Mar 27

PhD: Inverse problems of magnetic moments recovery from partial field data.

In geosciences and paleomagnetism, estimating the remanent magnetization in old rocks is an important issue to study past evolution of the Earth and other planets or bodies. However, the magnetization cannot be directly measured and only partial values of the magnetic field that it produces can be recorded.

We first consider the case of thin samples, modeled as a planar set S of R^2 x {0}, carrying a magnetization m (a 3-dimensional vector field supported on S). This setup is typical of scanning microscopy that was developed recently to measure a single component of a weak magnetic field, close to the sample. Specifically, one is given a record of b_3[m] (tiny: a few nano Teslas), the vertical component of the magnetic field produced by m, on a planar region Q of R^2 x {h} located at some fixed height h > 0 above the sample plane. We assume that both S and Q are smooth enough bounded connected open sets in their respective planes, and that the magnetization m belongs to [L^2(S)]^3, whence b_3[m] belongs to L^2(Q). Such magnetizations possess net moments <m> (belonging to R^3) defined as their integral on S.

Recovering the magnetization m or its net moment <m> from available measurements of b_3[m] are inverse problems for the Poisson-Laplace equation in the upper half-space  R^3_+ with right hand side in divergence form. Indeed, Maxwell’s equations in the quasi-static approximation identify the divergence of m with the Laplacian of a scalar magnetic potential in R^3_+ whose normal derivative on Q coincides with b_3[m]. Hence Neumann data b_3[m] are available on Q (subset of R^3_+), and we aim at recovering m or <m> on S. We thus face recovery issues on the boundary of the harmonicity domain from (partial) data available inside.

Such inverse problems are typically ill-posed and call for regularization. Indeed, magnetization recovery is not even unique, due to the existence of non identically zero silent sources such that b_3[m] = 0. And though such sources have vanishing moment so that net moment recovery is unique, estimation of the latter turns out to be unstable with respect to measurements errors.

The present topic is mainly concerned with moment recovery, which brings some very useful information for magnetization estimation. This will be approach using best constrained approximation (bounded extremal problems) that furnishes linear estimators for the components of the net moment. Against these functions, the scalar products with the data allow to estimates these components. Their numerical computations, in particular following an iterative algorithm, will be the main issue of the PhD.
They will be tested on actual data from our physicists partners. Indeed, this work will take place in our associate team Impinge (http://www-sop.inria.fr/apics/IMPINGE/) between our team Apics and the Laboratory “Earth, Atmospheric and Planetary Sciences” at MIT (MA, USA). Impinge stands for “Inverse Magnetization Problems In Geosciences”. People of the Center for Constructive Approximation at Vanderbilt University (TN, USA) are also associated to the project.

Further, comparisons will be made with available asymptotic formula. Next, situations of 3 dimensional rocks samples will be considered.

The candidates must have an excellent background and cursus in mathematics and/or applied mathematics, know programming languages for numerical computations (C++, matlab, …), have at interests and even better some knowledge in physical issues.

Short bibliography (the references available at http://www-sop.inria.fr/apics/IMPINGE/):

– L. Baratchart, S. Chevillard and J. Leblond. Silent and equivalent magnetic distributions on thin plates. To appear in Theta Series in Advanced Mathematics.

– L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff and B. P. Weiss. Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions. Inverse Problems, 29(1), 2013.

– E. A. Lima, B. P. Weiss, L. Baratchart, D. P. Hardin and E. B. Saff. Fast inversion of magnetic field maps of unidirectional planar geological magnetization. Journal of Geophysical Research: Solid Earth, 118(6), 2013.

Contacts: Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, team APICS, INRIA Sophia Antipolis (see http://team.inria.fr/apics/members/).