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

Multi-source wavefield reconstruction of distributed acoustic sensing data using compressive sensing and seismic interferometry

Seismic data recorded by distributed acoustic sensing (DAS) interrogator units on deployed optical fiber are being used for a variety of subsurface imaging and monitoring investigations. To reduce the costs of active-source DAS surveying applications, seismic interferometry can be applied to estimate inter-sensor wavefields from DAS records. However, recording long-term records for ambient interferometry requires considerable data storage and sections of DAS optical fibers may be unusable because of broadside sensitivity considerations from the DAS fiber orientation and due to localized coherent energy sources with amplitudes significantly larger than the ambient signal of interest. Compressive sensing, a wavefield reconstruction technique, can mitigate the problems of large data storage and unusable data. We apply compressive sensing-based multi-source wavefield reconstruction to estimate correlograms of ambient DAS records from a fiber array in Perth, Australia. The multi-source method uses all available virtual-source gathers for simultaneous wavefield reconstruction and is different from the conventional single-source method that separately reconstructs individual virtual-source gathers. Using the Fourier and curvelet transforms to sparsify interferometric wavefields, we show that multi-source reconstruction is applicable to the DAS data and that the Fourier multi-source reconstruction can improve the recovered wavefields by approximately 5-10 dB, compared to the Fourier and curvelet single-source wavefield reconstructions.

Ultrasparse Ultrasonic Synthetic Aperture Focus Imaging by Passive Sensing

Ultrasonic synthetic aperture focus techniques (SAFTs) using less than the total number of available array elements to transmit ("sparse" transmissions) have been recently used in both medical imaging and industrial nondestructive testing (NDT) imaging to increase test speed and simplify multiplexer hardware. The challenge of sparse arrays is to obtain a reasonable image quality given the reduced transmitter-receiver combinations available to the beamforming process. This article proposes a "ultrasparse" SAFT method that employs a minimum number of transmitter elements (from one to four elements only) to obtain an entire full-matrix capture (FMC) set of waveforms. Specifically, a "virtual" FMC is obtained from normalized cross-power spectra between each array element pair in an implementation of "passive" ultrasonic sensing. In order to maintain high image quality without sacrificing imaging speed (e.g., applying a minimal initial time delay and keeping a short time recording window), several key steps have to be taken in this "passive" imaging mode, specifically: 1) the use of carefully designed segment-averaged normalized cross-power spectrum (NCPS) for robust passive reconstruction of the ultrasonic impulse response function (IRF) between two receivers; 2) the use of both the causal and acausal portions of the passively reconstructed IRFs; and 3) the compounding of multiple wave modes in the beamforming process. These steps also ensure the elimination of the near-field blind zone hence potentially enabling near-field imaging. The article first reviews the theory of passive IRF reconstruction between two receivers, comparing time-averaged cross correlation versus segment-averaged NCPS, and then demonstrates the application to ultrasparse SAFT FMC imaging of drilled holes in an aluminum block using a linear transducer array where only one to four elements are used in transmission.

A noise-based high-resolution time-reversal method for acoustic defect localization in water pipes

Pressurized water supply pipeline systems (PWSPS) are quintessential to human development and sustenance, but suffer a multitude of unresolved defects (i.e., leaks, blockages, etc.) due to aging and inaccessibility. In this paper, the ubiquitous high-frequency background noise in PWSPS is harnessed to introduce a high-resolution, passive defect detection technique. The relation between the acoustic Green's and cross correlation functions for a pressurized water pipe is derived for the case of high frequency waves, i.e., acoustic wavelengths smaller than the pipe diameter. This relation is subsequently used to formulate a time-reversal technique for localizing anomalies such as small variations in pipe wall impedance and through-wall defects (i.e., leaks). It is shown that the derived relationship between the cross correlation and acoustic Green's functions enables very accurate defect detection and localization by measuring the background noise at two locations along a pipe. This is an important result given that (i) there is limited access to buried PWSPS to conduct high-frequency active defect detection, and (ii) traditional methods to actively probe pipes (e.g., valve maneuverer) are low-resolution (tens to hundreds of meters) and often result in pipe overloading and fatigue.

Exploring surface source contributions to ocean ambient noise interferometry with airgun shots

A seismic reflection survey conducted directly over two bottom-mounted hydrophones in the north-east Pacific Ocean is used to explore how surface source locations affect ambient noise interferometry for the two hydrophones. The airgun shots are used as an approximation of an impulsive sound source at a discrete location, which allows us to investigate spatial contributions to the cross correlation between the two hydrophones. Simulated and experimental results are presented. The contributions to the cross correlations are explained by different reflections off the surface or bottom of the ocean, and a discussion about what can and cannot be inferred about the emergence to the Green's function is presented.

Evaluation of Various Data Acquisition Scenarios for the Retrieval of Seismic Body Waves from Ambient Noise Seismic Interferometry Technique via Numerical Modeling

Seismic interferometry is often proposed as a cost-efficient technique for reservoir monitoring including CO2 sequestration due to its low cost and environmental advantages over active source imaging. Although many studies have demonstrated the ability of seismic interferometry to retrieve surface waves, body wave imaging remains challenging due to their generally lower amplitudes of body waves in seismic interferometry data. An optimum data acquisition strategy can help retrieve low amplitude body waves better, however, rare attempts have been made to evaluate various data acquisition strategies. In this study, we use numerical modeling to examine three different acquisition schemes to evaluate the retrievability of P waves from seismic interferometry data. From our numerical results, we observe that (1) positing receivers beneath the attenuated weathered layer improves the data quality and signal to noise ratio, but additional processing steps including predictive deconvolution and Radom transform filter are necessary to remove the downgoing surface multiples, artifacts that are generated from this data acquisition; (2) vertical seismic profiling (VSP) alongside with the conventional surface seismic acquisition improve the target zone detection; and (3) crosswell acquisition of seismic interferometry is an ineffective means to obtain reflection events due to the non-similarity of ray paths from the noise sources meaning that the required stationary phase theory is not fulfilled.

A Robust Workflow for Acquiring and Preprocessing Ambient Vibration Data from Small Aperture Ocean Bottom Seismometer Arrays to Extract Scholte and Love Waves Phase-Velocity Dispersion Curves

The phase-velocity dispersion curve (DC) is an important characteristic of the propagation of surface waves in sedimentary environments. Although the procedure for DC estimation in onshore environments using ambient vibration recordings is well established, the DC estimation in offshore environments using Ocean Bottom Seismometers (OBS) array recordings of ambient vibrations presents three additional challenges: (1) the localization of sensors, (2) the orientation of the OBS horizontal components, and (3) the clock error. Here, we address these challenges in an inherent preprocessing workflow to ultimately extract the Love and Scholte wave DC from small aperture OBS array measurements performed between 2018 and 2020 in Lake Lucerne (Switzerland). The arrays have a maximum aperture of 679 m and a maximum deployment water depth of 81 m. The challenges related to the OBS location on the lake floor are addressed by combining the multibeam bathymetry map and the backscatter image for the investigated site with the differential GPS coordinates of the OBS at recovery. The OBS measurements are complemented by airgun surveys. Airgun data are first used to estimate the misorientation of the horizontal components of the OBS and second to estimate the clock error. To assess the robustness of the preprocessing workflow, we use two array processing methods, namely the three-component high-resolution frequency-wavenumber and the interferometric multichannel analysis of surface waves, to estimate the dispersion characteristics of the propagating Scholte and Love waves for one of the OBS array sites. The results show the effectiveness of the preprocessing workflow. We observe the phase-velocity dispersion curve branches in the frequency range between 1.2 and 3.2 Hz for both array processing techniques.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s00024-021-02923-8.

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.

A novel principle to localize the sensitivity of waveform tomography using wave interferences at the observation boundary

When using waveform tomography to perform high-resolution imaging of a medium, it is vital to calculate the sensitivity in order to describe how well a model fits a given set of data and how the sensitivity changes with the spatial distribution of the heterogeneities. The traditional principle behind calculating the sensitivity-for detecting small changes-suffers from an inherent limitation in case other structures, not of interest, are present along the wave propagation path. We propose a novel principle that leads to enhanced localization of the sensitivity of the waveform tomography, without having to know the intermediate structures. This new principle emerges from a boundary integral representation which utilizes wave interferences observed at multiple points. When tested on geophysical acoustic wave data, this new principle leads to much better sensitivity localization and detection of small changes in seismic velocities, which were otherwise impossible. Overcoming the insensitivity to a target area, it offers new possibilities for imaging and monitoring small changes in properties, which is critical in a wide range of disciplines and scales.

Potential Pitfalls in the Analysis and Structural Interpretation of Seismic Data from the Mars InSight Mission

The Seismic Experiment for Interior Structure (SEIS) of the InSight mission to Mars, has been providing direct information on Martian interior structure and dynamics of that planet since it landed. Compared to seismic recordings on Earth, ground motion measurements acquired by SEIS on Mars are made under dramatically different ambient noise conditions, but include idiosyncratic signals that arise from coupling between different InSight sensors and spacecraft components. This work is to synthesize what is known about these signal types, illustrate how they can manifest in waveforms and noise correlations, and present pitfalls in structural interpretations based on standard seismic analysis methods. We show that glitches, a type of prominent transient signal, can produce artifacts in ambient noise correlations. Sustained signals that vary in frequency, such as lander modes which are affected by variations in temperature and wind conditions over the course of the Martian Sol, can also contaminate ambient noise results. Therefore, both types of signals have the potential to bias interpretation in terms of subsurface layering. We illustrate that signal processing in the presence of identified nonseismic signals must be informed by an understanding of the underlying physical processes in order for high fidelity waveforms of ground motion to be extracted. While the origins of most idiosyncratic signals are well understood, the 2.4 Hz resonance remains debated and the literature does not contain an explanation of its fine spectral structure. Even though the selection of idiosyncratic signal types discussed in this paper may not be exhaustive, we provide guidance on best practices for enhancing the robustness of structural interpretations.

Thickness and structure of the martian crust from InSight seismic data

A planet's crust bears witness to the history of planetary formation and evolution, but for Mars, no absolute measurement of crustal thickness has been available. Here, we determine the structure of the crust beneath the InSight landing site on Mars using both marsquake recordings and the ambient wavefield. By analyzing seismic phases that are reflected and converted at subsurface interfaces, we find that the observations are consistent with models with at least two and possibly three interfaces. If the second interface is the boundary of the crust, the thickness is 20 ± 5 kilometers, whereas if the third interface is the boundary, the thickness is 39 ± 8 kilometers. Global maps of gravity and topography allow extrapolation of this point measurement to the whole planet, showing that the average thickness of the martian crust lies between 24 and 72 kilometers. Independent bulk composition and geodynamic constraints show that the thicker model is consistent with the abundances of crustal heat-producing elements observed for the shallow surface, whereas the thinner model requires greater concentration at depth.

Seismic Interferometry from Correlated Noise Sources

It is a well-established principle that cross-correlating seismic observations at different receiver locations can yield estimates of band-limited inter-receiver Green’s functions. This principle, known as Green’s function retrieval or seismic interferometry, is a powerful technique that can transform noise into signals which enable remote interrogation and imaging of the Earth’s subsurface. In practice it is often necessary and even desirable to rely on noise already present in the environment. Theory that underpins many applications of ambient noise interferometry assumes that the sources of noise are uncorrelated in time. However, many real-world noise sources such as trains, highway traffic and ocean waves are inherently correlated in space and time, in direct contradiction to the these theoretical foundations. Applying standard interferometric techniques to recordings from correlated energy sources makes the Green’s function liable to estimation errors that so far have not been fully accounted for theoretically nor in practice. We show that these errors are significant for common noise sources, always perturbing or entirely obscuring the phase one wishes to retrieve. Our analysis explains why stacking may reduce the phase errors, but also shows that in commonly encountered circumstances stacking will not remediate the problem. This analytical insight allowed us to develop a novel workflow that significantly mitigates effects arising from the use of correlated noise sources. Our methodology can be used in conjunction with already existing approaches, and improves results from both correlated and uncorrelated ambient noise. Hence, we expect it to be widely applicable in ambient noise studies.

Laboratory seismic measurements for layer-specific description of fluid mud and for linking seismic velocities to rheological properties

The velocities of the seismic waves propagating in the fluid-mud layer are governed by the rheological properties and density of the fluid mud. Performing seismic transmission measurements inside the fluid mud can give good estimates of the seismic velocities and, thus, of the rheological properties and density. Laboratory ultrasonic transmission measurements of the wave velocities in the fluid-mud layer and their temporal evolution are shown. It is found that the shear-wave velocity and yield stress are positively correlated. Performing a seismic reflection survey for characterization of the fluid-mud layers could be more practical because it allows towing the sources and receivers above the top of fluid-mud layer. Interpretation of the results from a reflection survey, though, is influenced by the water layer above the fluid mud. Applying seismic interferometry to reflection measurements can eliminate the influence of the water layer and retrieve a reflection response from inside the fluid-mud layer. This eliminates the influence of the temperature and salinity of the water layer to obtain information about the seismic properties of the fluid-mud layer. To introduce the approach of retrieving and extracting the reflection response from inside the fluid-mud layer, data from laboratory measurements are used. The obtained compressional- and shear-wave velocities are validated by comparing them with values from current transmission measurements.

Virtual head waves in ocean ambient noise: Theory and modeling

The Green's function retrieval in media with horizontal boundaries usually only considers the extraction of direct and reflected waves but ignores the virtual head waves, which have been observed experimentally from ocean ambient noise and used to invert for geometric and environmental parameters. This paper derives the extraction of virtual head waves from ocean ambient noise using a vertically spaced sensor pair in a Pekeris waveguide. Ocean ambient noise in the water column is a superposition of direct, reflected, and head waves. The virtual head waves are produced by the cross-correlations between head waves and either reflected waves or other head waves. The locations of sources that contribute to the virtual head waves are derived based on the method of stationary phase. It is the integration over time of contributions from these sources that makes the virtual head waves observable. The estimation of seabed sound speed with virtual head waves using a vertical line array is also demonstrated. The slope of the virtual head waves is different from that of direct and reflected waves in the virtual source gather; it is therefore possible to constructively stack the virtual head waves. The predictions are verified with simulations.

Identifying the Frequency Dependent Interactions between Ocean Waves and the Continental Margin on Seismic Noise Recordings

This study presents an exploration into identifying the interactions between ocean waves and the continental margin in the origination of double-frequency (DF, 0.1–0.5 Hz) microseisms recorded at 33 stations across East Coast of USA (ECUSA) during a 10-day period of ordinary ocean wave climate. Daily primary vibration directions are calculated in three frequency bands and projected as great circles passing through each station. In each band, the great circles from all stations exhibit largest spatial density primarily near the continental slope in the western North Atlantic Ocean. Generation mechanisms of three DF microseism events are explored by comparing temporal and spatial variations of the DF microseisms with the migration patterns of ocean wave fronts in Wavewatch III hindcasts. Correlation analyses are conducted by comparing the frequency compositions of and calculating the Pearson correlation coefficients between the DF microseisms and the ocean waves recorded at selected buoys. The observations and analyses lead to a hypothesis that the continental slope causes wave reflection, generating low frequency DF energy and that the continental shelf is where high frequency DF energy is mainly generated in ECUSA. The hypothesis is supported by the primary vibration directions being mainly perpendicular to the strike of the continental slope.

Virtual-source imaging and repeatability for complex near surface

Based on seismic interferometry, the virtual source (VS) method is able to produce virtual gathers at buried receiver locations by crosscorrelating the direct-downgoing waves with corresponding reflected-upgoing waves from surface-source gathers. Theoretically, the VS records can improve seismic quality with less negative impact from overburdened complexities. However, shallow complex structures and weathering layers at near surface not only severely distort the wavepaths, but also introduce multiples, surface waves, scattering noise, and interference among different wave modes. These additional seismic responsescontaminate both direct-downgoing and reflected-upgoing wavefields. As a result, the VS gathers experience spurious events and unbalanced illuminations associated with distorted radiation patterns. Conventional stacking operator can produce significant artifacts for sources associated with ineffective-wavepath cancellation. We review three publications and summarize a comprehensive workflow to address these issues using data-driven offset stacking, wavelet-crosscorrelation filtering, and radiation-pattern correction. A data-driven offset stacking theme, with each individual source contribution is weighted by certain quality measures, is applied for available offsets. The wavelet crosscorrelation transforms time-offset data into local time-frequency and local time-frequency-wavenumber domains. Filters are designed for the power-spectrum in each domain. The radiation-pattern correction spatially alters the contaminated direct-wavefields using a zero-phase matched filter, such that the filtered wavefield is consistent with the model-based direct P-wavefields observed at buried receiver locations. Our proposed workflow produces significant improvement as demonstrated in the 13 time-lapse field surveys that included substantial repeatability problems across a 17-month survey gap.

Characterizing the seabed in the Straits of Florida by using acoustic noise interferometry and time warping

Interferometry of ambient and shipping noise in the ocean provides a way to estimate physical parameters of the seafloor and the water column in an environmentally friendly manner without employing any controlled sound sources. With noise interferometry, two-point cross-correlation functions of noise serve as the probing signals and replace the Green's function measured in active acoustic remote sensing. The amount of environmental information that can be obtained with passive remote sensing and the robustness of the estimates of the seafloor parameters increase when contributions of individual normal modes are resolved in the noise cross-correlation function. Using the data obtained in the 2012 noise-interferometry experiment in the Straits of Florida, dispersion curves of the first four normal modes are obtained in this paper by application of the time-warping transform to noise cross correlations. The passively measured dispersion curves are inverted for unknown geoacoustic properties of the seabed. Resulting thickness of the sediment layer and sound speed are consistent with the geoacoustic models obtained earlier by other means.

Monitoring cleaning cycles of fouled ducts using ultrasonic coda wave interferometry (CWI)

Fouling in heat exchangers is the buildup of deposits on the solid surfaces. These deposits reduce the eco-efficiency of the processing equipment and increase the risk of subsequent surface contamination with the formation of biofilms. In the agro-food and water supplier sectors, which are our main concern, fouling on the hot walls of processing heat exchangers is a common occurrence and requires frequent cleaning cycles to ensure hygiene requirements are met. This results in a considerable ecological footprint. Sensors and diagnostic tools for monitoring fouling are thus of utmost importance to ensure the rational validation of the cleaning end-point and to decrease the environmental impact of the cleaning cycles. In this paper, a non-destructive ultrasonic monitoring technique using coda waves and the associated signal processing was tested to monitor the evolution over time of a deposit layer on a solid wall during cleaning. To ascertain the feasibility of the method, a piece of wax of controlled thickness was deposited to simulate the initial fouling state and a cleaning cycle was launched. The decorrelation coefficient was used as an indicator to monitor fouling. This article presents the principle of this unprecedented technique for measuring the degree of fouling. The results of the experiments show that this non-destructive monitoring technology is sensitive to changes in fouling and that the decorrelation coefficient curves are in agreement with the cleaning kinetics captured using a video camera, thus ascertaining the pertinence of the diagnostic tool proposed.

Toward Creating a Subsurface Camera

In this article, the framework and architecture of a Subsurface Camera (SAMERA) are envisioned and described for the first time. A SAMERA is a geophysical sensor network that senses and processes geophysical sensor signals and computes a 3D subsurface image in situ in real time. The basic mechanism is geophysical waves propagating/reflected/refracted through subsurface enter a network of geophysical sensors, where a 2D or 3D image is computed and recorded; control software may be connected to this network to allow view of the 2D/3D image and adjustment of settings such as resolution, filter, regularization, and other algorithm parameters. System prototypes based on seismic imaging have been designed. SAMERA technology is envisioned as a game changer to transform many subsurface survey and monitoring applications, including oil/gas exploration and production, subsurface infrastructures and homeland security, wastewater and CO₂ sequestration, and earthquake and volcano hazard monitoring. System prototypes for seismic imaging have been built. Creating SAMERA requires interdisciplinary collaboration and the transformation of sensor networks, signal processing, distributed computing, and geophysical imaging.

Acoustic remote sensing for source localization and atmospheric tomography: Applications of the cross-correlation Green's function retrieval method

In this work, the application of the cross-correlation Green's function retrieval method for source localization and atmospheric acoustic tomography is presented. Open field experimental measurements of an acoustic source, with an impulsive waveform, are conducted for the evaluation of an array system. Of particular interest are the source localization and sound speed estimation capabilities of the array system. The cross-correlation delay-and-sum beamformer is used to estimate source directivity and sound speed. This beamformer inherently employs the cross-correlation Green's function retrieval method between a pair of receivers. The beamforming results adequately identify the various source directions as well as the scatterers along the propagation path. Reasonable sound speed estimates are obtained at the peak frequency of the retrieved Green's functions. In the case of atmospheric acoustic tomography, the estimated sound speed from the array system can serve as an average background sound speed in a tomographic inversion algorithm. Utilizing a tomographic inversion algorithm with radial basis functions and the estimated sound speed, the reconstruction of temperature and wind velocity profiles are demonstrated.

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.

The nature of Earth's correlation wavefield: late coda of large earthquakes

The seismic correlation wavefield constructed from the stacked cross-correlograms of the late coda of earthquake signals at stations across the globe provides a wealth of observed pulses as a function of inter-station distance. The interval from 3 to 10 h after the onset of major earthquakes is employed for the period range from 15 to 50 s. The observations can be well matched by synthetic seismograms for a radially stratified Earth. Many of the correlation phases have similar time behaviour to those in the regular wavefield, but others have no correspondence. All such correlation phases can be explained by the interaction of arrivals with a common slowness at the each of the stations being correlated. Using a generalized ray description of the seismic wavefield, the time-distance behaviour of these correlation phases arises from differences in accumulated phase on different propagation paths through the Earth. Distinct arrivals emerge from the correlation field when there are many ways in which combinations of seismic phases can arise with the same difference in propagation legs. The constituents of the late coda are dominated by steeply travelling waves, and in consequence features associated with multiple passages through the whole Earth emerge distinctly, such as high-order multiples of PKIKP .

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.

Acoustic noise interferometry in a time-dependent coastal ocean

Interferometry of underwater noise provides a way to estimate physical parameters of the water column and the seafloor without employing any controlled sound sources. In applications of acoustic noise interferometry to coastal oceans, the propagation environment changes appreciably during the averaging times that are necessary for the Green's functions to emerge from noise cross-correlations. Here, a theory is developed to quantify the effects of nonstationarity of the propagation environment on two-point correlation functions of diffuse noise. It is shown that temporal variability of the ocean limits from above the frequency range, where noise cross-correlations approximate the Green's functions. The theoretical predictions are in quantitative agreement with results of the 2012 noise interferometry experiment in the Florida Straits. The loss of coherence at high frequencies constrains the passive acoustic remote sensing to exploiting a low-frequency part of measured noise cross-correlations, thus limiting the resolution of deterministic inversions. On the other hand, the passively measured coherence loss contains information about statistical characteristics of the ocean dynamics at unresolved spatial and temporal scales.

Robust passive reconstruction of dynamic transfer function in dual-output systems

The focus of this paper is the estimation of the dynamic transfer function between two outputs of a linear system subjected to an uncontrolled and generally unknown excitation, and accounting for possible uncorrelated noise present at both outputs. Several applications of this case exist in the passive identification of dynamic systems including the health monitoring and/or non-destructive evaluation of structures subjected to natural "ambient" excitations. It is well known that noise-robust transfer function estimation of a single-input-single-output system can be achieved by a normalized cross-power spectrum operation. This paper shows that, for the subject case of a dual-output system, particular caution must be placed in the choice of the normalization factor to apply to the cross-power spectrum of the two outputs. In particular, an "inter-segment" averaging method is proposed for the normalization factor in combination with the classical "intra-segment" averaging of the cross-power spectrum in order to estimate the transfer function between the two outputs without the influence of the excitation spectrum and of the uncorrelated noise at the two receivers. Validating results are presented for synthetic signals and for experimental signals from an application to high-speed ultrasonic rail inspection exploiting the train wheels as the "ambient" excitation.

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.

Experimental estimation of in vacuo structural admittance using random sources in a non-anechoic room

Identification of unexploded ordinance buried in the sediment in the littoral waters throughout the world is a problem of great concern. When illuminated by low-frequency sonar some of these targets exhibit an elastic response that can be used to identify them. This elastic behavior is embodied and identified by a quantity called the in vacuo structural admittance matrix Y_s, a relationship between the sonar-induced forces and resulting vibration on its surface. When it is known it can be combined with surface impedances to predict the three-dimensional bistatic scattering in any fluid-like media and for any burial state (depth and orientation). At the heart of this is the measurement of Y_s and it is demonstrated in this paper that this can be accomplished by studying the target in a simple (acoustically unaltered) in-air laboratory environment. The target chosen in this study is a thick spherical shell that was illuminated by a nearly spatially isotropic array of remote loudspeakers. Y_s is constructed from ensemble averages of the cross-correlations of eight collocated accelerometers and microphones placed on the surface of the object. The structural admittance determined from the data showed excellent agreement with theory.

Extraction of scattering echo time by surf noise background subtracted autocorrelation

Extracting echo time is an important step in scatterer detection by ambient noise, while in general the scattered signal is weak and submerged in the background. An experiment of a Polyvinyl chloride pipe in a coastline surf noise environment is designed to extract the pipe's echo time by noise autocorrelation. As expected, the scattered wave of the pipe is submerged in an autocorrelation signal. A method called background subtracted autocorrelation is proposed in this paper, which can retrieve scattered echo time from autocorrelation signal effectively. And the biggest relative error of extracted echo time is less than 2% in the experiment.

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.

Ambient seismic wave field

The ambient seismic wave field, also known as ambient noise, is excited by oceanic gravity waves primarily. This can be categorized as seismic hum (1-20 mHz), primary microseisms (0.02-0.1 Hz), and secondary microseisms (0.1-1 Hz). Below 20 mHz, pressure fluctuations of ocean infragravity waves reach the abyssal floor. Topographic coupling between seismic waves and ocean infragravity waves at the abyssal floor can explain the observed shear traction sources. Below 5 mHz, atmospheric disturbances may also contribute to this excitation. Excitation of primary microseisms can be attributed to topographic coupling between ocean swell and seismic waves on subtle undulation of continental shelves. Excitation of secondary microseisms can be attributed to non-linear forcing by standing ocean swell at the sea surface in both pelagic and coastal regions. Recent developments in source location based on body-wave microseisms enable us to estimate forcing quantitatively. For a comprehensive understanding, we must consider the solid Earth, the ocean, and the atmosphere as a coupled system.

Effect of dispersion on the convergence rate for Green's function retrieval

Much information about wave propagation in a variety of structures has been obtained from Green's function retrieval by noise correlation. Here it is examined how dispersion affects Green's function retrieval and, in particular, its signal-to-noise ratio (SNR). On recalling how the inherent spread of a signal due to band limitation is augmented by spread due to dispersion and propagation distance, and how both affect amplitude, it is argued that SNR in highly dispersive media can be substantially lowered by strong dispersion. It is argued that this is most relevant for gravity waves over large propagation distances in the ocean or atmosphere. In particular, it is discussed that dispersion could explain recent retrieval failure from surface gravity wave noise in the ocean. Methods are considered to ameliorate the poor SNR due to dispersion. Numerical simulation is used to substantiate the analytic results.

Fluctuations in the cross-correlation for fields lacking full diffusivity: The statistics of spurious features

Inasmuch as ambient noise fields are often not fully diffuse the question arises as to how, or whether, noise cross-correlations converge to Green's function in practice. Well-known theoretical estimates suggest that the quality of convergence scales with the square root of the product of integration time and bandwidth. However, correlations from natural environments often show random features too large to be consistent with fluctuations from insufficient integration time. Here it is argued that empirical seismic correlations suffer in practice from spurious arrivals due to scatterers, and not from insufficient integration time. Estimates are sought for differences by considering a related problem consisting of waves from a finite density of point sources. The resulting cross-correlations are analyzed for their mean and variance. The mean is, as expected, Green's function with amplitude dependent on noise strength. The variance is found to have support for all times up to its maximum at the main arrival. The signal-to-noise ratio there scales with the square root of source density. Numerical simulations support the theoretical estimates. The result permits estimates of spurious arrivals' impact on identification of cross-correlations with Green's function and indicates that spurious arrivals may affect estimates of amplitudes, complicating efforts to infer attenuation.

Green's Function Retrieval with Absorbing Probes in Reverberating Cavities

The cross-correlation of a diffuse wave field converges toward the difference between the anticausal and causal Green's functions between two points. This property has paved the way to passive imaging using ambient noise sources. In this Letter, we investigate Green's function retrieval in electromagnetism. Using a model based on the fluctuation dissipation theorem, we demonstrate theoretically that the cross-correlation function strongly depends on the absorption properties of the receivers. This is confirmed in measurements within a reverberation chamber. In contrast to measurements with noninvasive probes, we show that only the anticausal Green's function can be retrieved with a matched antenna. Finally, we interpret this result as an equivalent time-reversal experiment with an electromagnetic sink.

Waveform modeling and inversion of ambient noise cross-correlation functions in a coastal ocean environment

Theoretical studies have shown that cross-correlation functions (CFs) of time series of ambient noise measured at two locations yield approximations to the Green's functions (GFs) that describe propagation between those locations. Specifically, CFs are estimates of weighted GFs. In this paper, it is demonstrated that measured CFs in the 20-70 Hz band can be accurately modeled as weighted GFs using ambient noise data collected in the Florida Straits at ∼100 m depth with horizontal separations of 5 and 10 km. Two weighting functions are employed. These account for (1) the dipole radiation pattern produced by a near-surface source, and (2) coherence loss of surface-reflecting energy in time-averaged CFs resulting from tidal fluctuations. After describing the relationship between CFs and GFs, the inverse problem is considered and is shown to result in an environmental model for which agreement between computed and simulated CFs is good.

Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion

Wave scattering provides profound insight into the structure of matter. Typically, the ability to sense microstructure is determined by the ratio of scatterer size to probing wavelength. Here, we address the question of whether macroscopic waves can report back the presence and distribution of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. In our analysis, monosized hard scatterers 5  μm in radius are immersed in lossless gelatin phantoms to investigate the effect of multiple reflections on the propagation of shear waves with millimeter wavelength. Steady-state monochromatic waves are imaged in situ via magnetic resonance imaging, enabling quantification of the phase velocity at a voxel size big enough to contain thousands of individual scatterers, but small enough to resolve the wavelength. We show in theory, experiments, and simulations that the resulting coherent superposition of multiple reflections gives rise to power-law dispersion at the macroscopic scale if the scatterer distribution exhibits apparent fractality over an effective length scale that is comparable to the probing wavelength. Since apparent fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent. Our results are generic to wave phenomena and carry great potential for sensing microstructure that exhibits intrinsic fractality, such as, for instance, vasculature.

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.

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.

Spatial filtering in ambient noise interferometry

Theoretically, the empirical Green's function between a pair of receivers can be extracted from the cross correlation of the received diffuse noise. The diffuse noise condition rarely is met in the ocean and directional sources may bias the Green's function. Here matrix-based spatial filters are used for removing unwanted contributions in the cross correlations. Two methods are used for solving the matrix filter design problem. First a matrix least-square problem is solved with a low-rank approximation of the pseudo-inverse, here, derived for linear and planar arrays. Second, a convex optimization approach is used to solve the design problem reformulated with ad hoc constraints. The spatial filter is applied to real-data cross correlations of elements from a linear array to attenuate the contribution of a discrete interferer. In the case of a planar array and simulated data, a spatial filter enables a passive upgoing/downgoing wavefield separation along with an efficient rejection of horizontally propagating noise. The impact of array size and frequency band on the filtered cross correlations is discussed.

Green's function retrieval through cross-correlations in a two-dimensional complex reverberating medium

Cross-correlations of ambient noise averaged at two receivers lead to the reconstruction of the two-point Green's function, provided that the wave-field is uniform azimuthally, and also temporally and spatially uncorrelated. This condition depends on the spatial distribution of the sources and the presence of heterogeneities that act as uncorrelated secondary sources. This study aims to evaluate the relative contributions of source distribution and medium complexity in the two-point cross-correlations by means of numerical simulations and laboratory experiments in a finite-size reverberant two-dimensional (2D) plate. The experiments show that the fit between the cross-correlation and the 2D Green's function depends strongly on the nature of the source used to excite the plate. A turbulent air-jet produces a spatially uncorrelated acoustic field that rapidly builds up the Green's function. On the other hand, extracting the Green's function from cross-correlations of point-like sources requires more realizations and long recordings to balance the effect of the most energetic first arrivals. When the Green's function involves other arrivals than the direct wave, numerical simulations confirm the better Green's function reconstruction with a spatially uniform source distribution than the typical contour-like source distribution surrounding the receivers that systematically gives rise to spurious phases.

Physical and non-physical energy in scattered wave source-receiver interferometry

Source-receiver interferometry allows Green's functions between sources and receivers to be estimated by means of convolution and cross-correlation of other wavefields. Source-receiver interferometry has been observed to work surprisingly well in practical applications when theoretical requirements (e.g., complete enclosing boundaries of other sources and receivers) are contravened: this paper contributes to explain why this may be true. Commonly used inter-receiver interferometry requires wavefields to be generated around specific stationary points in space which are controlled purely by medium heterogeneity and receiver locations. By contrast, application of source-receiver interferometry constructs at least kinematic information about physically scattered waves between a source and a receiver by cross-convolution of scattered waves propagating from and to any points on the boundary. This reduces the ambiguity in interpreting wavefields generated using source-receiver interferometry with only partial boundaries (as is standard in practical applications), as it allows spurious or non-physical energy in the constructed Green's function to be identified and ignored. Further, source-receiver interferometry (which includes a step of inter-receiver interferometry) turns all types of non-physical or spurious energy deriving from inter-receiver interferometry into what appears to be physical energy. This explains in part why source-receiver interferometry may perform relatively well compared to inter-receiver interferometry when constructing scattered wavefields.

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.

Retrieval of Green's function in the radiative transfer regime

The field-field correlation function of an imperfectly diffuse acoustic field is shown to equal the (time derivative of) Green's function times the specific intensity of the noise at the position of the pseudo-source directed toward the pseudo-receiver. The identity is established in a high frequency limit in which stations are separated by distances large compared to a wavelength and in which equal-time correlations vary smoothly in space. The specific intensity is governed by a radiative transport equation. This observation permits interpretation of correlation amplitudes and promises to facilitate the retrieval of attenuation, site amplification factors, and scattering strengths from noise correlations.

Cross-correlations of diffuse noise in an ocean environment using eigenvalue based statistical inference

Cross-correlations of diffuse noise fields can be used to extract environmental information. The influence of directional sources (usually ships) often results in a bias of the travel time estimates obtained from the cross-correlations. Using an array of sensors, insights from random matrix theory on the behavior of the eigenvalues of the sample covariance matrix (SCM) in an isotropic noise field are used to isolate the diffuse noise component from the directional sources. A sequential hypothesis testing of the eigenvalues of the SCM reveals eigenvalues dominated by loud sources that are statistical outliers for the assumed diffuse noise model. Travel times obtained from cross-correlations using only the diffuse noise component (i.e., by discarding or attenuating the outliers) converge to the predicted travel times based on the known array sensor spacing and measured sound speed at the site and are stable temporally (i.e., unbiased estimates). Data from the Shallow Water 2006 experiment demonstrates the effectiveness of this approach and that the signal-to-noise ratio builds up as the square root of time, as predicted by theory.

Enhancing the emergence rate of coherent wavefronts from ocean ambient noise correlations using spatio-temporal filters

Extracting coherent wavefronts between passive receivers using cross-correlations of ambient noise (CAN) provides a means for monitoring the seismoacoustic environment without using active sources. However, using cross-correlations between single receivers can require a long recording time in order to extract stable coherent arrivals from CAN. This becomes an issue if the propagation medium fluctuates significantly during the recording period. To address this issue, this article presents a general spatio-temporal filtering procedure to enhance the emergence rate for coherent wavefronts extracted from time-averaged ambient noise correlations between two spatially separated arrays. The robustness of this array-based CAN technique is investigated using ambient shipping noise recorded over 24 h in the frequency band [250-850 Hz] on two vertical line arrays deployed 143 m apart in shallow water (depth 20 m). Experimental results confirm that the array-based CAN technique can significantly reduce the recording duration (e.g., from 22 h to 30 min) required for extracting coherent wavefronts of sufficient amplitude (e.g., 20 dB over residual temporal fluctations) when compared to conventional CAN implementations between single pairs of hydrophones. These improvements of the CAN technique could benefit the development of noise-based ocean monitoring applications such as passive acoustic tomography.

The seismic noise wavefield is not diffuse

Passive seismology is burgeoning under the apparent theoretical support of diffuse acoustics. However, basic physical arguments suggest that this theory may not be applicable to seismic noise. A procedure is developed to establish the applicability of the diffuse field paradigm to a wavefield, based on testing the latter for azimuthal isotropy and spatial homogeneity. This procedure is then applied to the seismic noise recorded at 65 sites covering a wide variety of environmental and subsoil conditions. Considering the instantaneous oscillation vector measured at single triaxial stations, the hypothesis of azimuthal isotropy is rejected in all cases with high confidence, which makes the spatial homogeneity test unnecessary and leads directly to conclude that the seismic noise wavefield is not diffuse. However, such a conclusion has no practical effect on passive imaging, which is also possible in non-diffuse wavefields.

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.

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).

Source-receiver interferometry for seismic wavefield construction and ground-roll removal

Seismic interferometry describes the construction of unmeasured wavefield responses (or Green's functions) between two or more points by applying cross-correlation, deconvolution, or convolution to seismic data recordings. The practical implications are that, applying inter-receiver interferometry, a “virtual” (imaginary) source of energy can be created at the location to a real receiver by using energy recorded from surrounding sources. Similarly, by using inter-source interferometry, a virtual receiver can be created at the location of a real source by using energy recorded at surrounding receivers (Curtis et al., 2009). These two methods can be combined to create the new technique of source-receiver interferometry (Curtis, 2009; Curtis and Halliday, 2010; Halliday and Curtis, 2010) which synthesizes real-source to real-receiver Green's function estimates using only energy recorded at a surrounding boundary of receivers and from an additional surrounding boundary of sources. The boundary sources in each case can be active (such as an explosive or vibrating source) or passive (such as noise from anthropogenic activity or ocean waves). This paper describes the first real-data application of source-receiver interferometry, and demonstrates its potential to enhance existing methods of interferometric ground-roll suppression.

Lagrangian and energy forms for retrieving the impulse response of the Earth due to random electromagnetic forcing

We distinguish between trivial and nontrivial differences in retrieving the real or imaginary parts of the Green's function. Trivial differences come from different Green's function definitions. The energy and lagrangian forms constitute nontrivial differences. Magnetic noise sources suffice to extract the quasistatic electromagnetic-field Earth impulse response in the lagrangian form. This is of interest for Earth subsurface imaging. A numerical example demonstrates that all source vector components are necessary to extract a single-field vector component.

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.