Integration of a Priori Gravity Field Models in Boundary Element Formulations to Geodetic Boundary Value Problems
Current high resolution geopotential models of the Earth are based on a combination of satellite and terrestrial data. Satellite data are well-suited to recover the long-wavelength features of the geopotential up to some degree N, whereas terrestrial gravity and height data fix the medium and short wavelengths. Usually, the recovering of the medium and short-wavelengths from terrestrial data is formulated as a boundary value problem (BVP) for the difference between the Earth’s geopotential and the long-wavelength geopotential model as derived from satellite data commonly referred to as the disturbance potential. Since a number of geopotential coefficients of the satellite model cannot be improved by terrestrial data, we should fix them when solving the BVP. Then we are faced with a constrained (overdetermined) BVP for the Laplace equation. This has implications for the representation formula and/or the choice of the trial & test space in Galerkin boundary element methods.
We consider multipole representation, modified kernel functions, and modified trial spaces. The latter are the best choice for theoretical and numerical reasons. We propose a general method to construct a system of base functions that fix an a priori given set of geopotential coefficients. In addition, we address the problem of compression rates and stability, which implies the use of multiscale base functions. Various implementations are tested and compared for the altimetry-gravimetry II BVP.
KeywordsBoundary Element Method Geopotential Model Test Space Terrestrial Data Geopotential Coefficient
Unable to display preview. Download preview PDF.
- Hackbusch, W. (1995). Integral equations: theory and numerical treatment. Birkhäuser Verlag Basel Boston Berlin.Google Scholar
- Heck, B. (1997). Formulation and linearization of boundary value problems: from observables to a mathematical model. In Sansò, F. and Rummel, R., editors, Geodetic Boundary Value Problems in View of the One Centimeter Geoid, volume 65 of Lecture Notes in Earth Science. Springier Verlag, Berlin Heidelberg.Google Scholar
- Kleemann, B. H., Rathsfeld, A., and Schneider, R. (1996). Multiscale methods for boundary integral equations and their application to boundary value problems in scattering theory and geodesy. In W. Hackbusch, G. W., editor, Boundary Elements: Implementation and Analysis of Advanced Algorithms, volume 54 of Notes on Numerical Fluid Mechanics. Vieweg Verlag Braunschweig.Google Scholar
- Klees, R. (1997). Topics on boundary element methods. In F. Sansò, R.R., editor, Geodetic Boundary Value Problems in View of the One Centimeter Geoid, volume 65 of Lecture Notes in Earth Sciences. Vieweg Verlag Heidelberg.Google Scholar
- Klees, R., Lage, C, and Schwab, C. (1999). Fast numerical solution of the vector Molodensky problem. Proc. IV Hotine-Marussi Symposium on Mathematical Geodesy, Trento, Italy.Google Scholar
- Lehmann, R. (1997). Studies on the Use of Boundary Element Methods in Physical Geodesy. Publ. German Geodetic Commission, Series A, 113(70–84). Munich.Google Scholar
- Schneider, R. (1995). Multiskalen-und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung groβer vollbesetzter Gleichungssysteme. Habilitation thesis, Technical University Darmstadt.Google Scholar