Wavelet modelling of the spherical inverse source problem with application to geomagnetism

Abstract

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.

Nonparametric regression on the hyper-sphere with uniform design

“…This problem has been addressed by a proliferating literature leading to the creation of a wide variety of wavelet frames intrinsic to the sphere (see, for example, Freeden et al 1998;Freeden and Michel 2004). These spherical wavelet frames have since then found many applications in modeling the Earth's magnetic field (Holschneider et al 2003;Maier 2005;Panet et al 2005), atmospheric flows (Fengler 2005), oceanographic flows (Freeden et al 2005), or ionospheric currents (Mayer 2004). At the same time, Narcowich and Ward (1996) introduced spherical basis functions (SBFs) and used them to design multiresolution analysis MRA of spherical signals.…”

Spherical decompositions in a global and local framework: theory and an application to geomagnetic modeling

“…To gain solvability of this problem, we assume that these tangential currents are located on a sphere R , with fixed radius R > 0, which is situated somewhere in the ionospheric E-region, generally at an altitude of about 115 km above the Earth-centered sphere of mean Earth radius R E = 6,371.2 km. In this setting, we can now relate the magnetic field measured at some altitude R sat with a tangential current system at altitude R tan = R sat via the spherical Biot-Savart operator, as has been done, e.g., in Mayer 2004. Observing the result of Fukushima 1976, i.e., tangential surface-curl-free currents produce no magnetic effect inside the sphere on which they are present, one can only hope to recover an equivalent current system j equiv via the inversion of the spherical Biot-Savart operator (in this context, 'equivalent' current system means that j equiv produces the measured magnetic field but does not necessarily represent the true current system in the ionosphere).…”

Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions

“…We show that we can construct orthogonal sets in L 2 (S 2 , S 1 ) with spherical harmonics as starting point. The described approach relies on [18,7,15,16].…”