# Mathematical and numerical study of the inverse problem of electro-seismicity in porous media

#### Speciality : Mathématiques Appliquées

20/12/2017 - 15:00 Mr Qi Xue Salle 2 - RDC - Batiment IMAG

We study the inverse problem of the coupling phenomenon of electromagnetic (EM) and seismic waves. Partial differential equations governing the coupling phenomenon are composed of Maxwell and Biot equations. Since the coupling phenomenon is rather weak, in low frequency we only consider the transformation from EM waves to seismic waves. We use electroseismic model to refer to this transformation. In the model, the electric field becomes the source of Biot equations. A coupling coefficient is used to denote the efficiency of the transformation. To study the stability of the inverse problem, a Carleman estimate is derived for the electroseismic system. We proved that the reconstruction of electric parameters are Holder stable from boundary measurements. If our region of interest (a few hundred meters) is much less than the EM wavelength (thousands of meters), the electrostatic equation can be used instead of the whole Maxwell equations. In such a case, the inverse problem consists two step: inverse source problem for Biot equations and inverse parameters problem for electrostatic equation. We prove the stability of these two step inversion. At last we show some numerical experiments of the inverse source problem for Biot equations.

#### Directors:

- Mr Eric Bonnetier
*(Professeur - Université Grenoble Alpes )* - Mr Faouzi Triki
*(Maître de Conférénces - Université Grenoble Alpes )*

#### Raporteurs:

- Mr Gang Bao
*(Professeur - Université de Zhejiang )* - Mr Marteen De Hoop
*(Professor - Université Rice à Houston )*

#### Examinators:

- Mr Franck Boyer
*(Professeur - Université Toulouse III - Paul Sabatier )* - Mr Michel Dietrich
*(Directeur de recherche - CNRS )* - Mr Didier Auroux
*(Professeur - Université Nice Sophia Antipolis )*