ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 45B05 Michel, Volker, University of Kaiserslautern, Germany A New Way of Reconstructing the Density of the Earth From Measurements of the Gravitational Potential The gravimetry problem of physical geodesy is concerned with the inversion of Newton’s gravitational potential

V (y) = ∫_{\overline{B_{int}}} \frac{F (x)}{|x − y|} dx,

where \overline{B_{int}} := \{x ∈ ℝ^{3} | |x| ≤ β\}. Unfortunately the solution is not unique. More precisely there exists a decomposition L^{2}(\overline{B_{int}}) = H(\overline{B_{int}}) ⊕ H(\overline{B_{int}})^{⊥}, where H(\overline{B_{int}}) is the set of distributionally harmonic functions on \overline{B_{int}}. Functions in the orthogonal space are called anharmonic. It is already known that the corresponding integral operator is only injective on H(\overline{B_{int}}). Hence the presented way to solve the problem consists of two steps: Step 1: The harmonic part is reconstructed from V using new wavelet methods. Step 2: Based on a-priori informations the anharmonic part is determined in a subset of H(\overline{B_{int}})^{⊥} using a special spline concept.