Section: Scientific Foundations

Inverse Problems

  • Inverse scattering problems. 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 [77] . Moreover the precision in the reconstruction of the shape of an obstacle strongly depends on the quality of the given far-field pattern (FFP) measurements: the range of the measurements set and the level of noise in the data. Indeed, the numerical experiments (for example  [88] , [91] , [84] , [85] ) performed in the resonance region, that is, for a wavelength that is approximately equal to the diameter of the obstacle, tend to indicate that in practice, and at least for simple shapes, a unique and reasonably good solution of the IOP can be often computed using only one incident wave and full aperture far-field data (FFP measured only at a limited range of angles), as long as the aperture is larger than π. For smaller apertures the reconstruction of the shape of an obstacle becomes more difficult and nearly impossible for apertures smaller than π/4.

    This plus the fact that from a mathematical viewpoint the FFP can be determined on the entire sphere S1 from its knowledge on a subset of S1 because it is an analytic function, we propose [74] , [75] a solution methodology to extend the range of FFP data when measured in a limited aperture and not on the entire sphere S1. It is therefore possible to solve the IOP numerically when only limited aperture measurements are available. The objective of Magique-3D  is to extend this work to 3D problems of acoustic scatterin and to tackle the problem of elasto-acoustic scattering.

  • Depth Imaging in the context of DIP. The challenge of seismic imaging is to obtain the best representation of the subsurface from the solution of the full wave equation that is the best mathematical model according to the time reversibility of its solution. The most used technique of imaging is RTM (Reverse Time Migration), [76] , which is an iterative process based on the solution of a collection of wave equations. The high complexity of the propagation medium requires the use of advanced numerical methods, which allows one to solve several wave equations quickly and accurately. The research program DIP has been defined by researchers of Magique-3D  and engineers of Total jointly. It has been created with the aim of gathering researchers of Inria, with different backgrounds and the scientific programm will be coordinated by Magique-3D . In this context, Magique-3D  will contribute by working on the inverse problem and by continuing to develop new algorithms in order to improve the RTM.