Worst-case analysis of array beampatterns using interval arithmetic.

Over the past decade, interval arithmetic (IA) has been used to determine tolerance bounds of phased-array beampatterns. IA only requires that the errors of the array elements are bounded and can provide reliable beampattern bounds even when a statistical model is missing. However, previous research has not explored the use of IA to find the error realizations responsible for achieving specific bounds. In this study, the capabilities of IA are extended by introducing the concept of "backtracking," which provides a direct way of addressing how specific bounds can be attained. Backtracking allows for the recovery of the specific error realization and corresponding beampattern, enabling the study and verification of which errors result in the worst-case array performance in terms of the peak sidelobe level (PSLL). Moreover, IA is made applicable to a wider range of arrays by adding support for arbitrary array geometries with directive elements and mutual coupling in addition to element amplitude, phase, and positioning errors. Last, a simple formula for approximate bounds of uniformly bounded errors is derived and numerically verified. This formula gives insights into how array size and apodization cannot reduce the worst-case PSLL beyond a certain limit.

Wave equations for porous media described by the Biot model.

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator. Optimal correction factors are introduced into the wave equation to compensate for the discrepancies in the compressional wave dispersion and attenuation. Additionally, the method for incorporating the squirt flow mechanism into the wave equation via the Extended Biot poroviscoelastic model is described. The proposed wave equation has a physical basis and satisfies the passivity criterion.

Estimating tropospheric and stratospheric winds using infrasound from explosions.

The receiver-to-source backazimuth of atmospheric infrasound signals is biased when cross-winds are present along the propagation path. Infrasound from 598 surface explosions from over 30 years in northern Finland is measured with high spatial resolution on an array 178 km almost due North. The array is situated in the classical shadow-zone distance from the explosions. However, strong infrasound is almost always observed, which is most plausibly due to partial reflections from stratospheric altitudes. The most probable propagation paths are subject to both tropospheric and stratospheric cross-winds, and the wave-propagation modelling in this study yields good correspondence between the observed backazimuth deviation and cross-winds from the European Centre for Medium-Range Weather Forecasts Reanalysis (ERA)-Interim reanalysis product. This study demonstrates that atmospheric cross-winds can be estimated directly from infrasound data using propagation time and backazimuth deviation observations. This study finds these cross-wind estimates to be in good agreement with the ERA-Interim reanalysis.

Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion.

Wave scattering provides profound insight into the structure of matter. Typically, the ability to sense microstructure is determined by the ratio of scatterer size to probing wavelength. Here, we address the question of whether macroscopic waves can report back the presence and distribution of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. In our analysis, monosized hard scatterers 5 µm in radius are immersed in lossless gelatin phantoms to investigate the effect of multiple reflections on the propagation of shear waves with millimeter wavelength. Steady-state monochromatic waves are imaged in situ via magnetic resonance imaging, enabling quantification of the phase velocity at a voxel size big enough to contain thousands of individual scatterers, but small enough to resolve the wavelength. We show in theory, experiments, and simulations that the resulting coherent superposition of multiple reflections gives rise to power-law dispersion at the macroscopic scale if the scatterer distribution exhibits apparent fractality over an effective length scale that is comparable to the probing wavelength. Since apparent fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent. Our results are generic to wave phenomena and carry great potential for sensing microstructure that exhibits intrinsic fractality, such as, for instance, vasculature.

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.

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.

SURF imaging beams in an aberrative medium: Generation and postprocessing enhancement.

This paper presents numerical simulations of dual-frequency second-order ultrasound field (SURF) reverberation suppression transmit-pulse complexes. Such propagation was previously studied in a homogeneous medium. In this work, the propagation path includes a strongly aberrating body wall modeled by a sequence of delay screens. Each of the applied SURF transmit pulse complexes consists of a high-frequency 3.5-MHz imaging pulse combined with a low-frequency 0.5-MHz sound speed manipulation pulse. Furthermore, the feasibility of two signal postprocessing methods are investigated using the aberrated transmit SURF beams. These methods have previously been shown to adjust the depth of maximum SURF reverberation suppression within a homogeneous medium. The need for this study arises because imaging situations in which reverberation suppression is useful are also likely to produce pulse wave front distortion (aberration). Such distortions could potentially produce time delays that cancel the accumulated propagation time delay needed for the SURF reverberation suppression technique. Results show that both the generation of synthetic SURF reverberation suppression imaging transmit beams and the following postprocessing adjustments are attainable even when a body wall introduces time delays which are larger than previously reported delays measured on human body wall specimens.

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 Thomson's 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.

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.

Post-processing enhancement of reverberation-noise suppression in dual-frequency SURF imaging.

A post-processing adjustment technique to enhance dual-frequency second-order ultrasound field (SURF) reverberation-noise suppression imaging in medical ultrasound is analyzed. Two variant methods are investigated through numerical simulations. They both solely involve post-processing of the propagated high-frequency (HF) imaging wave fields, which in real-time imaging corresponds to post-processing of the beamformed receive radio-frequency signals. Hence, the transmit pulse complexes are the same as for the previously published SURF reverberation-suppression imaging method. The adjustment technique is tested on simulated data from propagation of SURF pulse complexes consisting of a 3.5-MHz HF imaging pulse added to a 0.5-MHz low-frequency soundspeed manipulation pulse. Imaging transmit beams are constructed with and without adjustment. The post-processing involves filtering, e.g., by a time-shift, to equalize the two SURF HF pulses at a chosen depth. This depth is typically chosen to coincide with the depth where the first scattering or reflection occurs for the reverberation noise one intends to suppress. The beams realized with post-processing show energy decrease at the chosen depth, especially for shallow depths where, in a medical imaging situation, a body-wall is often located. This indicates that the post-processing may further enhance the reverberation- suppression abilities of SURF imaging. Moreover, it is shown that the methods might be utilized to reduce the accumulated near-field energy of the SURF transmit-beam relative to its imaging region energy. The adjustments presented may therefore potentially be utilized to attain a slightly better general suppression of multiple scattering and multiple reflection noise compared with non-adjusted SURF reverberation-suppression imaging.

Nonlinear propagation acoustics of dual-frequency wide-band excitation pulses in a focused ultrasound system.

In this article, acoustic propagation effects of dual-frequency wide-band excitation pulses in a focused ultrasound system are demonstrated in vitro. A designed and manufactured dual-frequency band annular array capable of transmitting 0.9/7.5 MHz center frequency wide-band pulses was used for this purpose. The dual-frequency band annular array, has been designed using a bi-layer piezo-electric stack. Water tank measurements demonstrate the function of the array by activating the low- and high-frequency layers individually and simultaneously. The results show that the array works as intended. Activating the low- and high-frequency layers individually, results in less than -50 dB signal level from the high- and low-frequency layers respectively. Activating both layers simultaneously, produce a well defined dual-frequency pulse. The presence of the low-frequency pulse leads to compression, expansion, and a time delay of the high-frequency pulse. There is a phase shift between the low- and high-frequency pulse as it propagates from the array to the focus. This makes the latter described effects also dependent on the array configuration. By varying the low-frequency pressure, a shift of up to 0.5 MHz in center frequency of a 8.0 MHz transmitted high-frequency pulse is observed at the array focus. The results demonstrate the high propagation complexity of dual-frequency pulses.

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.

An annular array design proposal with multiple geometric pre-foci.

An annular array design method is described and used for definition of an array proposal. The geometric pre-focus, realized by curving or use of an acoustic lens, varies among the annuli. Element sizes and geometric pre-focus depths are determined by a maximum allowed phase-shift within the active depth region of each element on receive, resulting in fewer elements or larger apertures compared with the standard equal area design. The method allows combination of large aperture and high frequency with a large receive depth of field. The developed design rules are used to define an array for imaging within the frequency interval [7.5, 15] MHz. Its total aperture diameter is 22 mm and the thinnest element is 0.23 mm wide. Receive beams resulting from this array are simulated. The beams and their sidelobe-to-mainlobe energy ratios are compared with an ideally focused reference where there are no phase shifts over the elements. Within nearly the entire depth of field, the sidelobe-to-mainlobe energy ratio is less than 5 dB higher than for the ideal reference.