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Course, academic year 2023/2024
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Mathematical Methods for Study of the Gravitational Field and Figure of the Earth - NDGF026
Title: Matematické metody studia gravitačního pole a tvaru Země
Guaranteed by: Department of Geophysics (32-KG)
Faculty: Faculty of Mathematics and Physics
Actual: from 2013
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Petr Holota
Annotation -
Last update: T_KG (22.04.2013)
Data sources on the Earth's surface and in its exterior. General formulation of boundary value problems of potential theory in physical geodesy. Categories of problems. Perturbations of an initial model of the gravitational field and figure of the Earth. Classical and modern methods in the solution of linear geodetic boundary value problems. Geodetic interpretation of the results, history and the importance of the subject.
Aim of the course -
Last update: T_KG (22.04.2013)

The objectives of the subject are to give in the first place an explanation concerning the very content of studies on gravitational field and figure of the Earth with emphasis on their geodetic and geophysical importance and at the same time also to clarify the motivation and methods of classical and modern physical geodesy. Within the explanation the subject is presented in terms of its mathematical formulation. The mathematical nature of typical problems belonging to the topic under study is discussed and methods for their solution are shown. The subject also contains a review on contemporary domestic and international activities in the given field.

Course completion requirements -
Last update: prof. RNDr. František Gallovič, Ph.D. (10.06.2019)

Oral exam

Literature - Czech
Last update: T_KG (22.04.2013)

[1] Heiskanen W.A.-H. Moritz.: Physical Geodesy, W.H. Freeman and Company, San Francisco and London, 1967.

[2] Hobson E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, 1931.

[3] Holota P.: Isozenithals in the neighbourhood of an Earth's model and the boundary condition for the disturbing potential. Manuscripta geodaetica, Vol. 13 (1988): 257-266.

[4] Holota P.: On the Iteration Solution of the Geodetic Boundary-value Problem and Some Model Refinements. Travaux de l'Association Intl. de Géodésie, Tome 29, Paris, 1992: 260-289.

[5] Holota P.: Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. Journal of Geodesy, Vol. 71 (1997), No. 10: 640-651.

[6] Holota P.: Direct methods in physical geodesy. In: Schwarz K.-P. (ed.). Geodesy Beyond 2000 - The Challenges of the First Decade. IAG General Assembly, Birmingham, July 19-30, 1999, IAG Symposia, Vol. 121, Springer, Berlin - Heidelberg - New York, 2000: 163-170

[7] Holota P.: Variational methods in the recovery of the gravity field - Galerkin's matrix for an ellipsoidal domain. In: M.G. Sideris (ed.): Gravity, Geoid and Geodynamics 2000, GGG2000 IAG International Symposium, Banff, Alberta, Canada, July 31 - August 4, 2000. IAG Symposia, Vol. 123, Springer, Berlin-Heidelberg-New York, 2001: 277-283.

[8] Holota P.: Variational methods in the representation of the gravitational potential. In: Schäfer U. (ed.): Proc. of the Workshop on Analytical representation of potential field anomalies for Europe (AROPA), ?Institute d?Europe?, Münsbach Castle, Grand-Duchy of Luxembourg, 23-27 October 2001. Cahiers du Centre Européen de Géodynamique et de Séismologie, Vol. 20, Luxembourg, 2003: 3-11.

[9] Holota P.: Some topics related to the solution of boundary-value problems in geodesy. In: Sans? F. (ed.): V Hotine-Marussi Symposium on Mathematical Geodesy, Matera, Italy, June 17-21, 2002. IAG Symposia, Vol. 127, Springer, Berlin-Heidelberg, 2004: 189-200.

[10] Holota P.: Successive approximations in the solution of weakly formulated geodetic boundary-value problem. In: Sans?, F. (ed.). A Window on the Future of Geodesy. IAG Gen. Assembly, Sapporo, Japan, June 30 - July 11, 2003. IAG Symposia, Vol. 128, Springer, Heidelberg-New York, 2005: 452-458.

[11] Holota P. and Nesvadba O.: 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 Symp. of Theoretical and Computational Geodesy, Wuhan, China, 29 May - 2 June, 2006. IAG Symposia, Vol. 132, Springer, Berlin-Heidelberg-New York, 2007: 314-320.

[12] Hörmander L.: The Boundary Problems of Physical Geodesy. The Royal Inst. of Technology, Division of Geodesy, Stockhlom, 1975; also in: Archive for Rational Mechanics and Analysis 62(1976): 1-52.

[13] John O., Nečas J.: Rovnice matematické fyziky. SPN, Praha, 1972.

[14] Kufner A., John O., Fučík, S.: Function Spaces. Academia, Prague, 1977.

[15] Moritz H.: Advanced Physical Geodesy. Herbert Wichmann Vlg. Karlsruhe, Abacus Press Tunbridge Wells Kent, 1980.

[16] Nádeník Z.: Kulové funkce pro geodézii ? Matematická příprava ke studiu knihy W.A. Heiskanen - H. Moritz: Physical Geodesy, 1967. Published by VÚGTK, Zdiby, 2000.

[17] Rektorys K.: Variační metody v inženýrských problémech a v problémech matematické fyziky, SNTL, Praha, 1974; also in English: Variational methods. Reidel Co., Dordrecht-Boston, 1977.

[18] Sanso F., Rummel R. (eds.): Geodetic Boundary Value Problems in View of the One Centimetr Geoid. Lecture Notes in Earth Sciences 65, Springer, Berlin-Heidelberg-New York, 1997.

Teaching methods -
Last update: T_KG (22.04.2013)

Lecture

Syllabus -
Last update: T_KG (22.04.2013)

Surface data sources on the gravity potential of the Earth's. Information on the gravitational potential from space geodetic methods.

General formulation of boundary value problems of potential theory in physical geodesy. Choice of the system of coordinates and the representation of the figure of the Earth as an embedding of the unit sphere in Euclidean three-dimensional space. Geodetic fixed, free and mixed boundary value problems - Stokes' and Molodensky's problem, gravimetric boundary value problem, altimetry-gravimetry boundary value problems. A note on problems in airborne gravimetry and satellite gradiometry.

The linearization of geodetic boundary value problems, infinitesimal and finite perturbations of an initial model of the figure and gravity field of the Earth - position anomaly, deflections of the vertical, telluroid, quasi-geoid, geoid and the tie to systems of heights. The loss of smoothness in iteration solutions.

Classical and modern methods in the solution of linear geodetic boundary value problems - integral equations method, Green's function method, principle of methods based on the concept of Hilbert spaces - function bases, variational and collocation methods. Successive approximations in the representation of topography effects.

Geodetic and geophysical importance of the subject, its history and present state, international co-operation in the field.

 
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