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Error bounds for the spectral approximation of the potential of a homogeneous almost spherical body

Abstract

Several kinds of approximation of the gravitational potential of a homogeneous body by truncated spherical harmonics series are in use in physical geodesy. However, only one of them is capable of a representation converging to the true potential in the whole layer between the Brillouin sphere and the Bjerhammar sphere of the body. We aim at providing various majorizations, namely upper bounds, of the error with the double purpose of proving explicitly the convergence in the sense of different norms and of giving computable bounds, that might be used in numerical studies. The first aim is reached for all the norms. For the second, however, it turns out that among the bounds, when applied to the example of the terrain correction of the Earth, only those referring to the mean absolute error and the mean squared error at the level of Brillouin sphere of minimum radius give significant and useful results. In order to make the computation an easy exercise, a simple approximate formula has been developed requiring only the use of the distribution function of the heights of the surface of the body with respect to the Bjerhammar sphere.

References

  • Ågren J., 2004. The analytical continuation bias in geoid determination using potential coefficients and terrestrial gravity data. J. Geodesy, 78, 314–332, DOI: https://doi.org/10.1007/s00190-004-0395-0

    Article  Google Scholar 

  • Balmino G., 1994. Gravitational potential harmonics from the shape of an homogeneous body. Celest. Mech. Dyn. Astron., 60, 331–364

    Article  Google Scholar 

  • Balmino G., Vales N., Bonvalot S. and Briais A., 2012. Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J. Geodesy, 86, 499–520, DOI: https://doi.org/10.1007/s00190-011-0533-4

    Article  Google Scholar 

  • Bucha B. and Sansò F., 2021. Gravitational field modelling near irregularly shaped bodies using spherical harmonics: a case study for the asteroid (101955) Bennu. J. Geodesy, 95, Art.No. 56, DOI: https://doi.org/10.1007/s00190-021-01493-w

  • Bucha B., Hirt C. and Kuhn M., 2019. Divergence-free spherical harmonic gravity field modelling based on the Runge-Krarup theorem: a case study for the Moon. J. Geodesy, 93, 489–513, DOI: https://doi.org/10.1007/s00190-018-1177-4

    Article  Google Scholar 

  • Freeden W. and Schreiner M., 2009. Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup. Springer-Verlag, Berlin, Germany

    Book  Google Scholar 

  • Górski K.M., Bills B.G. and Konopliv A.S., 2018. A high resolution mars surface gravity grid. Planet. Space Sci., 160, 84–106, DOI: https://doi.org/10.1016/j.pss.2018.03.015

    Article  Google Scholar 

  • Hirt C. and 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, DOI: https://doi.org/10.1002/2013JB010900

    Article  Google Scholar 

  • Hirt C. and 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, DOI: https://doi.org/10.1002/2017JE005298

    Article  Google Scholar 

  • Hirt C., Reußner E., Rexer M. and 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, DOI: https://doi.org/10.1002/2016JB013249

    Article  Google Scholar 

  • Hirt C., Yang M., Kuhn M., Bucha B., Kurzmann A. and Pail R., 2019. SRTM2gravity: an ultrahigh resolution global model of gravimetric terrain corrections. Geophys. Res. Lett., 46, 4618–4627, DOI: https://doi.org/10.1029/2019GL082521

    Article  Google Scholar 

  • Hotine M., 1969. Mathematical Geodesy. ESSA Monograph 2. U.S. Environmental Science Services Administration, Washington, DC

    Google Scholar 

  • Jekeli C., 1983. A numerical study of the divergence of spherical harmonic series of the gravity and height anomalies at the Earth’s surface. Bull. Geod., 57, 10–28, DOI: https://doi.org/10.1007/BF02520909

    Article  Google Scholar 

  • Jekeli C., 2017. Spectral Methods in Geodesy and Geophysics. CRC Press, Boca Raton, FL

    Book  Google Scholar 

  • Martinec Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Lecture Notes in Earth Sciences, 73. Springer-Verlag, Berlin, Germany

    Google Scholar 

  • Moritz H., 1980. Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, Germany

    Google Scholar 

  • Reimond S. and Baur O., 2016. Spheroidal and ellipsoidal harmonic expansions of the gravitational potential of small solar system bodies. Case study: Comet 67P/Churyumov-Gerasimenko. J. Geophys. Res.-Planets, 121, 497–515, DOI: https://doi.org/10.1002/2015JE004965

    Article  Google Scholar 

  • Sansò F. and Sideris M.G. (Eds), 2012. Geoid Determination: Theory and Methods. Lecture Notes in Earth System Sciences, Springer-Verlag, Berlin, Germany

    Google Scholar 

  • Sjöberg L.E., 1977. On the Errors of Spherical Harmonic Developments of Gravity at the Surface of the Earth. Report 257. Department of Geodetic Sciences, The Ohio State University, Columbus, OH

    Book  Google Scholar 

  • Sjöberg L.E., 1999. On the downward continuation error at the Earth’s surface and the geoid of satellite derived geopotential models. Boll. Geod. Sci. Affini, 58, 215–229

    Google Scholar 

  • Takahashi Y. and Scheeres D., 2014. Small body surface gravity fields via spherical harmonic expansions. Celest. Mech. Dyn. Astron., 119, 169–206, DOI: https://doi.org/10.1007/s10569-014-9552-9

    Article  Google Scholar 

  • Wieczorek M.A., 2015. Gravity and topography of the terrestrial planets. In: Schubert G. and Spohn T. (Eds), Physics of Terrestrial Planets and Moons. Treatise on Geophysics (Second Edition), 10, 153–193, Elsevier, Amsterdam, The Netherlands

    Google Scholar 

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Correspondence to Lorenzo Rossi.

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Bucha, B., Rossi, L. & Sansò, F. Error bounds for the spectral approximation of the potential of a homogeneous almost spherical body. Stud Geophys Geod 65, 235–260 (2021). https://doi.org/10.1007/s11200-021-0730-4

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  • DOI: https://doi.org/10.1007/s11200-021-0730-4

Keywords

  • homogeneous bodies
  • gravity field
  • spectral approximation
  • convergence theory
  • numerical bounds