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Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings

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Abstract

We report on testing the UNB (University of New Brunswick) software suite for accurate regional geoid model determination by use of Stokes-Helmert’s method against an Australian Synthetic Field (ASF) as “ground truth”. This testing has taken several years and has led to discoveries of several significant errors (larger than 5mm in the resulting geoid models) both in the UNB software as well as the ASF. It was our hope that, after correcting the errors in UNB software, we would be able to come up with some definite numbers as far as the achievable accuracy for a geoid model computed by the UNB software. Unfortunately, it turned out that the ASF contained errors, some of as yet unknown origin, that will have to be removed before that ultimate goal can be reached. Regardless, the testing has taught us some valuable lessons, which we describe in this paper. As matters stand now, it seems that given errorless gravity data on 1′ by 1′ grid, a digital elevation model of a reasonable accuracy and no topographical density variations, the Stokes-Helmert approach as realised in the UNB software suite is capable of delivering an accuracy of the geoid model of no constant bias, standard deviation of about 25 mm and a maximum range of about 200 mm. We note that the UNB software suite does not use any corrective measures, such as biases and tilts or surface fitting, so the resulting errors reflect only the errors in modelling the geoid.

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References

  • Ågren J, 2004. Regional Geoid Determination Methods for the Era of Satellite Gravimetry: Numerical Investigations Using Synthetic Earth Gravity Models. PhD Thesis, Department of Infrastructure, Royal Institute of Technology (KTH), Stockholm, Sweden, 246 pp.

    Google Scholar 

  • Ardalan A.A. and Grafarend E.W., 2001. Ellipsoidal geoidal undulations (ellipsoidal Bruns formula): case studies. J. Geodesy, 75, 544–552, DOI: 10.1007/s001900100212.

    Article  Google Scholar 

  • Avalos-Naranjo D., Sosa-Gaytan M. and Munoz-Abundes R., 2011. Aspectos metodologicos de las cartas y modelos gravimetricos 2010. Instituto Nacional de Estadística y Geografía, Aguascalientes, Mexico, October 2011 (in Spanish).

    Google Scholar 

  • Baran I., Kuhn M., Claessens S.J., Featherstone W.F., Holmes S.A. and Vaníček P., 2006. A synthetic Earth gravity model designed specifically for testing regional gravimetric geoid determination algorithms. J. Geodesy, 80, 1–16, DOI: 10.1007/s00190-005-0002-z.

    Article  Google Scholar 

  • Blitzkow D., de Matos A.C.O.C., Fairhead J.D., Pacino M.C., Lobianco M.C.B. and Campos I.O., 2012. The progress of the geoid model computation for South America under GRACE and EGM2008 models. In: Kenyon S., Pacino M.C. and Marti U. (Eds.), Geodesy for Planet Earth. International Association of Geodesy Symposia 136, Springer Verlag, Heidelberg, Germany, 893–899, DOI: 10.1007/978-3-642-20338-1_112.

    Chapter  Google Scholar 

  • Claessens S.J., 2003. A Synthetic Earth Model. MSc Thesis. Delft University Press, Delft, The Netherlands, 61 pp. (http://www.cage.curtin.edu.au/~sten/pubs/clas2002.pdf).

    Google Scholar 

  • Ellmann A. and Vaníček P., 2007. UNB application of Stokes-Helmert’s approach to geoid computation. J. Geodyn., 43, 200–213, DOI: 10.1016/j.jog.2006.09.019.

    Article  Google Scholar 

  • Farr T.G., Rosen P.A., Caro E., Crippen R., Duren R., Hensley S., Kobrick M., Paller M., Rodriguez E., Roth L., Seal D., Shaffer S., Shimada J., Umland J., Werner M., Oskin M., Burbank D. and Alsdorf D., 2007. The Shuttle Radar Topography Mission. Rev. Geophys., 45, RG2004, DOI: 10.1029/2005RG000183.

    Article  Google Scholar 

  • Featherstone W.E., 2002. Tests of two forms of Stokes’s integral using a synthetic gravity field based on spherical harmonics. In: Grafarend E.W., Krumm F.W. and Schwarze V.S., (Eds.), Geodesy — The Challenge of the Third Millennium. Springer Verlag, Berlin, Heidelberg, Germany, 163–171.

    Google Scholar 

  • Featherstone W.E., 2003. Software for computing five existing types of deterministically modified integration kernel for gravimetric geoid determination. Comput. Geosci., 29, 183–193, DOI: 10.1016/S0098-3004(02)00074-2.

    Article  Google Scholar 

  • Featherstone W.E. and Kuhn M., 2006. Height systems and vertical datums: a review in the Australian context. J. Spat. Sci., 51, 21–42, DOI: 10.1080/14498596.2006.9635062.

    Article  Google Scholar 

  • Featherstone W.E., Kirby J.F., Kearsley A.H.W., Gilliland J.R., Johnston G.M., Steed J., Forsberg R. and Sideris M.G., 2001. The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data. J. Geodesy, 75, 313–330, DOI: 10.1007/s001900100177.

    Article  Google Scholar 

  • Featherstone W.E., Holmes S.A., Kirby J.F. and Kuhn M., 2004. Comparison of remove-compute-restore and University of New Brunswick techniques to geoid determination over Australia, and inclusion of Wiener-type filters in reference field contribution. J. Surv. Eng.-ASCE, 130, 40–47, DOI: 10.1061/(ASCE)0733-9453(2004)130:1(40).

    Article  Google Scholar 

  • Fellner J.J., Kuhn M. and Featherstone W.E., 2011. Development of a Synthetic Earth Gravity Model by 3D mass optimisation based on forward modelling. Earth Planets Space, 63, 1–8.

    Article  Google Scholar 

  • Filmer M.S., Featherstone W.E. and Kuhn M., 2010. The effect of EGM2008-based normal, normal-orthometric and Helmert orthometric height systems on the Australian levelling network. J. Geodesy, 84, 501–513, DOI: 10.1007/s00190-010-0388-0.

    Article  Google Scholar 

  • Haagmans R., 2000. A synthetic Earth model for use in geodesy. J. Geodesy, 74, 503–511, DOI: 10.1007/s001900000112.

    Article  Google Scholar 

  • Hadamard J., 1923. Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven.

    Google Scholar 

  • Hastings D.A. and Dunbar P.K., 1998. Development & assessment of the global land one-km base elevation digital elevation model (GLOBE). ISPRS Archives, 32(4), 218–221 (http://www.ngdc.noaa.gov/mgg/topo/globe.html).

    Google Scholar 

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

    Google Scholar 

  • Helmert F.R., 1884. Die mathematischen und physicalishen Theorien der höheren Geodäsie, Vol. 2. B.G. Teubner, Leipzig, Germany (in German, reprinted in 1962 by Minerva GMBH, Frankfurt/Main).

    Google Scholar 

  • Hirt C., 2011. Mean kernels to improve gravimetric geoid determination based on modified Stokes’s integration. Comput. Geosci., 37, 1836–1842, DOI: 10.1016/j.cageo.2011.01.005.

    Article  Google Scholar 

  • Hirt C., Featherstone W.E. and Claessens S.J., 2011. On the accurate numerical evaluation of geodetic convolution integrals. J. Geodesy, 85, 519–538, DOI: 10.1007/s00190-011-0451-5.

    Article  Google Scholar 

  • Holota P. and Nesvadba O., 2008. Model refinements and numerical solutions of weakly formulated boundary-value problems in physical geodesy. In: Xu P., Liu J. and Dermanis A. (Eds.): VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy. International Association of Geodesy Symposia 132, Springer Verlag, Berlin, Heidelberg, Germany, 320–326.

    Chapter  Google Scholar 

  • Hörmander L., 1976. The boundary problems of physical geodesy. Arch. Ration. Mech. Anal., 62, 1–52, DOI: 10.1007/BF00251855.

    Article  Google Scholar 

  • Janák J., Pašteka R., Záhorec P. and Loviska Z., 2006. Terrain correction in extremely disturbed terrain. Contrib. Geophys. Geod., 36, 41–52.

    Google Scholar 

  • Jekeli C., 1988. The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscripta Geodaetica, 13, 106–113.

    Google Scholar 

  • Kadir M.A., Fashir H.H. and Omar K., 1999. A regional gravimetric co-geoid over South East Asia. Geomatics Research Australasia, 71, 37–56.

    Google Scholar 

  • Kingdon R. and Vaníček P., 2011. Poisson downward continuation solution by the Jacobi method. J. Geod. Sci., 1, 74–81, DOI: 10.2478/v10156-010-0009-0.

    Google Scholar 

  • Kingdon R., Vaníček P., Santos M., Ellmann A. and Tenzer R., 2005. Toward an improved height system for Canada. Geomatica, 59, 241–249.

    Google Scholar 

  • Kuhn M., 2000. Geoidbestimmung unter Verwendung verschiedener Dichtehypothesen. Deutsche Geodätische Kommission, Reihe C, No. 520, München, Germany (in German).

    Google Scholar 

  • Kuhn M., 2003. Geoid determination with density hypotheses from isostatic models and geological information. J. Geodesy, 77, 50–65, DOI: 10.1007/s00190-002-0297-y.

    Article  Google Scholar 

  • Kuhn M. and Featherstone W.E., 2005. Construction of a synthetic Earth gravity model by forward gravity modelling. In: Sansò F. (Ed.), A Window on the Future of Geodesy. International Association of Geodesy Symposia 128, Springer Verlag, Berlin, Heidelberg, Germany, 350–355.

    Chapter  Google Scholar 

  • Kuhn M., Featherstone W.E. and Kirby J.F., 2009. Complete spherical Bouguer gravity anomalies over Australia. Austr. J. Earth Sci., 56, 213–223, DOI: 10.1080/08120090802547041.

    Article  Google Scholar 

  • Li X. and Wang Y.-M., 2011. Comparisons of geoid models over Alaska computed with different Stokes’s kernel modifications. J. Geod. Sci., 1, 136–142, DOI: 10.2478/v10156-010-0016-1.

    Google Scholar 

  • Lemoine F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson R.G., Pavlis E.C., Rapp R.H. and Olson T.R., 1998. The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. NASA Technical Report TP-1998-206861, National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, USA.

    Google Scholar 

  • Martinec Z., 1993. Effect of Lateral Density Variations of Topographical Masses in View of Improving Geoid Model Accuracy over Canada. Contract Report No. 93-002. Energy, Mines and Resources (now Geomatics Canada), Geodetic Survey Division, Ottawa, Canada, 112 pp.

    Google Scholar 

  • Martinec Z., 1996. Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J. Geodesy, 70, 805–828, DOI: 10.1007/BF00867158.

    Google Scholar 

  • Martinec Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Springer-Verlag, Berlin, Germany.

    Google Scholar 

  • Martinec Z. and Vaníček P., 1994a. The indirect effect of Stokes-Helmert’s technique for a spherical approximation of the geoid. Manuscripta Geodaetica, 19, 213–219.

    Google Scholar 

  • Martinec Z. and Vaníček P., 1994b. Direct topographical effect of Helmert’s condensation for a spherical approximation of the geoid. Manuscripta Geodaetica, 19, 257–268.

    Google Scholar 

  • Molodensky M.S., 1945. Fundamental problems of geodetic geavimetry. Trudy TsNIIGAIK, 42, Geodezizdat, Moscow, Russia (in Russian).

  • Molodensky M.S., Yeremeev V.F. and Yurkina M.I., 1960. Methods for study of the external gravitational field and figure of the Earth. TRUDY Ts NIIGAiK, 131, Geodezizdat, Moscow, Russia (English translat.: Israel Program for Scientific Translation, Jerusalem 1962).

  • Novák P., Vaníček P., Veronneau M., Holmes S.A. and Featherstone W., 2001. On the accuracy of modified Stokes’ integration in high-frequency gravimetric geoid determination. J. Geodesy, 74, 644–654, DOI: 10.1007/s001900000126.

    Article  Google Scholar 

  • Pail R., 2000. Synthetic Global Gravity Model for Planetary Bodies and Applications in Satellite Gravity Gradiometry. Ph.D. Thesis. Mitteilungen der geodi adischen Institute der Technischen Universität Graz, Folge 85, Austria.

    Google Scholar 

  • Paul M.K., 1973. A method of evaluating the truncation error coefficients for geoidal height. Bulletin Géodésique, 47, 413–425, DOI: 10.1007/BF02521951.

    Article  Google Scholar 

  • Pavlis N.K., Holmes S.A., Kenyon S.C. and Factor J.F., 2012. The development and evaluation of Earth Gravitational Model (EGM2008). J. Geophys. Res., 117, B04406, DOI: 10.10292011JB008916.

    Article  Google Scholar 

  • Rapp R.H., 1997. Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference. J. Geodesy, 71, 282–289, DOI: 10.1007/s001900050096.

    Article  Google Scholar 

  • Santos M.C., Vaníček P., Featherstone W.E., Kingdon R., Ellmann A., Martin B.A., Kuhn M. and Tenzer R., 2006. The relation between the rigorous and Helmert’s definitions of orthometric heights. J. Geodesy, 80, 691–704, DOI: 10.1007/s00190-006-0086-0.

    Article  Google Scholar 

  • Sjöberg L.E., 2003. Ellipsoidal corrections to order e 2 of geopotential coefficients and Stokes’ formula. J. Geodesy, 77, 139–147, DOI: 10.1007/s00190-003-0321-x.

    Article  Google Scholar 

  • Smith D.A., 1998. There is no such thing as “The” EGM96 geoid: Subtle points on the use of a global geopotential model. International Geoid Service Bulletin, 8, 17–28.

    Google Scholar 

  • Stokes G.G., 1849. On the variation of gravity at the surface of the Earth. Transactions of the Cambridge Philosophical Society, 8, 672–695.

    Google Scholar 

  • Sun W. and Vaníček P., 1998. On some problems of the downward continuation of the 5′×5′mean Helmert gravity disturbance. J. Geodesy, 72, 411–420, DOI: 10.1007/s001900050180.

    Article  Google Scholar 

  • Tenzer R., Vaníček P., Santos M.C., Featherstone W.E. and Kuhn M., 2005. The rigorous determination of orthometric heights. J. Geodesy, 79, 82–92, DOI: 10.1007/s00190-005 -0445-2.

    Article  Google Scholar 

  • Tikhonov A.N. and Arsenin V.Y., 1977. Solutions of Ill-Posed Problems. John Wiley & Sons, New York.

    Google Scholar 

  • Tziavos I.N., 1996. Comparisons of spectral techniques for geoid computations over large regions. J. Geodesy, 70, 357–373, DOI: 10.1007/s001900050027.

    Google Scholar 

  • Valty P., Duquenne H. and Panet I., 2012. Auvergne dataset: testing several geoid computation methods. In: Kenyon S., Pacino M.C. and Marti U. (Eds.), Geodesy for Planet Earth. International Association of Geodesy Symposia 136, Springer Verlag, Heidelberg, Germany, 465–472, DOI: 10.1007/978-3-642-20338-1_56.

    Chapter  Google Scholar 

  • Vaníček P. and Kleusberg A., 1987. The Canadian geoid — Stokesian approach. Manuscripta Geodaetica, 12, 86–98.

    Google Scholar 

  • Vaníček P. and Sjöberg L.E., 1991. Reformulation of Stokes’s theory for higher than second-degree reference field and modification of integration kernels. J. Geophys. Res., 96(B4), 6529–6339, DOI: 10.1029/90JB02782.

    Article  Google Scholar 

  • Vaníček P. and Martinec Z., 1994. Stokes-Helmert scheme for the evaluation of a precise geoid. Manuscripta Geodaetica, 19, 119–128.

    Google Scholar 

  • Vaníček P., Ong P. and Zhang C., 1991. New gravimetric geoid for Canada: the “UNB90” solution. In: Rapp R.H. and Sansò F. (Eds.), Determination of the Geoid: Present and Future. International Association of Geodesy Symposia 106, Springer Verlag, Heidelberg, Germany, 214–219.

    Chapter  Google Scholar 

  • Vaníček P., Najafi M., Martinec Z., Harrie L. and Sjöberg L.E., 1995. Higher-degree reference field in the generalised Stokes-Helmert scheme for geoid computation. J. Geodesy, 70, 176–182, DOI: 10.1007/BF00943693.

    Article  Google Scholar 

  • Vaníček P., Sun W., Ong P., Martinec Z., Najafi M., Vajda P. and Horst B., 1996. Downward continuation of Helmert’s gravity. J. Geodesy, 71, 21–34, DOI: 10.1007/s001900050072.

    Article  Google Scholar 

  • Vaníček P., Huang J., Novák P., Pagiatakis S.D., Véronneau M., Martinec Z. and Featherstone W.E., 1999 Determination of the boundary values for the Stokes-Helmert problem. J. Geodesy, 73, 160–192, DOI: 10.1007/s001900050235.

    Google Scholar 

  • Vaníček P., Novák P. and Martinec Z., 2001. Geoid, topography, and the Bouguer plate or shell. J. Geodesy, 75, 210–215, DOI: 10.1007/s001900100165.

    Article  Google Scholar 

  • Vaníček P., Kingdon R. and Santos M., 2012. Geoid versus quasi-geoid: a case of physics versus geometry. Contrib. Geophys. Geod., 42, 101–119.

    Google Scholar 

  • Véronneau M. and Huang J., 2011. The reference surface and maintenance of the up-coming height system in Canada. American Geophysical Union, Fall Meeting 2011, San Francisco, CA, Abstract #G53A-0891.

    Google Scholar 

  • Vermeer M., 1998. The geoid as a product. Proceedings, Second Continental Workshop on the Geoid in Europe, March 10–14, 1998. Report 98:4, Finnish Geodetic Institute, Masala, Finland, 63–69.

    Google Scholar 

  • Wessel P. and Smith W.H.F., 1998. New, improved version of the generic mapping tools released. Eos Trans. AGU, 79(47), 579, DOI: 1029/98EO00426.

    Article  Google Scholar 

  • Wong J., 2002. On Picard Criterion and the Well-Posed Nature of Harmonic Downward Continuation. Technical Report 213, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, Canada.

    Google Scholar 

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Author notes

  1. Robert Kingdon

    Present address: Fugro Airborne Surveys, 2191 Thurston Drive, Ottawa, Ontario, K1G 6C9, Canada

  2. David Avalos-Naranjo

    Present address: Instituto Nacional de Estadística y Geografía, Av. Héroe de Nacozari Sur 2301, C.P. 20276, Aguascalientes, Mexico

Authors and Affiliations

  1. Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, E3B 5A3, Canada

    Petr Vaníček, Robert Kingdon, Marcelo C. Santos & David Avalos-Naranjo

  2. Western Australian Centre for Geodesy & The Institute for Geoscience Research, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia

    Michael Kuhn, Will E. Featherstone & Christian Hirt

  3. Faculty of Civil Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086, Tallinn, Estonia

    Artu Ellmann

  4. Geophysics Section, Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland

    Zdeněk Martinec

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  1. Petr Vaníček
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Vaníček, P., Kingdon, R., Kuhn, M. et al. Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings. Stud Geophys Geod 57, 369–400 (2013). https://doi.org/10.1007/s11200-012-0270-z

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