Weak localization of seismic waves

Reflection Measurement of the Scattering Mean Free Path at the Onset of Multiple Scattering

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.

Experimental Implementation of Wave Propagation in Disordered Time-Varying Media

Here, we study and implement the temporal analog in time disordered sytems. A spatially homogeneous medium is endowed with a time structure composed of randomly distributed temporal interfaces. This is achieved through electrostriction between water surface and an electrode. The wave field observed is the result of the interferences between reflected and refracted waves on the interfaces. Although no eigenmode can be associated with the wave field, several common features between space and time emerge. The waves grow exponentially depending on the disorder level in agreement with a 2D matrix evolution model such as in the spatial case. The relative position of the momentum gap appearing in the time modulated systems plays a central role in the wave field evolution. When tuning the excitation to compensate for the damping, transient waves, localized in time, appear on the liquid surface. They result from a particular history of the multiple interferences produced by a specific sequence of time boundaries.

Ultra slow acoustic energy transport in dense fish aggregates

A dramatic slowing down of acoustic wave transport in dense fish shoals is observed in open-sea fish cages. By employing a multi-beam ultrasonic antenna, we observe the coherent backscattering phenomenon. We extract key parameters of wave transport such as the transport mean free path and the energy transport velocity of diffusive waves from diffusion theory fits to the experimental data. The energy transport velocity is found to be about 10 times smaller than the speed of sound in water, a value that is exceptionally low compared with most observations in acoustics. By studying different models of the fish body, we explain the basic mechanism responsible for the observed very slow transport of ultrasonic waves in dense fish shoals. Our results show that, while the fish swim bladder plays an important role in wave scattering, other organs have to be considered to explain ultra-low energy transport velocities.

Topological Effects of a Vorticity Filament on the Coherent Backscattering Cone

In this Letter, we report on the effects of a vorticity filament on the coherent backscattering cone. Using ultrasonic waves in a strongly reverberating cavity, we experimentally show that the discrete number of loops of acoustic paths around a pointlike vortex located at the center of the cavity drives the cancellation and the potential rebirth of the coherent backscattering enhancement. The vorticity filament behaves, then, as a topological anomaly for wave propagation that provides some new insight between reciprocity and weak localization.

Observation of transverse coherent backscattering in disordered photonic structures

Coherent backscattering, also referred to as weak localization, is an exciting multidisciplinary phenomenon that appears in disordered systems of multiple coherent-wave scattering. Providing proper scattering conditions in (2 + 1) dimensional randomized photonic systems, we optically implement, observe, and analyse transverse coherent backscattering. Ensembles of disordered wave-guide structures are prepared by random-intensity nondiffracting writing entities according to the beam’s intensity distribution. The structure size of the induced potentials naturally define an effective mobility edge, and thus, we identify a crucial impact of the plane probe waves’ spatial frequency on the strength and shape of the spectral coherent backscattering signal. We additionally observe transverse elastic scattering as a precursor of weak localization. To testify the coherent character as a fundamental condition for coherent backscattering, we propose a scheme to continuously reduce the spatial coherence of the probe beam which directly reduces the degree of localization and coherent backscattering. With our experiments, we propose a testing platform that allows comprehensive examination of coherent backscattering with a broad set of preparation parameters and under uncritical laboratory conditions. Our results are directly transferable to more complex systems of disordered wave potentials, not being restricted to photonic systems.

Focusing inside Disordered Media with the Generalized Wigner-Smith Operator

We introduce a wave front shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specific experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength-ranging from strongly focusing on the designated target to completely bypassing it. Our protocol works without optimization or phase conjugation and we expect it to be particularly attractive for optical imaging in disordered media.

Return to the Origin as a Probe of Atomic Phase Coherence

We report on the observation of the coherent enhancement of the return probability ["enhanced return to the origin" (ERO)] in a periodically kicked cold-atom gas. By submitting an atomic wave packet to a pulsed, periodically shifted, laser standing wave, we induce an oscillation of ERO in time that is explained in terms of a periodic, reversible dephasing in the weak-localization interference sequences responsible for ERO. Monitoring the temporal decay of ERO, we exploit its quantum-coherent nature to quantify the decoherence rate of the atomic system.

Far-field coherent backscatter enhancement from random aggregations of scatterers and comparisons to backscattering from single isolated spheres

Coherent backscatter enhancement (CBE), a multiple scattering phenomenon, may cause an enhancement of up to a factor of two in the average intensity backscattered from a random aggregation of scatterers. In the ocean, CBE may occur when a fish school or a bubble cloud is remotely illuminated. The research reported here explored the possibility that CBE might be used to remotely discriminate between an aggregation of many scatterers and a single isolated scattering object. For this investigation, the far-field harmonic acoustic pressure backscattered from aggregations of randomly placed omnidirectional point scatterers was determined from numerical solution of the equations from Foldy [(1945) Phys. Rev. 67(3,4), 107-119], and compared to equivalent results from single spherical scatterers having hard surfaces, pressure-release surfaces, or aggregation-matched effective-medium properties. Interestingly, CBE causes a spherical aggregation to backscatter as much or more sound than a single perfectly reflecting sphere of the same size when (ka)^1/2(ks)^-4/5(kσ_s^1/2)^3/4 ≥ 2.3, where k is the acoustic wave number, a is the aggregation radius, s is the average spacing between scatterers, and σ_s is a scatterer's cross section. And, backscattered intensity samples (in dB) from all simulated aggregations followed an extreme value distribution, a finding that supports the conventional use of backscatter statistics for remote aggregation-versus-single-object discrimination.

Anderson Mobility Gap Probed by Dynamic Coherent Backscattering

We use dynamic coherent backscattering to study one of the Anderson mobility gaps in the vibrational spectrum of strongly disordered three-dimensional mesoglasses. Comparison of experimental results with the self-consistent theory of localization allows us to estimate the localization (correlation) length as a function of frequency in a wide spectral range covering bands of diffuse transport and a mobility gap delimited by two mobility edges. The results are corroborated by transmission measurements on one of our samples.

Coherent Backscattering Reveals the Anderson Transition

We develop an accurate finite-time scaling analysis of the angular width of the coherent backscattering (CBS) peak for waves propagating in 3D random media. Applying this method to ultracold atoms in optical speckle potentials, we show how to determine both the mobility edge and the critical exponent of the Anderson transition from the temporal behavior of the CBS width. Our method could be used in experiments to fully characterize the 3D Anderson transition.

Simulating acoustic coherent backscattering enhancement from random aggregations of omnidirectional scatterers

Coherent backscatter enhancement (CBE) is a multiple scattering phenomenon that can lead to a doubling of the backscattered field intensity from a random aggregation of scatterers. It may be useful for remote sensing of scatterer aggregations, such as fish schools. This paper presents simulations of acoustic CBE from randomly placed omnidirectional point scatterers based on Foldy's field equations. The simulations are verified and validated through comparisons with Bragg scattering and Foldy's effective-medium theory, assessments of acoustic energy conservation, and comparisons with prior optical and acoustical CBE results. To make CBE comparisons with prior optics results, a CBE coherence function was postulated to account for resolution differences between the optics and simulation studies. For the higher-resolution optics studies, the postulated coherence function yields a CBE of 1.68, which matches optical CBE measurements. For the lower-resolution simulations, the same coherence function yields a CBE of 1.034, which agrees with appropriately extrapolated CBE simulation results, 1.030 ± 0.005. Assuming comparable resolution, the acoustics experiment and simulations both produce a CBE of approximately 1.5. The CBE peak is found to increase approximately monotonically with (k(2)σs)(1/4)(ks)(-1), where k is the wave number, s is the average spacing between scatterers, and σs is a scatterer's cross section.

Wave propagation in disordered fractured porous media

Extensive computer simulations have been carried out to study propagation of acoustic waves in a two-dimensional disordered fractured porous medium, as a prelude to studying elastic wave propagation in such media. The fracture network is represented by randomly distributed channels of finite width and length, the contrast in the properties of the porous matrix and the fractures is taken into account, and the propagation of the waves is studied over broad ranges of the fracture number density ρ and width b. The most significant result of the study is that, at short distances from the wave source, the waves' amplitude, as well as their energy, decays exponentially with the distance from the source, which is similar to the classical problem of electron localization in disordered solids, whereas the amplitude decays as a stretched exponential function of the distance x that corresponds to sublocalization, exp(-x(α)) with α < 1. Moreover, the exponent α depends on both ρ and b. This is analogous to electron localization in percolation systems at the percolation threshold. Similar results are obtained for the decay of the waves' amplitude with the porosity of the fracture network. Moreover, the amplitude decays faster with distance from the source x in a fractured porous medium than in one without fractures. The mean speed of wave propagation decreases linearly with the fractures' number density.

Comparison between experimental and 2-D numerical studies of multiple scattering in Inconel600 by means of array probes

Ultrasonic non-destructive testing of polycrystalline structures can be disturbed by scattering at grain boundaries. Understanding and modeling this so-called "structural noise" is crucial for characterization as well as detection purposes. Structural noise can be considered as a fingerprint of the material under investigation, since it contains information about its microstructure. The interpretation of experimental data necessitates an accurate comprehension of complex phenomena that occur in multiple scattering media and thus robust scattering models. In particular, numerical models can offer the opportunity to realize parametrical studies on controlled microstructures. However, the ability of the model to simulate wave propagation in complex media must be validated. In that perspective, the main objective of the present work is to evaluate the ability of the finite-element code ATHENA 2D to reproduce typical features of multiple wave scattering in the context of ultrasonic non-destructive evaluation, with an array of sources and receivers. Experiments were carried out with a 64-element array, around 2 MHz. The sample was a mock-up of Inconel600 exhibiting a coarse grain structure with a known grain size distribution. The numerical model of this microstructure is based on Voronoi diagrams. Two physical parameters were used to compare numerical and experimental data: the coherent backscattering peak, and the singular value distribution of the array response matrix. Though the simulations are 2-D, a good agreement was found between simulated and experimental data.

Coherent backscattering of ultracold atoms

We report on the direct observation of coherent backscattering (CBS) of ultracold atoms in a quasi-two-dimensional configuration. Launching atoms with a well-defined momentum in a laser speckle disordered potential, we follow the progressive build up of the momentum scattering pattern, consisting of a ring associated with multiple elastic scattering, and the CBS peak in the backward direction. Monitoring the depletion of the initial momentum component and the formation of the angular ring profile allows us to determine microscopic transport quantities. We also study the time evolution of the CBS peak and find it in fair agreement with predictions, at long times as well as at short times. The observation of CBS can be considered a direct signature of coherence in quantum transport of particles in disordered media. It is responsible for the so called weak localization phenomenon, which is the precursor of Anderson localization.

Weak localization in mesoscopic hole transport: berry phases and classical correlations

We consider phase-coherent transport through ballistic and diffusive two-dimensional hole systems based on the Kohn-Luttinger Hamiltonian. We show that intrinsic heavy-hole-light-hole coupling gives rise to clear-cut signatures of an associated Berry phase in the weak localization which renders the magnetoconductance profile distinctly different from electron transport. Nonuniversal classical correlations determine the strength of these Berry phase effects and the effective symmetry class, leading even to antilocalization-type features for circular quantum dots and Aharonov-Bohm rings in the absence of additional spin-orbit interaction. Our semiclassical predictions are confirmed by numerical calculations.

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 setup 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 megahertz 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.

Anomalous fluctuations of vertical velocity of Earth and their possible implications for earthquakes

High-quality measurements of seismic activities around the world provide a wealth of data and information that are relevant to understanding of when earthquakes may occur. If viewed as complex stochastic time series, such data may be analyzed by methods that provide deeper insights into their nature, hence leading to better understanding of the data and their possible implications for earthquakes. In this paper, we provide further evidence for our recent proposal [P. Mansour, Phys. Rev. Lett. 102, 014101 (2009)10.1103/PhysRevLett.102.014101] for the existence of a transition in the shape of the probability density function (PDF) of the successive detrended increments of the stochastic fluctuations of Earth's vertical velocity V_{z} , collected by broadband stations before moderate and large earthquakes. To demonstrate the transition, we carried out extensive analysis of the data for V_{z} for 12 earthquakes in several regions around the world, including the recent catasrophic one in Haiti. The analysis supports the hypothesis that before and near the time of an earthquake, the shape of the PDF undergoes significant and discernable changes, which can be characterized quantitatively. The typical time over which the PDF undergoes the transition is about 5-10 h prior to a moderate or large earthquake.

Turbulencelike behavior of seismic time series

We report on a stochastic analysis of Earth's vertical velocity time series by using methods originally developed for complex hierarchical systems and, in particular, for turbulent flows. Analysis of the fluctuations of the detrended increments of the series reveals a pronounced transition in their probability density function from Gaussian to non-Gaussian. The transition occurs 5-10 hours prior to a moderate or large earthquake, hence representing a new and reliable precursor for detecting such earthquakes.

Anderson localization of ultrasound in plates: further experimental results

Diffuse wave transport is studied in a thick plate with densely machined-in multiple scatterers. As anticipated by theory, energy at short wavelengths diffuses across the structure. Energy localizes at longer wavelengths for which lambda2pi is comparable to the mean free path.

Temporal changes in the lunar soil from correlation of diffuse vibrations

It was recently demonstrated that one can reconstruct the impulse response between passive sensors by cross-correlating diffuse waves or ambient noise. Using seismic waves recorded on the moon, we show here that not only direct waves can be retrieved, but also late arrivals that have been scattered before reaching the seismometers. As these late arrivals propagate for a longer time, they are more sensitive to weak perturbations of the medium such as velocity changes. This high sensitivity of scattered waves is used to monitor periodic velocity changes in the lunar soil by measuring small delays of the passively retrieved coda waves. The velocity changes result from temperature variations due to periodic heating of the lunar surface by the sun.

Diffusive transport of light in a two-dimensional disordered packing of disks: analytical approach to transport mean free path

We study photon diffusion in a two-dimensional random packing of monodisperse disks as a simple model of granular media and wet foams. We assume that the intensity reflectance of disks is a constant r . We present an analytic expression for the transport mean free path l;{*} in terms of the velocity of light in the disks and host medium, radius R and packing fraction of the disks, and the intensity reflectance. For glass beads immersed in air or water, we estimate transport mean free paths about half the experimental ones. For air bubbles immersed in water, l;{*}R is a linear function of 1epsilon , where epsilon is the liquid volume fraction of the model wet foam. This throws light on the empirical law of Vera [Appl. Opt. 40, 4210 (2001)] and promotes more realistic models.

Short wavelength approximation of a boundary integral operator for homogeneous and isotropic elastic bodies

A short wavelength approximation of a boundary integral operator for two-dimensional isotropic and homogeneous elastic bodies is derived from first principles starting from the Navier-Cauchy equation. Trace formulas for elastodynamics are deduced connecting the eigenfrequency spectrum of an elastic body to the set of periodic rays where mode conversion enters as a dynamical feature.

Ultrasonic imaging of highly scattering media from local measurements of the diffusion constant: separation of coherent and incoherent intensities

As classical imaging fails with diffusive media, one way to image a multiple-scattering medium is to achieve local measurements of the dynamic transport properties of a wave undergoing diffusion. This paper presents a method to obtain local measurements of the diffusion constant D in a multiple-scattering medium. The experimental setup consists in an array of programmable transducers placed in front of the multiple-scattering medium to be imaged. By achieving Gaussian beamforming both at emission and reception, an array of virtual sources and receivers located in the near field is constructed. The time evolution of the incoherent component of the intensity backscattered on this virtual array is shown to represent directly the growth of the diffusive halo as sqrt[Dt]. A matrix treatment is proposed to separate the incoherent intensity from the coherent backscattering peak. Once the incoherent contribution is isolated, a local measurement of the diffusion constant is possible. The technique is applied to image the long-scale variations of D in a random-scattering sample made of two parts with a different concentration of cylindrical scatterers. This experimental result is obtained with ultrasonic waves around 3 MHz. It illustrates the possibility of imaging diffusive media from local measurements of the diffusion constant, based on coherent Gaussian beamforming and a matrix "antisymmetrization," which creates a virtual antireciprocity.

Effect of the photon's Brownian Doppler shift on the weak-localization coherent-backscattering cone

We report the first observation of the dependence of the coherent-backscattering (CBS) enhanced cone with the frequency of the backscattered photon. The experiment is performed on a diffusing liquid suspension and the Doppler broadening of light is induced by the Brownian motion of the scatterers. Heterodyne detection on a CCD camera is used to measure the complex field (i.e., the hologram) of the light that is backscattered at a given frequency. The analysis of the holograms yield the frequency and the propagation direction of the backscattered photons. We observe that the angular CBS cone becomes more narrow in the tail of the Brownian spectrum. The experimental results are in good agreement with a simple theoretical model.

Shape of a wave front in a heterogenous medium

Wave propagation in a heterogeneous medium, characterized by a distribution of local elastic moduli, is studied. Both acoustic and elastic waves are considered, as are spatially random and power-law correlated distributions of the elastic moduli with nondecaying correlations. Three models--a continuum scalar model, and two discrete models--are utilized. Numerical simulations indicate the existence, at all times, of the relation, alpha = H, where alpha is the roughness exponent of the wave front in the medium, and H is the Hurst exponent that characterizes the spatial correlations in the distribution of the local elastic moduli. Hence, a direct relation between the static morphology of an inhomogeneous correlated medium and its dynamical properties is established. In contrast, for a wave front in random media, alpha = 0 (logarithmic growth) at short times, followed by a crossover to the classical value, alpha = 1/2, at long times.

Observation of multiple scattering of kHz vibrations in a concrete structure and application to monitoring weak changes

In this paper, we present two experimental studies of mechanical wave propagation in a concrete building around 1 kHz. The first experiment is devoted to the observation of the coherent backscattering enhancement, which demonstrates the presence of multiple diffractions in the late part of the wave records. An application of multiple diffraction and reverberations is proposed in a second experiment. Thanks to their sensitivity to weak changes of the medium, the late records are used to monitor weak change in concrete wave velocity induced by thermal variations. The velocity change measurements have a precision of deltac/c=10(-4). Such a precision is difficult to obtain with direct waves. This experiment is the first step to other applications like stress, damage, aging, or crack monitoring in concrete structures.

Diffusion of monochromatic classical waves

We study the diffusion of monochromatic classical waves in a disordered acoustic medium by scattering theory. In order to avoid artifacts associated with mathematical point scatterers, we model the randomness by small but finite insertions. We derive expressions for the configuration-averaged energy flux, energy density, and intensity for one-, two-, and three-dimensional (3D) systems with an embedded monochromatic source using the ladder approximation to the Bethe-Salpeter equation. We study the transition from ballistic to diffusive wave propagation and obtain results for the frequency dependence of the medium properties such as mean free path and diffusion coefficient as a function of the scattering parameters. We discover characteristic differences of the diffusion in 2D as compared to the conventional 3D case, such as an explicit dependence of the energy flux on the mean free path and quite different expressions for the effective transport velocity.

Deterministic weak localization in periodic structures

In some perfect periodic structures classical motion exhibits deterministic diffusion. For such systems we present the weak localization theory. As a manifestation for the velocity autocorrelation function a universal power law decay is predicted to appear at four Ehrenfest times. This deterministic weak localization is robust against weak quenched disorders, which may be confirmed by coherent backscattering measurements of periodic photonic crystals.

Passive retrieval of Rayleigh waves in disordered elastic media

When averaged over sources or disorder, cross correlation of diffuse fields yields the Green's function between two passive sensors. This technique is applied to elastic ultrasonic waves in an open scattering slab mimicking seismic waves in the Earth's crust. It appears that the Rayleigh wave reconstruction depends on the scattering properties of the elastic slab. Special attention is paid to the specific role of bulk to Rayleigh wave coupling, which may result in unexpected phenomena, such as a persistent time asymmetry in the diffuse regime.

Localization of elastic waves in heterogeneous media with off-diagonal disorder and long-range correlations

Using the Martin-Siggia-Rose method, we study propagation of acoustic waves in strongly heterogeneous media which are characterized by a broad distribution of the elastic constants. Gaussian-white distributed elastic constants, as well as those with long-range correlations with nondecaying power-law correlation functions, are considered. The study is motivated in part by a recent discovery that the elastic moduli of rock at large length scales may be characterized by long-range power-law correlation functions. Depending on the disorder, the renormalization group (RG) flows exhibit a transition to localized regime in any dimension. We have numerically checked the RG results using the transfer-matrix method and direct numerical simulations for one- and two-dimensional systems, respectively.