Abstract
The purpose of this paper is to discuss the construction of the reproducing kernel of Hilbert’s space of functions that are harmonic in the exterior of an oblate ellipsoid of revolution. The motivation comes from the weak solution concept applied to Neumann’s problem for Laplace’s partial differential equation in gravity field studies. The use of the reproducing kernel enables the construction of a function basis that is suitable for the approximation representation of the solution and offers a straightforward way leading to entries in Galerkin’s matrix of the respective linear system for unknown scalar coefficients. The serious problem, however, is the summation of the series that represents the kernel. It is difficult to reduce the number of summation indices since in the ellipsoidal case there is no straightforward analogue to the addition theorem known for the spherical situation. This makes the computation of the kernel and the set of the entries in Galerkin’s matrix rather demanding, even by means of high performance computer facilities. Therefore, the reproducing kernel and its series representation are analyzed. The apparatus of hypergeometric functions and series is used. The kernel is split into parts. Some of the resulting series may be summed relatively easily, except for some technical tricks. For the remaining series, however, the summation needs more complex tools. In particular, the summation was converted to elliptic integrals. This approach leads to an effective numerical treatment of the kernel. The results are presented. Finally, the relation of the reproducing kernel to Green’s function of the second kind (Neumann’s function) is discussed with a special view to physical geodesy applications.
Similar content being viewed by others
References
Bateman H. and Erdélyi A., 1953. Higher Transcendental Functions. Volume 1. McGraw-Hill Book Company, Inc., New York-Toronto-London.
Bers L., John F. and Schechter M., 1964. Partial Differential Equations. Interscience Publishers, New York — London — Sydney.
Brovar V.V., Kopeikina Z.S. and Pavlova M.V., 2001. Solution of the Dirichlet and Stokes exterior boundary problems for the Earth’s ellipsoid. J. Geodesy, 74, 767–772.
Carlson B.C., 1979. Computing elliptic integrals by duplication. Numerische Mathematik, 33, 1–16.
Čuřík F., 1944. Matematika. Technický prvodce, XLIX, No. 196. Česká matice technická, Prague, Czech Republic (in Czech).
Fei Z.L. and Sideris M.G., 2000. A new method for computing the ellipsoidal correction for Stokes’s formula. J. Geodesy, 74, 223–231.
Fei Z.L. and Sideris M.G., 2001. Corrections to “A new method for computing the ellipsoidal correction for Stokes’s formula”. J. Geodesy, 74, 671.
Grafarend E.W., Ardalan A. and Sideris M.G., 1999. The spheroidal fixed-free two-boundary-value problem for geoid determination (the spheroidal Brun’s transform). J. Geodesy, 73, 513–533.
Heck B. and Seitz K., 2003. Solution of the linearized geodetic boundary value problem for an ellipsoidal boundary to order e 3. J. Geodesy, 77, 182–192.
Heiskanen W.A. and Moritz H., 1967. Physical Geodesy. W.H. Freeman and Company, San Francisco, CA.
Hobson E.W., 1931. The Theory of Spherical and Ellipsoidal Harmonics. University Press, Cambridge, U.K.
Hofmann-Wellenhof B. and Moritz H., 2005. Physical Geodesy. Springer, Wien — New York.
Holota P. 1992a. On the iteration solution of the geodetic boundary-value problem and some model refinements. Travaux de l’Association Internationale de Geodesie, 29, 260–289.
Holota P. 1992b. Integral representation of the disturbing potential: Effects involved, iteration technique and its convergence. In: Holota P. and Vermeer M. (Eds), First Continental Workshop on the Geoid in Europe: Towards a Precise Pan-european Reference Geoid for the Nineties, Prague, May 11–14, 1992. Research Institute of Geodesy, Topography and Cartography, Zdiby, Czech Republic, 402–419, ISBN: 8090131921, 9788090131927.
Holota P., 1997. Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. J. Geodesy, 71, 640–651.
Holota P., 2003a. Variational methods in the representation of the gravitational potential. In: Schäfer U. (Ed.), Proceedings of the Workshop: Analytical Representation of Potential Field Anomalies for Europe (AROPA): October 23th to 27th, 2001, Münsbach-Castle, Münsbach, Grand-Duchy of Luxembourg. Centre européen de géodynamique et de séismologie, Luxembourg, Cahiers du Centre Européen de Géodynamique et de Séismologie, 20, 3–11.
Holota P., 2003b. Gronwall’s inequality in an approximate computation of ellipsoidal harmonics. In: Em. Univ.-Prof. Dipl.-Ing. Dr.h.c.mult. Dr.techn. Helmut Moritz Festschrift zum 70. Geburtstag Institut für Geodäsie, Technische Universität Graz, Graz, Austria, 111–122.
Holota P., 2004. Some topics related to the solution of boundary-value problems in geodesy. In: Sansò F. (Ed.), V Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, 127, Springer-Verlag, Berlin, Germany, 189–200.
Holota P., 2011. Reproducing kernel and Galerkin’s matrix for the exterior of an ellipsoid: Application in gravity field studies. Stud. Geophys. Geod., 55, 397–413.
Holota P. and Nesvadba O., 2007. 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, Germany, 314–320.
Holota P. and Nesvadba O., 2012. Method of successive approximations in solving geodetic boundary value problems — Analysis and numerical experiments. In: Sneeuw N., Novák P., Crespi M. and Sansò F. (Eds), VII Hotine-Marussi Symposium. International Association of Geodesy Symposia, 137, Springer-Verlag, Berlin, Germany, 189–198.
Holota P. and Nesvadba O., 2014. Analytical continuation in physical geodesy constructed by means of tools and formulas related to an ellipsoid of revolution. Geophys. Res. Abs., 16, EGU2014–16953 (http://meetingorganizer.copernicus.org/EGU2014/EGU2014-16953.pdf).
Hotine M., 1969. Mathematical Geodesy. ESSA Monographs. U.S. Environmental Science Services Administration, Washington, D.C.
Huang J., Véronneau M. and Pagiatakis S.D. 2003. On the ellipsoidal correction to the spherical Stokes solution of the gravimetric geoid. J. Geodesy, 77, 171–181.
Jahnke E. and Emde F., 1945. Tables of Functions with Formulae and Curves. Dover Publications, New York.
Koch K.-R. 1968. Solution of the Geodetic Boundary Value Problem in Case of Reference Ellipsoid. Report 104, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH.
Koch K.-R. 1969. Solution of the geodetic boundary value problem for a reference ellipsoid. J. Geophys. Res., 74, 3796–3803.
Kratzer A. and Franz W., 1960. Transzendente Funktionen. Akademische Verlagsgesellschaft, Leipzig, Germany (in German).
Martinec Z. 1998. Contribution of Green’s function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition. J. Geodesy, 72, 460–471.
Martinec Z. and Grafarend E.W. 1997a. Solution of the Stokes boundary-vaue problem on an ellipsoid of revolution. Stud. Geophys. Geod., 41, 103–129.
Martinec Z. and Grafarend E.W. 1997b. Construction of Green’s function to the external Dirichlet boundary-vaue problem for the Laplace equation on an ellipsoid of revolution. J. Geodesy, 71, 562–570.
Mazurova E.M. and Yurkina M.I. 2011. Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth. Stud. Geophys. Geod., 55, 455–464.
Meschkowski H., 1962. Hilbertsche Räume mit Kernfunktion. Springer-Verlag, Berlin, Germany (in German).
Michlin S.G., 1964. Variational Methods in Mathematical Physics. Pergamon Press, New York.
Moritz H., 1980. Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, Germany.
Moritz H., 1992. Geodetic Reference System 1980. In: Tscherning C.C. (Ed.) The Geodesist’s Handbook 1992. Bulletin Géodésique, 66, 187–192.
Nečas J., 1967. Les méthodes directes en théorie des équations elliptiques. Academia, Prague, Czech Republic (in French).
Nesvadba O., 2009. Numerical problems in evaluating high degree and order associated Legendre functions. 3, 11, EGU2009–1225 (http://meetingorganizer.copernicus.org/EGU2009/EGU2009-1225.pdf).
Nesvadba O., 2010. Reproducing kernels in harmonic spaces and their numerical implementation. Geophys. Res. Abs., 12, EGU2010–14298 (http://meetingorganizer.copernicus.org/EGU2010/EGU2010-14298.pdf).
Nesvadba O., Holota P. and Klees R., 2007. A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P. and Rizos C. (Eds.), Dynamic Planet — Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools. International Association of Geodesy Symposia, 130, Springer-Verlag, Berlin, Germany, 370–376.
Nesvadba O., Holota P. and Lederer M., 2013. Quasigeoid and the relation of ETRS to the Bpv system in the Czech Republic. International Association of Geodesy Scientific Assembly, http://www.iag2013.org/IAG_2013/Program_files/abstracts_iag_2013_2808.pdf.
Neyman Yu.M., 1979. A Variational Method of Physical Geodesy. Nedra Publishers, Moscow, Russia (in Russian).
Pellinen L.P. 1982. Effects of Earth’s ellipticity on solving geodetic boundary value problem. Bollettino di Geodesia e Scienze Affini, 41, 89–103.
Pick M., Pícha J. and Vyskoil V., 1973. Theory of the Earth’s Gravity Field. Academia, Prague, Czech Republic.
Prudnikov A.P., Brychkov Yu.A. and Marichev O.I., 1981. Integrals and Series, Elementary Functions. Nauka Publishers, Moscow, Russia (in Russian).
Rektorys K., 1977. Variational Methods. Reidel Co., Dordrecht, The Netherlands.
Sansò F., 1986. Statistical methods in physical geodesy. In: Sünkel H. (Ed.), Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, 7. Springer-Verlag, Berlin, Germanz, 49–155.
Sansò F. and Sona G., 2001. ELGRAM: an ellipsoidal gravity model manipulator. Bollettino di Geodesia e Scienze Affini, 60, 215–226.
Sjöberg L.E. 2004. The ellipsoidal corrections to the topographic geoid effects. J. Geodesy, 77, 804–808.
Smirnov V.I., 1965. Lectures in Higher Mathematics. Vol. I. Nauka Publishers, Moscow, Russia (in Russian).
Smirnov V.I., 1958. Lectures in Higher Mathematics. Vol. III, Part 2. Nauka Publishers, Moscow, Russia (in Russian).
Sona G., 1995. Numerical problems in the computation of ellipsoidal harmonics. J. Geodesy, 70, 117–126.
Thong N.C., 1992. Spheroidal harmonic analysis and integral formula of the Earth’s gravitational field based on gravimetric data. In: Holota P. and Vermeer M. (Eds), First Continental Workshop on the Geoid in Europe: Towards a Precise Pan-european Reference Geoid for the Nineties, Prague, May 11-14, 1992. Research Institute of Geodesy, Topography and Cartography, Zdiby, Czech Republic, 393–401, ISBN: 8090131921, 9788090131927.
Thong N.C. and Grafarend E.W., 1989. A spherical harmonic model of the terrestrial gravitational field. Manuscripta Geodaetica, 14, 285–304.
Tscherning C.C., 1975. Application of collocation. Determination of a local approximation to the anomalous potential of the Earth using “exact” astro-gravimetric collocation. In: Brosowski B. and Martensen E. (Eds), Methoden und Verfahren der Mathematischen Physik, 14, 83–110.
Tscherning C.C., 2004. A discussion on the use of spherical approximation or no approximation in gravity field modelling with emphasis on unsolved problems in least-squares collocation. In: Sansò F. (Ed.), V Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, 127. Springer-Verlag, Berlin, Germanz, 184–188.
Yu J., Jekeli C. and Zhu M. 2003. Analytical solution of the Dirichlet and Neumann boundary-value problems with an Ellipsoidal Boundary. J. Geodesy, 76, 653–667.
Yurkina M.I., 1996. Use of Green’s function for determining the disturbing potential of an ellipsoidal earth. Fizicheskaya Geodeziya, 2, TsNIIGAiK, 165–178 (in Russian).
Zagrebin D.V., 1956. Die Theorie des regularisierten Geoids. Veröffentlichungen des Geodätischen Instituts in Potsdam, No. 9, Akademie-Verlag, Berlin, Germany (in German).
Zhidkov N.P., 1977. Linear Approximations of Functionals. Moscow University Publishers, Moscow, Russia (in Russian).
Zhu Z.W. 1981. The Stokes Problem for the Ellipsoid Using Ellipsoidal Kernels. Report 319. Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Holota, P., Nesvadba, O. Reproducing kernel and Neumann’s function for the exterior of an oblate ellipsoid of revolution: application in gravity field studies. Stud Geophys Geod 58, 505–535 (2014). https://doi.org/10.1007/s11200-013-0861-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11200-013-0861-3