be efficiently performed using an analytical continuation based on the gradient approach, provided that formulae for radial derivatives of the truncation coefficients are available. To demonstrate the new cap-modified spectral technique, we forward model the Earth’s degree-360 topography, obtaining near- and far-zone effects on gravity disturbances expanded up to degree 3600. The computation is carried out on the Earth’s surface and the results are validated against an independent spatial-domain Newtonian integration (\(1\,\upmu \mathrm {Gal}\) RMS agreement). The new technique is expected to assist in mitigating the spectral filter problem of residual terrain modelling and in the efficient construction of full-scale global gravity maps of highest spatial resolution.

Keywords

Spectral gravity forward modelling Spherical harmonics Molodensky’s truncation coefficients Topographic potential Gradient approach Residual terrain modelling 

Notes

Acknowledgements

Blažej Bucha was supported by the ProjectVEGA 1/0954/15 and acknowledges the computational resources made available by the HPC centres at the Slovak University of Technology in Bratislava and at the Slovak Academy of Sciences, which are parts of the Slovak Infrastructure of High Performance Computing (SIVVP project, ITMS code 26230120002, funded by the European region development funds, ERDF). Christian Hirt would like to thank the German National Research Foundation (DFG) for providing funding under Grant Agreement Hi 1760/01. The spatial-domain Newtonian integration was performed using the supercomputing resources kindly provided by Western Australia’s Pawsey Supercomputing Center. The maps were produced using the Generic Mapping Tools (Wessel and Smith 1998).

References

  1. Balmino G (1994) Gravitational potential harmonics from the shape of an homogeneous body. Celest Mech Dyn Astron 60:331–364CrossRefGoogle Scholar
  2. Balmino G, Vales N, Bonvalot S, Briais A (2012) Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J Geod 86:499–520.  https://doi.org/10.1007/s00190-011-0533-4 CrossRefGoogle Scholar
  3. 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.  https://doi.org/10.1016/j.cageo.2014.02.005 CrossRefGoogle Scholar
  4. Bucha B, Janák J, Papčo J, Bezděk A (2016) High-resolution regional gravity field modelling in a mountainous area from terrestrial gravity data. Geophys J Int 207:949–966.  https://doi.org/10.1093/gji/ggw311 CrossRefGoogle Scholar
  5. Bucha B, Hirt C, Kuhn M (2018) Runge–Krarup-type gravity field solutions to avoid divergence in traditional external spherical harmonic modelling. J Geod (submitted)Google Scholar
  6. Comtet L (1974) Advanced combinatorics: the art of finite and infinite expansions, revised and enlarged edn. D. Reidel Publishing Company, Dordrecht, p 343Google Scholar
  7. Eshagh M (2009) On satellite gravity gradiometry. Ph.D. thesis, Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden, p 222Google Scholar
  8. Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geod 83:595–619.  https://doi.org/10.1007/s00190-008-0275-0 CrossRefGoogle Scholar
  9. Featherstone WE (2013) Deterministic, stochastic, hybrid and band-limited modifications of Hotine’s integral. J Geod 87:487–500.  https://doi.org/10.1007/s00190-013-0612-9 CrossRefGoogle Scholar
  10. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Report No. 355, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, p 129Google Scholar
  11. Freeden W, Schneider F (1998) Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory. Math Methods Appl Sci 21:129–163CrossRefGoogle Scholar
  12. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87:645–660.  https://doi.org/10.1007/s00190-013-0636-1 CrossRefGoogle Scholar
  13. Grombein T, Seitz K, Heck B (2017) On high-frequency topography-implied gravity signals for a height system unification using GOCE-based global geopotential models. Surv Geophys 38:443–477.  https://doi.org/10.1007/s10712-016-9400-4 CrossRefGoogle Scholar
  14. Heiskanen WA, Moritz H (1967) Physical geodesy. W. H. Freeman and Company, San Francisco, p 364Google Scholar
  15. Hirt C (2012) Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach. J Geod 86:729–744.  https://doi.org/10.1007/s00190-012-0550-y CrossRefGoogle Scholar
  16. Hirt C, Kuhn M (2014) Band-limited topographic mass distribution generates full-spectrum gravity field: gravity forward modeling in the spectral and spatial domains revisited. J Geophys Res Solid Earth 119:3646–3661.  https://doi.org/10.1002/2013JB010900 CrossRefGoogle Scholar
  17. Hirt C, Kuhn M (2017) Convergence and divergence in spherical harmonic series of the gravitational field generated by high-resolution planetary topography—a case study for the Moon. J Geophys Res Planets 122:1727–1746.  https://doi.org/10.1002/2017JE005298 CrossRefGoogle Scholar
  18. Hirt C, Rexer M (2015) Earth 2014: 1 arc-min shape, topography, bedrock and ice-sheet models—available as gridded data and degree-10,800 spherical harmonics. Int J Appl Earth Obs Geoinf 39:103–112.  https://doi.org/10.1016/j.jag.2015.03.001 CrossRefGoogle Scholar
  19. Hirt C, Featherstone WE, 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 84:557–567.  https://doi.org/10.1007/s00190-010-0395-1 CrossRefGoogle Scholar
  20. Hirt C, Claessens S, Fecher T, Kuhn M, Pail R, Rexer M (2013) New ultrahigh-resolution picture of Earth’s gravity field. Geophys Res Lett 40:4279–4283.  https://doi.org/10.1002/grl.50838 CrossRefGoogle Scholar
  21. 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.  https://doi.org/10.1016/j.cageo.2014.09.001 CrossRefGoogle Scholar
  22. Hirt C, Reußner E, Rexer M, Kuhn M (2016) Topographic gravity modeling for global Bouguer maps to degree 2160: validation of spectral and spatial domain forward modeling techniques at the 10 microgal level. J Geophys Res Solid Earth 121:6846–6862.  https://doi.org/10.1002/2016JB013249 CrossRefGoogle Scholar
  23. Hoffmann-Wellenhof B, Moritz H (2005) Physical Geodesy. Springer, New York, p 403Google Scholar
  24. Holmes SA (2003) High degree spherical harmonic synthesis for simulated earth gravity modelling. Ph.D. thesis, Department of Spatial Sciences, Curtin University of Technology, Perth, Australia, p 171Google Scholar
  25. 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 90:883–902.  https://doi.org/10.1007/s00190-016-0917-6 CrossRefGoogle Scholar
  26. Makhloof AA, Ilk KH (2008) Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography. J Geod 82:613–635.  https://doi.org/10.1007/s00190-008-0214-0 CrossRefGoogle Scholar
  27. Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Springer, Berlin, p 223Google Scholar
  28. Mikuška J, Pašteka R, Marušiak I (2006) Estimation of distant relief effect in gravimetry. Geophysics 71:J59–J69.  https://doi.org/10.1190/1.2338333 CrossRefGoogle Scholar
  29. Moazezi S, Zomorrodian H, Siahkoohi HR, Azmoudeh-Ardalan A, Gholami A (2016) Fast ultrahigh-degree global spherical harmonic synthesis on nonequispaced grid points at irregular surfaces. J Geod 90:853–870.  https://doi.org/10.1007/s00190-016-0915-8 CrossRefGoogle Scholar
  30. Molodensky MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitational field and figure of the Earth. Israel Program for Scientific Translations, Jerusalem, p 248, translated from Russian (1960)Google Scholar
  31. Moreaux G, Tscherning CC, Sanso F (1999) Approximation of harmonic covariance functions on the sphere by non-harmonic locally supported functions. J Geod 73:555–567CrossRefGoogle Scholar
  32. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560CrossRefGoogle Scholar
  33. Novák P, Vaníček P, Martinec Z, Véronneau M (2001) Effects of the spherical terrain on gravity and the geoid. J Geod 75:491–504CrossRefGoogle Scholar
  34. Paul MK (1973) A method of evaluating the truncation error coefficients for geoidal height. Bull Géod 110:413–425CrossRefGoogle Scholar
  35. Paul MK (1983) Recurrence relations for the truncation error coefficients for the extended Stokes function. Bull Géod 57:152–166CrossRefGoogle Scholar
  36. Pavlis NK (1991) Estimation of geopotential differences over intercontinental locations using satellite and terrestrial measurements. Report No. 409, Department of Geodetic Science, The Ohio State University, Ohio, USA, p 155Google Scholar
  37. Pohánka V (1988) Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys Prospect 35:733–751CrossRefGoogle Scholar
  38. Rexer M, Hirt C (2015) 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 36:803–830.  https://doi.org/10.1007/s10712-015-9345-z CrossRefGoogle Scholar
  39. Rexer M, Hirt C, Bucha B, Holmes S (2018) Solution to the spectral filter problem of residual terrain modelling RTM. J Geod.  https://doi.org/10.1007/s00190-017-1086-y Google Scholar
  40. Shepperd SW (1982) A recursive algorithm for evaluating Molodenskii-type truncation error coefficients at altitude. Bull Géod 56:95–105CrossRefGoogle Scholar
  41. Sjöberg LE (2005) A discussion on the approximations made in the practical implementation of the remove-compute-restore technique in regional geoid modelling. J Geod 78:645–653.  https://doi.org/10.1007/s00190-004-0430-1 CrossRefGoogle Scholar
  42. Sneeuw N (1994) Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective. Geophys J Int 118:707–716CrossRefGoogle Scholar
  43. Šprlák M, Hamáčková E, Novák P (2015) Alternative validation method of satellite gradiometric data by integral transform of satellite altimetry data. J Geod 89:757–773.  https://doi.org/10.1007/s00190-015-0813-5 CrossRefGoogle Scholar
  44. Tenzer R, Novák P, Vajda P, Ellmann A, Abdalla A (2011) Far-zone gravity field contributions corrected for the effect of topography by means of Molodensky’s truncation coefficients. Stud Geophys Geod 55:55–71CrossRefGoogle Scholar
  45. Thalhammer M (1995) Regionale Gravitationsfeldbestimmung mit zukünftigen Satellitenmissionen (SST und Gradiometrie). Reihe C, Heft Nr. 437, Deutsche Geodätische Kommission, München, Germany, p 96 (in German)Google Scholar
  46. Wessel P, Smith WHF (1998) New, improved version of generic mapping tools released. EOS Trans Am Geophys Union 79:579.  https://doi.org/10.1029/98EO00426 CrossRefGoogle Scholar
  47. Wieczorek MA, Phillips RJ (1998) Potential anomalies on a sphere: applications to the thickness of the lunar crust. J Geophys Res 103:1715–1724CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Theoretical GeodesySlovak University of Technology in BratislavaBratislavaSlovak Republic
  2. 2.Institute for Astronomical and Physical Geodesy, Institute for Advanced StudyTechnische Universität MünchenMunichGermany
  3. 3.Western Australian Geodesy Group, School of Earth and Planetary SciencesCurtin UniversityPerthAustralia

Personalised recommendations