Definition
Boundary Value Problem : the combination of a differential equation, i.e., the Laplace equation, given on a domain which has a boundary together with geodetically reflected observations for the solution at this boundary.
Introduction
In the past, the determination of the gravity potential in the exterior of the Earth from a boundary function could not be considered a classical boundary value problem of potential theory as the boundary surface itself was unknown. A way out was the choice of an approximate surface. This led to two alternative approaches of the so-called geodetic boundary value problem . The traditional concept was conceived by G. G. Stokes (1849), the other by M. S. Molodensky (1960). Stokes proposed reducing the data from the Earth’s surface to the geoid (see Figure 1). The difference between the reduced gravity on the geoid and the reference gravity on an ellipsoid is called the gravity anomaly. It constitutes the boundary data (see Figure 1). The...
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References
Freeden, W., and Gerhards, C., 2012. Geomathematically Oriented Potential Theory. Boca Raton: Chapman & Hall/CRC Press.
Freeden, W., and Kersten, H., 1980. The geodetic boundary-value problem using the known surface of the earth. Veröffentlichungen des Geodätischen Instituts der Rheinisch-Westfälischen Technischen Hochschule Aachen, 29.
Freeden, W., and Kersten, H., 1981. A constructive approximation theorem for the oblique derivative problem in potential theory. Mathematical Methods in the Applied Sciences, 3, 104–114.
Freeden, W., and Mayer, C., 2006. Multiscale solution for the Molodensky problem on regular telluroidal surfaces. Acta Geodaetica et Geophysica Hungarica, 41, 55–86.
Grafarend, E. W., Klapp, M., and Martinec, Z., 2015. Spacetime modelling of the Earth’s gravity field by ellipsoidal harmonics. In Freeden, W., Nashed, Z., and Sonar, T. (eds.), Handbook of Geomathematics, 2nd edn., p. 381–496, Heidelberg: Springer.
Grothaus, M., and Raskop, T., 2015. Oblique stochastic boundary-value problem. In Freeden, W., Nashed, Z., and Sonar, T. (eds.), Handbook of Geomathematics, 2nd edn., p. 2285–2316, Heidelberg: Springer.
Hofmann-Wellenhof, B., and Moritz, H., 2006. Physical Geodesy, 2nd edn. Wien/New York: Springer.
Hörmander, L., 1976. The boundary problems of physical geodesy. Archive for Rational Mechanics and Analysis, 62, 1–52.
Koch, K. R., and Pope, A. J., 1972. Uniqueness and existence for the geodetic boundary value problem using the known surface of the Earth. Bulletin Géodésique, 106, 467–476.
Molodensky, M. S., Eremeev, V. F., and Yurkina, M. I., 1960. Methods for study of the external gravitational field and figure of the Earth. Trudy TsNIIGAiK, Geodezizdat, Moscow, 131, 1960. (English translat.: Israel Program for Scientific Translation, Jerusalem, 1962).
Moritz, H., 1989. Advanced Physical Geodesy, 2nd edn. Karlsruhe: Wichmann.
Rummel, R., 1992. Geodesy. In Nierenberg, W. A. (ed.), Encyclopedia of Earth System Science. San Diego: Academic Press, Vol. 2, pp. 253–262.
Sansò, F., 2015. Geodetic boundary value problem. In Freeden, W., Nashed, Z., and Sonar, T. (eds.), Handbook of Geomathematics, 2nd edn., p. 291–320, Heidelberg: Springer.
Stokes, G. G., 1849. On the variation of gravity on the surface of the Earth. Transactions of the Cambridge Philosophical Society, 8, 672–695.
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Augustin, M., Freeden, W. (2015). A Survey on Classical Boundary Value Problems in Physical Geodesy. In: Grafarend, E. (eds) Encyclopedia of Geodesy. Springer, Cham. https://doi.org/10.1007/978-3-319-02370-0_117-1
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DOI: https://doi.org/10.1007/978-3-319-02370-0_117-1
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