## Section: New Results

### Inverse Problems

#### Reconstruction of an elastic scatterer immersed in a homogeneous fluid

Participants : Hélène Barucq, Rabia Djellouli, Élodie Estecahandy.

The determination of the shape of an obstacle from its effects on known acoustic or electromagnetic waves is an important problem in many technologies such as sonar, radar, geophysical exploration, medical imaging and nondestructive testing. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because it is ill-posed and nonlinear. Its investigation requires as a prerequisite the fundamental understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods.

In this work, we are interested in retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics of the surface of the obstacle. The corresponding direct elasto-acoustic scattering problem consists in the scattering of time-harmonic acoustic waves by an elastic obstacle ${\Omega}^{s}$ embedded in a homogeneous medium ${\Omega}^{f}$, that can be formulated as follows:

$\begin{array}{cc}\hfill \Delta p+({\omega}^{2}/{c}_{f}^{2})\phantom{\rule{0.166667em}{0ex}}p=0& \phantom{\rule{14.22636pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega}^{f}\\ \hfill \nabla \xb7\sigma \left(u\right)+{\omega}^{2}{\rho}_{s}\phantom{\rule{0.166667em}{0ex}}u=0& \phantom{\rule{14.22636pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega}^{s}\\ \hfill {\omega}^{2}{\rho}_{f}u\xb7n=\partial p/\partial n+\partial {e}^{i\phantom{\rule{0.166667em}{0ex}}(\omega /{c}_{f})\phantom{\rule{0.166667em}{0ex}}x\xb7d}/\partial n& \phantom{\rule{14.22636pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\Gamma \\ \hfill \sigma \left(u\right)n=-pn-{e}^{i\phantom{\rule{0.166667em}{0ex}}(\omega /{c}_{f})\phantom{\rule{0.166667em}{0ex}}x\xb7d}n& \phantom{\rule{14.22636pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\Gamma \\ \hfill \underset{r\to +\infty}{lim}r\left(\partial p/\partial r-i\phantom{\rule{0.166667em}{0ex}}(\omega /{c}_{f})\phantom{\rule{0.166667em}{0ex}}p\right)=0\end{array}$ | (1) |

where $p$ is the fluid pressure in ${\Omega}^{f}$ whereas $u$ is the displacement field in ${\Omega}^{s}$, and $\sigma \left(u\right)$ represents the stress tensor of the elastic material.

This boundary value problem has been investigated mathematically and results pertaining to the existence, uniqueness and regularity can be found in [86] and the references therein, among others. We propose a solution methodology based on a regularized Newton-type method for solving the IOP. The proposed method is an extension of the regularized Newton algorithm developed for solving the case where only Helmholtz equation is involved, that is the acoustic case by impenetrable scatterers [79] . The direct elasto-acoustic scattering problem defines an operator $F:\Gamma \to {p}_{\infty}$ which maps the boundary $\Gamma $ of the scatterer ${\Omega}^{s}$ onto the far-field pattern ${p}_{\infty}$. Hence, given one or several measured far-field patterns ${\tilde{p}}_{\infty}\left(\widehat{x}\right)$, corresponding to one or several given directions $d$ and wavenumbers $k$, one can formulate IOPs as follows:

We propose a solution methodology based on a regularized Newton-type method to solve this inverse obstacle problem. At each Newton iteration, we solve the forward problem using a finite element solver based on discontinuous Galerkin approximations, and equipped with high-order absorbing boundary conditions. We have first characterized the Fréchet derivatives of the scattered field. They are solution to the same boundary value problem as the direct problem with other transmission conditions. This work has been presented both in FACM11 and in WAVES 2011. A paper has been submitted.

#### $hp$-adaptive inversion of magnetotelluric measurements

Participants : Hélène Barucq, Julen Alvarez Aramberri, David Pardo.

The magnetotelluric (MT) method is a passive electromagnetic (EM) exploration technique that allows to determine the resistivity distribution in the subsurface of the area of interest on scales varying from few meters to hundreds of kilometers. Commercial uses include hydrocarbon (oil and gas) exploration, geothermal exploration, and mining exploration, as well as hydrocarbon and groundwater monitoring. MT measurements are governed by the electromagnetic phenomena, which can be described by Maxwell's equations. We solve those equations by a goal-oriented hp-adaptivity Finite Element Method (FEM).

In order to estimate the resistivity distribution in the Earth's subsurface, we solve an Inverse Problem. We define a Misfit Function that represents the difference between the measured and computed data for a particular resistivity distribution. By minimizing this misfit function using a gradient based approach with model reduction techniques, and hence solving the inverse problem, we are able to determine the properties of the subsurface materials.