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

Rapid emergence of empirical Green's functions from cross-correlations of ambient sound on continental shelfa)

Applications of acoustic noise interferometry to passive remote sensing of the ocean rely on retrieval of empirical Green's functions (EGFs) from cross-correlations of ambient sound at spatially separated points. At ranges of tens of ocean depths, obtaining stable and accurate EGF estimates usually requires noise averaging periods of hours or days. Using data acquired in the Shallow Water 2006 experiment on the continental shelf off New Jersey, it is found that at ranges of 40-70 ocean depths, the EGFs can be retrieved with noise averaging times as short as 64 s. The phenomenon is observed for various receiver pairs but does not occur simultaneously in all azimuthal directions. The rapidly emerging EGFs have a wider frequency band and a richer normal mode content than the EGFs obtained in previous studies using long averaging times and are better suited for monitoring physical processes in the water column. Available acoustic and environmental data is examined to understand the conditions leading to rapid EGF emergence from diffuse noise. Strong intermittency is observed in the horizontal directionality of ambient sound. Rapid emergence of EGF in shallow-water waveguide is found to occur when the directionality of diffuse ambient noise is favorable.

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.

Enhancing ambient noise correlation processing using vector sensors

Ambient noise cross-correlations between separated sensors can yield estimates of the Green's function between them. Vector sensors (which record both pressure and acoustic velocity vector components) can leverage their directionality to reject ambient noise sources that do not contribute to the emergence of the Green's function, thus improving performance over standard omnidirectional hydrophones. To quantify this performance gain, a time-domain analytical expression for the correlation between each component of a vector sensor in the presence of an isotropic ambient noise field is derived. Improvement of the velocity channel correlations relative to pressure channel correlations is examined for varying bandwidth, sensor separation distance, and additive channel noise levels. Last, the experimentally measured reduction in variance for the velocity channels correlations vs pressure correlations, using drifting vector sensors deployed in the Long Island Sound, were found to be comparable to the theoretical prediction. Overall, both theoretical and experimental results indicate modest gains are obtained when extracting the Green's function from velocity correlations over using pressure correlations. Thus, vector sensors can be used to reduce the required averaging time for this noise correlation processing, which may be especially useful, for instance, in a fluctuating environment or for drifting sensors.

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.

Physical and computer-based modeling in internal temperature reconstruction by the method of passive acoustic thermometry

The purpose of this work was to investigate experimentally the capacity of passive acoustic thermometry (PAT) for the reconstruction of 1D, time-variable distributions of the internal temperature. Because in the PAT a noise signal is measured, a considerable integration time (about one minute) is required to attain an acceptable error level (0.5-1K). To optimize the time, an algorithm was proposed to take account of the fact that the temperature satisfied the heat equation. The problem was reduced to that of determining two parameters (initial temperature and thermal diffusivity) of the object under study. The desired parameters were considered constant and were not determined anew after each measurement; instead, their values were refined using all the previous measurements. The proposed algorithm was tested experimentally (where the temperature was reconstructed in a model object, a slab of polytetrafluoroethylene) and investigated by means of computer modeling. The duration of one measurement was about 5.5s. As a result, an error of the temperature reconstruction of about 0.5K, acceptable for medical applications, was attained after 30-60s (depending on the depth) from the beginning of the measurements. After that, temperature distributions can be reconstructed after each measurement without loss of the reconstruction accuracy. The proposed method can be used to control the temperature under a local hyperthermia, lasting 1 min and more, of the human body.

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.

Passive estimation of internal temperatures making use of broadband ultrasound radiated by the body

The internal temperatures of plasticine models and the human forearm in vivo were determined, based on remote measurements of their intrinsic ultrasonic radiation. For passive detection of the thermal ultrasonic radiation an acoustic radiometer was developed, based on a broadband 0.8-3.3 MHz disk-shaped ultrasonic detector with an 8 mm aperture. To reconstruct temperature profiles using the experimentally measured spectra of thermal acoustic radiation a priori information was used regarding the temperature distribution within the objects being investigated. The temperature distribution for heated plasticine was considered to be a monotonic function. The distribution for the human forearm was considered to fit a heat equation incorporating blood flow parameters. Using sampling durations of 45 s the accuracy of temperature measurement inside a plasticine model was 0.5 K. The measured internal temperature of the forearm in vivo, at 36.3 °C, corresponded to existing physiological data. The results obtained verify the applicability of this passive method of wideband ultrasonic thermometry to medical applications that involve local internal heating of biological tissue.

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.

Extracting the Green's function from measurements of the energy flux

Existing methods for Green's function extraction give the Green's function from the correlation of field fluctuations recorded at those points. In this work it is shown that the Green's function for acoustic waves can be retrieved from measurements of the integrated energy flux through a closed surface taken from three experiments where two time-harmonic sources first operate separately, and then simultaneously. This makes it possible to infer the Green's function in acoustics from measurements of the energy flux through an arbitrary closed surface surrounding both sources. The theory is also applicable to quantum mechanics where the Green's function can be retrieved from measurement of the flux of scattered particles through a closed surface.

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).

Resonance blocking and passing effects in two-dimensional elastic waveguides with obstacles

Resonance localization of wave energy in two-dimensional (2D) waveguides with obstacles, known as a trapped mode effect, results in blocking of wave propagation. This effect is closely connected with the allocation of natural resonance poles in the complex frequency plane, which are in fact the spectral points of the related boundary value problem. With several obstacles the number of poles increases in parallel with the number of defects. The location of the poles in the complex frequency plane depends on the defect's relative position, but the gaps of transmission coefficient plots generally remain in the same frequency ranges as for every single obstacle separately. This property gives a possibility to extend gap bands by a properly selected combination of various scatterers. On the other hand, a resonance wave passing in narrow bands associated with the poles is also observed. Thus, while a resonance response of a single obstacle works as a blocker, the waveguide with several obstacles becomes opened in narrow vicinities of nearly real spectral poles, just as it is known for one-dimensional (1D) waveguides with a finite number of periodic scatterers. In the present paper the blocking and passing effects are analyzed based on a semi-analytical model for wave propagation in a 2D elastic layer with cracks or rigid inclusions.

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.

Cross-correlation function of acoustic fields generated by random high-frequency sources

Long-range correlations of noise fields in arbitrary inhomogeneous, moving or motionless fluids are studied in the ray approximation. Using the stationary phase method, two-point cross-correlation function of noise is shown to approximate the sum of the deterministic Green's functions describing sound propagation in opposite directions between the two points. Explicit relations between amplitudes of respective ray arrivals in the noise cross-correlation function and the Green's functions are obtained and verified against specific problems allowing an exact solution. Earlier results are extended by simultaneously accounting for sound absorption, arbitrary distribution of noise sources in a volume and on surfaces, and fluid inhomogeneity and motion. The information content of the noise cross-correlation function is discussed from the viewpoint of passive acoustic characterization of inhomogeneous flows.

Influence of the noise sources motion on the estimated Green's functions from ambient noise cross-correlations

It has been demonstrated theoretically and experimentally that an estimate of the Green's function between two receivers can be obtained by cross-correlating acoustic (or elastic) ambient noise recorded at these two receivers. Coherent wavefronts emerge from the noise cross-correlation time function due to the accumulated contributions over time from noise sources whose propagation path pass through both receivers. Previous theoretical studies of the performance of this passive imaging technique have assumed that no relative motion between noise sources and receivers occurs. In this article, the influence of noise sources motion (e.g., aircraft or ship) on this passive imaging technique was investigated theoretically in free space, using a stationary phase approximation, for stationary receivers. The theoretical results were extended to more complex environments, in the high-frequency regime, using first-order expansions of the Green's function. Although sources motion typically degrades the performance of wideband coherent processing schemes, such as time-delay beamforming, it was found that the Green's function estimated from ambient noise cross-correlations are not expected to be significantly affected by the Doppler effect, even for supersonic sources. Numerical Monte-Carlo simulations were conducted to confirm these theoretical predictions for both cases of subsonic and supersonic moving sources.

Emergence of deterministic Green's functions from noise generated by finite random sources

Two-point correlation functions of sufficiently diffuse wave fields generated by uncorrelated random sources are known to approximate deterministic Green's functions between the two points. This property is utilized increasingly for passive imaging and remote sensing of the environment. Here we show that the relation between the Green's functions and the noise cross-correlation function holds under much less restrictive conditions than previously thought. It can even hold when ambient noise sources have correlation ranges large compared to the wavelength. Admissible correlation ranges are limited from above by the size of the Fresnel zone at wave propagation between the points where noise cross correlation is evaluated.

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.

On the correlation of non-isotropically distributed ballistic scalar diffuse waves

Theorems indicating that a fully equipartitioned random wave field will have correlations equivalent to the Green's function that would be obtained in an active measurement are now legion. Studies with seismic waves, ocean acoustics, and laboratory ultrasound have confirmed them. So motivated, seismologists have evaluated apparent seismic travel times in correlations of ambient seismic noise and tomographically constructed impressive maps of seismic wave velocity. Inasmuch as the random seismic waves used in these evaluations are usually not fully equipartitioned, it seems right to ask why it works so well, or even if the results are trustworthy. The error, in apparent travel time, due to non-isotropic specific intensity is evaluated here in a limit of large receiver-receiver separation and for the case in which the source of the noise is in the far field of both receivers. It is shown that the effect is small, even for cases in which one might have considered the anisotropy to be significant, and even for station pairs separated by as little as one or two wavelengths. A formula is derived that permits estimations of error and corrections to apparent travel time. It is successfully compared to errors seen in synthetic waveforms.

Green's function approximation from cross-correlation of active sources in the ocean

Green's function approximation via ocean noise cross-correlation, referred to here as ocean acoustic interferometry, has been demonstrated experimentally for passive noise sources. Active sources offer the advantages of higher frequencies, controllability, and continuous monitoring. Experimental ocean acoustic interferometry is described here for two active source configurations: a source lowered vertically and one towed horizontally. Results are compared and contrasted with cross-correlations of passive noise. The results, in particular, differences between the empirical Green's function estimates and simulated Green's functions, are explained with reference to theory and simulations. Approximation of direct paths is shown to be consistently good for each source configuration. Secondary (surface reflection) paths are shown to be more accurate for hydrophones with a greater horizontal separation.