Abstract
In this paper an overdetermined Geodetic Boundary Value Problem (GBVP) approach for telluroid and quasi-geoid computations is presented. The presented GBVP approach can solve the problem of potential value computation on the surface of the Earth, which when applied to a mapping scheme, e.g., here minimum distance mapping, provides a point-wise approach to telluroid computation. Besides, we have succeeded in reducing the number of equations and unknowns of the minimum distance telluroid mapping by one. The sufficient condition of minimum distance telluroid mapping is also recapitulated. Since the introduced GBVP approach has the advantage of implementing various gravity observables simultaneously as input boundary data, it can be regarded as a data fusion technique that exploits all available gravity data. The developed GBVP is used for the computation of the quasi-geoid within a test area in Southwest Finland.
Similar content being viewed by others
References
Adam J, Denker H (1991) Test computations for a local quasigeoid in Hungry using FFT. Acta Geodetica Geophysica et Montanistica Hung 26: 33–43
Amos MJ, Featherstone WE (2009) Unification of New Zealand’s local vertical datums: iterative gravimetric quasigeoid computations. J Geod 83: 57–68
Ardalan AA (2000) High-resolution regional geoid computation in the World Geodetic Datum 2000, based upon collocation of linearized observationals of the type GPS, gravity potential and gravity intensity. Ph.D. thesis, Department of geodesy and geoinformatics, Stuttgart University, Stuttgart
Ardalan AA, Grafarend EW (2004) High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid. J Geod 78: 138–156
Ardalan AA, Grafarend EW, Ihde J (2002) Molodensky potential telluroid based on a minimum-distance map. Case study: the quasi-geoid of East Germany in the World Geodetic Datum 2000. J Geod 76: 127–138
Barzaghi R, Brovelli MA, Sona G, Manzino A, Sguerso D (1996) The new Italian quasigeoid: ITALGEO95. Bollettino di Geodesia e Scienze Affini 55: 57–72
Baxley J, Moorhouse J (1984) Lagrange multiplier problems in economics. Am Math Mon 91: 404–412
Denker H (1998) Evaluation and improvement of the EGG97 quasigeoid model for Europe by GPS and leveling data. Reports of the Finnish Geodetic Institute 98:4, 53–61, Masala
Dore P, Simonsen O (1936a) Minutes of the work of the meetings of the section of levelling. Joint meeting of section II and IV, 11 September, A.M., on levelling and gravity. Bulletine Geodesique 52: 9–13
Dore P, Simonsen O (1936b) Minutes of the work of the meetings of the section of levelling. Joint meeting with the association of physical oceanography, 12 September, A.M.m Bulletine Geodesique 52: 13–17
Duquenne H (1999) Comparison and combination of a gravimetric quasigeoid with a leveled GPS data set by statistical analysis. Physics and Chemistry of the Earth. Part A: Solid Earth Geodesy 24: 79–83
Engelis T (1987) Spherical harmonic expansion of the Levitus Sea Surface Topography. Report No. 385, The Ohio State University, Research Foundation
Featherstone WE, Kirby JF (1998) Estimates of the separation between the geoid and quasigeoid over Australia. Geomatics Res Australasia 68: 79–90
Grafarend EW (1976) Geodetic applications of stochastic processes. Phys Earth Planetary Interiors 12: 151–179
Grafarend EW, Ardalan AA (1999) World Geodetic Datum 2000. J Geod 73: 611–623
Grafarend EW, Ardalan A, Sideris MG (1999) The Spheroidal fixed-free two boundary-value problem for geoid determination (the Spheroidal Brun’s transform). J Geod 73: 513–533
Grafarend EW (2006) Linear and nonlinear models: fixed effects, random effects or mixed models. de Gruyter Verlag, Berlin
Hands DW (2004) Introductory mathematical economics, 2nd edn. Oxford University Press, London (1st ed., Heath, 1991)
Holota P (1983a) The altimetry gravimetry boundary value problem I: linearization, Friedrich’s inequality. Boll Geod Sci Afini XLII: 13–32
Holota P (1983b) The altimetry gravimetry boundary value problem II: weak solution, V-ellipticity. Boll Geod Sci Afini XLII: 69–84
Huang J (2002) Computational methods for the discrete downward continuation of the Earth gravity and effects of lateral topographical mass density variation on gravity and the geoid, A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Geodesy and Geomatics Engineering, The University of New Brunswick, May, 2002
Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76: 259–268
Kostelecky J, Kostelecky JRJ, Pesek I, Simek J, Svabensky O, Weigel J, Zeman A (2004) Quasigeoids for the territory of the Czech Republic. Stud Geophys Geod 48: 503–518
Lysaker DI, Omang OCD, Pettersen BR, Solheim D (2007) Quasugeoid evaluation with improved leveled height data for Norway. J Geod 81: 617–627
Martinec Z, Grafarend EW (1997) Construction of Green’s function to an external Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution. J Geod 71: 562–570
Molodensky MS (1945) Main problem of geodetic gravimetry. Trans. centr. res. Inst. G., A. & C., p 42
Molodensky MS (1948) External gravitational field and the figure of the Earth’s physical surface. Information of the USSR academy of sciences, geographical and geophysical series 13, No. 3
Molodensky MS, Eremeev VF, Yurkina MI (1960) Methods for study of the external gravitational field and figure of the Earth. Translated from Russian by Israel Program for Scientific Translations for the Office of Technical Services, Department of Commerce, Washington, DC, USA (1962)
Nahavandchi H, Soltanpour A (2006) Improved determination of heights a conversion surface by combining gravimetric Quasi-Geoid/Geoid and GPS-Levelling height differences. Stud Geophys Geod 50: 165–180
Novak P, Kern M, Schwarz KP, Heck B (2001) On the determination of a band-limited gravimetric geoid from airborne gravimetry. Department of Geomatics Engineering, The University of Calgary, Technical Report 30013: 1–222
Novak P, Kostelecky J, Klokocnik J (2009) Testing global geopotential models through comparison of a local quasi-geoid model with GPS/leveling data. Stud Geophys Geod 53: 39–60
Ono S (1990) Determination of quasigeoid heights. J Geod Soc Jpn 36: 37–50
Prutkin I, Klees R (2008) On the non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies. J Geod 82: 147–156
Reigber C, Schmidt R, Flechtner F, Koenig R, Meyer U, Neumayer KH, Schwintzer P, Yuan Zhu S (2006) An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN_GRACE02S. J Geodyn 39: 1–10
Safari A, Ardalan AA, Grafarend EW (2005) A new ellipsoidal gravimetric, satellite altimetry and astronomiac boundary value problem, a case study: The geoid of Iran. J Geodyn 39: 545–568
Sjoberg LE (1995) On the quasigeoid to geoid separation. Manuscr Geod 20: 182–192
Soltanpour A, Nahavandchi H, Featherstone WE (2006) The use of second-generation wavelets to combine a gravimetric Quasigeoid model with GPS-levelling data. J Geod 80: 82–93
Sommer M, Breuer R, Bommehardt H, Krohan HJ (1996) An approximation of the surface of quasigeoids in areas of the new German states. Nachrichten aus dem Karten-und Vermessungswesen, Reihe l, 114, pp 5–21
Svensson SL (1983) Solution of the altimetry–gravimetry problem. Bull Geod 57: 332–353
Svensson SL (1988) Some remarks on the altimetry- gravimetry problem. Manuscr Geod 13: 63–74
Tenzer R, Novak P, Moore P, Kuhn M, Vanicek P (2006) Explicit Formula for the Geoid-Quasigeoid separation. Stud Geophys Geod 50: 607–618
Torge W, Denker H (1999) One the use of the European Gravimetric (Quasi) geoid EGG97 in Germany. Zeitschrift für Vermessungswesen 124: 154–166
Vanicek P, Krakiwsky E (1986) Geodesy the concept. Book Elsevier Science publishers, Amsterdam
Xu P, Shen Y, Fukuda Y, Lio Y (2006) Variance Component Estimation in Linear Inverse Ill-posed Models. J Geod 80: 69–81
Zhang C (2005) Content and precision-determination of difference between geoid and quasigeoid. Geomat Inf Sci Wuhan Univ 30: 471–473
Zhang L, Li F, Chen W, Zhang C (2009) Height datum unification between Shenzhen and Hong Kong using the solution of the linearized fixed-gravimetric boundary value problem. J Geod 83: 411–417
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to M. S. Molodensky for his innovative contributions to geodesy.
Rights and permissions
About this article
Cite this article
Ardalan, A.A., Karimi, R. & Bilker-Koivula, M. An overdetermined geodetic boundary value problem approach to telluroid and quasi-geoid computations. J Geod 84, 97–104 (2010). https://doi.org/10.1007/s00190-009-0347-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-009-0347-9