Some inverse problems of paleomagnetism

Some ancient rocks and meteorites possess remanent magnetization and thus store valuable record of the past magnetic field on Earth and other planets or satellites. Thanks to recent advances in magnetometry (SQUID microscope technique) offering the possibility to measure magnetic fields of very low intensities, extraction of this relict magnetic information has become reality. Deducing magnetization of a geosample hinges on processing the measurements of the weak magnetic field available in its nearest neighborhood. An endeavor to develop a robust and efficient method for an appropriate treatment of these data leads to a number of challenging problems such as effective extension of the restricted measurement data and extraction of certain features of the magnetization (typically, its mean value over the whole sample) without solving the entire inverse problem. 
In the present talk, I will mainly focus on the issues of estimation of the net moment of the planar compactly supported sample addressing it by means of Fourier analysis and Kelvin transforms. If time permits, I will also touch on the field extension issue leading to interesting eigenvalue problems for integral operators. Some numerical illustrations of the results will be provided.
This is a joint work with Juliette Leblond, Laurent Baratchart (APICS team, INRIA Sophia Antipolis - Mediterranee) and Eduardo Lima (Department of Earth, Atmospheric and Planetary Sciences, MIT).