Abstract
The aim of the paper is to implement the Green’s function method for the solution of the Linear Gravimetric Boundary Value Problem. The approach is iterative by nature. A transformation of spatial (ellipsoidal) coordinates is used that offers a possibility for an alternative between the boundary complexity and the complexity of the coefficients of Laplace’s partial differential equation governing the solution. The solution domain is carried onto the exterior of an oblate ellipsoid of revolution. Obviously, the structure of Laplace’s operator is more complex after the transformation. It was deduced by means of tensor calculus and in a sense it reflects the geometrical nature of the Earth’s surface. Nevertheless, the construction of the respective Green’s function is simpler for the solution domain transformed. It gives Neumann’s function (Green’s function of the second kind) for the exterior of an oblate ellipsoid of revolution. In combination with successive approximations it enables to meet also Laplace’s partial differential equation expressed in the system of new (i.e. transformed) coordinates.
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References
Bjerhammar A, Svensson L (1983) On the geodetic boundary-value problem for a fixed boundary surface - satellite approach. Bull Geod 57:382–393
Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. Springer, Berlin
Grafarend EW (1989) The geoid and the gravimetric boundary-value problem. Rep 18 Dept Geod. The Royal Institute of Technology, Stockholm
Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman and Company, San Francisco
Holota P (1985) A new approach to iteration solutions in solving geodetic boundary value problems for real topography. In: Proc. 5th Int. Symp. Geod. and Phys. of the Earth, GDR, Magdeburg, Sept. 23rd–29th, 1984, Part II. Veroff. d. Zentr. Inst. f. Phys. d. Erde, Nr. 81, Teil II, pp 4–15
Holota P (1986) Boundary value problems in physical geodesy: present state, boundary perturbation and the Green-Stokes representation. In: Proc. 1st Hotine-Marussi Symp. on Math. Geodesy, Rome, 3–5 June 1985, vol 2. Politecnico di Milano, pp 529–558
Holota P (1989) Laplacian versus topography in the solution of the Molodensky problem by means of successive approximations. In: Kejlso E, Poder K, Tscherning CC (eds) Festschrift to Torben Krarup, Geodaetisk Inst., Meddelelse No. 58, Kobenhavn, pp 213–227
Holota P (1992a) On the iteration solution of the geodetic boundary-value problem and some model refinements. Contribution to Geodetic Theory and Methodology. In: XXth General Assembly of the IUGG, IAG-Sect. IV, Vienna, 1991. Politecnico di Milano, 1991, pp 31–60; also in: Travaux de l’Association Internationale de Geodesie, Tome 29, Paris: 260–289
Holota P (1992b) Integral representation of the disturbing potential: effects involved, iteration technique and its convergence. In: Holota P, Vermeer M (eds) Proc. First continental workshop on the geoid in Europe, Prague, May 11–14, 1992. Research Inst. of Geod., Topog. and Cartog., Prague, in co-operation with IAG-Subcommis. for the Geoid in Europe, Prague, pp 402–419
Holota P (1997) Coerciveness of the linear gravimetric boundary value problem and geometrical interpretation. J Geod 71:640–651
Holota P (2003) Green’s function and external masses in the solution of geodetic boundary-value problems. In: Tziavos IN (ed) Gravity and Geoid, 3rd Meeting of the Intl. Gravity and Geoid Commission, Thessaloniki, Greece, August 26–30, 2002. Ziti Editions, Thessaloniki, pp 108–113
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, Matera, Italy, June 17–21, 2002. International Association of Geodesy Symposia, vol 127. Springer, Berlin, pp 189–200
Holota P (2011) Reproducing kernel and Galerkin’s matrix for the exterior of an ellipsoid: application in gravity field studies. Studia geophysica et geodaetica 55(3):397–413
Holota P (2016) Domain transformation and the iteration solution of the linear gravimetric boundary value problem. In: Freymueller J, Sánchez L (eds) International symposium on earth and environmental sciences for future generations. Proceedings of the IAG General Assembly, Prague, Czech Republic, June 22–July 2, 2015. International Association of Geodesy Symposia, vol 147. Springer, Cham, pp 47–52. https://doi.org/10.1007/1345_2016_236
Holota P, Nesvadba O (2014) Reproducing kernel and Neumann’s function for the exterior of an oblate ellipsoid of revolution: application in gravity field studies. Studia geophysica et geodaetica 58(4):505–535
Holota P, Nesvadba O (2016) Small modifications of curvilinear coordinates and successive approximations applied in geopotential determination. 2016 AGU Fall Meeting, Session G21B (Scientific and Practical Challenges of Replacing NAD 83, NAVD 88, and IGLD 85), San Francisco, USA, 12–16 December 2016, poster. https://agu.confex.com/agu/fm16/meetingapp.cgi/Paper/189936
Holota P, Nesvadba O (2018a) Neumann’s function and its derivatives constructed for the exterior of an ellipsoid and adapted to an iteration solution of the linear gravimetric boundary value problem. In: Geophysical Research Abstracts, vol 20, EGU2018-18558
Holota P, Nesvadba O (2018b) Boundary complexity and kernel functions in classical and variational concepts of solving geodetic boundary value problems. In: Freymueller J, Sánchez L (eds) International symposium on advancing geodesy in a changing world. International Association of Geodesy Symposia, vol 149. Springer, Cham, pp 31–41. https://doi.org/10.1007/1345_2018_34
Koch KR, Pope AJ (1972) Uniqueness and existence for the geodetic boundary-value problem using the known surface of the Earth. Bull Geod 106:467–476
Lyusternik LA, Sobolev VI (1965) Foundations of functional analysis. Nauka Publishers, Moscow. (in Russian)
Moritz H (1992) Geodetic reference system 1980. In: Tscherning CC (ed) The Geodesist’s Handbook 1992. Bulletin Géodésique, vol 66, no 2, pp 187–192
Roach GF (1982) Green’s functions, 2nd edn. Cambridge University Press, Cambridge
Sansò F, Sideris MG (2013) Geoid determination - theory and methods. Springer, Berlin
Sokolnikoff IS (1971) Tensor analysis. Theory and applications to geometry and mechanics of continua. Nauka Publishers, Moscow. (in Russian)
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The work on this paper was supported by the Ministry of Education, Youth and Sports of the Czech Republic through Project No. LO1506. This support is gratefully acknowledged.
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Holota, P., Nesvadba, O. (2019). Green’s Function Method Extended by Successive Approximations and Applied to Earth’s Gravity Field Recovery. In: Novák, P., Crespi, M., Sneeuw, N., Sansò, F. (eds) IX Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 151. Springer, Cham. https://doi.org/10.1007/1345_2019_67
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