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
Balmino G., 1994. Gravitational potential harmonics from the shape of an homogeneous body. Celest. Mech. Dyn. Astron., 60, 331–364
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
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
Freeden W. and Schreiner M., 2009. Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup. Springer-Verlag, Berlin, Germany
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
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
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
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
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
Hotine M., 1969. Mathematical Geodesy. ESSA Monograph 2. U.S. Environmental Science Services Administration, Washington, DC
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
Jekeli C., 2017. Spectral Methods in Geodesy and Geophysics. CRC Press, Boca Raton, FL
Martinec Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Lecture Notes in Earth Sciences, 73. Springer-Verlag, Berlin, Germany
Moritz H., 1980. Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, Germany
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
Sansò F. and Sideris M.G. (Eds), 2012. Geoid Determination: Theory and Methods. Lecture Notes in Earth System Sciences, Springer-Verlag, Berlin, Germany
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
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
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
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provided a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11200-021-0730-4
Keywords
- homogeneous bodies
- gravity field
- spectral approximation
- convergence theory
- numerical bounds