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#### Keywords

- Biot-Savart Operator (1)
- Biot-Savart operator (1)
- Inverses Problem (1)
- Molodensky Problem (1)
- Molodensky problem (1)
- Regularisierung (1)
- Vektorkugelfunktionen (1)
- Vektorwavelets (1)
- Wavelet (1)
- Wavelet Analysis auf regulären Flächen (1)
- harmonic scaling functions and wavelets (1)
- multiscale approximation on regular telluroidal surfaces (1)
- vector spherical harmonics (1)
- vectorial wavelets (1)

Based on the well-known results of classical potential theory, viz. the limit and jump relations for layer integrals, a numerically viable and e±cient multiscale method of approximating the disturbing potential from gravity anomalies is established on regular surfaces, i.e., on telluroids of ellipsoidal or even more structured geometric shape. The essential idea is to use scale dependent regularizations of the layer potentials occurring in the integral formulation of the linearized Molodensky problem to introduce scaling functions and wavelets on the telluroid. As an application of our multiscale approach some numerical examples are presented on an ellipsoidal telluroid.

The article is concerned with the modelling of ionospheric current systems from induced magnetic fields measured by satellites in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered vectorial operator equation. Based on the knowledge of the singular system a regularization technique in terms of certain product kernels and corresponding convolutions can be formed. In order to reconstruct ionospheric current systems from satellite magnetic field data, an inversion of the Biot-Savart's law in terms of multiscale regularization is derived. The corresponding operator is formulated and the singular values are calculated. The method is tested on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.