Multiple scattering of ultrasound in weakly inhomogeneous media: application to human soft tissues

Waves scattered by a weakly inhomogeneous random medium contain a predominant single scattering contribution as well as a multiple scattering contribution which is usually neglected, especially for imaging purposes. A method based on random matrix theory is proposed to separate the single and multiple scattering contributions. The experimental set up uses an array of sources/receivers placed in front of the medium. The impulse responses between every couple of transducers are measured and form a matrix. Single-scattering contributions are shown to exhibit a deterministic coherence along the antidiagonals of the array response matrix, whatever the distribution of inhomogeneities. This property is taken advantage of to discriminate single from multiple-scattered waves. This allows one to evaluate the absorption losses and the scattering losses separately, by comparing the multiple scattering intensity with a radiative transfer model. Moreover, the relative contribution of multiple scattering in the backscattered wave can be estimated, which serves as a validity test for the Born approximation. Experimental results are presented with ultrasonic waves in the MHz range, on a synthetic sample (agar-gelatine gel) as well as on breast tissues. Interestingly, the multiple scattering contribution is found to be far from negligible in the breast around 4.3 MHz.Comment: 35 pages, 11 figures, final version, contains the appendix of the original articl

Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $\ell^*$, scattering phase function $f$ and anisotropy factor $g$. Discarding the operator term in the wave equation is shown to have a significant impact on $f$ and $g$, yet limited to the low-frequency regime i.e., when the correlation length of the disorder $\ell_c$ is smaller than or comparable to the wavelength $\lambda$. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path $\ell^*$ whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around $300\,\%$ in the low-frequency regime, and still above $30\,\%$ for $\ell_c/\lambda=10^3$ no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations

Fluctuations of correlations and Green's function reconstruction: role of scattering

Correlations of ambient seismic or acoustic vibrations are now widely used to reconstruct the impulse response between two passive receivers as if a source was placed at one of them. This provides the opportunity to do imaging without a source, or \textsl{passive imaging}. Applications include terrestrial and solar seismology, underwater acoustics, and structural health monitoring, to cite only a few. Nevertheless, for a given set of data, correlations do not only yield the Green's function between the sensors. They also contain residual fluctuations that result from an imperfect time or source averaging that might eventually blur the images. In this article, we propose a heuristic model to describe the level of fluctuations of the correlations in the case of non-stationary wavefields, and more particularly in the case of scattering media. The work includes theoretical derivations and numerical simulations. The role of multiple scattering is quantitatively evaluated. The level of fluctuations decreases when the duration and intensity of the diffuse waves increase. The role of absorption is also discussed: absorption is properly retrieved by correlation, but the level of fluctuations is greater, thus degrading the Green's function reconstruction. Discrepancies of our simple model in the case of strong multiple scattering ($k\ell^*\leq 18$) are discussed

Recurrent scattering and memory effect at the Anderson localization transition

We report on ultrasonic measurements of the propagation operator in a strongly scattering mesoglass. The backscattered field is shown to display a deterministic spatial coherence due to a remarkably large memory effect induced by long recurrent trajectories. Investigation of the recurrent scattering contribution directly yields the probability for a wave to come back close to its starting spot. The decay of this quantity with time is shown to change dramatically near the Anderson localization transition. The singular value decomposition of the propagation operator reveals the dominance of very intense recurrent scattering paths near the mobility edge.Comment: 5 pages, 4 figure

Detection and imaging in a random medium: A matrix method to overcome multiple scattering and aberration

We present an imaging technique particularly suited to the detection of a target embedded in a strongly scattering medium. Classical imaging techniques based on the Born approximation fail in this kind of configuration because of multiply scattered echoes and aberration distortions. The experimental set up we consider uses an array of programmable transmitters/receivers. A target is placed behind a scattering medium. The impulse responses between all array elements are measured and form a matrix. The core of the method is to separate the single-scattered echo of the target from the multiple scattering background. This is possible because of a deterministic coherence along the antidiagonals of the array response matrix, which is typical of single scattering. Once this operation is performed, target detection is achieved by applying the DORT method (French acronym for decomposition of the time reversal operator). Experimental results are presented in the case of wide-band ultrasonic waves around 3 MHz. A 125-element array is placed in front of a collection of randomly distributed steel rods (diameter 0.8mm). The slab thickness is three times the scattering mean free path. The target is a larger steel cylinder (diameter 15 mm) that we try to detect and localize. The quality of detection is assessed theoretically based on random matrix theory and is shown to be significantly better than what is obtained with classical imaging methods. Aside from multiple scattering, the technique is also shown to reduce the aberrations induced by an heterogeneous layer.Comment: 48 pages, 18 figures, corrected typos, v