Multiscale analysis of wave propagation in random media. Application to correlation-based imaging

We consider sensor array imaging with the purpose to image reflectors embedded in a medium. Array imaging consists in two steps. In the first step waves emitted by an array of sources probe the medium to be imaged and are recorded by an array of receivers. In the second step the recorded signals are processed to form an image of the medium. Array imaging in a scattering medium is limited because coherent signals recorded at the receiver array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. If, however, an auxiliary passive (receiver) array can be placed between the reflector to be imaged and the scattering medium then the cross correlations of the incoherent signals on this array can also be used to image the reflector. This situation is important in particular in oil reservoir monitoring when auxiliary receivers can be implemented in wells and its study requires a multiscale analysis of wave propagation in random media. In this review we describe the results obtained in two recent papers using multiscale analysis of wave propagation in random media. In [J. Garnier and G. Papanicolaou, Inverse Problems 28 (2012), 075002] we show that if (i) the source array is infinite, (ii) the scattering medium is modeled by either an isotropic random medium in the paraxial regime or a randomly layered medium, and (iii) the medium between the auxiliary array and the object to be imaged is homogeneous, then imaging with cross correlations completely eliminates the effects of the random medium. It is as if we imaged with an active array, instead of a passive one, near the object. In [J. Garnier and G. Papanicolaou, SIAM J. Imaging Sci. 7 (2014), 1210] we analyze the resolution of the image when both the source array and the passive receiver array are finite. We show that for isotropic random media in the paraxial regime, imaging not only is not affected by the inhomogeneities but the resolution can in fact be enhanced. This is because the random medium can increase the diversity of the illumination. We also show analytically that this does not happen in a randomly layered medium, and there may be some loss of resolution in this case. ∗Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris Diderot, 75205 Paris Cedex 13, France garnier@math.univ-paris-diderot. fr Séminaire Laurent-Schwartz — EDP et applications Institut des hautes études scientifiques, 2013-2014 Exposé no XIII, 1-19