Elastic wave thermal fluctuations, ultrasonic waveforms by correlation of thermal phonons

Energy partitioning among compressional and shear waves in three-dimensional attenuating elastic media

In strongly scattering elastic media without attenuation and dispersion, the wavefield is dominated by shear waves, and in three dimensions, the ratio of the S to P energy is given by ES/EP=2(vP/vS)3. This study investigates how this ratio is influenced by attenuation. Both the case of the ringdown mode, where the energy evolves from initial values, and the case of energy equilibrium, where the attenuation is balanced by energy injection sources, are treated. It is shown that in ringdown mode, the energy ratio ES/EP satisfies a Ricatti equation in time: hence, the energy ratio is not an exponential function of time. It is also shown that the long-time energy ratio differs from the value in non-attenuating media when the attenuation coefficients for P and S waves are different. In the case of energy equilibrium, the energy ratio only is equal to the value in non-attenuating media when (1) the time scale of P- and S-wave equilibration is much smaller than the attenuation time or (2) the energy injection rate for each wave type is balanced by the dissipation for that wave type. The latter situation happens when the wavefield is excited by thermal fluctuations in thermal equilibrium.

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.

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.

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.

Measuring the effect of ambient noise directionality and split-beam processing on the convergence of the cross-correlation function

Measurements of ambient noise have been used to infer information about the ocean acoustic environment. In recent years the correlation of ambient noise has been shown to give estimates of the travel time of acoustic paths between the sensors recording the noise. A number of issues affect the results of the noise correlation. This paper presents the results of noise correlation of the two horizontally separated arrays of sensors in the 2010 ambient noise experiment. Using the experimental data, the effects on the convergence of the noise correlation are examined with respect to the size and shape of the arrays, the length of time used, and the directionality of the noise field.

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-correlation in band-limited ocean ambient noise fields

Observations of ambient noise in the ocean are generally band limited, because of the natural spectral shape of the noise or the restricted bandwidth of the detection system. Either way, the noise may be regarded as white noise to which a band-limiting filter has been applied. An analysis of the two-point cross-correlation function of such filtered noise is presented for two cases, isotropic and surface-generated noise. The most pronounced effects occur with high-pass and bandpass filters when the low-frequency cut-off falls well above the first few zeros in the coherence function. In this situation, the sensor separation is very many times the longest acoustic wavelength (associated with the lowest frequency) in the passband. The filtering then produces sharp pulses at correlation delays equal to the numerical value of the acoustic travel time between the sensors. Although these pulses are narrow, they have a finite width, within which a fine structure appears in the form of multiple rapid oscillations, due to the differentiating action of the filter. The number of such oscillations increases as the low-frequency roll-off of the filter becomes steeper. This fine structure is evident in several recently published experimental determinations of the cross-correlation function of band-limited ocean ambient noise.

Coherence and interference in diffuse noise: on the information and statistics associated with spatial wave correlations in directional noise fields

Broadband noise correlation methods for the passive extraction of information about the propagation of waves between distant sensor locations have received considerable attention in the literature. For the case of an isotropic ambient field distribution, there is a well-defined relationship between the expectation value of the wave coherence over the sensors and point-to-point wave propagation. Experimental applications, however, must contend with ambient field anisotropy as well as the performance limitations associated with stochastic fluctuations. This paper explores the influence of ambient field directionality on both (1) the connection between the measured wave coherence and sensor-to-sensor propagation and on (2) the rate at which measurements stochastically converge to the expectation value of the underlying wave coherence. Due to diffraction, the relationship between the measured wave coherence and sensor-to-sensor propagation is shown to be robust to even highly directional ambient field features. While the fluctuations of a stochastic system are generally known to depend on bandwidth and measurement duration, the rate of stochastic convergence depends additionally on the cross-spectral power density (coherent power) relative to the power-spectral density (total incident power). Practical experimental implications of these results are discussed.

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.

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.

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

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

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

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

Extracting the local Green's function on a horizontal array from ambient ocean noise

Using only ocean ambient noise recordings it is possible to approximate the local time domain Green's function (TDGF) and extract the time delays associated with different ray path between the elements of a bottom hydrophone array. Comparing the strength of the noise correlation function taken over increasing time windows with residual fluctuations points to an optimum time window to use in the noise correlation function. Through comparison with computer simulations the resulting time series is shown to accurately approximate noise responses in the environment. Analysis of the TDGF gives accurate environmental detail, specifically the critical angle at the water-sediment interface.

Emergence of the acoustic Green's function from thermal noise

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

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.

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

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

Time reversal of ocean noise

It has recently been shown [Roux et al., "Extracting coherent wave fronts from acoustic ambient noise in the ocean," J. Acoust. Soc. Am. 116, 1995-2003 (2004)] that the time-averaged correlation of ocean noise between two points yields a deterministic waveguide arrival structure embedded in the time-domain Green's function. By performing a set of these correlations between a vertical receive array and a single receiver, transfer functions necessary for time reversal can be extracted from ocean noise. Theory and simulation demonstrate this process and data of opportunity from the North Pacific Acoustic Laboratory experiment confirm the expected performance of a noise-based time reversal mirror.