Frequency-dependent Phase Velocity Research Articles

A Single-Sensor Approach for Noninvasively Tracking Phase Velocity in Tendons during Dynamic Movement.

Shear wave tensiometry is a noninvasive method for directly measuring wave speed as a proxy for force in tendons during dynamic activities. Traditionally, tensiometry has used broadband excitation pulses and measured the wave travel time between two sensors. In this work, we demonstrate a new method for tracking phase velocity using shaped excitations and measurements from a single sensor. We observed modulation of phase velocity in the Achilles tendon that was generally consistent with wave speed measures obtained via broadband excitation. We also noted a frequency dependence of phase velocity, which is expected for dispersive soft tissues. The implementation of this method could enhance the use of noninvasive wave speed measures to characterize tendon forces. Further, the approach allows for the design of smaller shear wave tensiometers usable for a broader range of tendons and applications.

Damping-induced dispersion in simple waveguides

The goal is to gain a non-dimensional formulation of the complex wavenumber knowing the onedimensional partial differential equation for wave propagation and the harmonic wave approach. This can be disassembled into its components, namely the imaginary part, showing the decay of a wave, and the real part, showing its spatial propagation. Having an eye on damping in waveguides including internal and external damping, we discuss the resulting consequences, which are frequency-dependent phase and group velocities (dispersion), as well as the energy transport and dissipation. This bears relevance for many domains of physics such as seismic and electromagnetic waves.

Ultrasonic bandgaps in viscoelastic 1D-periodic media: Mechanical modeling and experimental validation

Multi-material additive manufacturing is receiving increasing attention in the field of acoustics, in particular towards the design of micro-architectured periodic media used to achieve programmable ultrasonic responses. To unravel the effect of the material properties and spatial arrangement of the printed constituents, there is an unmet need in developing wave propagation models for prediction and optimization purposes. In this study, we propose to investigate the transmission of longitudinal ultrasound waves through 1D-periodic biphasic media, whose constituent materials are viscoelastic. To this end, Bloch–Floquet analysis is applied in the frame of viscoelasticity, with the aim of disentangling the relative contributions of viscoelasticity and periodicity on ultrasound signatures, such as dispersion, attenuation, and bandgaps localization. The impact of the finite size nature of these structures is then assessed by using a modeling approach based on the transfer matrix formalism. Finally, the modeling outcomes, i.e., frequency-dependent phase velocity and attenuation, are confronted with experiments conducted on 3D-printed samples, which exhibit a 1D periodicity at length-scales of a few hundreds of micrometers. Altogether, the obtained results shed light on the modeling characteristics to be considered when predicting the complex acoustic behavior of periodic media in the ultrasonic regime.

Ultrasound characterization of the viscoelastic properties of additively manufactured photopolymer materials.

Photopolymer-based additive manufacturing has received increasing attention in the field of acoustics over the past decade, specifically towards the design of tissue-mimicking phantoms and passive components for ultrasound imaging and therapy. While these applications rely on an accurate characterization of the longitudinal bulk properties of the materials, emerging applications involving periodic micro-architectured media also require the knowledge of the transverse bulk properties to achieve the desired acoustic behavior. However, a robust knowledge of these properties is still lacking for such attenuating materials. Here, we report on the longitudinal and transverse bulk properties, i.e., frequency-dependent phase velocities and attenuations, of photopolymer materials, which were characterized in the MHz regime using a double through-transmission method in oblique incidence. Samples were fabricated using two different printing technologies (stereolithography and polyjet) to assess the impact of two important factors of the manufacturing process: curing and material mixing. Overall, the experimentally observed dispersion and attenuation could be satisfactorily modeled using a power law attenuation to identify a reduced number of intrinsic ultrasound parameters. As a result, these parameters, and especially those reflecting transverse bulk properties, were shown to be very sensitive to slight variations of the manufacturing process.

Surface wave tomography using active sources in engineering applications: A 3D experiment at a test site of known conditions

In this study a specific workflow is proposed for the application of Surface wave tomography in near surface applications using active sources. The proposed methodology can be implemented in 2D or 3D acquisition schemes, resulting in high resolution phase and group surface wave velocity maps that can be used for Vs estimation in the survey area. Our methodology starts with the implementation of a pre-processing procedure in which a series of common steps for the quality control and optimization of the data is applied. Then, a reliable reference phase velocity dispersion curve can be estimated internally from our data. This reference curve is used for the automatic assignment of phase and group delays between nearby stations in the time domain. Afterwards, high precision frequency-dependent phase and group velocity maps can be produced through inversion with the Eikonal equation. The proposed methodology is evaluated by application in a 3D acquisition example in a controlled test of known conditions. The resulting frequency-dependent phase and group velocity maps delineate efficiently the expected lateral and depth variations in the survey area. The comparison of the results with available geotechnical information and other geophysical methods confirms the method's validity, demonstrating the effectiveness of the technique and its value in near surface applications as an attractive tool for the calculation of quasi-3D shear wave velocity for a survey area.

Measurement of the ultrasound attenuation and dispersion in 3D-printed photopolymer materials from 1 to 3.5 MHz.

Over the past decade, the range of applications in biomedical ultrasound exploiting 3D printing has rapidly expanded. For wavefront shaping specifically, 3D printing has enabled a diverse range of new, low-cost approaches for controlling acoustic fields. These methods rely on accurate knowledge of the bulk acoustic properties of the materials; however, to date, robust knowledge of these parameters is lacking for many materials that are commonly used. In this work, the acoustic properties of eight 3D-printed photopolymer materials were characterised over a frequency range from 1 to 3.5 MHz. The properties measured were the frequency-dependent phase velocity and attenuation, group velocity, signal velocity, and mass density. The materials were fabricated using two separate techniques [PolyJet and stereolithograph (SLA)], and included Agilus30, FLXA9960, FLXA9995, Formlabs Clear, RGDA8625, RGDA8630, VeroClear, and VeroWhite. The range of measured density values across all eight materials was 1120-1180 kg · m-3, while the sound speed values were between 2020 to 2630 m · s-1, and attenuation values typically in the range 3-9 dB · MHz-1· cm-1.

Analysis of S waves singularity points in porous rock with two orthogonal sets of mesoscale fractures

SUMMARY The shear waves phase velocity surfaces in orthorhombic (ORT) and lower symmetry anisotropic models touch each other in one or more points resulting in so called singularity points or acoustic axes. These singularity points result in dramatic changes of velocities, amplitudes and polarizations creating problems in seismic data processing and analysis. Considering the frequency-dependent anisotropy due to mesoscale fractures in Chapman's model, we describe the singularity points in porous rock with two orthogonal sets of mesoscale fractures. First, we give the equations for frequency-dependent phase velocities of P, S1 and S2 waves in this anelastic ORT media. Then, we derive the expressions for frequency-dependent singularity points within the symmetry planes and discuss the conditions to detect the existence of singularity point. Finally, the influences of frequency, porosity, fracture density, fracture scale and saturating fluid style on the positions of singularity points within the symmetry plane are investigated.

Acoustic Propagation in a Medium with Spatially Distributed Relaxation Processes and a Possible Explanation of a Frequency Power Law Attenuation

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.

Frequency-dependent anisotropy in partially saturated porous rock with multiple sets of mesoscale fractures

SUMMARY Accurate modelling of the frequency-dependence of seismic wave velocity related to fracture system and fluid content is crucial to the quantitative interpretation of seismic data in fractured reservoirs. Both mesoscale fractures and patchy saturation effects can cause significant velocity dispersion and attenuation in the seismic frequency band due to wave-induced fluid flow (WIFF) mechanism. Considering the coupled impact of ‘mesoscale fractures’ and ‘patchy saturation’, we derive expressions for the frequency-dependent anisotropy in partially saturated porous rock containing two fracture sets with different orientations, sizes and connectivities. Especially, we simplify the rock-physics model as an orthorhombic (ORT) media by assuming the mesoscale fractures to be orthogonal and give the explicit expressions for frequency-dependent elastic constants. Finally, we give the expressions for the frequency-dependent phase velocity in patchy saturated and fractured ORT media and investigate the effect of patchy saturation on P-wave velocity at different polar and azimuth angles. In this paper, we investigate the effects of fluid saturation and fluid pressure on frequency-dependent velocities and Thomsen anisotropy parameters. Also, the effect of the relative permeability is very noticeable. The relaxation frequency can be lower in partially saturated fractured rocks compared with the fully saturated case, which makes the rock have a larger stiffness. The non-monotonic relationships between frequency-dependent anisotropy and fluid saturation add complexity to seismic forward modelling and inversion in reservoirs with complex fracture patterns.

Shallow Damage Zone Structure of the Wasatch Fault in Salt Lake City from Ambient-Noise Double Beamforming with a Temporary Linear Array

Abstract We image the shallow structure across the East Bench segment of the Wasatch fault system in Salt Lake City using ambient noise recorded by a month-long temporary linear seismic array of 32 stations. We first extract Rayleigh-wave signals between 0.4 and 1.1 s period using noise cross correlation. We then apply double beamforming to enhance coherent cross-correlation signals and at the same time measure frequency-dependent phase velocities across the array. For each location, based on available dispersion measurements, we perform an uncertainty-weighted least-squares inversion to obtain a 1D VS model from the surface to 400 m depth. We put all piece-wise continuous 1D models together to construct the final 2D VS model. The model reveals high velocities to the east of the Pleistocene Lake Bonneville shoreline reflecting thinner sediments and low velocities particularly in the top 200 m to the west corresponding to the Salt Lake basin sediments. In addition, there is an ∼400-m-wide low-velocity zone that narrows with depth adjacent to the surface trace of the East Bench fault, which we interpret as a fault-related damage zone. The damage zone is asymmetric, wider on the hanging wall (western) side and with greater velocity reduction. These results provide important constraints on normal-fault earthquake mechanics, Wasatch fault earthquake behavior, and urban seismic hazard in Salt Lake City.

Observing traveling waves in glaciers with remote sensing: new flexible time series methods and application to Sermeq Kujalleq (Jakobshavn Isbræ), Greenland

Abstract. The recent influx of remote sensing data provides new opportunities for quantifying spatiotemporal variations in glacier surface velocity and elevation fields. Here, we introduce a flexible time series reconstruction and decomposition technique for forming continuous, time-dependent surface velocity and elevation fields from discontinuous data and partitioning these time series into short- and long-term variations. The time series reconstruction consists of a sparsity-regularized least-squares regression for modeling time series as a linear combination of generic basis functions of multiple temporal scales, allowing us to capture complex variations in the data using simple functions. We apply this method to the multitemporal evolution of Sermeq Kujalleq (Jakobshavn Isbræ), Greenland. Using 555 ice velocity maps generated by the Greenland Ice Mapping Project and covering the period 2009–2019, we show that the amplification in seasonal velocity variations in 2012–2016 was coincident with a longer-term speedup initiating in 2012. Similarly, the reduction in post-2017 seasonal velocity variations was coincident with a longer-term slowdown initiating around 2017. To understand how these perturbations propagate through the glacier, we introduce an approach for quantifying the spatially varying and frequency-dependent phase velocities and attenuation length scales of the resulting traveling waves. We hypothesize that these traveling waves are predominantly kinematic waves based on their long periods, coincident changes in surface velocity and elevation, and connection with variations in the terminus position. This ability to quantify wave propagation enables an entirely new framework for studying glacier dynamics using remote sensing data.

Holographic extraction of plane waves from an ultrasound beam for acoustic characterization of an absorbing layer of finite dimensions.

For the acoustic characterization of materials, a method is proposed for interpreting experiments with finite-sized transducers and test samples in terms of the idealized situation in which plane waves are transmitted through an infinite plane-parallel layer. The method uses acoustic holography, which experimentally provides complete knowledge of the wave field by recording pressure waveforms at points on a surface intersected by the acoustic beam. The measured hologram makes it possible to calculate the angular spectrum of the beam to decompose the field into a superposition of plane waves propagating in different directions. Because these waves cancel one another outside the beam, the idealized geometry of an infinite layer can be represented by a sample of finite size if its lateral dimensions exceed the width of the acoustic beam. The proposed method relies on holograms that represent the acoustic beam with and without the test sample in the transmission path. The method is described theoretically, and its capabilities are demonstrated experimentally for silicone rubber samples by measuring their frequency-dependent phase velocities and absorption coefficients in the megahertz frequency range.

Broadband finite-difference Q-compensated reverse time migration algorithm for tilted transverse isotropic media

[Formula: see text]-compensated reverse time migration (Q-RTM) is effective for improving the seismic imaging quality degraded by low-[Formula: see text] anomalies. However, it is difficult to apply existing pseudospectral-based Q-RTM methods to large-scale problems due to the obstacles to high-efficient parallelization posed by the global pseudospectral operators. On the other hand, finite-difference Q-RTM is intrinsically appropriate for domain decomposition and parallel computation, thus being suitable for industrial-sized problems but facing a twofold challenge: (1) to effectively compensate the phase in a broad-bandwidth sense during the wave back-propagation process and (2) to accurately handle the tilted transverse isotropic (TTI) medium with attenuation. We have developed a new framework of finite-difference Q-RTM algorithm by expanding the linear viscoacoustic constitutive relation to a series of integer-order differential terms and a unique integral term that can decouple the amplitude and the phase to allow accurate compensation in a broader frequency range. This framework has two typical implementations: (1) optimizing the frequency-dependent phase velocity (while fixing the negative constant [Formula: see text]) and (2) optimizing the [Formula: see text] value (while fixing the frequency-dependent phase velocity). We generalize this broadband finite-difference Q-RTM algorithm to TTI media, where an artificial [Formula: see text] is applied to suppress the shear-wave artifacts induced by the acoustic TTI approximation. Numerical examples demonstrate that this Q-RTM method accurately compensates the phase and amplitude in a broad frequency range of 5–70 Hz and produces high-quality images. Due to the local nature of finite-difference operators, this algorithm is expected to outperform the existing pseudospectral-based Q-RTM methods in terms of computational efficiency and implementation convenience for real-world Q-RTM projects.

Combined spatiotemporal and frequency-dependent shear wave elastography enables detection of vulnerable carotid plaques as validated by MRI

Fatal cerebrovascular events are often caused by rupture of atherosclerotic plaques. However, rupture-prone plaques are often distinguished by their internal composition rather than degree of luminal narrowing, and conventional imaging techniques might thus fail to detect such culprit lesions. In this feasibility study, we investigate the potential of ultrasound shear wave elastography (SWE) to detect vulnerable carotid plaques, evaluating group velocity and frequency-dependent phase velocities as novel biomarkers for plaque vulnerability. In total, 27 carotid plaques from 20 patients were scanned by ultrasound SWE and magnetic resonance imaging (MRI). SWE output was quantified as group velocity and frequency-dependent phase velocities, respectively, with results correlated to intraplaque constituents identified by MRI. Overall, vulnerable lesions graded as American Heart Association (AHA) type VI showed significantly higher group and phase velocity compared to any other AHA type. A selection of correlations with intraplaque components could also be identified with group and phase velocity (lipid-rich necrotic core content, fibrous cap structure, intraplaque hemorrhage), complementing the clinical lesion classification. In conclusion, we demonstrate the ability to detect vulnerable carotid plaques using combined SWE, with group velocity and frequency-dependent phase velocity providing potentially complementary information on plaque characteristics. With such, the method represents a promising non-invasive approach for refined atherosclerotic risk prediction.

3D S-wave velocity imaging of Reykjanes Peninsula high-enthalpy geothermal fields with ambient-noise tomography

Tomographic imaging based on ambient seismic noise measurements has shown to be a powerful tool, especially in areas like Iceland, where the microseism illumination is excellent. In this paper, we produce a 3D S-wave tomographic image over the western Reykjanes Peninsula high-enthalpy geothermal fields and evaluate the reliability of the tomographic results for different resolutions through simulated and real data. We use 30 broadband stations operating for approximately one-and-a-half year and apply ambient noise seismic interferometry for each station-pair. This results in empirical Green's functions in which especially the ballistic surface waves (BSW) are well resolved. The retrieved BSW exhibit a high signal-to-noise ratio between 0.1 and 0.5 Hz, and the beamforming analysis indicates an apparent surface-wave velocity of 3 km/s over a broad azimuthal range. For the tomographic inversion, we invert the estimated phase velocities between all station pairs to frequency-dependent phase velocity maps in four different resolutions (1, 2, 3, and 4 km) using a Tikhonov regularisation. With the estimated regularisation parameter per frequency per resolution, we invert simulated data for checkerboard sensitivity tests per frequency for different combinations of velocity anomaly sizes and resolutions.Finally, after the inversion to depth, we detect S-wave velocity anomalies with variations between −15% and 15% with reference to an estimated average velocity using 1 km and 3 km of lateral resolutions and 1 km of vertical resolution. This study shows the potential of ambient noise tomography as complementary seismological tool for reservoir characterization.

Correlations of the Phase Velocity and the Dispersion with the Bone Mineral Density and the Architectural Parameters in Bovine Trabecular Bone

The present study aims to investigate the correlations of the phase velocity and the dispersion (the dispersion rate and the frequency dependence of the phase velocity) with the bone mineral density (BMD) and the architectural parameters (the bone volume fraction, the trabecular thickness, and the trabecular separation) in bovine trabecular bone in vitro. The frequency-dependent phase velocity, the dispersion rate, and the frequency dependence were measured over a frequency range from 0.2 to 0.6 MHz in 22 bovine femoral trabecular bone samples. The phase velocity and the dispersion rate showed significant correlations with the BMD and the architectural parameters. In contrast, the frequency dependence of the phase velocity was found to exhibit very weak correlations with the BMD and the architectural parameters. The stratified model for wave propagation in alternating parallel solid-fluid layers successfully predicted the variations of the phase velocity with the architectural parameters observed in the 22 bone samples.

Surface wave analysis: improving the accuracy of the shear-wave velocity profile through the efficient joint acquisition and Full Velocity Spectrum (FVS) analysis of Rayleigh and Love waves

ABSTRACTSurface wave propagation can be exploited to investigate subsurface conditions in terms of shear wave velocities in a number of possible applications (geotechnical site characterisation, seismic-risk assessment, crustal studies and non-destructive testing). Nowadays, one of the most common methods adopted to analyse the dispersion of surface waves is based on the determination of the Rayleigh wave frequency-dependent phase velocities obtained from multichannel active data. The obtained values represent the dispersion curve, which is inverted to determine the vertical shear-wave velocity (VS) profile. After briefly recalling some fundamental facts and problems regarding surface wave propagation and analysis, we present the Full Velocity Spectrum (FVS) approach which here is used to jointly invert the velocity spectra of the Rayleigh and Love waves acquired by means of a set of horizontal geophones only (Rayleigh waves are in fact analysed while considering their radial component). It is shown that for non-trivial data sets for which modal dispersion curves cannot be soundly extracted, the joint FVS approach may represent an efficient way to properly analyse the data, thus eventually obtaining a robust VS profile free from significant ambiguities that would otherwise inevitably affect the results obtained by following the ordinary approach based on the modal dispersion curve(s) of just one component.

Shear Waves Reveal Viscoelastic Changes in Skeletal Muscles After Hemispheric Stroke.

We investigated alterations in material properties such as elasticity and viscoelasticity of stroke-affected muscles using ultrasound induced shear waves and mechanical models. We used acoustic radiation force to generate shear waves along fascicles of biceps muscles and measured their propagation velocity. The shear wave data were collected in muscles of 13 hemiplegic stroke survivors under passive conditions at 90°, 120°, and 150° elbow flexion angles. In a viscoelastic medium, as opposed to a purely elastic medium, the shear wave propagation velocity depends on the frequency content of the induced wave. Therefore, in addition to the shear wave group velocity (GpV), we also estimated a frequency-dependent phase velocity (PhV). We found significantly higher GpVs and PhVs in stroke-affected muscles ( ). The velocity data were used to estimate shear elasticity and viscosity using an elastic and viscoelastic material models. A pure elastic model showed increased shear elasticity in stroke-affected muscles ( ). The Voigt model estimates of viscoelastic properties were also significantly different between the stroke-impaired and non-impaired muscles. We observed significantly larger model-estimated viscosity values on the stroke-affected side at elbow flexion angles of 120° and 150°. Furthermore, the creep behavior (tissue strain resulting from the application of sudden constant stress) of the model was also different between muscles of the paretic and non-paretic side. We speculate that these changes are associated with the structural disruption of muscles after stroke and may potentially affect force generation from muscle fibers as well as transmission of force to tendons.

A state-of-the-art review of passive MASW survey for subsurface profiling

Passive multichannel analysis of surface wave (MASW) is a nondestructive technique capable of exploring soil subsurface to a great depth by utilizing ambient long-wavelength passive surface waves without requiring an active source. The method yields subsurface information in terms of shear wave velocity (Vs) profile of the surveyed area. Vs is a vital dynamic property of soil which has numerous applications in various geotechnical and earthquake engineering problems. The accuracy of final Vs profile obtained by MASW technique primarily depends upon the resolution of the dispersion curve which is plot of frequency-dependent phase velocity of the propagating Rayleigh wave. The present study critically reviews the effects and influences of various geometrical parameters (such as the shape and size of receiver arrays) and the recording and processing parameters (such as acquisition time, sampling frequency, and vertical stacking) that control the quality of raw wavefield data and subsequent dispersion imaging. The study reveals that there is no fixed recommendation for any of the mentioned parameters to be used to obtain a high-resolution dispersion image in case of passive MASW survey. Therefore, extensive experimentations are required to further understand and develop standard guidelines for field experimentation and data processing so as to reinforce the method as a stand-alone technique.

Ultrasonic characterization of porous gyroid scaffolds for bone tissue engineering: Mechanical modelling and numerical validation

Bone substitutes can be used for pre-implant surgery in presence of volumetric bone defects. In this context, the ultrasound characterization of the bone substitute is a key issue. To this end, we model the implant as a 3D porous structure and we study its ultrasonic behavior. In the framework of artificial bone substitutes, microstructured scaffolds are widely used. In the literature, several geometrical configurations have been tested: among them, the gyroid-shaped scaffolds turn out to be an excellent choice, thanks to its ability to reproduce the behavior and the porous structure of trabecular bone. This study is focused on the mechanical modelling and numerical validation of wave propagation in a porous implant substitute. In particular, 3D finite-difference time-domain (FDTD) simulations were performed on a gyroid-shaped scaffold of saturated with water to validate the continuum mechanical model. Ultrasound excitations at different central frequencies were used in order to investigate the frequency-dependent behavior phase and group velocities.