On the emergence of the Green's function in the correlations of a diffuse field: pulse-echo using thermal phonons

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.

Diffuse field cross-correlations: Scattering theory and electromagnetic experiments

The passive estimation of impulse responses from ambient noise correlations arouses increasing interest in seismology, acoustics, optics, and electromagnetism. Assuming the equipartition of the noise field, the cross-correlation function measured with noninvasive receiving probes converges towards the difference of the causal and anticausal Green's functions. Here, we consider the case when the receiving field probes are antennas which are well coupled to a complex medium-a scenario of practical relevance in electromagnetism. We propose a general approach based on the scattering matrix formalism to explore the convergence of the cross-correlation function. The analytically derived theoretical results for chaotic systems are confirmed in microwave measurements within a mode-stirred reverberation chamber. This study provides fundamental insight into the Green's function retrieval technique and paves the way for a new technique to characterize electromagnetic antennas.

Instantaneous Phase Coherence Imaging for Near-Field Defects by Ultrasonic Phased Array Inspection

This paper describes an imaging method for near-field defect detection in aluminum plates based on Green's function recovery and application of instantaneous phase coherence weighting factors. The directly acquired acoustic information of near-field defects is usually obscured by the nonlinear effects due to the physical limitation of the acquisition system. Using the diffuse field to recover the Green's function can effectively retrieve the early time information. However, averaging operations of finite number in this process produces an imperfect imaging result. In order to improve the image quality, two kinds of instantaneous phased coherence weighting factors are used to weight the Green's function to reduce the background noise and improve the signal-to-noise ratio: the instantaneous phase coherence factor (IPCF), and the instantaneous phase weighting factor (IPWF). Experiments are conducted on two aluminum plates with two and four near-field defects, respectively. As a result, the background noise of amplitude images weighted by IPCF and IPWF is less than that of the conventional total focusing method (TFM). In addition, the IPCF image achieves a better signal-to-noise ratio (SNR) than that of IPWF, and the phase discontinuity in an IPWF image is suppressed through the IPCF.

Wavenumber Imaging of Near-Surface Defects in Rails using Green's Function Reconstruction of Ultrasonic Diffuse Fields

Wavenumber imaging with Green's function reconstruction of ultrasonic diffuse fields is used to realize fast imaging of near-surface defects in rails. Ultrasonic phased array has been widely used in industries because of its high sensitivity and strong flexibility. However, the directly measured signal is always complicated by noise caused by physical limitations of the acquisition system. To overcome this problem, the cross-correlations of the diffuse field signals captured by the probe are performed to reconstruct the Green's function. These reconstructed signals can restore the early time information from the noise. Experiments were conducted on rails with near-surface defects. The results confirm the effectiveness of the cross-correlation method to reconstruct the Green's function for the detection of near-surface defects. Different kinds of ultrasonic phased array probes were applied to collect experimental data on the surface of the rails. The Green's function recovery is related to the number of phased array elements and the excitation frequency. In addition, the duration and starting time of the time-windowed diffuse signals were explored in order to achieve high-quality defect images.

Sparse-TFM Imaging of Lamb Waves for the Near-Distance Defects in Plate-Like Structures

The ultrasonic phased array total focusing method (TFM) has the advantages of high imaging resolution and high sensitivity to small defects. However, it has a long imaging time and cannot realize near-distance defect imaging, which limits its application for industrial detection. A sparse-TFM algorithm is adopted in this work to solve the problem regarding rapid imaging of near- distance defects in thin plates. Green’s function is reconstructed through the cross-correlation of the diffuse full matrix captured by the ultrasonic phased array. The reconstructed full matrix recovers near-distance scattering information submerged by noise. A sparse array is applied to TFM for rapid imaging. In order to improve the imaging resolution, the location of active array elements in the sparse array can be optimized using the genetic algorithm (GA). Experiments are conducted on three aluminium plates with near-distance defects. The experimental results confirm that the sparse-TFM algorithm of Lamb waves can be used for near-distance defects imaging, which increases the computational efficiency by keeping the imaging accuracy. This paper provides a theoretical guidance for Lamb wave non-destructive testing of the near-distance defects in plate-like structures.

Asymptotic analysis for dispersion relations and travel times in noise cross-correlations: spherically symmetric case

The cross-correlations of ambient noise or earthquake codas are massively used in seismic tomography to measure the dispersion curves of surface waves and the travel times of body waves. Such measurements are based on the assumption that these kinematic parameters in the cross-correlations of noise coincide with those in Green's functions. However, the relation between the cross-correlations of noise and Green's functions deserves to be studied more precisely. In this paper, we use the asymptotic analysis to study the dispersion relations of surface waves and the travel times of body waves, and come to the conclusion that for the spherically symmetric Earth model, when the distribution of noise sources is laterally uniform, the dispersion relations of surface waves and the travel times of SH body-wave phases in noise correlations should be exactly the same as those in Green's functions.

Estimating the Green's function using a single channel dual-beam interferometer

Cross-correlation of independent, equipartitioned wavefields is a well-established method to estimate the elastic Green's function, commonly termed seismic interferometry. In this article, the sum of a wavefield recorded at two locations in a single channel is used to estimate the Green's function via the autocorrelation; the result contains some predicted artefacts. The underlying theory and hardware required to estimate the Green's function is presented and compared to traditional seismic interferometry. This technique is used to estimate the elastic Green's function between two locations on an aluminum block with surface scatterers. Wavefields excited via rapid thermoelastic expansion of the surface using a pulsed laser are detected by a dual-beam heterodyne interferometer. The detector is capable of directly recording the sum of a wavefield measured at two locations in a single channel. This method could be an effective, low cost, and non-contacting technique for structural monitoring, particularly where ambient noise has established equipartitioned wavefields in the structure.

Diffuse field full matrix capture for near surface ultrasonic imaging

This article reports a technique for near-surface ultrasonic array imaging. Information equivalent to an undelayed full matrix of inter-element responses is produced through cross-correlation of a later time diffuse full matrix. This reconstructed full matrix lacks the nonlinear effects of early time saturation present in a directly acquired response. Consequently the near-surface material information usually obscured by this effect is retrieved. Furthermore it is shown that a hybrid full matrix formed through a temporally weighted sum of coherent and reconstructed matrices allows for effective near-surface and bulk material imaging from a single direct-contact experimental realisation.

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.

Diffuse shear wave imaging: toward passive elastography using low-frame rate spectral-domain optical coherence tomography

Optical coherence tomography (OCT) can map the stiffness of biological tissue by imaging mechanical perturbations (shear waves) propagating in the tissue. Most shear wave elastography (SWE) techniques rely on active shear sources to generate controlled displacements that are tracked at ultrafast imaging rates. Here, we propose a noise-correlation approach to retrieve stiffness information from the imaging of diffuse displacement fields using low-frame rate spectral-domain OCT. We demonstrated the method on tissue-mimicking phantoms and validated the results by comparison with classic ultrafast SWE. Then we investigated the in vivo feasibility on the eye of an anesthetized rat by applying noise correlation to naturally occurring displacements. The results suggest a great potential for passive elastography based on the detection of natural pulsatile motions using conventional spectral-domain OCT systems. This would facilitate the transfer of OCT-elastography to clinical practice, in particular, in ophthalmology or dermatology.

Experimental study of passive defect localization in plates using ambient noise

Passive listening methodology has been shown to be a practical and effective method for passive structural health monitoring. In this work, this approach is applied experimentally to monitor the occurrence of defects in thin aluminum plates. A correlation matrix is estimated from noise vibrations recorded on a transducer array. A defect is localized by applying a beamforming algorithm to the difference between the correlation matrices obtained with and without the defect. We successfully detect defects for different kinds of noise sources. Moreover, we show that this technique is robust to detect massive inclusions, holes, and cracks. With a vibrometer, we observe that the fidelity of the estimated transient responses strongly depends on the number of uncorrelated noise sources. Finally, we show that the defect is successfully localized even if the noise source distribution is not uniform, provided that it remains spatially stationary between the states with and without defect. A simple theoretical framework is proposed to interpret these results.

Passive ultrasonics using sub-Nyquist sampling of high-frequency thermal-mechanical noise

Monolithic integration of capacitive micromachined ultrasonic transducer arrays with low noise complementary metal oxide semiconductor electronics minimizes interconnect parasitics thus allowing the measurement of thermal-mechanical (TM) noise. This enables passive ultrasonics based on cross-correlations of diffuse TM noise to extract coherent ultrasonic waves propagating between receivers. However, synchronous recording of high-frequency TM noise puts stringent requirements on the analog to digital converter's sampling rate. To alleviate this restriction, high-frequency TM noise cross-correlations (12-25 MHz) were estimated instead using compressed measurements of TM noise which could be digitized at a sampling frequency lower than the Nyquist frequency.

Virtual source method for diffuse optical imaging

The Green's function for diffusive wave propagation can be obtained by utilizing the representation theorems of the convolution type and the correlation type. In this work, the Green's function is retrieved by making use of the Robin boundary condition and the representation theorems for diffusive media. The diffusive Green's function between two detectors for photon flux is calculated by combining detector readings due to point light sources and utilizing virtual light sources at the detector positions in optical tomography. Two dimensional simulations for a circular region with eight sources and eight detectors located on the boundary are performed using a finite element method to demonstrate the feasibility of virtual sources. The most important potential application would be the replacement of noisy measurements with synthetic measurements that are provided by the virtual sources. This becomes an important issue in small animal and human studies. In addition, the same method may also be used to reduce the imaging time.

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.

Thermal-mechanical-noise-based CMUT characterization and sensing

When capacitive micromachined ultrasonic transducers (CMUTs) are monolithically integrated with custom-designed low-noise electronics, the output noise of the system can be dominated by the CMUT thermal-mechanical noise both in air and in immersion even for devices with low capacitance. Because the thermal-mechanical noise can be related to the electrical admittance of the CMUTs, this provides an effective means of device characterization. This approach yields a novel method to test the functionality and uniformity of CMUT arrays and the integrated electronics when a direct connection to CMUT array element terminals is not available. Because these measurements can be performed in air at the wafer level, the approach is suitable for batch manufacturing and testing. We demonstrate this method on the elements of an 800-μm-diameter CMUT-on-CMOS array designed for intravascular imaging in the 10 to 20 MHz range. Noise measurements in air show the expected resonance behavior and spring softening effects. Noise measurements in immersion for the same array provide useful information on both the acoustic cross talk and radiation properties of the CMUT array elements. The good agreement between a CMUT model based on finite difference and boundary element methods and the noise measurements validates the model and indicates that the output noise is indeed dominated by thermal-mechanical noise. The measurement method can be exploited to implement CMUT-based passive sensors to measure immersion medium properties, or other parameters affecting the electro-mechanics of the CMUT structure.

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.

Unified Green's function retrieval by cross correlation

It has been shown by many authors that the cross correlation of two recordings of a diffuse wave field at different receivers yields the Green's function between these receivers. Recently the theory has been extended for situations where time-reversal invariance does not hold (e.g., in attenuating media) and where source-receiver reciprocity breaks down (in moving fluids). Here we present a unified theory for Green's function retrieval that captures all these situations and, because of the unified form, readily extends to more complex situations, such as electrokinetic Green's function retrieval in poroelastic or piezoelectric media. The unified theory has a wide range of applications in "remote sensing without a source."

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.

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.

Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.

Extracting the Green's function from the correlation of coda waves: a derivation based on stationary phase

The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.