## Section: New Results

### Inverse problems

#### Inverse problems in submarine acoustics

Participants : Ă‰douard Canot, Samih Zein.

In the fourth part of the IFREMER contract (see section 8.1.6 ), we consider a new seismic inverse problem; we seek to invert seismic traveltime measures from an available seismic survey to find the wave velocity underground. Our inversion method is based on ray tracing and genetic algorithms.

We studied the seismic ray theory which may be applied to wave propagation problems by doing some approximations. We developed a code (EIKOLIN) which is more suitable for our application than another existing code available from Internet (ANRAY). EIKOLIN is not expensive in computation because only linear interpolations are used (instead of cubic B-Spline for ANRAY). This feature let us process the inversion using genetic algorithms which requires a great number of simulations.

This code has been applied to wide angle seismic data taken from a campaign made in Morocco (Ifremer 2002). Figure 1 shows the wave paths (seismic rays) in a cross section of the underground. These rays are refracted when they cross the interface between the two layers (direct and reflected rays are not considered). Note that the scale is not the same in the x- and z- directions. In the lower layer, the rays are curved due to the presence of a velocity gradient. Figure 2 represents the solution of our inverse problem: it shows the velocity distribution in a cross section of the underground.

A SVD-based sensitivity analysis has been carried out. It gives the identifiable parameter of the velocity from the seismic inversion: we found that only the wave velocity of the lower layer and near the interface can be recovered.

Report [57] is related to the previous part of this contract whereas publications [54] , [53] and [58] are related to this work.

#### Inverse problems in geodesy

Participants : Amine Abdelmoula, Bernard Philippe.

This work was done in the context of the STIC/Tunisia project (see 8.3.1 ), in collaboration with ENIT. The geoid is the level surface of the earth attraction at the sea level. That surface is obtained as a correction of a regular surface by fitting existing measures. The problem ends up with a large structured generalized least squares problem. Therefore, we plan to apply our algorithms on QR factorizations ( 6.1.4 ). During that year, a Matlab chain of treatments has been developed from existing techniques. Its reliability is presently tested on public data. One of the goals is to apply it to data from Tunisian services in order to recover parts of the Tunisian geoid [17] . The main research direction on which we now focus, is the determination of an equivalent mass system which can generate a given geoid. That type of inverse problems meets common interest of the SAGE team and of the LAMSIN team as well.

#### Inverse problems in electrocardiography

Participant : Jocelyne Erhel.

This work was done in the context of the STIC/Tunisia project (see 8.3.1 ), in collaboration with ENIT. We have designed two different methods for recovering missing data on the heart surface, from electrical measures done on the body surface. Both methods recover the epicardial potential and flux. One method is based on a self-regularizing energy-like error functional, uses two direct problems and their adjoint to compute the gradient of the functional. The other method is based on a classical measurement to computations misfit function, uses a boundary integral model and a regularized generalized least-squares formulation. Numerical experiments highlight the efficiency and the robustness of the two methods (paper in preparation).