This work presents experimental and theoretical results concerning the dispersion and attenuation caused by scattering during the propagation of ultrasonic waves on the surface of a polycrystal. Rayleigh and head waves are measured in the case of two Inconel® 600 samples with different average grain sizes. The coherent, i.e., ensemble-averaged, waves are estimated, as well as their frequency-dependent phase velocities and scattering mean-free paths. The results obtained from a contactless laser setup are compared to those obtained from a transducer array placed on the surface of the sample. The influence of contact is highlighted, particularly at low frequency and in the small-grained sample, where the attenuation by scattering is lower. Moreover, the two-point correlation (TPC) functions of both samples are estimated, and it is shown that neither is exponential. Standard theoretical models are adapted to these particular TPCs and yield effective bulk wavenumbers, from which effective surface wavenumbers can be calculated via a simple and approximate method. The theoretical results are then compared to the experimental ones.

We present a physically intuitive matrix approach for quantitative imaging of complex media. While standard reflection imaging methods generally rely on confocal focusing operations, matrix imaging consists of decoupling the location of the incident and received focal spots [1]. Following this principle, a self-portrait of the focusing process can be obtained around each point of the medium. The Gouy phase shift exhibited by each focal spot can be leveraged to finely monitor the wave velocity distribution inside the medium [2]. A local multiple scattering rate can also be evaluated and its depth evolution can lead to a local measurement of the scattering mean free path, independently from absorption losses [3]. The approach is here demonstrated with ultrasound for a controllable phantom system before being applied in vivo to liver. The wave velocity and the scattering mean free path are quantitative markers for biomedical diagnosis but they are also important monitoring parameters for nondestructive testing and geophysical applications. This work thus opens important perspectives for quantitative imaging of heterogeneous media in all fields of acoustics. [1] W. Lambert etal., Phys. Rev. X 10, 021048 (2020). [2] F. Bureau et al., arXiv:2409.13901 (2024). [3] A. Goicoechea et al., Phys. Rev. Lett. 133, 176301 (2024).

Multiple scattering of waves presents challenges for imaging complex media but offers potential for their characterization. Its onset is actually governed by the scattering mean free path ℓ_{s} that provides crucial information on the medium microarchitecture. Here, we introduce a reflection matrix method designed to estimate this parameter from the time decay of the single scattering rate. Our method is first validated by an ultrasound experiment on a tissue-mimicking phantom before being applied in vivo to a human liver. This Letter opens important perspectives for quantitative imaging of heterogeneous media with waves, whether it be for nondestructive testing, biomedical, or geophysical applications.

In a heterogeneous medium, the wave field can be decomposed as an infinite series known as the Born expansion. Each term of the Born expansion corresponds to a scattering order, it is thus theoretically possible to discriminate single and multiple scattering contribution to the field. Experimentally, what is actually measured is the total field in which all scattering orders interfere. Conventional imaging methods usually rely on the assumption that the multiple scattering contribution can be disregarded. In a back-scattering configuration, this assumption is valid for small depths, and begins to fail for depths larger than the scattering mean-free path ℓ_{s}. It is therefore a key issue to estimate the relative amount of single and multiple scattering in experimental data. To this end, a single-scattering estimator ρ[over ̂] computed from the reflection matrix has been introduced in order to assess the weight of single scattering in the backscattered wave field. In this paper, the meaning of this estimator is investigated and a particular attention is given to recurrent scattering. In a diffraction-limited experiment, a multiple scattering sequence is said to be recurrent if the first and last scattering events occur in the same resolution cell. Recurrent scattering is shown to be responsible for correlations between single scattering and higher scattering orders of the Born expansion, inducing a bias to the estimator ρ[over ̂] that should rather be termed confocal scattering ratio. Interestingly, a more robust estimator is built by projecting the reflection matrix in a focused basis. The argument is sustained by numerical simulations as well as ultrasonic data obtained around 1.5 MHz in a model medium made of nylon rods immersed in water. From a more general perspective, this work raises fundamental questions about the impact of recurrent scattering on wave imaging.

SUMMARYReflection seismic imaging usually suffers from a loss of resolution and contrast because of the fluctuations of the wave velocities in the Earth’s crust. In the literature, phase distortion issues are generally circumvented by means of a background wave velocity model. However, it requires a prior tomography of the wave velocity distribution in the medium, which is often not possible, especially in depth. In this paper, a matrix approach of seismic imaging is developed to retrieve a 3-D image of the subsoil, despite a rough knowledge of the background wave velocity. To do so, passive noise cross-correlations between geophones of a seismic array are investigated under a matrix formalism. They form a reflection matrix that contains all the information available on the medium. A set of matrix operations can then be applied in order to extract the relevant information as a function of the problem considered. On the one hand, the background seismic wave velocity can be estimated and its fluctuations quantified by projecting the reflection matrix in a focused basis. It consists in investigating the response between virtual sources and detectors synthesized at any point in the medium. The minimization of their cross-talk can then be used as a guide star for approaching the actual wave velocity distribution. On the other hand, the detrimental effect of wave velocity fluctuations on imaging is overcome by introducing a novel mathematical object: The distortion matrix. This operator essentially connects any virtual source inside the medium with the distortion that a wavefront, emitted from that point, experiences due to heterogeneities. A time reversal analysis of the distortion matrix enables the estimation of the transmission matrix that links each real geophone at the surface and each virtual geophone in depth. Phase distortions can then be compensated for any point of the underground. Applied to passive seismic data recorded along the Clark branch of the San Jacinto fault zone (SJFZ), the present method is shown to provide an image of the fault until a depth of 4 km over the frequency range 10–20Hz with an horizontal resolution of 80 m. Strikingly, this resolution is almost one eighth below the diffraction limit imposed by the geophone array aperture. The heterogeneities of the subsoil play the role of a scattering lens and of a transverse waveguide which increase drastically the array aperture. The contrast is also optimized since most of the incoherent noise is eliminated by the iterative time reversal process. Beyond the specific case of the SJFZ, the reported approach can be applied to any scales and areas for which a reflection matrix is available at a spatial sampling satisfying the Nyquist criterion.

Reflection seismic imaging usually suffers from a loss of resolution and contrast because of the fluctuations of the wave velocities in the Earth's crust. In the literature, phase distortion issues are generally circumvented by means of a background wave velocity model. However, it requires a prior tomography of the wave velocity distribution in the medium, which is often not possible, especially in depth. In this paper, a matrix approach of seismic imaging is developed to retrieve a three-dimensional image of the subsoil without any prior knowledge of the background wave speed. To do so, passive noise cross-correlations between geophones of a seismic array are investigated under a matrix formalism. They form a reflection matrix that can be leveraged to synthesize virtual sources and detectors at any point in the medium. From this focused reflection matrix, a novel mathematical object is introduced: the distortion matrix. This operator essentially connects any virtual source inside the medium with the distortion that a wavefront, emitted from that point, experiences due to heterogeneities. A time reversal analysis of the distortion matrix enables the estimation of the transmission matrix that links each real geophone at the surface and each virtual geophone in depth. Phase distortions can then be compensated for any point of the underground. Applied to seismic data recorded along the Clark Branch of the San Jacinto Fault Zone, the present method is shown to provide an image of the fault until a depth of 4 km over the frequency range 10-20 Hz with a transverse resolution of 80 m. Strikingly, this resolution is almost one eighth below the diffraction limit imposed by the geophone array aperture. The heterogeneities of the subsoil play the role of a scattering lens which increases drastically the array aperture. The contrast is also optimized since most of the incoherent noise is eliminated by the iterative time reversal process. Beyond the specific case of the San Jacinto Fault Zone, the reported approach can be applied to any scales and areas for which a reflection matrix is available at a spatial sampling satisfying the Nyquist criterion.

The ultrasonic inspection of polycrystalline coarse gain structures suffers from attenuation noise due to multiple scattering at grain boundaries within these materials. In this paper, we present the current state of affairs of our research in multiple scattering filtering, based on a method proposed by Aubry et al. [1] initially developed and validated on artificial scattering media consisting of fluids with a large number of cylindrical rods, and subsequently applied to polycrystalline alloys [2].

Multiple scattering of seismic waves is often seen as a nightmare for conventional migration techniques that generally rely on a ballistic or a single‐scattering assumption. In heterogeneous areas such as volcanoes, the multiple‐scattering contribution limits the imaging‐depth to one scattering mean free path, the mean distance between two successive scattering events for body waves. In this Letter, we propose a matrix approach of passive seismic imaging that pushes back this fundamental limit by making an efficient use of scattered body waves drowned into a noisy seismic coda. As a proof of concept, the case of the Erebus volcano in Antarctica is considered. The Green's functions between a set of geophones placed on top of the volcano are first retrieved by the cross correlation of coda waves induced by multiple icequakes. This set of impulse responses forms a reflection matrix. By combining a matrix discrimination of singly scattered waves with iterative time reversal, we are able to push back the multiple scattering limit beyond 10 scattering mean free paths. The matrix approach reveals the internal structure of the Erebus volcano: A chimney‐shaped structure at shallow depths, a magma reservoir at 2,500 m and several cavities at sea level and below it. The matrix approach paves the way toward a greatly improved monitoring of volcanic structures at depth. Beyond this specific case, the matrix approach of seismic imaging can generally be applied to all scales and areas where multiple scattering events undergone by body waves prevent in‐depth imaging of the Earth's crust.

Ultrasonic evaluation of coarse-grain materials generates multiple scattering at high frequency and large depth. Recent academic experiments with array probes showed the ability of a random matrix method [multiple scattering filter (MSF)] to reduce multiple scattering, hence improving detection. Here, MSF is applied to an industrial nickel-based alloy with coarse-grain structure. Two samples with average grain sizes 90 ± 60 μm and 750 ± 400 μm are inspected with wide-band 64-element arrays at central frequencies 2, 3, and 5 MHz. They contain cylindrical through-holes (1-mm radius) at various depths. The array transfer matrix is recorded and post-processed both in the flawless area and for eleven positions above each defect, which allows for a statistical analysis. MSF is compared with two conventional imaging techniques: the total focusing method (TFM) and the decomposition of the time-reversal operator (DORT). Several parameters to assess the performance of detection techniques are proposed and discussed. The results show the benefit of MSF, especially at high frequencies and for deep defects: at 5 MHz and 70 mm depth, i.e., more than three scattering mean-free paths, the detection rate for MSF ranges between 55% and 100% while it is found to be 0% both for TFM and DORT.

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 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 ℓ^{*}, 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 ℓ_{c} is smaller than or comparable to the wavelength λ. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path ℓ^{*} 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 ℓ_{c}/λ=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.

The ultrasonic inspection of polycrystalline media remains a challenge. The high noise levels due to interaction between the wave and the microstructure limit the efficiency of classical ultrasonic techniques to detect a defect in a coarse grain structure. The aim of this work is to reduce the influence of multiple scattering in order to increase the information obtained from the defect. The technique introduced here is based on array probes for the acquisition of the medium's response matrix by full matrix capture, after which a filter based on random matrix theory is applied. Here an improvement of this technique is applied on nickel-based alloy mock-ups that present an unfavourable grain structure and well known bulk and plane defects. The results in normal incidence and with an angle array probe of 128 elements and 5 MHz of central frequency are compared to classical phased array probe techniques.

We present theoretical calculations of the ensemble-averaged (or effective or coherent) wave field propagating in a heterogeneous medium considered as one realization of a random process. In the literature, it is usually assumed that heterogeneity can be accounted for by a random scalar function of the space coordinates, termed the potential. Physically, this amounts to replacing the constant wave speed in Helmholtz' equation by a space-dependent speed. In the case of acoustic waves, we show that this approach leads to incorrect results for the scattering mean free path, no matter how weak the fluctuations. The detailed calculation of the coherent wave field must take into account both a scalar and an operator part in the random potential. When both terms have identical amplitudes, the correct value for the scattering mean free paths is shown to be more than 4 times smaller (13/3, precisely) in the low-frequency limit, whatever the shape of the correlation function. Based on the diagrammatic approach of multiple scattering, theoretical results are obtained for the self-energy and mean free path within Bourret's and on-shell approximations. They are confirmed by numerical experiments.

Working with bone mimicking phantoms is necessary for the development of ultrasonic methods for bone characterization. The phantoms currently available for trabecular bone do not mimic some of the bone ultrasonic properties accurately. We report the fabrication of anisotropic phantoms, constituted of square rods, using a modified stereolithographic printer. Ultrasound pulse propagation was performed in the phantoms and the fast and slow waves could be observed for a propagation along the main orientation of the rods. This work demonstrates the feasibility of printing anisotropic structures with a typical size comparable to the size of bone trabeculae. Further work will be directed towards high resolution printing of full trabecular micro-structures.

Frequency-resolved experimental measurements of ultrasonic diffusivity in the MHz range are presented. The samples under study are two-dimensional random arrangements of parallel steel rods immersed in water and exhibit high-order multiple scattering. Their physical characteristics, particularly the density and pair-correlation functions of the scatterers, are well controlled. These synthetic samples are used as phantoms for actual inhomogeneous materials. The resonant nature of the scatterers has a strong effect on diffusivity, which is shown to vary significantly with frequency. This may affect the result of broadband measurements of apparent diffusivity, which can be expected to depend on time and sample thickness, whereas diffusivity is intrinsically an intensive parameter. Moreover, the transport speed is shown to vary drastically with frequency, sometimes by more than 50%, due to a very narrow resonance that slows down transport. Interestingly, this sharp resonance could only be revealed by experiments performed with coda waves, and not with ballistic or coherent waves whose frequency resolution is intrinsically limited from an experimental point of view.

In this paper, we are interested in the 2D computation of the pressure scattered by an elliptic scatterer using a semi-analytical method based on a decomposition of the solutions on a basis of cylindrical waves. This approach is perfectly adapted to circular scatterers, and has been extended to scatterers of arbitrary shape [F. Chati et al. (2004)]. We will see that this extended formulation yields some very difficult numerical issues, particularly in our context of a flat and small elliptic scatterer. The use of arbitrary precision mathematics appears as a possible workaround, even if the cost in terms of the computation time may be prohibitive.

In this work, we are interested in modeling the propagation of an ultrasonic field in the complex trabecular bone structure. For this purpose, we use a simplified formulation, based on a representation of the bone tissue as thin and flat elliptic scatterers. With an ultrasonic inspection frequency of 1 MHz, the largest dimension of the ellipse is about half a wavelength, while the smallest dimension is 7–10 times smaller. The computation of the scattered field is based on a modal decomposition, generalized to the case of scatterers of arbitrary geometry (Chati et al., “Modal theory applied to the acoustic scattering by elastic cylinders of arbitrary cross section,” J. Acoust. Soc. Am. 116, 2004). Due to the very thin and flat geometry, this approach suffers huge numerical difficulties that require the use of enhanced precision and very long computation times. In this paper, we illustrate these issues. An alternative has been proposed (Liu et al., “Conformal mapping for the Helmholtz equation: Acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid,” J. Acoust. Soc. Am. 131, 2012), based on a conformal mapping of the ellipse. We compare the two formulations and show how the conformal mapping allows reducing drastically the numerical issues resulting from the standard approach.