Strukturierung des Studienplans für das Bachelorstudium (Entwurf: 18.01.2007)

1. Considerations about some methodological concepts in highly precise gravimetric geoid determination Bernhard Heck Geodetic Institute, University of Karlsruhe Englerstr. 7 D – 76128 Karlsruhe, Germany heck@gik.uni-karlsruhe.de

2. Strukturierung des Studienplans für das Bachelorstudium (Entwurf: 18.01.2007) Contents Historical remarks Geodetic boundary value problems Geoid determination and GBVP Towards a mathematically rigorous concept of geoid determination Errors in gravity anomaly data sets Questions – instead of final conclusions

3. Strukturierung des Studienplans für das Bachelorstudium (Entwurf: 18.01.2007) Historical remarks

4. Isaac Newton (1642 – 1727) : Front page of the first edition of the Principia (1686) Physical arguments: Earth in hydrostatic equilibrium  Ellipsoid of revolution

5. The Earth as a geoid: C.F. Gauß (1843/1846) G.G. Stokes (1849) J.B. Listing (1873) New concept: Reference surface = equipotential surface On the Variation of Gravity at the Surface of the Earth. By G.G. Stokes, M.A., Fellow of Pembroke College, Cambridge

7. The Earth with irregular boundary surface Boundary surface = topographic surface of the Earth Solution of a GBVP M.S. Molodenskii (1945 – 1960) Molodenskii, M.S.; Eremeev, V.F.; Yurkina, M.I: Methods for Study of the External Gravitational Field and Figure of the Earth. Transl. from Russian by the Israel Program for Scientific Translations for the Office of Technical Services, Jerusalem 1962 Further developments: Hirvonen, Moritz, Krarup, Sanso, Hörmander, Holota, Grafarend, …

8. Geodetic Boundary Value Problems

9. Geodetic boundary value problems Gravity field: gravity potential W W = V + Z V gravitational potential Z centrifugal potential Differential equation Lap V = - 4G Laplace-Poisson equation GBVP: Given: W - Wo and gravity vector grad W on the boundary surface S Unknown: W in external space of S and eventually geometry of S

10. „Fixed“ GBVP • S known (GPS positioning) • Given: grad W on S • Unknown: W in space external of S • (2) „Free“ GBVP • a) Vectorial free GBVP“ • S completely unknown • Given: W - Wo and grad W on S • Unknown: W in space external of S • and position vector of S • b) „Scalar free“ GBVP • ,  known (horizontal coordinates) • Given: W - Wo and • on S • Unknown: W in space external of S • and vertical coordinate (h) Classification of the GBVP: „free“, „non-linear“, „oblique“

11. Solution scheme for the GBVP Approximations: normal potential U, telluroid Approximation: Approximation: l X l ~ R = const. Analytical solution (integral formula) Free, non-linear GBVP Linearisation Linear GBVP Spherical approximation Linear GBVP in spherical approximation Constant radius approximation Spherical GBVP

12. The scalar free GBVP „Geodetic“ variant of the Molodensky problem Given on S: W(P) - Wo: Levelling + gravity Unknown: W (X) in space external of S W = V + Z Lap V = 0 h=HN+

13. The scalar free GBVP Reference for linearisation: U Normal potential of a level ellipsoid HNNormal height; postulate: Wo - W(P) = Uo - U(Q) (telluroid mapping) HN (P) = h(Q) Decomposition W = U + T T: Disturbing potential h = HN+ : height anomaly (=quasigeoidal height) h=HN+

14. The scalar free GBVP Linearisation:  = T(Q)/ (Q) Bruns formula Fundamental equation Analytical solution (series expansion) ~ terrain correction h=HN+

15. Analytical approximation errors in the GBVP Linearisation Non-linear terms in the boundary condition Spherical approximation ellipsoidal terms in the boundary condition topographical terms in the boundary condition Planar approximation omission of terms of order (h/R) ~ 10-3 Constant radius approximation ~ downward continuation effect, Molodensky‘s series terms

16. Evaluation of the non-linear boundary condition (North America) • True field ~ EIGEN_GL04C; Nmax = 360 • Topography model: GTOPO30 • Runtime: user 1d 18h (K. Seitz) • Output: • non-linear BC • non-linear effects in the BC • Coordinates of the telluroid points • (input for ellipsoidal effects) Statistics [mGal]Min Max Mean L1 L2 Linear BC -244.885 229.076 -8.246 20.011 25.884 Non-linear BC -245.197 229.235 -8.246 20.018 25.895 Non-linear effect -0.326 0.259 0.000 0.011 0.018 Zeta (P-Q) [m] -62.631 13.516 -29.888 29.967 32.066

20. Ellipsoidal correction δNE = δTE(rE(φ,λ), φ, λ))/γ(φ) in m, 0 ≤ m ≤ n ≤ 360 (Hammer equal-area projection) Heck, B. and Seitz, K. (2003): Solutions of the linearized geodetic boundary value problem for an ellipsoidal boundary to order e3. JGeod, 77, 182-192. DOI 10.1007/s00190-002-0309-y.

21. Power spectrum of δNE(in m2) and T (in m4s-4) Heck, B. and Seitz, K. (2003): Solutions of the linearized geodetic boundary value problem for an ellipsoidal boundary to order e3. JGeod, 77, 182-192. DOI 10.1007/s00190-002-0309-y.

22. Numerical approximation errors Evaluation of surface integrals: - Stokes integral - Terrain correction - Molodensky‘s series terms of higher order - Poisson integral and derivatives - ……… Truncation error Integration over spherical cap, neglection of outer zone Modified integral kernels Numerical evaluation by FFT (gridded data) Finite region - boundary effects, periodic continuation (zero padding) 2D FFT - neglection of sphericity (1D FFT for large regions) Aliasing, etc.