Power-law Attenuation Research Articles

Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations.

The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations. To demonstrate some of the similarities and differences that are observed in three related time-fractional partial differential equations, time-domain Green's functions are calculated numerically for the power law wave equation, the Szabo wave equation, and for the Caputo wave equation. These Green's functions are evaluated for water with a power law exponent of y = 2, breast with a power law exponent of y = 1.5, and liver with a power law exponent of y = 1.139. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident very close to the source and that these causal and noncausal time-domain Green's functions converge to the same result away from the source. When noncausal time-domain Green's functions are convolved with a short pulse, no evidence of noncausal behavior remains in the time-domain, which suggests that these causal and noncausal time-fractional models are equally effective for these numerical calculations.

Prediction of multivalued waveforms in media with power-law attenuation

The lossless Burgers equation predicts a multivalued waveform beyond a certain propagation distance. Inclusion of thermoviscous attenuation, which increases as frequency squared, prevents the occurrence of multivalued waveforms. The same is true for any attenuation law that is proportional to frequency raised to an exponent greater than unity. For exponents less than unity the situation is less clear. For example, when attenuation is constant with frequency (exponent equal zero) there is a critical value of the attenuation coefficient below which a multivalued waveform is predicted and above which it is not. To investigate the prediction of multivalued waveforms for power-law attenuation with exponents between zero and unity, a Burgers equation with the loss term expressed as a fractional derivative is used [Prieur and Holm, J. Acoust. Soc. Am. 130, 1125 (2011)]. Transformation of the equation into intrinsic coordinates following Hammerton and Crighton [J. Fluid Mech. 252, 585 (1993)] permits numerical solutions to be obtained that are used to determine the parameter space in which initially sinusoidal plane waves are predicted to evolve into multivalued waveforms for power-law attenuation with exponents less than unity. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

Spatial Dispersion of Elastic Waves in a Bar Characterized by Tempered Nonlocal Elasticity

Abstract We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kröner-Eringen’s model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.

Tidal dissipation in creeping ice and the thermal evolution of Europa

Abstract The thermal and mechanical evolution of Europa and comparable icy satellites—the physics behind creating and sustaining a subsurface water ocean—depends almost entirely on the mechanical dissipation of tidal energy in ice to produce heat, the mechanism(s) of which remain poorly understood. In deformation experiments, we combine steady–state creep and low-frequency, small-strain periodic loading, similar conditions in which tectonics and tidal flexing are occurring simultaneously. The data reveal that the relevant, power-law attenuation in ice (i) is non-linear, depending on strain amplitude, (ii) is independent of grain size, and (iii) exceeds in absorption the prediction of the Maxwell solid model by an order of magnitude. The Maxwell solid model is widely used to model the dynamics of planetary ice shells, so this discrepancy is important. The prevalent understanding of damping in the geophysical context is that it is controlled by chemical diffusion on grain boundaries, which renders attenuation strongly dependent on grain size. In sharp contrast, our results indicate instead the importance of intracrystalline dislocations and their spatial interactions as the critical structural variable affecting dissipation. These dislocation structures are controlled by stress and realized by accumulated plastic strain. Thus, tectonics and attenuation are coupled, which, beyond the icy satellite/subsurface ocean problem, has implications also for understanding the attenuation of seismic waves in deforming regions of the Earth's upper mantle.

Nonlocal Diffusion in Porous Media: A Spatial Fractional Approach

AbstractOne-dimensional diffusion problems in bounded porous media characterized by the presence of nonlocal interactions are investigated by assuming a Darcy’s constitutive equation of convolution integral type. A power law attenuation function is implemented: Analogies and differences of the flow-rate-pressure law with respect to other nonlocal and fractal models are outlined. By means of the continuity relationship, the fractional diffusion equation is then derived. It involves spatial Riemann-Liouville derivatives with a noninteger order consisting of between 1 and 2. The solution is obtained numerically using fractional finite differences, and results are presented in both the transient and the steady-state regimes. Eventually, the physical meaning of fractional operators is discussed and potential applications of the analysis are suggested.

QLg tomography in Gujarat, Western India

We propose a novel Lg attenuation tomography model (QLg tomography) for the state of Gujarat, Western India, using earthquake data recorded by the Gujarat Seismic Network, operated by the Institute of Seismological Research in Gandhinagar. The waveform dataset consist of 400 3-component recordings, produced by 60 earthquakes with magnitude (ML) spanning from 3.6 to 5.1, recorded at 60 seismic stations having epicentral distances spanning between 200 and 500 km. Spectral amplitude decays for Lg wave displacement were obtained by generalized inversion at 17 frequencies spanning between 0.9 and 9 Hz. Lg wave propagation efficiency was measured by Lg/Pn spectral ratio categorizing as efficient ratio ≥6 for 86%, intermediate ratio of 3–6 for 10% and inefficient ratio <3 for 4% paths of total 400 ray paths. The earthquake size and quality of waveform recorded at dense network found sufficient to resolve lateral variation of QLg in Gujarat.Average power-law attenuation relationship obtained for Gujarat as QLg(f) = 234f0.64, which corresponds to high attenuation in comparison to peninsular India shield region and other several regions around the world. QLg tomography resolves the highly attenuating crust of extremely fractured Saurashtra region and tectonically active Kachchh region. The Gujarat average attenuation is also lying in between them. The low attenuation in Cambay and Narmada rift basins and extremely low attenuation in patch of Surendranagar area is identified. This study is the first attempt and can be utilized as pivotal criteria for scenario hazard assessment, as maximum hazard has been reported in highly attenuating tectonically active Kachchh region and in low attenuating Cambay, Narmada and Surendranagar regions. The site and source terms are also obtained along with the QLg inversion. The estimated site responses are comparable with observed local geological condition and agree with the previously reported site amplifications at the same sites. The source terms are comparable with local magnitude estimated from Network. The Mw (Lg) is nearly equivalent to ML (GSN) and the slight differences are noted for larger magnitude events.

Frequency power-law attenuation and dispersion in marine sediments

Acoustic waves in marine sediments, including the fine-grained muds and clays, often exhibit an attenuation in the form of a frequency power law in which the exponent is close to unity. The frequency dependence of the wave speed, or dispersion, associated with such a power law has long been a subject of debate. Causality arguments, often in the form of the Kramers-Kronig dispersion relations, are usually applied to the problem, but these are not sufficient to characterize the dispersion fully. Following an alternative approach, a wave equation will be introduced which predicts the dispersion when the attenuation is of the frequency power law type. Based on this wave equation, it will be shown that only certain values of the frequency exponent are allowed, otherwise causality is violated, indicating that the associated attenuation and dispersion are not physically realizable. In effect, for these prohibited values of the frequency exponent, the Fourier components in the dispersion and attenuation cannot combine to give a zero response for negative times. Emerging from the theory is an expression for the complex wavenumber that is complete and exact, and from which all the properties of the dispersion and attenuation may be derived.

Four ways to interpret temporal memory operators in the wave equation

Arbitrary power law attenuation and dispersion in medical ultrasound and elastography can be modeled with temporal memory operators in the wave equation. Here four alternative interpretations are given. Power laws for attenuation with exponents other than 2 correspond to the use of convolution operators expressible as fractional derivatives. The corresponding constitutive equation can be verified by time domain measurements. A framework with multiple relaxation processes can be formulated via the complex compressibility. The constitutive equation can also be expressed with higher order derivatives.

On the Nonlinear Effects in Focused Ultrasound Beams with Frequency Power Law Attenuation

When finite amplitude ultrasound propagation is considered, changes in spatial features of focused ultrasound beams can be observed. These nonlinear effects typically appear in thermoviscous fluids as focal displacements, beam-width variations or gain changes. However, in soft-tissue media, the frequency dependence of the attenuation doesn’t obey a squared law. In this way, these complex media response leads to weak dispersion that prevents the cumulative processes of energy transfer to higher harmonics. In this work we explore the influence of different frequency power law attenuation responses and its influence on the self-defocusing effects in focused ultrasound beams. Thus, we numerically explore the spatial field distributions produced by low-Fresnel number devices and High Intensity Focused Ultrasound (HIFU) radiating trough different soft-tissue media.

Research on the method and experiment of path rainfall intensity inversion using a microwave link

Accurate measurement of rainfall with high spatial and temporal resolution have important significance in meteorology, hydrology, agriculture, environment, flood warning and weather forecasting, etc. Based on the rain-induced power-law attenuation, an inversion method of the path rainfall intensity is proposed by using a microwave link. Starting from the atmospheric gas absorption attenuation model, a correction model of rainfall effective attenuation and a rainfall inversion model for line-of-sight microwave links are proposed, based on the microwave rain attenuation characteristics and raindrop size distribution statistics. A line-of-sight microwave link is designed and used to measure the rainfall, and the path average rain rate is inversed by means of rainfall inversion model, which is compared with a disdrometer. Results show that the correlation coefficient of rain rate inversed by microwave link with that of disdrometer is higher than 0.6 mostly, and the maximum value is 0.9647; the error of the accumulated rain amount is less than 0.5 mm, and the minimum value is 0.0827 mm; the relative error of the accumulated rain amount is less than 15%, the minimum value is 3.3415%. Experiments confirm the feasibility and accuracy of rainfall inversion obtained using a microwave link.

A Simple Approximation of the Aggregate Interference From a Cluster of Many Interferers With Correlated Shadowing

We examine the statistical distribution of the interference produced by a cluster of many co-channel interferers: e.g., a sensor network, or a city full of active wireless devices and access points. We consider a clustered interferer layout and analyze the interference as experienced at a given point outside (and not immediately near to) the interferer area. We model the propagation paths as experiencing power law attenuation and lognormal correlated shadowing, with pairwise correlations depending on the relative positions of the two interferers. It has already been shown in literature that adding correlation to the shadowing model can give significantly different, and more realistic, results. Our solution is mostly analytical, with only a small amount of numerical integration required. Whereas both simulation and other analytical methods become very computationally demanding as the number of interferers grows, our method's complexity is independent of the number of interferers, and its precision actually improves when their number increases.

Attenuated Fractional Wave Equations With Anisotropy.

This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in complex media. Analytical expressions that describe power law attenuation and anomalous dispersion in each direction are derived for these fractional calculus models.

Analysis of ionic fragment size distribution in collision of nanosecond laser with the C60 molecule

At the critical point of fragmentation phase transition occurring in C60 system, we found that the yield distribution of ionic fragments Cn+ (n=3-16) originating from the multi-fragmentation pattern can be approximated fairly well by a power-law attenuation form of n−λ. Under the framework of Fisher model, the experimental result implies that the ionization-degree distribution of fragments obeys a power law enhancement form as the size n.

Acoustic precursor wave propagation in viscoelastic media

Precursor field theory has been developed to describe the dynamics of electromagnetic field evolution in causally attenuative and dispersive media. In Debye dielectrics, the so-called Brillouin precursor exhibits an algebraic attenuation rate that makes it an ideal pulse waveform for communication, sensing, and imaging applications. Inspired by these studies in the electromagnetic domain, the present paper explores the propagation of acoustic precursors in dispersive media, with emphasis on biological media. To this end, a recently proposed causal dispersive model is employed, based on its interpretation as the acoustic counterpart of the Cole¿Cole model for dielectrics. The model stems from the fractional stress¿strain relation, which is consistent with the empirically known frequency power-law attenuation in viscoelastic media. It is shown that viscoelastic media described by this model, including human blood, support the formation and propagation of Brillouin precursors. The amplitude of these precursors exhibits a sub-exponential attenuation rate as a function of distance, actually being proportional to z(-p), where z is the distance traveled within the medium and 0.5 <p < 1. The precursors identified in this work facilitate the design of optimal waveforms for propagation in complex media, creating new possibilities for acoustic-pulse-based communication and imaging systems.

Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones.

Shock Waves in Weakly Compressed Granular Media

We experimentally probe nonlinear wave propagation in weakly compressed granular media and observe a crossover from quasilinear sound waves at low impact to shock waves at high impact. We show that this crossover impact grows with the confining pressure P0, whereas the shock wave speed is independent of P0-two hallmarks of granular shocks predicted recently. The shocks exhibit surprising power law attenuation, which we model with a logarithmic law implying that shock dissipation is weak and qualitatively different from other granular dissipation mechanisms. We show that elastic and potential energy balance in the leading part of the shocks.

Numerical model of a causal and fractional all-frequency wave equation for lossy media

A numerical algorithm, to simulate a lossy acoustic wave equation with fractional time derivative terms, is presented. The inclusion of fractional derivatives yields a causal acoustic wave equation which can model and predict power law attenuation for which the level of absorption is proportional to frequency raised to a non-integer power. The fractional Zener wave equation is derived from the fractional Zener stress-strain constitutive relation and contains two absorption terms. One term which includes the Laplacian, similar to the traditional wave equation in viscous fluids, has a coefficient, which is proportional to a relaxation time. The other term is a fractional time derivative, higher than second order, and is proportional to the creep or retardation time. Both relaxation and retardation parameters will affect the frequency behavior of the attenuation. The inclusion of two terms enables the modeling of power-law attenuation in all frequency ranges. It results in more stable numerical simulations as both the overall mass and stiffness matrices of the finite element algorithm are changed according to the level of absorption. Results show the animation of diffraction patterns of focused and planar acoustic waves by inclusions with different values for the two different parameters, retardation time and relaxation time.

Model-based discrete relaxation process representation of band-limited power-law attenuation

Frequency-dependent acoustical loss due to a multitude of physical mechanisms is commonly modeled by multiple relaxations. For discrete relaxation distributions, such models correspond with causal wave equations of integer-order temporal derivatives. It has also been shown that certain continuous distributions may give causal wave equations with fractional-order temporal derivatives. This paper demonstrates analytically that if the wave-frequency ω satisfies ΩL≪ω ≪ΩH, a continuous relaxation distribution populating only Ω∈[ΩL,ΩH] gives the same effective wave equation as for a fully populated distribution. This insight sparks the main contribution: the elaboration of a method to determine discrete relaxation parameters intended for mimicking a desired attenuation behavior for band-limited waves. In particular, power-law attenuation is discussed as motivated by its prevalence in complex media, e.g., biological tissue. A Mittag-Leffler function related distribution of relaxation mechanisms has previously been shown to be related to the fractional Zener wave equation of three power-law attenuation regimes. Because these regimes correspond to power-law regimes in the relaxation distribution, the idea is to sample the distribution's compressibility contributions evenly in logarithmic frequency while appropriately taking the stepsize into account. This work thence claims to provide a model-based approach to determination of discrete relaxation parameters intended to adequately model attenuation power-laws.

FRACTIONAL WAVE EQUATIONS WITH ATTENUATION.

Fractional wave equations with attenuation have been proposed by Caputo [5], Szabo [27], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

On a fractional Zener elastic wave equation

Abstract This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.