We are interested here in the reconstruction of inclusions in highly heterogeneous unknown media. The medium is probed by using classical acoustic waves. When fluctuations become
too strong, inversion methodologies based on a microscopic
description of wave propagation become strongly dependent on the unknown details of the
medium. In some situations, it is therefore preferable to use a
macroscopic model for a quantity that is quadratic in the wave
fields. Here, such macroscopic models take the form of radiative
transfer equations. We then replace an inverse problem based on the microscopic wave equation by an inverse problem based on the macroscopic transport equations. In particular:
Imaging models and reconstructions
- We show that transport and diffusion equations indeed accurately describe the propagation of the energy of high-frequency waves in a random medium and correctly captures the presence of an inclusion. A stress is put on the statistical stability, that is how much the energy depends on the realizations of the random medium .
- Imaging models based on the wave energy, or on the correlation of the wavefield in presence of an inclusion and the wavefield in absence of an inclusion are constructed. We also discuss the effect of small inclusions on the data .
- It is shown that time reversal measurements enjoy a larger signal-to-noise ratio in the presence of background noise than do direct energy measurements .
- We perform reconstructions of inclusions in several settings, in particular one for which the inclusion is hidden behind a blocker with no line-of-sight .