Recovering the Green's function from field-field correlations in an open scattering medium

Full-field noise-correlation elastography for in-plane mechanical anisotropy imaging

Elastography contrast imaging has great potential for the detection and characterization of abnormalities in soft biological tissues to help physicians in diagnosis. Transient shear-waves elastography has notably shown promising results for a range of clinical applications. In biological soft tissues such as muscle, high mechanical anisotropy implies different stiffness estimations depending on the direction of the measurement. In this study, we propose the evolution of a noise-correlation elastography approach for in-plane anisotropy mapping. This method is shown to retrieve anisotropy from simulation images before being validated on agarose anisotropic tissue-mimicking phantoms, and the first results on in-vivo biological fibrous tissues are presented.

Passive reconstruction of dispersion curve in plates via flow-induced random vibration: The effect of boundary reflection and sensor placement

Flow-induced random vibration propagates throughout a plate-like structure and thus contains information about the structural properties and states. Previous research has shown that the group velocity under various frequencies can be estimated by the cross-correlation between two measurement points, in which the prominent peak signifies the travel time of the random guided wave. However, it is found that the performance of this method depends on the sensor placement and the plate size; in some cases, the cross-correlation is contaminated such that the extraction of dispersion curve is problematic. This is because the wave package in the cross-correlation corresponding to the direct propagation between the two sensors overlaps with the coda wave packages associated with complex boundary reflections. This hypothesis is justified by a passive experiment on an ellipse-shaped plate: in the cross-correlation between the foci of the ellipse, the imprints of the direct propagation and various orders of boundary reflections can be clearly separated and precisely predicted by the ray method. Based on these findings, a sensor placement criterion is proposed, by which a robust passive reconstruction of dispersion curve using flow-induced random vibration is guaranteed.

Geothermal and structural features of La Palma island (Canary Islands) imaged by ambient noise tomography

La Palma island is located in the NW of the Canary Islands and is one of the most volcanically active of the archipelago, therefore the existence of geothermal resources on the island is highly probable. The main objective of this work is to detect velocity anomalies potentially related to active geothermal reservoirs on La Palma island, by achieving a high-resolution seismic velocity model of the first few kilometres of the crust using Ambient Noise Tomography (ANT). The obtained ANT model is merged with a recent local earthquake tomography model. Our findings reveal two high-velocity zones in the island's northern and southern parts, that could be related to a plutonic intrusion and old oceanic crust materials. Conversely, four low-velocity zones are imaged in the southern part of the island. Two of them can be related to hydrothermal alteration zones located beneath the Cumbre Vieja volcanic complex. This hypothesis is reinforced by comparing the S-wave velocity model with the seismicity recorded during the pre-eruptive phase of the 2021 Tajogaite eruption, which revealed an aseismic volume coinciding with these low-velocity zones. Another low-velocity zone is observed in the southern part of the island, which we interpret as highly fractured rocks which could favour the ascent of hot fluids. A last low-velocity zone is observed in the central part of the island and associated with loose deposits generated by the Aridane valley mega landslide.

Wave-field representations with Green's functions, propagator matrices, and Marchenko-type focusing functions

Classical acoustic wave-field representations consist of volume and boundary integrals, of which the integrands contain specific combinations of Green's functions, source distributions, and wave fields. Using a unified matrix-vector wave equation for different wave phenomena, these representations can be reformulated in terms of Green's matrices, source vectors, and wave-field vectors. The matrix-vector formalism also allows the formulation of representations in which propagator matrices replace the Green's matrices. These propagator matrices, in turn, can be expressed in terms of Marchenko-type focusing functions. An advantage of the representations with propagator matrices and focusing functions is that the boundary integrals in these representations are limited to a single open boundary. This makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary.

Imaging of Thermal Effects during High-Intensity Ultrasound Treatment in Liver by Passive Elastography: A Preliminary Feasibility in Vitro Study

High-intensity focused ultrasound is a non-invasive modality for thermal ablation of tissues through locally increased temperature. Thermal lesions can be monitored by elastography, following the changes in the elastic properties of the tissue as reflected by the shear-wave velocity. Most studies on ultrasound elastography use shear waves created by acoustic radiation force. However, in the human body, the natural noise resulting from cardiac activity or arterial pulsatility can be used to characterize elasticity through noise-correlation techniques, in the method known as passive elastography. The objective of this study was to investigate the feasibility of monitoring high-intensity ultrasound treatments of liver tissue using passive elastography. Bovine livers were heated to 80°C using a high-intensity planar transducer and imaged with a high-frame-rate ultrasound imaging device. The dynamics of lesion formation are captured through tissue stiffening and lesion expansion.

Bone-conducted sound in a dolphin's mandible: Experimental investigation of elastic waves mediating information on sound source position

Mammals use binaural or monaural (spectral) cues to localize acoustic sources. While the sensitivity of terrestrial mammals to changes in source elevation is relatively poor, the accuracy achieved by the odontocete cetaceans' biosonar is high, independently of where the source is. Binaural/spectral cues are unlikely to account for this remarkable skill. In this paper, bone-conducted sound in a dolphin's mandible is studied, investigating its possible contribution to sound localization. Experiments are conducted in a water tank by deploying, on the horizontal and median planes of the skull, ultrasound sources that emit synthetic clicks between 45 and 55 kHz. Elastic waves propagating through the mandible are measured at the pan bones and used to localize source positions via either binaural cues or a correlation-based full-waveform algorithm. Exploiting the full waveforms and, most importantly, reverberated coda, it is possible to enhance the accuracy of source localization in the vertical plane and achieve similar resolution of horizontal- vs vertical-plane sources. The results noted in this paper need to be substantiated by further experimental work, accounting for soft tissues and making sure that the data are correctly mediated to the internal ear. If confirmed, the results would favor the idea that dolphin's echolocation skills rely on the capability to analyze the coda of biosonar echoes.

Green's function extraction by crosscorrelation and multidimensional deconvolution for outdoor sound propagation

In this work, the Green's function is estimated from outdoor measurements of controlled-sources. Crosscorrelation and multidimensional deconvolution have successfully been employed for Green's function retrieval. Crosscorrelation assumes a lossless medium and equipartitioned wavefield; when these assumptions are not satisfied it may result in a Green's function smeared with the source point-spread function. Multidimensional deconvolution removes the point-spread function from the retrieved Green's function. Both methods are employed to estimate the Green's function between two array stations for a single and multiple controlled-sources. The results demonstrate that the source-to-center radius has a negligible effect on the retrieved Green's function, if the source-to-center radius is larger than the distance between the two array stations.

Passive guided waves measurements using fiber Bragg gratings sensors

Guided elastic waves are often studied as an effective solution for Structural Health Monitoring (SHM) systems of plate-like structures thanks to the capacity to propagate on large distances. In typical applications such as monitoring delaminations in aircraft fuselage, a network made of piezoelectric transducer (PZT) is used to emit and receive such waves in the structure. Fiber Bragg grating (FBG) sensors on optical fibers are a promising alternative to PZT for guided waves measurements in practical applications due to the capacity for dense multiplexing and robustness with respect to the environment. However, unlike conventional PZT transducers, FBG sensors cannot emit waves. It is demonstrated here that FBG sensors can be used in combination with a passive diffuse noise cross-correlation technique in order to extract the coherent guided waves propagating between two sensors. This could lead to a system using only FBG sensors in the near future. The reconstructed signals can then be analyzed with usual guided waves algorithms, like in active SHM systems, keeping all the advantages of this kind of monitoring in terms of fine diagnosis. The experimental demonstration shown in this paper is performed at ultrasonic frequencies (20-100 kHz) typically used in guided waves based SHM systems showing the potential of the approach.

Temporally weighting a time varying noise field to improve Green function retrieval

The authors consider the retrieval of Green functions G from the correlations of non-stationary non-fully diffuse noise incident on an array of sensors. Multiple schemes are proposed for optimizing the time-varying weights with which correlations may be stacked. Using noise records created by direct numerical simulation of waves in a two-dimensional multiply scattering medium, cases are shown in which conventional stacking does a poor job and for which the proposed schemes substantially improve the recovered G, rendering it more causal and/or more symmetric, and more similar to the actual G. It is found that the schemes choose weights such that the effective incident intensity distribution is closer to isotropic.

Virtual acoustics in inhomogeneous media with single-sided access

A virtual acoustic source inside a medium can be created by emitting a time-reversed point-source response from the enclosing boundary into the medium. However, in many practical situations the medium can be accessed from one side only. In those cases the time-reversal approach is not exact. Here, we demonstrate the experimental design and use of complex focusing functions to create virtual acoustic sources and virtual receivers inside an inhomogeneous medium with single-sided access. The retrieved virtual acoustic responses between those sources and receivers mimic the complex propagation and multiple scattering paths of waves that would be ignited by physical sources and recorded by physical receivers inside the medium. The possibility to predict complex virtual acoustic responses between any two points inside an inhomogeneous medium, without needing a detailed model of the medium, has large potential for holographic imaging and monitoring of objects with single-sided access, ranging from photoacoustic medical imaging to the monitoring of induced-earthquake waves all the way from the source to the earth’s surface.

Passive detection and localization of fatigue cracking in aluminum plates using Green's function reconstruction from ambient noise

Recent theoretical and experimental studies have demonstrated that a local Green's function can be retrieved from the cross-correlation of ambient noise field. This technique can be used to detect fatigue cracking in metallic structures, owing to the fact that the presence of crack can lead to a change in Green's function. This paper presents a method of structural fatigue cracking characterization method by measuring Green's function reconstruction from noise excitation and verifies the feasibility of crack detection in poor noise source distribution. Fatigue cracks usually generate nonlinear effects, in which different wave amplitudes and frequency compositions can cause different nonlinear responses. This study also undertakes analysis of the capacity of the proposed approach to identify fatigue cracking under different noise amplitudes and frequency ranges. Experimental investigations of an aluminum plate are conducted to assess the cross-correlations of received noise between sensor pairs and finally to detect the introduced fatigue crack. A damage index is proposed according to the variation between cross-correlations obtained from the pristine crack closed state and the crack opening-closure state when sufficient noise amplitude is used to generate nonlinearity. A probability distribution map of damage is calculated based on damage indices. The fatigue crack introduced in the aluminum plate is successfully identified and oriented, verifying that a fatigue crack can be detected by reconstructing Green's functions from an imperfect diffuse field in which ambient noise sources exist locally.

Reflecting boundary conditions for interferometry by multidimensional deconvolution

In an acoustical context, interferometry takes advantage of existing (ambient) wavefield recordings by turning receivers into so-called "virtual sources." The medium's response to these virtual sources can be harnessed to image that medium. Most interferometric applications, however, suffer from the fact that the retrieved virtual-source responses deviate from the true medium responses. The accrued artefacts are often predominantly due to a non-isotropic illumination of the medium of interest, and prohibit accurate interferometric imaging. Recently, it has been shown that illumination-related artefacts can be removed by means of a so-called multidimensional deconvolution (MDD) process. However, the current MDD formulation, and hence method, relies on separation of waves traveling inward and outward through the boundary of the medium of interest. As a consequence, it is predominantly useful when receivers are illuminated from one side only. This puts constraints on the applicability of the current MDD formulation to omnidirectional wavefields. In this paper, a modified formulation of the theory underlying interferometry by MDD is presented. This modified formulation eliminates the requirement to separate inward and outward propagating wavefields and, consequently, holds promise for the application of MDD to non-isotropic, omnidirectional wavefields.

Coda reconstruction from cross-correlation of a diffuse field on thin elastic plates

This study contributes to the evaluation of the robustness and accuracy of Green's function reconstruction from cross-correlation of strongly dispersed reverberated signals, with disentangling of the respective roles of ballistic and reverberated ("coda") contributions. We conduct a suite of experiments on a highly reverberating thin duralumin plate, where an approximately diffuse flexural wave field is generated by taking advantage of the plate reverberation and wave dispersion. A large number of impulsive sources that cover the whole surface of the plate are used to validate ambient-noise theory through comparison of the causal and anticausal (i.e., positive- and negative-time) terms of the cross-correlation to one another and to the directly measured Green's function. To quantify the contribution of the ballistic and coda signals, the cross-correlation integral is defined over different time windows of variable length, and the accuracy of the reconstructed Green's function is studied as a function of the initial and end times of the integral. We show that even cross-correlations measured over limited time windows converge to a significant part of the Green's function. Convergence is achieved over a wide time window, which includes not only direct flexural-wave arrivals, but also the multiply reverberated coda. We propose a model, based on normal-mode analysis, that relates the similarity between the cross-correlation and the Green's function to the statistical properties of the plate. We also determine quantitatively how incoherent noise degrades the estimation of the Green's function.

A single-sided representation for the homogeneous Green's function of a unified scalar wave equation

A unified scalar wave equation is formulated, which covers three-dimensional (3D) acoustic waves, 2D horizontally-polarised shear waves, 2D transverse-electric EM waves, 2D transverse-magnetic EM waves, 3D quantum-mechanical waves and 2D flexural waves. The homogeneous Green's function of this wave equation is a combination of the causal Green's function and its time-reversal, such that their singularities at the source position cancel each other. A classical representation expresses this homogeneous Green's function as a closed boundary integral. This representation finds applications in holographic imaging, time-reversed wave propagation and Green's function retrieval by cross correlation. The main drawback of the classical representation in those applications is that it requires access to a closed boundary around the medium of interest, whereas in many practical situations the medium can be accessed from one side only. Therefore, a single-sided representation is derived for the homogeneous Green's function of the unified scalar wave equation. Like the classical representation, this single-sided representation fully accounts for multiple scattering. The single-sided representation has the same applications as the classical representation, but unlike the classical representation it is applicable in situations where the medium of interest is accessible from one side only.

Unified double- and single-sided homogeneous Green's function representations

In wave theory, the homogeneous Green's function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green's function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green's function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green's function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green's function retrieval.

Localization of acoustic sensors from passive Green's function estimation

A number of methods have recently been developed for passive localization of acoustic sensors, based on the assumption that the acoustic field is diffuse. This article presents the more general case of equipartition fields, which takes into account reflections off boundaries and/or scatterers. After a thorough discussion on the fundamental differences between the diffuse and equipartition models, it is shown that the method is more robust when dealing with wideband noise sources. Finally, experimental results show, for two types of boundary conditions, that this approach is especially relevant when acoustic sensors are close to boundaries.

Retrieving time-dependent Green's functions in optics with low-coherence interferometry

We report on the passive measurement of time-dependent Green's functions in the optical frequency domain with low-coherence interferometry. Inspired by previous studies in acoustics and seismology, we show how the correlations of a broadband and incoherent wave field can directly yield the Green's functions between scatterers of a complex medium. Both the ballistic and multiple scattering components of the Green's function are retrieved. This approach opens important perspectives for optical imaging and characterization in complex scattering media.

The "round trip" theory for reconstruction of Green's functions at passive locations

An expression for the Green's function at an arbitrary set of passive locations (no applied force) is derived and validated by experiment. Three sets of points are involved, the passive reconstruction points, c, which lie on a virtual boundary and two sets of auxiliary points, denoted a and b, located either side. The reconstruction is achieved using Green's functions forming a "round trip" from and to the reconstruction points via a and b. A two stage measurement procedure is described involving excitation at b and a but with no excitation required at the reconstruction points. A known "round trip" relationship is first introduced which is theoretically exact for points on a multi-point interface between two linear, time invariant subsystems. Experimental results for frequency response functions of a beam-plate structure show that this relationship gives good results in practice. It is then shown that the theory provides an Nth order approximation for the Green's function at arbitrary points, where N is the number of points at b. The expression is validated by reconstructing point and transfer frequency response functions at two passive points on an aluminum plate.

Green's function retrieval and passive imaging from correlations of wideband thermal radiations

We present an experimental demonstration of electromagnetic Green's function retrieval from thermal radiations in anechoic and reverberant cavities. The Green's function between two antennas is estimated by cross correlating milliseconds of decimeter noise. We show that the temperature dependence of the cross-correlation amplitude is well predicted by the blackbody theory in the Rayleigh-Jeans limit. The effect of a nonuniform temperature distribution on the cross-correlation time symmetry is also explored. Finally, we open a new way to image scatterers using ambient thermal radiations.

A unified optical theorem for scalar and vectorial wave fields

The generalized optical theorem is an integral relation for the angle-dependent scattering amplitude of an inhomogeneous scattering object embedded in a homogeneous background. It has been derived separately for several scalar and vectorial wave phenomena. Here a unified optical theorem is derived that encompasses the separate versions for scalar and vectorial waves. Moreover, this unified theorem also holds for scattering by anisotropic elastic and piezoelectric scatterers as well as bianisotropic (non-reciprocal) EM scatterers.

Analyzing the coda from correlating scattered surface waves

The accuracy of scattered Rayleigh waves estimated using an interferometric method is investigated. Summing the cross correlations of the wave fields measured all around the scatterers yields the Green's function between two excitation points. This accounts for the direct wave and the scattered field (coda). The correlations themselves provide insights into the location of the scatterers, as well as which scatterer is responsible for particular parts of the coda. Furthermore, these measurements confirm a constant-time arrival in the correlations, not part of the Green's function, but which has previously been derived as a result of the generalized optical theorem.

A passive inverse filter for Green's function retrieval

Passive methods for the recovery of Green's functions from ambient noise require strong hypotheses, including isotropic distribution of the noise sources. Very often, this distribution is nonisotropic, which introduces bias in the Green's function reconstruction. To minimize this bias, a spatiotemporal inverse filter is proposed. The method is tested on a directive noise field computed from an experimental active seismic data set. The results indicate that the passive inverse filter allows the manipulation of the spatiotemporal degrees of freedom of a complex wave field, and it can efficiently compensate for the noise wavefield directivity.

Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.

Noise interferometry in an inhomogeneous environment in the geometric limit

An approximation to the transient Green's function G(x(a)∣x(b),t) between points x(a) and x(b) can be estimated by taking the time derivative of the correlation function C(ab)(t) of records of ambient noise measured at locations x(a) and x(b). From the general relationship between C(ab)(t) and G(x(a)∣x(b),t) it is shown, using a stationary-phase-like argument, that in an inhomogeneous environment in the geometric limit C(ab)(t) consists of a superposition of signed step functions and two-sided logarithmic singularities that are delayed in time by the travel times of the rays connecting x(a) and x(b).

Acoustic emission localization in complex dissipative anisotropic structures using a one-channel reciprocal time reversal method

This paper presents an imaging method for the localization of the impact point in complex anisotropic structures with diffuse field conditions, using only one passive transducer. The proposed technique is based on the reciprocal time reversal approach (inverse filtering) applied to a number of waveforms stored into a database containing the experimental Green's function of the structure. Unlike most acoustic emission monitoring systems, the present method exploits the benefits of multiple scattering, mode conversion, and boundaries reflections to achieve the focusing of the source with high resolution. Compared to a standard time reversal approach, the optimal refocusing of the back propagated wave field at the impact point is accomplished through a "virtual" imaging process. The robustness of the inverse filtering technique is experimentally demonstrated on a dissipative stiffened composite panel and the source position can be retrieved with a high level of accuracy in any position of the structure. Its very simple configuration and minimal processing requirements make this method a valid alternative to the conventional imaging Structural Health Monitoring systems for the acoustic emission source localization.

On the two-point cross-correlation function of anisotropic, spatially homogeneous ambient noise in the ocean and its relationship to the Green's function

It is well established that the free-space Green's function can be recovered from the two-point cross-correlation function of a random noise field if the noise is white and isotropic. Ambient noise in the ocean rarely satisfies either of these conditions. However, a non-uniform spectrum could be pre-whitened by the application of a suitable filter but anisotropy cannot be so readily eliminated. To investigate the effects of vertical anisotropy, three azimuthally uniform, spatially homogeneous noise fields are analyzed, two of which are idealized, while the third is representative of ambient noise in the deep ocean. In each case, the coherence function, the cross-correlation function, and the derivative of the latter with respect to the correlation delay, are derived for vertical and horizontal alignments of the sensor pair. With vertical sensors, any step-function discontinuity in the directional density function is mapped into a delta function at an appropriate time delay in the derivative (with respect to time delay) of the cross-correlation function. No such mapping occurs with horizontal sensors. In this case, only horizontally traveling noise can generate delta functions in the derivative of the cross-correlation function, and these always appear at the retarded time on either side of the origin.

Accuracy of the deterministic travel time retrieval from cross-correlations of non-diffuse ambient noise

Measurements of long-range cross-correlations of ambient noise underlie acoustic noise interferometry, a promising technique for passive remote sensing of the environment. Previously established simple, exact relations between deterministic Green's functions and the cross-correlation function of perfectly diffuse noise do not necessarily hold for noise fields in the ocean and atmosphere. Here, the method of a stationary phase is applied to study the information content of the cross-correlation function of non-diffuse noise and to quantify the accuracy of passive measurements of the acoustic travel times.

Retrieval of Green's functions of elastic waves from thermal fluctuations of fluid-solid systems

Fluctuation-dissipation and flow reversal theorems are used to study long-range correlation of thermal phonons in a stationary heterogeneous mechanical system comprised of arbitrary inhomogeneous fluid flow and anisotropic solid. At thermal equilibrium, with an appropriate choice of physical observables to characterize thermal fluctuations within the fluid and within the solid, the general integral expression for the two-point correlation function of the fluctuations reduces to a linear combination of deterministic Green's functions, which describe wave propagation in opposite directions between the two points. It is demonstrated that the cross-correlation of thermal noise contains as much information about the environment as can be obtained in active reciprocal transmission experiments with transceivers placed at the two points. These findings suggest a possible application of ambient noise cross-correlation to passive acoustic characterization of inhomogeneous flows in fluid-solid systems in laboratory and geophysical settings.

Green's function approximation from cross-correlations of 20-100 Hz noise during a tropical storm

Approximation of Green's functions through cross-correlation of acoustic signals in the ocean, a method referred to as ocean acoustic interferometry, is potentially useful for estimating parameters in the ocean environment. Travel times of the main propagation paths between hydrophone pairs were estimated from interferometry of ocean noise data that were collected on three L-shaped arrays off the New Jersey coast while Tropical Storm Ernesto passed nearby. Examination of the individual noise spectra and their mutual coherence reveals that the coherently propagating noise is dominated by signals of less than 100 Hz. Several time and frequency noise normalization techniques were applied to the low frequency data in order to determine the effectiveness of each technique for ocean acoustic applications. Travel times corresponding to the envelope peaks of the noise cross-correlation time derivatives of data were extracted from all three arrays, and are shown to be in agreement with the expected direct, surface-reflected, and surface-bottom-reflected interarray hydrophone travel times. The extracted Green's function depends on the propagating noise. The Green's function paths that propagate horizontally are extracted from long distance shipping noise, and during the storm the more vertical paths are extracted from breaking waves.

Linearly filtered estimation of the time-domain Green's function from measurements of ambient noise

It is possible to estimate the time-domain Green's function of a channel based on measurements of ambient noise by sensors at either end of the channel. This paper presents theoretical results for the impact of filtering on this problem. These results lead to the development of two experimental rules-of-thumb. It is shown that there exists a relationship between system bandwidth and sensor separation, which determines the resolvability of the measurements. The relationship between high-pass filtering and differentiation is discussed, contributing to the debate about whether or not differentiation is required to estimate the time-domain Green's function.

Multichannel array diagnosis using noise cross-correlation

A practical application of noise cross-correlation for the diagnosis of a multichannel ocean hydrophone array is derived. Acoustic data were recorded on a horizontal line array on the New Jersey Shelf while Tropical Storm Ernesto passed through. Results obtained from active source measurements reveal that signals from several hydrophones, which were recorded on certain channels before the storm, are recorded on different channels after the storm. Noise cross-correlation of data recorded during the storm show when, and in what manner, these changes took place.

Reconstruction of Rayleigh-Lamb dispersion spectrum based on noise obtained from an air-jet forcing

The time-domain cross correlation of incoherent and random noise recorded by a series of passive sensors contains the impulse response of the medium between these sensors. By using noise generated by a can of compressed air sprayed on the surface of a plexiglass plate, we are able to reconstruct not only the time of flight but the whole wave forms between the sensors. From the reconstruction of the direct A(0) and S(0) waves, we derive the dispersion curves of the flexural waves, thus estimating the mechanical properties of the material without a conventional electromechanical source. The dense array of receivers employed here allow a precise frequency-wavenumber study of flexural waves, along with a thorough evaluation of the rate of convergence of the correlation with respect to the record length, the frequency, and the distance between the receivers. The reconstruction of the actual amplitude and attenuation of the impulse response is also addressed in this paper.

Retrieving reflection responses by crosscorrelating transmission responses from deterministic transient sources: application to ultrasonic data

By crosscorrelating transmission recordings of acoustic or elastic wave fields at two points, one can retrieve the reflection response between these two points. This technique has previously been applied to measured elastic data using diffuse wave-field recordings. These recordings should be relatively very long. The retrieval can also be achieved by using deterministic transient sources with the advantage of using short recordings, but with the necessity of using many P-wave and S-wave sources. Here, it is shown how reflections were retrieved from the cross correlation of transient ultrasonic transmission data measured on a heterogeneous granite sample.

Ocean acoustic interferometry

Ocean acoustic interferometry refers to an approach whereby signals recorded from a line of sources are used to infer the Green's function between two receivers. An approximation of the time domain Green's function is obtained by summing, over all source positions (stacking), the cross-correlations between the receivers. Within this paper a stationary phase argument is used to describe the relationship between the stacked cross-correlations from a line of vertical sources, located in the same vertical plane as two receivers, and the Green's function between the receivers. Theory and simulations demonstrate the approach and are in agreement with those of a modal based approach presented by others. Results indicate that the stacked cross-correlations can be directly related to the shaded Green's function, so long as the modal continuum of any sediment layers is negligible.

Extracting the Green's function of attenuating heterogeneous acoustic media from uncorrelated waves

The Green's function of acoustic or elastic wave propagation can, for loss-less media, be retrieved by correlating the wave field that is excited by random sources and is recorded at two locations. Here the generalization of this idea to attenuating acoustic waves in an inhomogeneous medium is addressed, and it is shown that the Green's function can be retrieved from waves that are excited throughout the volume by spatially uncorrelated injection sources with a power spectrum that is proportional to the local dissipation rate. For a finite volume, one needs both volume sources and sources at the bounding surface for the extraction of the Green's functions. For the special case of a homogeneous attenuating medium defined over a finite volume, the phase and geometrical spreading of the Green's function is correctly retrieved when the volume sources are ignored, but the attenuation is not.

Unified Green's function retrieval by cross-correlation; connection with energy principles

It has been shown theoretically and observationally that the Green's function for acoustic and elastic waves can be retrieved by cross-correlating fluctuations recorded at two locations. We extend the concept of the extraction of the Green's function to a wide class of scalar linear systems. For systems that are not invariant under time reversal, the fluctuations must be excited by volume sources in order to satisfy the energy balance (equipartitioning) that is needed to extract the Green's function. The general theory for retrieving the Green's function is illustrated with examples that include the diffusion equation, Schrödinger's equation, a vibrating string, the acoustic wave equation, a vibrating beam, and the advection equation. Examples are also shown of situations where the Green's function cannot be extracted from ambient fluctuations. The general theory opens up new applications for the extraction of the Green's function from field correlations that include flow in porous media, quantum mechanics, and the extraction of the response of mechanical structures such as bridges.

Emergence of the acoustic Green's function from thermal noise

The fluctuation-dissipation theorem is used to show how acoustic Green's functions corresponding to sound propagation in opposite directions between any two given points can be extracted from time series of thermal noise recorded at these points. The result applies to arbitrarily inhomogeneous, moving or motionless fluids with time-independent parameters, and demonstrates that the two-point correlation function of thermal noise contains as much information about the environment as can be obtained acoustically by placing transceivers at the two points.

Retrieving the Green's function of the diffusion equation from the response to a random forcing

It is known that the Green's function for nondissipative acoustic or elastic wave propagation can be extracted by correlating noise recorded at different receivers. This property is often related to the invariance for time reversal of the acoustic or elastic wave equations. The diffusion equation is not invariant for time reversal. It is shown in this work that the Green's function of the diffusion equation can also be retrieved by correlating solutions of the diffusion equation that are excited randomly and are recorded at different locations. This property can be used to retrieve the Green's function for diffusive systems from ambient fluctuations. Potential applications include the fluid pressure in porous media, electromagnetic fields in conducting media, the diffusive transport of contaminants, and the intensity of multiply scattered waves.

Recovering the acoustic Green's function from ambient noise cross correlation in an inhomogeneous moving medium

We study long-range correlation of diffuse acoustic noise fields in an arbitrary inhomogeneous, moving fluid. The flow reversal theorem is used to show that the cross-correlation function of ambient noise provides an estimate of a combination of the Green's functions corresponding to sound propagation in opposite directions between the two receivers. Measurements of the noise cross correlation allow one to quantify flow-induced acoustic nonreciprocity and evaluate both spatially averaged flow velocity and sound speed between the two points.

Microscale multiple scattering of coherent surface acoustic wave packets probed with gigahertz time-reversal acoustics

The multiple scattering of coherent surface acoustic wave packets in a microstructure is studied using an ultrafast optical technique. By recording a set of acoustic transfer functions, we show that it is possible to implement time-reversal acoustics and refocus the wave packets up to the GHz range, two orders of magnitude higher than usual. Many applications in time-reversal acoustics are thus transposable to correspondingly smaller structures, opening the way to efficient nondestructive characterization and manipulation of multiple scattering on the microscale.

Equivalence of the virtual-source method and wave-field deconvolution in seismic interferometry

Seismic interferometry and the virtual-source method are related approaches for extracting the Green's function that accounts for wave propagation between receivers by making suitable combinations of the waves recorded at these two receivers. These waves can either be excited by active, controlled, sources, or by natural incoherent sources. We compare this technique with the deconvolution of the wave field recorded at different receivers. We show that the deconvolved wave field is a solution of the same wave equation as that for the physical system, but that the deconvolved wave forms may satisfy different boundary conditions than those of the original system. We apply this deconvolution approach to the wave motion recorded at various levels in a building after an earthquake, and show how to extract the building response for different boundary conditions. Extracting the response of the system with different boundary conditions can be used to enhance, or suppress, the normal-mode response. In seismic exploration this principle can be used for the suppression of surface-related multiples.

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.

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.

Fluctuations in diffuse field-field correlations and the emergence of the Green's function in open systems

Recent intense interest in diffuse field correlation functions, with applications to passive imaging in underwater acoustics and seismology, has raised questions about the degree with which a retrieved waveform can be expected to conform to the Green's function, and in particular the degree with which a ray arrival may be discerned. On considering a simple scalar wave model consisting of fields with distributed random sources, the difffuse field-field correlation function R is defined as a sum of correlation integrals, one for each of the many distinct distributed sources. It is then shown that this ensemble of fields has a correlation function with expectation (R) equal to the Green's function. This model also lends itself to calculations of the variance of R, and thus to estimates of the degree to which an R calculated using finite amounts of data will conform to the Green's function. The model predicts that such conformation is strongest at low frequencies. Ray arrivals are detectable if sufficient data have been collected, but the amount of data needed scales in three dimensions with the square of the source-receiver separation, and the square of the frequency. Applications to seismology are discussed.

Modeling of wave propagation in inhomogeneous media

We present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time-reversal invariance and reciprocity in generally inhomogeneous media. The approach relies on a representation theorem of the wave equation to express the Green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium. Following a predictable initial computational effort, Green's functions between arbitrary points in the medium can be computed as needed using a simple cross-correlation algorithm.

High-Resolution Surface-Wave Tomography from Ambient Seismic Noise

Cross-correlation of 1 month of ambient seismic noise recorded at USArray stations in California yields hundreds of short-period surface-wave group-speed measurements on interstation paths. We used these measurements to construct tomographic images of the principal geological units of California, with low-speed anomalies corresponding to the main sedimentary basins and high-speed anomalies corresponding to the igneous cores of the major mountain ranges. This method can improve the resolution and fidelity of crustal images obtained from surface-wave analyses.

Arrival-time structure of the time-averaged ambient noise cross-correlation function in an oceanic waveguide

Coherent deterministic arrival times can be extracted from the derivative of the time-averaged ambient noise cross-correlation function between two receivers. These coherent arrival times are related to those of the time-domain Green's function between these two receivers and have been observed experimentally in various environments and frequency range of interest (e.g., in ultrasonics, seismology, or underwater acoustics). This nonintuitive result can be demonstrated based on a simple time-domain image formulation of the noise cross-correlation function, for a uniform distribution of noise sources in a Pekeris waveguide. This image formulation determines the influence of the noise-source distribution (in range and depth) as well as the dependence on the receiver bandwidth for the arrival-time structure of the derivative of the cross-correlation function. These results are compared with previously derived formulations of the ambient noise cross-correlation function. Practical implications of these results for sea experiments are also discussed.