Abstract. We study an inverse problem for the wave equation where localized wave sources in random scattering media are to be determined from time resolved measurements of the waves at an array of receivers. The sources are far from the array, so the measurements are affected by cumulative scattering in the medium, but they are not further than a transport mean free path, which is the length scale characteristic of the onset of wave diffusion that prohibits coherent imaging. The inversion is based on the Coherent Interferometric (CINT) imaging method which mitigates the scattering effects by introducing an appropriate smoothing operation in the image formation. This smoothing stabilizes statistically the images, at the expense of their resolution. We complement the CINT method with a convex (l 1 ) optimization in order to improve the source localization and obtain quantitative estimates of the source intensities. We analyze the method in a regime where scattering can be modeled by large random wavefront distortions, and quantify the accuracy of the inversion in terms of the spatial separation of individual sources or clusters of sources. The theoretical predictions are demonstrated with numerical simulations. Key words. waves in random media, coherent interferometric imaging, l 1 optimization, mutual coherence. 1. Introduction. Waves measured by a collection of nearby sensors, called an array of receivers, carry information about their source and the medium through which they travel. We consider a typical remote sensing regime with sources of small (point-like) support, and study the inverse problem of determining them from the array measurements. When the waves travel in a known and non-scattering (e.g. homogeneous) medium, the sources can be localized with reverse time migration [4, 5] also known as backprojection [20] . This estimates the source locations as the peaks of the image formed by superposing the array recordings delayed by travel times from the receivers to the imaging points. The accuracy of the estimates depends on the array aperture, the distance of the sources from the array, and the temporal support of the signals emitted by the sources. It may be improved under certain conditions by using l 1 optimization, which seeks to invert the linear mapping from supposedly sparse vectors of the discretized source amplitude on some mesh, to the array measurements. The fast growing literature of imaging with l 1 optimization in homogeneous media includes compressed sensing studies such as [19, 18] , synthetic radar imaging studies like [1, 8] , array imaging studies like [14] , and the resolution study [7] . In this paper we assume that the waves travel in heterogeneous media with fluctuations of the wave speed caused by numerous inhomogeneities. The amplitude of the fluctuations is small, meaning that a single inhomogeneity is a weak scatterer. However, there are many inhomogeneities that interact with the waves on their way from the sources to the receivers, and their scattering effect accumulates. Because in applications it is impossible to know the inhomogeneities in detail, and these cannot be estimated from the array measurements as part of the inversion, the fluctuations of the wave speed are uncertain. We model this uncertainty with a random process, and thus study inversion in random media. In this stochastic framework, the actual heterogeneous medium in which the waves propagate is one realization of the random model. The data measured at the array are uncertain and the question is how to mitigate the uncertainty to get images that are robust with respect to arbitrary medium realizations, i.e., they are statistically stable. The mitigation of uncertainty in the wave propagation model becomes important when the sources are further than a few scattering mean free paths from the array. The scattering mean free path is the length scale on which the waves randomize [23] , meaning that their fluctuations from one medium realization to another are large in comparison with their coherent (statistical expectation) part. The random wave distortions registered at the array are very different from additive and uncorrelated noise assumed usually in inversion. They are more difficult to mitigate and lead to poor and unreliable source reconstructions by coherent methods like reverse time migration or standard l 1 optimization. The Coherent Interferometric (CINT) method [9, 6] is designed to deal efficiently with such random distortions, as long as there is some residual coherence in the array measurements. This holds when the sources are separated from the array by distances (ranges) that are large with respect to the scattering mean free path, but do not exceed a transport mean free path, which is the distance at which the waves forget their initial direction [23] . The
It is shown in REF that the resolution of CINT images may be improved using convex optimization.
Imaging in random media with convex optimization
arXiv: Mathematical Physics
Mathematics Physics Computer Science