Sven Peter Näsholm

On a fractional Zener elastic wave equation

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.

Linking multiple relaxation, power-law attenuation, and fractional wave equations

The acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584-1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation. This work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling.

A causal and fractional all-frequency wave equation for lossy media

This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation.

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.

Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations

It is argued that fractional acoustic wave equations come in two kinds. The first kind is constructed ad hoc to have loss operators that fit power law measurements. The second kind is more fundamental as they in addition are based on underlying physical equations. Here that means constitutive equations. These equations are the fractional Kelvin-Voigt and the more general fractional Zener stress-strain relationships as well as a fractional version of the Fourier heat law. The properties of the wave equations are given in terms of attenuation, and phase/group velocities for low-, intermediate- and high-frequency regions. In the most general case, the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant, finite value for high frequencies. It is also shown that the fractional Zener wave equation is equivalent to the multiple relaxation model for attenuation.

Transmit beams adapted to reverberation noise suppression using dual-frequency SURF imaging

A method that uses dual-frequency pulse complexes of widely separated frequency bands to suppress noise caused by multiple scattering or multiple reflections in medical ultrasound imaging is presented. The excitation pulse complexes are transmitted to generate a second order ultrasound field (SURF) imaging synthetic transmit beam. This beam has reduced amplitude near the transducer, which illustrates the multiple scattering suppression ability of the imaging method. Field simulations solving a nonlinear wave equation are used to calculate SURF imaging beams, which are compared with beams for pulse inversion (PI) and fundamental imaging. In addition, a combined SURF and PI beam generation is described and compared with the beams mentioned above. A quality ratio, relating the energy within the near-field to that within the imaging region, is defined and used to score the multiple scattering and multiple reflection suppression abilities when imaging with the different beams. The realized combined SURF-PI beam scores highest, followed by SURF, PI (that score equally well), and the fundamental. The amplitude in the imaging region and therefore also the SNR is highest for the fundamental followed by SURF, PI, and SURF-PI. The work hence indicates that when substituting PI for SURF, one may trade increased SNR into use of increased imaging frequencies without loss of multiple scattering and multiple reflection noise suppression.

Probabilistic infrasound propagation using realistic atmospheric perturbations

This study demonstrates probabilistic infrasound propagation modeling using realistic perturbations. The ensembles of perturbed analyses, provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), include error variances of both model and assimilated observations. Ensemble spread profiles indicate a yearly mean effective sound speed variation of up to 8 ms?1 in the stratosphere, exceeding occasionally 25 ms?1 for a single ensemble set. It is shown that errors in point estimates of effective sound speed are dominated by variations in wind strength and direction. One year of large mining explosions in the Aitik mine, northern Sweden, observed at infrasound array IS37 in northern Norway are simulated using 3-D ray tracing. Probabilistic propagation modeling using the ensembles demonstrates that small-scale fluctuations are not always necessary to improve the match between predictions and observations.

Feasibility of Second Harmonic Imaging in Active Sonar: Measurements and Simulations

Nonlinear acoustics allows for applications like tissue harmonic imaging in medicine and parametric arrays in underwater acoustics. Mainstream sonars transmit and receive signals at the same frequency and up to now energy transferred to higher harmonic frequencies has been mainly seen as a disturbance for target strength estimation, e.g., in fishery research. This paper investigates the feasibility of utilizing the part of the signal generated around the second harmonic frequency band by nonlinear propagation of sound in water. It presents the potential enhancements the second harmonic signal may provide for target imaging as well as multifrequency target recognition. It compares measurements of the pressure field radiated by commercial transducers in water at 121 and 200 kHz up to a range of 12 m with numerical simulations. The detected levels of higher harmonic signals agree with simulations of nonlinear wave propagation. This verifies the implementation of the simulator and allows a comparison of the beam characteristics at longer ranges when filtered around the fundamental or second harmonic frequencies. An example of pulse-echo imaging with spherical targets is also shown using signals at the fundamental and second harmonic frequencies where the second harmonic signal can detect one of the targets that the fundamental signal cannot. Using the active sonar equation to estimate the maximum range, simulations based on a simple model including ambient noise and volume reverberation confirm that with a source level of 228 dB and a detection threshold of 12 dB the fundamental signal at 200 kHz can detect a fish of target strength -36 dB to approximately 343 m while the detection range of the second harmonic signal is approximately 243 m. The combined use of the signal components in the second harmonic and fundamental frequency bands provides a high-resolution image at short range and a long-range imaging capability at a lower resolution as well as a multifrequency characterization of targets.

Applying Thomson's multitaper approach to reduce speckle in medical ultrasound imaging

To reduce the variance of speckle in coherent imaging systems, one must average images with different speckle realizations. Traditionally, these images have been formed by observing the target region from slightly different angles (spatial compounding) or by varying the involved temporal frequencies (frequency compounding). In this paper, we investigate a third option based on Thomsons multitaper approach to power spectrum estimation. The tapers are applied spatially, as array weights. Our investigations, based on both recorded ultrasound data and simulations, verify that the multitaper approach can be used for speckle reduction at a rate comparable to that of the more traditional method of spatial compounding. Because of the spectral concentration of the tapers, an added benefit is reduced side lobe levels, which can result in steeper edges and better definition of cyst-like structures.

Coherent plane-wave compounding and minimum variance beamforming

Achieving increased frame rate without compromising the image quality is desirable in medical ultrasound imaging. Coherent plane-wave compounding has recently been suggested as an approach to achieve this. This work proposes to generate coherent compound plane-wave images using a minimum variance adaptive beamformer. Through simulations of point scatterers and cyst phantoms, a threefold increase in frame rate is shown.