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An alternative geoid model for Africa using the shallow-layer method

Abstract

The aim of the current investigation is to determine an alternative geoid model for Africa using the shallow-layer method. The shallow-layer method, following the basic definition of the geoid, differs essentially from the traditional geoid determination techniques (Stokes and Molodensky) that it doesn’t need real gravity data. It comes from the definition of the geoid. Here, the shallow-layer method is used to determine a 5′ × 5′ geoid model for Africa covering the latitudes between −36°N and 39°N and longitudes from −20°E to 53°E The Earth Gravitational Model (EGM2008), the global topographic model (DTM2006.0), the global crustal model (CRUST1.0) and the Danish National Space Center data set (DNSC08) global models have been used to construct and define the shallow layer and its interior structure. A combination of prism and tesseroid modelling methods have been utilized to determine the gravitational potential produced by the shallow-layer masses. The validation and tests of the computed shallow-layer geoid have been done at two different levels. First, a comparison between the computed shallow-layer geoid and the recently developed AFRgeo2019 gravimetric geoid for Africa (based on real gravity data) has been carried out. Second, a comparison of the computed shallow-layer geoid with several geoid models computed using different global geopotential models has been performed. The results show that the computed shallow-layer geoid behaves similarly to those determined by the global geopotential models. Differences between the shallow-layer and the AFRgeo2019 gravimetric geoids are generally small (below 0.5 m) at most of the African continent

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References

  • Abd-Elmotaal H.A., 2015. Validation of GOCE models in Africa. Newton’s Bull., 5, 149–162

    Google Scholar 

  • Abd-Elmotaal H.A., Seitz K., Kühtreiber N. and Heck B., 2018 AFRGDB V2.0: the gravity database for the geoid determination in Africa. In: Freymueller J.T. and Sánchez L. (Eds), International Symposium on Advancing Geodesy in a Changing World. International Association of Geodesy Symposia, 149, 61–70, Springer, Cham, Switzerland, DOI: https://doi.org/10.1007/1345_2018_29

    Chapter  Google Scholar 

  • Abd-Elmotaal H.A., Seitz K., Kühtreiber N. and Heck B., 2019. AFRgeo v1.0: a geoid model for Africa. KIT Scientific Working Papers, 125, DOI: https://doi.org/10.5445/IR/1000097013

  • Abd-Elmotaal H.A., Kühtreiber N., Seitz K. and Heck B., 2020a. The new AFRGDB V2.2 gravity database for Africa. Pure Appl. Geophys., 177, 4365–4375, DOI: https://doi.org/10.1007/s00024-020-02481-5

    Article  Google Scholar 

  • Abd-Elmotaal H.A., Kühtreiber N., Seitz K., Heck B., 2020b. A precise geoid model for Africa: AFRgeo2019. In: International Association of Geodesy Symposia, Springer, Berlin, Heidelberg, DOI: https://doi.org/10.1007/1345_2020_122

    Google Scholar 

  • Andersen O.B. and Knudsen P., 2009. DNSC08 mean sea surface and mean dynamic topography models. J. Geophys. Res.-Oceans, 114, C11001, DOI: https://doi.org/10.1029/2008JC005179

    Article  Google Scholar 

  • Andersen O.B., Knudsen P. and Berry P.A., 2010. The DNSC08GRA global marine gravity field from double retracked satellite altimetry. J. Geodesy, 84, 191–199

    Article  Google Scholar 

  • Anderson E.G., 1976. The Effect of Topography on Solutions of Stokes’ Problem. School of Surveying, University of New South Wales, Kensington, NSW, Canada (https://trove.nla.gov.au/work/9479860)

    Google Scholar 

  • Brockmann J.M., Zehentner N., Höck E., Pail R., Loth I., Mayer-Gürr T. and Schuh W.D., 2014. EGM TIM RL05: an independent geoid with centimeter accuracy purely based on the GOCE mission. Geophys. Res. Lett., 41, 8089–8099

    Article  Google Scholar 

  • Bruinsma S., Förste C., Abrikosov O., Lemoine J.M., Marty J.C., Mulet S., Rio M.H. and Bonvalot S., 2014. ESA’s satellite-only gravity field model via the direct approach based on all GOCE data. Geophys. Res. Lett., 41, 7508–7514, DOI: 110.1002/2014GL062045

    Article  Google Scholar 

  • Burša M., Kenyon S., Kouba J., Šíma Z., Vatrt V., Vítek V. and Vojtíšková M., 2007. The geopotential value W0 for specifying the relativistic atomic time scale and a global vertical reference system. J. Geodesy, 81, 103–110

    Article  Google Scholar 

  • Driscoll J.R. and Healy D.M., 1994. Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math., 15, 202–250

    Article  Google Scholar 

  • Förste C., Bruinsma S., Abrikosov O., Lemoine J.M., Schaller T., Götze H.J., Ebbing J., Marty J.C., Flechtner F., Balmino G. and Biancale R., 2014. EIGEN-6C4: The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. GFZ Data Services, DOI: https://doi.org/10.5880/icgem.2015.1

  • Gilardoni M., Reguzzoni M. and Sampietro D., 2016. GECO: a global gravity model by locally combining GOCE data and EGM2008. Stud. Geophys. Geod., 60, 228–247

    Article  Google Scholar 

  • Grafarend E.W., 1994. What is a geoid? In: Vaníček P. and Christou N.T. (Eds), Geoid and Its Geophysical Interpretations. CRC Press, Boca Raton, FL, 3–32

    Google Scholar 

  • Grombein T., Seitz K. and Heck B., 2010. Modelling topographic effects in GOCE gravity gradients. Geotechnologien Science Report, 17, 84–93

    Google Scholar 

  • Heck B. and Seitz K., 2007. A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J. Geodesy, 81, 121–136

    Article  Google Scholar 

  • Heiskanen W.A. and Moritz H., 1967. Physical Geodesy. Freeman, San Francisco, CA

    Google Scholar 

  • Hofmann-Wellenhof B. and Moritz H., 2006. Physical Geodesy. Springer-Verlag, Vienna, Austria

    Google Scholar 

  • Holmes S.A. and Featherstone W.E., 2002. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J. Geodesy, 76, 279–299

    Article  Google Scholar 

  • Kenyon S, Factor J, Pavlis N, Holmes S (2007) Toward the next Earth gravitational model. SEG Technical Program Expanded Abstracts 2007. Society of Exploration Geophysicists, Tulsa, OK, 733–735

    Chapter  Google Scholar 

  • Kiamehr R. and Sjöberg L., 2005. Effect of the SRTM global DEM on the determination of a high-resolution geoid model: a case study in Iran. J. Geodesy, 79, 540–551

    Article  Google Scholar 

  • Laske G., Masters G., Ma Z. and Pasyanos M., 2012. CRUST1.0: An updated global model of Earths crust. Geophys. Res. Abs., 14, EGU2012–3743–1

    Google Scholar 

  • Laske G., Masters G., Ma Z. and Pasyanos M., 2013. Update on CRUST1.0 a 1-degree global model of Earths crust. Geophys. Res. Abs., 15, EGU2013–2658

    Google Scholar 

  • Liang W., Xu X., Li J. and Zhu G., 2018. The determination of an ultra-high gravity field model SGG-UGM-1 by combining EGM2008 gravity anomaly and GOCE observation data. Acta Geodaetica et Cartographica Sinica, 47, 425–434, DOI: https://doi.org/10.11947/j.AGCS.2018.20170269 (in Chinese)

    Google Scholar 

  • Listing J., 1872. Regarding our present knowledge of the figure and size of the Earth. Rep. Roy. Soc. Sci. Gottingen, 1–66

  • Meissl P., 1971. Preparation for the Numerical Evaluation of Second Order Molodensky-Type Formulas. Report 163. Department of Geodetic Science and Surveying, Ohio State University, Columbus, OH

    Google Scholar 

  • Merry C.L., 2003. The African geoid project and its relevance to the unification of African vertical reference frames. 2-nd FIG Regional Conference, Marrakech, Morocco (https://www.fig.net/resources/proceedings/fig_proceedings/morocco/proceedings/TS9/TS9_3_merry.pdf

  • Merry C.L., Blitzkow D., Abd-Elmotaal H.A., Fashir H., John S., Podmore F. and Fairhead J., 2005. A preliminary geoid model for Africa. In: Sansò F. (Ed.), A Window on the Future of Geodesy. International Association of Geodesy Symposia, 128. Springer-Verlag, Heidelberg, Germany, 374–379, DOI: https://doi.org/10.1007/3-540-27432-4_64

    Chapter  Google Scholar 

  • Mohr P.J., Taylor B.N. and Newell D.B., 2008. CODATA recommended values of the fundamental physical constants: 2006. Rev. Mod. Phys., 80, 633–730

    Article  Google Scholar 

  • Molodensky M.S., Eremeev V.F. and Yurkina M.I., 1962. Methods for Study of the External Gravity Field and Figure of the Earth. Israel Program of Scientific Translations, Jerusalem, Israel

    Google Scholar 

  • Nagy D., Papp G. and Benedek J., 2000. The gravitational potential and its derivatives for the prism. J. Geodesy, 74, 552–560

    Article  Google Scholar 

  • Nagy D., Papp G. and Benedek J., 2002. Corrections to the gravitational potential and its derivatives for the prism. J. Geodesy, 76, 475–475

    Article  Google Scholar 

  • Pail R., Goiginger H., Mayrhofer R., Schuh W.D., Brockmann J.M., Krasbutter I., Höck E. and Fecher T., 2010. GOCE gravity field model derived from orbit and gradiometry data applying the time-wise method. In: Lacoste-Francis H. (Ed.), ESA Living Planet Symposium. ESA Spec. Publ. SP-686, ESA Publications Division, ESTEC, Noordwijk, The Netherlands (https://mediatum.ub.tum.de/doc/1368842/1368842.pdf)

    Google Scholar 

  • Pail R., Gruber T. and Fecher T., 2016. The combined gravity model GOCO05c — GFZ data services. Geophys. Res. Abs., 19, EGU2016–7696 (https://meetingorganizer.copernicus.org/EGU2016/EGU2016-7696.pdf)

    Google Scholar 

  • Papp G. and Kalmár J., 1996. Toward the physical interpretation of the geoid in the pannonian basin using 3-d model of the lithosphere. IGeS Bulletin, 5, 63–87

    Google Scholar 

  • Pavlis N.K., Holmes S.A., Kenyon S.C. and Factor J.K., 2012. The development and evaluation of the Earth gravitational model 2008 (EGM2008). J. Geophys. Res.-Solid Earth, 117, B04406, DOI: https://doi.org/10.1029/2011JB008916

    Article  Google Scholar 

  • Schall J., Eicker A. and Kusche J., 2014. The ITG-GOCE02 gravity field model from GOCE orbit and gradiometer data based on the short arc approach. J. Geodesy, 88, 403–409

    Article  Google Scholar 

  • Shen W.B. and Han J., 2013a. Improved geoid determination based on the shallow-layer method: A case study using EGM08 and CRUST2.0 in the Xinjiang and Tibetan regions. Terr. Atmos. Ocean. Sci., 24, 591–604, DOI: https://doi.org/10.3319/TAO.2012.11.12.01(TibXS)

    Article  Google Scholar 

  • Shen W.B. and Han J., 2013b. Global Geoid modeling and evaluation. In: Jin S. (Ed.), Geodetic Sciences — Observations, Modeling and Applications. IntechOpen, London, U.K., DOI: https://doi.org/10.5772/54649

    Google Scholar 

  • Shen W.B., Li J., Li J., Ning J. and Chao D., 2008. Applications of the fictitious compress recovery approach in physical geodesy. Geo-Spatial Inf. Sci., 11, 162–167

    Article  Google Scholar 

  • Stokes G.G., 1849. On the variations of gravity on the surface of the Earth. Trans. Cambridge Phil. Soc., 8, 672–695

    Google Scholar 

  • Tsoulis D., Novák P. and Kadlec M., 2009. Evaluation of precise terrain effects using high-resolution digital elevation models. J. Geophys. Res.-Solid Earth, 114, B02404, DOI: https://doi.org/10.1029/2008JB005639

    Article  Google Scholar 

  • Wang Z., 1998. Geoid and Crustal Structure in Fennoscandia. Finnish Geodetic Institute, Helsinki, Finland

    Google Scholar 

  • Xu X., Zhao Y., Reubelt T. and Tenzer R., 2017. A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models. Geodesy Geodyn., 8, 260–272

    Article  Google Scholar 

  • Zingerle P., Pail R. and Gruber T., 2019. High-resolution combined global gravity field modelling — towards a combined d/o 10800 model. Geophys. Res. Abs., 21, EGU2019–5425

    Google Scholar 

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Acknowledgements

The authors would like to express their sincere gratitude to professor Petr Holota, the Editor of the current paper, and the anonymous reviewers for their insightful comments and suggestions, which greatly helped to improve the quality of the manuscript. We would like to express our thanks to Prof. Gábor Papp for his valuable comments and critical suggestions towards enhancing the article. This study was supported by the National Natural Science Foundation of China (grants Nos 41631072, 42030105, 41721003, 41574007, and 41804012), the Discipline Innovative Engineering Plan of Modern Geodesy and Geodynamics (grant No. B17033), and the DAAD Thematic Network Project (grant No. 57173947).

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Correspondence to Wen-Bin Shen.

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Ashry, M., Shen, WB. & Abd-Elmotaal, H.A. An alternative geoid model for Africa using the shallow-layer method. Stud Geophys Geod 65, 148–167 (2021). https://doi.org/10.1007/s11200-020-0301-0

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  • DOI: https://doi.org/10.1007/s11200-020-0301-0

Keywords

  • shallow-layer method
  • African geoid
  • CRUST1.0
  • EGM2008
  • DNSC2008
  • AFRgeo2019