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In a heterogeneous medium, the wavefield can be decomposed as an infiniteseries known as the Born expansion. Each term of the Born expansion correspondsto a scattering order, it is thus theoretically possible to discriminate singleand multiple scattering field components. Experimentally, what is actuallymeasured is the total field in which all scattering orders interfere.Conventional imaging methods usually rely on the assumption that the multiplescattering contribution can be disregarded. In a back-scattering configuration,this assumption is valid for small depths, and begins to fail for depths largerthan the scattering mean-free path $\ell_s$. It is therefore a key issue toestimate the relative amount of single and multiple scattering in experimentaldata. To this end, a single scattering estimator $\hat{\rho}$ computed from thereflection matrix has been introduced in order to assess the weight of singlescattering in the backscattered wavefield. In this article, the meaning of thisestimator is investigated and a particular attention is given to recurrentscattering. In a diffraction-limited experiment, a multiple scattering sequenceis said to be recurrent if the first and last scattering events occur in thesame resolution cell. Recurrent scattering is shown to be responsible forcorrelations between single scattering and higher scattering orders of the Bornexpansion, inducing a bias to the estimator $\hat{\rho}$ that should rather betermed confocal scattering ratio. Interestingly, a more robust estimator isbuilt by projecting the reflection matrix in a focused basis. The argument issustained by numerical simulations as well as ultrasonic data obtained around1.5~MHz in a model medium made of nylon rods immersed in water. From a moregeneral perspective, this work raises fundamental questions about the impact ofrecurrent scattering on wave imaging.

Weight of single and recurrent scattering in the reflection matrix of complex media