Observation of equipartition of seismic waves

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.

Onset of energy equipartition among surface and body waves

We derive a radiative transfer equation that accounts for coupling from surface waves to body waves and the other way around. The model is the acoustic wave equation in a two-dimensional waveguide with reflecting boundary. The waveguide has a thin, weakly randomly heterogeneous layer near the top surface, and a thick homogeneous layer beneath it. There are two types of modes that propagate along the axis of the waveguide: those that are almost trapped in the thin layer, and thus model surface waves, and those that penetrate deep in the waveguide, and thus model body waves. The remaining modes are evanescent waves. We introduce a mathematical theory of mode coupling induced by scattering in the thin layer, and derive a radiative transfer equation which quantifies the mean mode power exchange. We study the solution of this equation in the asymptotic limit of infinite width of the waveguide. The main result is a quantification of the rate of convergence of the mean mode powers toward equipartition.

The 2020 coronavirus lockdown and seismic monitoring of anthropic activities in Northern Italy

In March/April 2020 the Italian government drastically reduced vehicle traffic and interrupted all non-essential industrial activities over the entire national territory. Italy thus became the first country in the world, with the exception of Hubei, to enact lockdown measures as a consequence of the COVID-19 outbreak and the need to contain it. Italy is also a seismically active area, and as such is monitored by a dense permanent network of seismic stations. We analyse continuous seismic data from many stations in northern and central Italy, and quantify the impact of the lockdown on seismic ambient noise, as a function of time and location. We find that the lockdown reduces ambient noise significantly in the 1–10 Hz frequency range; because natural sources of seismic noise are not affected by the lockdown, the seismic signature of anthropic noise can be characterised with unprecedented clarity, by simply comparing the signal recorded before and after the lockdown. Our results correlate well with independent evaluations of the impact of the lockdown (e.g., cell phone displacements), and we submit that ambient-noise seismology is a useful tool to monitor containment measures such as the coronavirus lockdowns.

Opposite sense ground rotations of a pair of Cavendish balances in earthquakes

Among the intermingled translational and rotational effects occurring in earthquakes, the translational effects are rather well understood. Recent experiments have been performed to investigate the rotational effects which have been observed for centuries that remain intriguing and less well understood. Although rotational seismology is of interest in a wide range of disciplines, rotational ground motions remain challenging to detect directly, especially their sense of rotation. To avoid a possible random response of a single balance, we locate two Cavendish balances in an ultra-low-noise laboratory. For the two successive 2012 Italian earthquakes in Emilia detected in exactly the same direction, opposed counterclockwise and clockwise responses of the two balances are recorded at the same site. Despite the complex combinations of Rayleigh and Love surface waves in the far-field, the two circular fundamental eigenmodes of the gravity-free torsion balances permit the determination of the ground rotation senses, which are indirectly confirmed by the phase shifted acceleration components in the balance frequency bandwidth, as well as the corresponding opposite driving angular impulses. The versatility of the Cavendish balances suggests that they could be used as inexpensive rotational sensor arrays in seismic areas to follow the propagation of ground rotations from the epicentres.

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.

Nondiffuse elastic and anelastic passive imaging

The property at the basis of passive acoustic imaging is that, taken any two points, one of them can be seen as the source of the waves and the other as the recording station. This property, which was shown to hold also in nondiffuse fields, is here exploited: (1) to allow an undistorted passive imaging through the simple use of the statistical mode to estimate wave velocity, (2) to determine the azimuth of the instantaneous Huygens sources of the noise wavefield, and (3) to measure, provided that the noise bandwidth is wide with respect to that of the local system, the material dissipation constant as a function of frequency. The authors applied this theory to study the seismic noise field in the Ravenna, North-Central Italy, shore area and found it capable to provide velocity dispersion curves matching those of independent surveys, to track the sources of seismic noise to a few major firms in Ravenna port, with the prevailing source switching at the time scale of seconds, and to measure the dissipation quality factor Q at approximately 20 independent of frequency in the range 1-30 Hz.

Steady state and time-dependent energy equilibration in two-dimensional random elastic slabs

The static and dynamic transport properties of elastic wave propagation through two-dimensional random slabs without internal reflection were studied at two different scattering parameters: one for Rayleigh scattering and the other for Rayleigh-Gans scattering. The spatial distribution and temporal evolution of shear (s-) and compressional (p-) wave energy densities inside the slabs were calculated by solving the radiative transfer equation and the generalized diffusion equation (GDE). The comparison of their results can determine the region of validity of the GDE. The process of energy equilibration between the two wave modes was demonstrated explicitly as well as the process of diffusion. The depth inside a slab that is needed to reach energy equilibration or diffusive behavior is found to be dependent on source polarization. The results also show that the bulk equilibration ratio can be found inside a sample only when the sample is sufficiently thick. Deviations of the equilibration ratio from its bulk value are found near the output surface due to the absence of in-flow energy flux. The behavior of the deviations is sensitive to the scattering parameter but independent of source polarization.

Large enhancement of phononic gap in periodic and quasiperiodic elastic composites by using air inclusions

We find that a phononic gap in a periodic or quasiperiodic elastic composite can be significantly enhanced by inserting air inclusions into the systems. The positions of the insertion are chosen to suppress the shear potential energy of the acoustical branches and lower their frequencies. This is demonstrated in two dimensions. Gap positions and sizes as functions of the radii of the air cylinders for systems of aluminum cylinders in epoxy and steel cylinders in epoxy are presented for both triangular and 12-fold quasiperiodic lattices.

Two perspectives on equipartition in diffuse elastic fields in three dimensions

The elastodynamic Green function can be retrieved from the cross correlations of the motions of a diffuse field. To extract the exact Green function, perfect diffuseness of the illuminating field is required. However, the diffuseness of a field relies on the equipartition of energy, which is usually described in terms of the distribution of wave intensity in direction and polarization. In a full three dimensional (3D) elastic space, the transverse and longitudinal waves have energy densities in fixed proportions. On the other hand, there is an alternative point of view that associates equal energies with the independent modes of vibration. These two approaches are equivalent and describe at least two ways in which equipartition occurs. The authors gather theoretical results for diffuse elastic fields in a 3D full-space and extend them to the half-space problem. In that case, the energies undergo conspicuous fluctuations as a function of depth within about one Rayleigh wavelength. The authors derive diffuse energy densities from both approaches and find they are equal. The results derived here are benchmarks, where perfect diffuseness of the illuminating field was assumed. Some practical implications for the normalization of correlations for Green function retrieval arise and they have some bearing for medium imaging.

Nonlinear effects for coda-type elastic waves in stressed granular media

Experimental results and their interpretations are presented on the nonlinear acoustic effects of multiple scattered elastic waves in unconsolidated granular media. Short wave packets with a central frequency higher than the so-called cutoff frequency of the medium are emitted at one side of the statically stressed slab of glass beads, and received at the other side after multiple-scattering and nonlinear elastic effects. Typical signals are strongly distorted compared to their initially radiated shape both due to nonlinearity and scattering. It is shown that acoustic waves with a deformation amplitude much lower than the mean static deformation of the contacts in the medium can modify the elastic properties of the medium. This addresses the problem of acoustic wave action on granular matter during and after acoustic excitation, which is necessary to understand in the nondestructive testing of the elastic properties of granular media by acoustic methods. Coda signal analysis is shown to be a powerful time-resolved tool in monitoring slight modifications in the elastic response of an unconsolidated granular structure.

Generalized eigenfunctions of layered elastic media and application to diffuse fields

The spectral decomposition of the elastic wave operator in a layered isotropic half-space is derived by means of standard functional analysis methods. Particular attention is paid to the coupled P-SV waves. The problem is formulated directly in terms of displacements which leads to a 2 x 2 Sturm-Liouville system. The resolvent kernel (Green's function) is expressed in terms of simple plane-wave solutions. Application of Stone's formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity. The generalized eigenfunction expansion is employed to define a diffuse field as a white noise process in modal space. By means of a Wigner transform, we calculate vertical to horizontal kinetic energy ratios in layered media, as a function of depth and frequency. Several illustrative examples are considered including energy ratios near a free surface, in the presence of a soft layer. Numerical comparisons between the generalized eigenfunction summation and a classical locked-mode approximation demonstrate the validity of the approach. The impact of the local velocity structure on the energy partitioning of a diffuse field is illustrated.

Temporal changes in the lunar soil from correlation of diffuse vibrations

It was recently demonstrated that one can reconstruct the impulse response between passive sensors by cross-correlating diffuse waves or ambient noise. Using seismic waves recorded on the moon, we show here that not only direct waves can be retrieved, but also late arrivals that have been scattered before reaching the seismometers. As these late arrivals propagate for a longer time, they are more sensitive to weak perturbations of the medium such as velocity changes. This high sensitivity of scattered waves is used to monitor periodic velocity changes in the lunar soil by measuring small delays of the passively retrieved coda waves. The velocity changes result from temperature variations due to periodic heating of the lunar surface by the sun.

Diffusive transport of light in a two-dimensional disordered packing of disks: analytical approach to transport mean free path

We study photon diffusion in a two-dimensional random packing of monodisperse disks as a simple model of granular media and wet foams. We assume that the intensity reflectance of disks is a constant r . We present an analytic expression for the transport mean free path l;{*} in terms of the velocity of light in the disks and host medium, radius R and packing fraction of the disks, and the intensity reflectance. For glass beads immersed in air or water, we estimate transport mean free paths about half the experimental ones. For air bubbles immersed in water, l;{*}R is a linear function of 1epsilon , where epsilon is the liquid volume fraction of the model wet foam. This throws light on the empirical law of Vera [Appl. Opt. 40, 4210 (2001)] and promotes more realistic models.

Passive imaging in nondiffuse acoustic wavefields

A main property of diffuse acoustic wavefields is that, taken any two points, each of them can be seen as the source of waves and the other as the recording station. This property is shown to follow simply from array azimuthal selectivity and Huygens principle in a locally isotropic wavefield. Without time reversal, this property holds approximately also in anisotropic azimuthally uniform wavefields, implying much looser constraints for undistorted passive imaging than those required by a diffuse field. A notable example is the seismic noise field, which is generally nondiffuse, but is found to be compatible with a finite aperture anisotropic uniform wavefield. The theoretical predictions were confirmed by an experiment on seismic noise in the mainland of Venice, Italy.

Fluctuations of local density of states and C0 speckle correlations are equal

We establish a conceptual relation between the fluctuations of the local density of states (LDOS) and the intensity correlations in speckle patterns resulting from the multiple scattering of waves in random media. We show that among the known types of speckle correlations (C1, C2, C3, and C0) only contributes to LDOS fluctuations in the infinite medium. We propose to exploit the equivalence of LDOS fluctuations and the C0 intensity correlation as a "selection rule" for scattering processes contributing to C0.

Observation of multiple scattering of kHz vibrations in a concrete structure and application to monitoring weak changes

In this paper, we present two experimental studies of mechanical wave propagation in a concrete building around 1 kHz. The first experiment is devoted to the observation of the coherent backscattering enhancement, which demonstrates the presence of multiple diffractions in the late part of the wave records. An application of multiple diffraction and reverberations is proposed in a second experiment. Thanks to their sensitivity to weak changes of the medium, the late records are used to monitor weak change in concrete wave velocity induced by thermal variations. The velocity change measurements have a precision of deltac/c=10(-4). Such a precision is difficult to obtain with direct waves. This experiment is the first step to other applications like stress, damage, aging, or crack monitoring in concrete structures.

Diffusion of monochromatic classical waves

We study the diffusion of monochromatic classical waves in a disordered acoustic medium by scattering theory. In order to avoid artifacts associated with mathematical point scatterers, we model the randomness by small but finite insertions. We derive expressions for the configuration-averaged energy flux, energy density, and intensity for one-, two-, and three-dimensional (3D) systems with an embedded monochromatic source using the ladder approximation to the Bethe-Salpeter equation. We study the transition from ballistic to diffusive wave propagation and obtain results for the frequency dependence of the medium properties such as mean free path and diffusion coefficient as a function of the scattering parameters. We discover characteristic differences of the diffusion in 2D as compared to the conventional 3D case, such as an explicit dependence of the energy flux on the mean free path and quite different expressions for the effective transport velocity.

Passive retrieval of Rayleigh waves in disordered elastic media

When averaged over sources or disorder, cross correlation of diffuse fields yields the Green's function between two passive sensors. This technique is applied to elastic ultrasonic waves in an open scattering slab mimicking seismic waves in the Earth's crust. It appears that the Rayleigh wave reconstruction depends on the scattering properties of the elastic slab. Special attention is paid to the specific role of bulk to Rayleigh wave coupling, which may result in unexpected phenomena, such as a persistent time asymmetry in the diffuse regime.

High-Resolution Surface-Wave Tomography from Ambient Seismic Noise

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

Codalike multiple scattering of elastic waves in dense granular media

We study the multiple scattering of short-wavelength ultrasound through the force networks in dry and wet glass bead packings under stress. Over long distance scales, the diffusion approximation is shown to describe adequately the transport of elastic waves dominated by shear waves. The recovered transport mean path reveals a short-range correlation of the force chains. Also we observe the drastic effect of wetting liquids on the energy dissipation in the granular medium. The relevance of these experimental findings for the seismological applications is discussed.

Generalized Berry conjecture and mode correlations in chaotic plates

We consider a modification of the Berry conjecture for eigenmode statistics in wave-bearing systems. The eigenmode correlator is conjectured to be proportional to the imaginary part of the Green's function. The generalization is applicable not only to scalar waves in the interior of homogeneous isotropic systems where the correlator is a Bessel function, but to arbitrary points of heterogeneous systems as well. In view of recent experimental measurements, expressions for the intensity correlator in chaotic plates are derived.

Weak localization of seismic waves

We report the observation of weak localization of seismic waves in a natural environment. It emerges as a doubling of the seismic energy around the source within a spot of the width of a wavelength, which is several tens of meters in our case. The characteristic time for its onset is the scattering mean-free time that quantifies the internal heterogeneity.

Extracting the Green function from diffuse, equipartitioned waves

A ballistic pulse launched in a strongly scattering random medium becomes diffusive after a few mean-free times. In this regime of diffusive propagation there is a net flux of energy away from the source. Eventually the flux goes to zero, in the equipartitioned regime, in which the signal consists of equal amounts of energy propagating in all directions. In this regime the two-point, two-time correlation of the wave-field should equal the sum of the advanced and retarded Green functions associated with the average medium. We observe the emergence of the Green function from this correlation at about 9 mean-free times in a highly heterogeneous rock sample.

Green function retrieval and time reversal in a disordered world

We apply the theory of multiple wave scattering to two contemporary, related topics: imaging with diffuse correlations and stability of the time reversal of diffuse waves, using equipartition, coherent backscattering, and frequency speckles as fundamental concepts.

Diffusive and localization behavior of electromagnetic waves in a two-dimensional random medium

In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan et al. [J. Opt. Soc. Am. B 10, 391 (1993)]. A set of self-consistent equations is presented, accounting for the multiple scattering in the system, and is then solved numerically. A strong localization regime is discovered in the frequency domain. The transport properties within, near the edge of, and nearly outside the localization regime are investigated for different parameters such as filling factor and system size. The results show that within the localization regime, waves are trapped near the transmitting source. Meanwhile, the diffusive waves follow an intuitive but expected picture. That is, they increase with traveling path as more and more random scattering incurs, followed by a saturation, then start to decay exponentially when the travelling path is large enough, signifying the localization effect. For the cases where the frequencies are near the boundary of or outside the localization regime, the results of diffusive waves are compared with the diffusion approximation, showing less encouraging agreement as in other systems [Asatryan et al., Phys. Rev. E 67, 036605 (2003)].

Coda-wave interferometry in finite solids: recovery of P-to-S conversion rates in an elastodynamic billiard

We study the temperature dependence of diffuse reverberant ultrasound in elastic bodies. Transient wave forms are found to undergo an almost pure dilation of 0.0277% per degree, related to the temperature dependence of wave speeds. The wave forms also suffer a distortion that, we argue, depends on the rate of conversion between the dilatational (P) and shear (S) waves. Distortion is found to scale in a manner consistent with theoretical arguments but also appears to be a function of the degree of ray chaos in the body, indicating that the mixing rates are slower in more regular bodies.

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

It is widely recognized that acoustic degrees of freedom coupled to a thermal bath have amplitudes which fluctuate with a mean square proportional to temperature; this is the basis for the Debye theory of the heat capacity of insulating solids. It is shown here that these elastic wave thermal phonons have correlation functions identical to the system's ultrasonic Green's function, and furthermore that thermal noise in ultrasonic detectors should have correlation functions equivalent to conventional waveforms obtained by active transmission and reception. This suggests the possibility of doing ultrasonics without a source. Theory for the identity is presented, and several room temperature laboratory confirmations are conducted in the frequency range 0.1-1.0 MHz. The thermal nature of the origin of these correlations is established by comparing their strength with theoretical expectations. Applications are discussed.

Information transfer through disordered media by diffuse waves

We consider the information content h of a scalar multiple-scattered, diffuse wave field psi(r) and the information capacity C of a communication channel that employs diffuse waves to transfer the information through a disordered medium. Both h and C are shown to be directly related to the mesoscopic correlations between the values of psi(r) at different positions r in space, arising due to the coherent nature of the wave. For the particular case of a communication channel between two identical linear arrays of n>>1 equally spaced transmitters or receivers (receiver spacing a), we show that the average capacity <C> proportional, variant n and obtain explicit analytic expressions for <C>/n in the limit of n--> infinity and kl--> infinity, where k=2pi/lambda, lambda is the wavelength, and l is the mean free path. Modification of the above results in the case of finite but large n and kl is discussed as well. If the signal to noise ratio S/N exceeds some critical value (S/N)(c), <C>/n is a nonmonotonic function of a, exhibiting maxima at ka=mpi (m=1,2, em leader ). For smaller S/N, ka=mpi correspond to local minima, while the absolute maximum of <C>/n is reached at some ka approximately (S/N)(1/2)<pi. We define the maximum average information capacity <C>(max) as <C> maximized over the receiver spacing a and the optimal normalized receiver spacing (ka)(opt) as the spacing maximizing <C>. Both <C>(max)/n and (ka)(opt) scale as (S/N)(1/2) for S/N<(S/N)(c), while (ka)(opt)=mpi and <C>(max)/n approximately log(S/N) for S/N>(S/N)(c).

Long-range correlations in the diffuse seismic coda

The late seismic coda may contain coherent information about the elastic response of Earth. We computed the correlations of the seismic codas of 101 distant earthquakes recorded at stations that were tens of kilometers apart. By stacking cross-correlation functions of codas, we found a low-frequency coherent part in the diffuse field. The extracted pulses have the polarization characteristics and group velocities expected for Rayleigh and Love waves. The set of cross-correlations has the symmetries of the surface-wave part of the Green tensor. This seismological example shows that diffuse waves produced by distant sources are sufficient to retrieve direct waves between two perfectly located points of observation. Because it relies on general properties of diffuse waves, this result has potential applications in other fields.

Influence of ballistics to diffusion transition in primary wave propagation on parametric antenna operation in granular media

A model of parametric transmitting antenna in granular media is developed, which takes into account velocity dispersion, frequency-dependent absorption, and frequency-dependent scattering of acoustic waves in granular media. The latter process may induce a transition from the ballistics to the diffusion regime of pump (primary) high-frequency wave propagation with increasing frequency. The conditions under which the transition from ballistics to diffusion manifests itself in the change of the demodulated (rectified) low-frequency acoustic pulse profile are established. It is demonstrated that parametric low-frequency radiation contains information on both absorption and scattering of high-frequency acoustic waves.

Quasi-two-dimensional transfer of elastic waves

A theory for multiple scattering of elastic waves is presented in a heterogeneous plate bounded by two flat free surfaces, whose horizontal size is infinite and whose transverse size is smaller than the mean free path of the waves. We derive a time-dependent, quasi-two-dimensional radiative transfer equation (i.e., two horizontal dimensions with a finite number of vertical mode) that describes the coupling of the eigenmodes of the layer (surface Rayleigh waves, shear horizontal waves, and Lamb waves). The fundamentally different element is that the traction-free boundary conditions are treated on the level of the wave equation, whereas at the same time elastic transfer can be considered over macroscopic horizontal distances. Expressions are found that relate the small-scale fluctuations to the lifetime of the modes and to their mode-coupling rates. We discuss the diffusion approximation that simplifies the mathematics of this model significantly, and which should apply at large lapse times. Finally, this model facilitates a study of coherent backscattering near the plate surface for different sources and for different detection configurations.

Multiple scattering of seismic waves

We present theory and numerical simulations to model seismic wave propagation in the Earth crust. We compare them to observations made in Mexico.

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

It is shown that a diffuse field is not devoid of phase information, but has a correlation function equal to the Green's function. More specifically, the cross-correlation between diffuse signals in two transducers is very nearly equal to the direct response of one transducer to an impulse applied to the other. This is true whether the diffuse field is one that was created by a distant source, or (if the detectors are sufficiently sensitive) created by thermal fluctuations in the specimen. Here we outline and review proofs, and laboratory demonstrations, from three recent archival publications.

Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies

Noise generated in an ultrasonic receiver circuit consisting of transducer and amplifier is usually ignored, or treated as a nuisance. Here it is argued that acoustic thermal fluctuations, with displacement amplitudes of 3 fm, contain substantial ultrasonic information. It is shown that the noise autocorrelation function is the waveform that would be obtained in a direct pulse/echo measurement. That thesis is demonstrated in experiments in which direct measurements are compared to correlation functions. The thermal nature of the elastodynamic noise that generates these correlations is confirmed by an absolute measurement of their strength, essentially a measurement of the sample temperature.