Frequency-wavenumber Domain Research Articles

Post-stack seismic data interpolation using a fast non-local similarity matching algorithm

In the process of seismic data acquisition, there are often missing seismic traces in seismic records, so it is necessary to reconstruct the missing data to provide high-quality data for subsequent seismic data migration and reservoir inversion. Traditional interpolation methods for post-stack seismic data are based on the sparse constraint in the frequency-wavenumber (f-k) domain. However, the data completed using the interpolation method usually leads to the loss of some weak signals when the dip of the post-stack seismic profile is complex. In this paper, the missing data could be regarded as the result of irregular noise with the same waveform and the original signal but with the opposite polarity. The non-local similarity in the denoising algorithm is introduced as a low-rank promoting transform of the low-rank regularization term, and an interpolation method based on non-local similarity is proposed (NLS-WNNM). Furthermore, a fast matching algorithm is developed to search and match the non-local similarity of missing seismic traces (abbreviation FNLS-WNNM), which reduces the loss of weak signals during interpolation. The traditional interpolation method based on f-k domain is compared with the NLS-WNNM to highlight the advancement of the method. Finally, the interpolation test applied to field data confirmes the robustness of the proposed method.

Experimental detection of the volume of the drip irrigation soil wetted body using Ground Penetrating Radar

The accurate detection of the volume of the soil wetted body (SWB) is the theoretical basis for the optimal design of a drip irrigation system. In view of the shortcomings of traditional detection methods, such as low efficiency, high cost, and poor in situ detection, this paper explores the rapid, in situ, and nondestructive estimation techniques of SWB volume using Ground Penetrating Radar (GPR). Through drip irrigation of different water volumes and geotechnical experiments, the SWBs with different sizes were formed in air-dried soil and natural soil, respectively. The GPR scan lines on the ground were arranged to obtain the SWB radargram, and the profile shape of the wetted body was restored by the F-K transformation process (frequency-wave domain migration algorithm) that can transform the wave equation in the time-space domain into the frequency-wavenumber domain to achieve offset homing. On this basis, the SWB volume estimation model was constructed by three methods: regression analysis, geometric calculation, and profile triple integration (profile area accumulation). Additionally, the accuracies of the models were verified by the measured data of the SWB excavation. For each of these methodologies, we illustrate the quality of the output with the measured data and discuss the possibilities and limitations. The results show that the profile triple integration model is not accurate. The regression analysis model has the greatest accuracy. However, owing to the difference in soil characteristics, the model input parameters must be changed. Relatively, the geometric calculation model is the best model to detect the SWB volume. From the results, we can draw a conclusion that the method combining F-K transformation process and the geometric calculation model can accurately measure the volume of SWB, and it has better adaptability in different soils. We demonstrate the GPR as a rapid geophysical tool that can be used successfully to explore the volume of the SWB formed under drip irrigation. This study expands the agricultural application of GPR and provides a new method for in situ nondestructive rapid detections of the SWB volume.

Dynamic track-soil interaction—calculations and measurements of slab and ballast tracks

The dynamic behaviour of slab and ballast tracks was investigated using measurements and calculations. Hammer impacts and train passages were analysed and measurements were made using geophones (velocity transducers) which had been time-integrated to displacements. The calculations were carried out in the frequency-wavenumber domain for multi-beam-on-continuous soil models. The characteristics of the different tracks and track elements were established in theory and by experiment. The frequency-dependent compliances (displacement transfer functions) showed clear rail-on-railpad resonances or highly damped track-soil resonances. Compared to the rail and sleeper, the track slab had much lower amplitudes. The slab track usually had the highest rail amplitudes due to soft railpads. Train passage yielded track displacements which were a superposition of the axle loads from the two neighbouring axles of a bogie and from the two bogies of two neighbouring carriages. This global behaviour was characteristic of the track slab of the slab track, whereas the rails of the slab and the ballast tracks behaved more locally with only one bogie of influence. The measurements agreed very well with the theory of continuous soil in the case of the six measured slab tracks and acceptably well for the six measured ballast tracks. The measurements allowed us to find appropriate model parameters and to check the models. For example, the Winkler model of the soil was found to be less appropriate because it reacted more locally.

An Analytic Algorithm for Dipole Electromagnetic Field in Fully Anisotropic Planar-Stratified Media

This article presents an efficient analytic algorithm to calculate the electromagnetic (EM) fields radiated from orthogonal magnetic and electric dipoles in the fully anisotropic planar-stratified media. The EM fields are transformed into the frequency–wavenumber domain by the double Fourier transform, which yields a set of ordinary differential equations, consisting of a system matrix, a field vector, and a source vector. The eigenvalues of the system matrix are uniquely determined, but the corresponding eigenvectors can be multiplied by an arbitrary constant or be normalized by energy. With these eigenvalues and eigenvectors, the field vectors are expressed as mode fields consisting of the up-going and down-going fields. The propagation of the mode fields in each layer is described by the generalized reflection/transmission method. To accelerate the evaluation of the inverse Fourier transform of EM field from spectral domain to spatial domain, an approximated primary field was subtracted from total field. Numerical results from a finite element method (FEM) validate the proposed method. Then, we apply this new algorithm to a variety of real applications in geophysical prospecting.

Robust and baseline-free full-field defect detection in complex composite parts through weighted broadband energy mapping of mode-removed guided waves

In this study, a non-destructive testing approach is investigated for finding damages in fiber reinforced polymers using broadband piezoelectric excitation and scanning laser Doppler vibrometer measurements.An efficient broadband filter in wavenumber-frequency domain is considered, in order to calculate a mode-removed broadband weighted root mean square (WRMS) energy map of the guided waves. Compensation of the frequency-dependent wave attenuation is implemented. Because this energy map relates exclusively to abnormalities in the wave field, it shows high sensitivity to all kinds of internal damages. The proposed damage map construction method is automated, and does not require any a priori information on the inspected part or measurement conditions. Further, it is not limited to application on flat coupon samples, but can be equally applied on complex composite parts.The methods are tested on an academic case of a CFRP plate with flat bottom holes and on an industrial case of a large stiffened CFRP aircraft panel with distributed manufacturing defects. It is shown that the broadband mode-removed WRMS energy map is highly sensitive to various kinds of damages, even if they are small and deep inside the material.

Modelling track and ground vibrations for a slab ballastless track as an infinitely long periodic structure subject to a moving harmonic load

This paper presents a model for predicting ground vibration generated from a slab ballastless track, subject to a moving harmonic load. The track consists of four layers: two rails, track slabs, bed plates and a track base resting on the ground. Different from existing models for slab tracks, the one developed in this paper takes into account both the discrete supports to the rail and the discontinuity of the slabs and bed plates, and treats the track/ground system as an infinitely long periodic structure. The rail and the base are represented as, respectively, an infinitely long Timoshenko beam and an infinitely long Euler-Bernoulli beam. The slabs and bed plates are modelled as Euler-Bernoulli beams of finite length, of which the responses are formulated using the mode superposition method. By using square window functions, the differential equations of motion of each track slab and bed plate are defined for the entire track length. The ground is idealised to be a horizontally layered structure. The steady-state response of the track/ground system is sought and formulated in the frequency-wavenumber domain. By using the periodic condition of a periodic structure, it is shown that the steady-state response of the track/ground system can be determined by the modal coefficients of the slabs and plates in the 0th unit cell (the 0th bay). Formulae are derived for these modal coefficients. Results including receptances of the rail, displacement spectra of the ground and critical speeds of an axle load are produced for two sets of track/ground parameters and compared with those when the slab and bed plates are assumed to be continuous beams, demonstrating when the later treatment works. The role of veering in dispersion curves of a periodic track in critical speed is identified. Formulations in this paper can be directly applied to the Chinese CRTS II and III tracks on a layered ground. They provide a basis for developing a complete ground vibration prediction model that incorporates a moving train, an infinitely long periodic track and a homogeneous or horizontally layered ground. They can also be used to assess if a periodic track/ground system can be simplified to be invariant in the track direction.

Slab Track Behaviour under Train Passage and Hammer Impact - Measurements at Different Sites and Calculated Track Interaction with Continuous Soils

This contribution intends to give an overview on the vibration behaviour of slab tracks in comparison of measurements and calculations and -6mmalso-6mm by comparison of different track types at more than ten different measuring sites. In theory, tracks on -6mmcontinuous-6mm soil are calculated by the frequency-wavenumber domain method. In experiment, geophone measurements are transformed to displacement results. Two aspects of track behaviour are considered, the frequency-dependant compliance of the track, measured by hammer impact, and the deflection under a passing axle load. In theory, the response to a single axle can be calculated, whereas in experiment, only the passage of the whole train can be measured. For comparison of theory and experiment, the calculated deflection under a single axle is superposed to get the response of the whole train. As a result, the slab track characteristics are completely different from the ballast track characteristics where each axle can be seen in the time histories. The slab track has a more global behaviour where only a whole bogie can be found in the track response and moreover, the two neighbouring bogies are not completely separated. The measurement of the different track elements (rail, sleeper, track plate, base layer) and the frequency-dependant compliances with possible resonances yield further information about the properties of the track elements. The calculations show that the soil has the dominant influence on the amplitudes and the width of the track-plate displacements. In the measurement results, the following parameters are analysed: slab track vs.~ballast track, different types of slab tracks, damaged slab tracks, different trains, switches at different measuring points, voided sleepers, an elastic layer, the mortar layer, and different soils at different places. Finally, a good agreement between measured and calculated results is found for the normal and some special (damaged, floating) slab tracks.

A partial convolution-based deep-learning network for seismic data regularization1

Spatial undersampling is a common problem in actual seismic data due to limitations in seismic survey environments, which can be satisfactorily solved by data regularization. The convolution-based deep-learning reconstruction methods require fewer assumptions than the conventional reconstruction methods (e.g., Curvelet-domain and F-X domain data regularization methods). However, the traditional convolution methods are not suitable for the large percentages of missing data. In this study, we propose an improved partial convolution-based (PConv-based) deep-learning network to reconstruct the missing data, which is evolved from the conventional convolution-based (CConv-based) method. The U-net is used as deep learning network to analyze both PConv-based method and CConv-based method. The PConv-based method adopts a hierarchical, regional-learning mechanism to dynamically update the constrained convolution results for the sample matrix. Hence, the problem of poor amplitude preservation in the data reconstruction has been addressed when multiple consecutive traces are missing. The influence of data loss ratio on reconstruction algorithm is also discussed in this study. The numerical test demonstrates that the trained network is able to process a sample dataset with 50% data lost and largely eliminate the noises in the frequency-wavenumber domain caused by the missing data. This proposed method is further evaluated by actual data, and the results are better than those obtained from the Curvelet-domain method. Moreover, the dataset reconstructed by the PConv-based deep-learning network has a great agreement with the original dataset in terms of amplitude.

Comparison of Time Domain and Frequency-Wavenumber Domain Ultrasonic Array Imaging Algorithms for Non-Destructive Evaluation.

Ultrasonic array imaging algorithms have been widely developed and used for non-destructive evaluation (NDE) in the last two decades. In this paper two widely used time domain algorithms are compared with two emerging frequency domain algorithms in terms of imaging performance and computational speed. The time domain algorithms explored here are the total focusing method (TFM) and plane wave imaging (PWI) and the frequency domain algorithms are the wavenumber algorithm and Lu’s frequency-wavenumber domain implementation of PWI. In order to make a fair comparison, each algorithm was first investigated to choose imaging parameters leading to overall good imaging resolution and signal-to-noise-ratio. To reflect the diversity of samples encountered in NDE, the comparison is made using both a low noise material (aluminium) and a high noise material (copper). It is shown that whilst wavenumber and frequency domain PWI imaging algorithms can lead to fast imaging, they require careful selection of imaging parameters.

Effects of modeling strategies for a slab track on predicted ground vibrations

In the paper, the effect of modeling strategies regarding the dynamic behavior of a railway slab track on a layered half-space is studied. The track is modeled with various degrees of accuracy through the use of either beam theory, shell finite elements or solid finite elements. The underlying soil response is included through a dynamic stiffness, obtained via the Green’s function for a horizontally layered visco-elastic half-space in the frequency–wavenumber domain. The effect of different assumptions regarding the track cross-section behavior and the track–soil interface conditions on the resulting free-field vibrations are studied for a harmonic load moving along the track. First, only the out-of-plane displacements of the slab–soil interface are coupled, i.e. only the vertical contact pressure is accounted for. Secondly, the effect of coupling the slab–soil in-plane displacements on the free-field vibrations is studied. It is found that the in-plane slab–soil coupling significantly affects the vertical vibration in the free-field. It is also found that a beam model of the track yields accurate response levels compared to a solid continuum model in the case of a thick slab, whereas considerable differences are obtained for a thin slab.

Vertical Velocity Profiles in Convectively Coupled Equatorial Waves and MJO: New Diagnoses of Vertical Velocity Profiles in the Wavenumber–Frequency Domain

Abstract A new diagnostic framework is developed and applied to ERA-Interim to quantitatively assess vertical velocity (omega) profiles in the wavenumber–frequency domain. Two quantities are defined with the first two EOF–PC pairs of omega profiles in the tropical ocean: a top-heaviness ratio and a tilt ratio. The top-heaviness and tilt ratios are defined, respectively, as the cospectrum and quadrature spectrum of PC1 and PC2 divided by the power spectrum of PC1. They represent how top-heavy an omega profile is at the convective maximum, and how much tilt omega profiles contain in the spatiotemporal evolution of a wave. The top-heaviness ratio reveals that omega profiles become more top-heavy as the time scale (spatial scale) becomes longer (larger). The MJO has the most top-heavy profile while the eastward inertio-gravity (EIG) and westward inertio-gravity (WIG) waves have the most bottom-heavy profiles. The tilt ratio reveals that the Kelvin, WIG, EIG, and mixed Rossby–gravity (MRG) waves, categorized as convectively coupled gravity waves, have significant tilt in the omega profiles, while the equatorial Rossby (ER) wave and MJO, categorized as slow-moving moisture modes, have less tilt. The gross moist stability (GMS), cloud–radiation feedback, and effective GMS were also computed for each wave. The MJO with the most top-heavy omega profile exhibits high GMS, but has negative effective GMS due to strong cloud–radiation feedbacks. Similarly, the ER wave also exhibits negative effective GMS with a top-heavy omega profile. These results may indicate that top-heavy omega profiles build up more moist static energy via strong cloud–radiation feedbacks, and as a result, are more preferable for the moisture mode instability.

Lamb wave mode decomposition and reconstruction based on the viscoelastic propagation model

The accurate information of Lamb wave packets, such as amplitude, time of flight, and waveform, are essential for damage detection and evaluation. Due to the dispersive, multimodal, and attenuative characteristics of Lamb waves propagating in viscoelastic media, the desired mode component is difficult to be extracted from response signals. To address this issue, a technique based on the viscoelastic propagation model of Lamb waves is proposed in this article. Different from the conventional frequency–wavenumber domain filtering, the proposed technique achieves mode extraction by estimating the parameters of the viscoelastic propagation model, that is, complex amplitude am(r, ω) and effective complex wavenumber ξm(ω). Benefiting from that, it enables to decompose and reconstruct the desired mode from spatial–temporal measurement data. First, the frequency-dependent effective complex wavenumbers are extracted using the matrix pencil method, where the real parts represent the dispersion and the imaginary parts represent the effective attenuation. After that, the complex amplitudes are obtained by solving linear least-square equations. Finally, the complex amplitude, effective complex wavenumber, and frequency range of the desired mode are picked out to reconstruct the mode component in the time domain. The finite element simulation and experimental results verify the effectiveness of our technique. Furthermore, the performance of the proposed technique in noisy circumstances is analyzed through synthetic signals, which shows a small reconstruction error and a good convergence.

A methodology based on 2.5D FEM-BEM for the evaluation of the vibration energy flow radiated by underground railway infrastructures

In this paper, a comprehensive numerical approach for modelling track/tunnel/soil systems in the context of ground-borne railway-induced vibration problems considering a full-space model of the soil is proposed. All the approach is formulated in the wavenumber-frequency domain and it consists of a coupled finite element–boundary element model of the tunnel/soil system, a semi-analytical model of the track, a multibody model for the vehicle and a model for the vibration propagation in the soil based on semi-analytical solutions of a cylindrical cavity in a full-space. This comprehensive approach has been developed with the aim of computing the vibration energy flow radiated upwards by underground railway tunnels. An axisymmetric formulation to deal with circular underground railway tunnels is included in the approach in order to improve the computational speed of the methodology. This formulation can also be used for other types of railway tunnels if a circular boundary of the boundary element mesh is considered. Since this methodology uses finite elements to model the tunnel structure, its modelling detail is higher than the previously developed methodologies dedicated to compute the vibration energy flow radiated by underground railway infrastructures, since they are based on semi-analytical modelling of the tunnel structure. The present methodology has been specifically designed to be used in general assessment studies about ground-borne underground railway-induced vibrations where decisions on the type of track and/or the application of mitigation measures at the source, as soft rail-pads, under-ballast or under-slab mats, have to be made. Moreover, this methodology can be used for the study of the vibration radiation patterns of railway tunnels.

On the three-dimensional sound fields from a moving monopole source above a non-locally reacting ground.

This study derives an accurate analytic solution for sound fields due to a moving point monopole source above a non-locally reacting ground. Based on the property of the invariant phase function, the acoustic pressure and normal particle velocity are matched across the air/ground interface in the frequency-wave number domain. Expressions for the integral form of the sound fields in three different frames are used to analyze and compare the original physical frame in the reception time geometry, the Lorentz frame, and the emission time frame. For ease of interpretation, the final form of the asymptotic solution is expressed in the geometry of the emission time frame. The accuracy of the asymptotic solution is validated using the direct numerical integration technique. The three-dimensional point source solution derived from the current study is compared with that of the two-dimensional line source solution. Explicit comparisons of the cylindrical and spherical wave reflection coefficients are available, allowing a better understanding of the impact of the ground wave term in the vicinity of a non-locally reflecting plane. Finally, contour plots are presented that demonstrate the need to employ the asymptotic formula for the non-locally reacting ground instead of approximating it with a locally reacting property.

Semi-automatic detection of buried rebar in GPR data using a genetic algorithm

This paper proposes a new method for semi-automatic detection of buried rebar in ground penetrating radar (GPR) data based on searching the reflected waveform of target by the genetic algorithm, which aims to overcome the inefficiency and instability of the manual process. Before the detection step, a binary image is generated by collecting all the local maximum points from A-scan GPR data to turn the searching for waveform image into curves. Meanwhile, the possible locations of the targets are preliminarily selected from the gray-scale frequency-wavenumber domain migration image of GPR data by setting a gray gradient threshold in the vertical direction. Then based on the genetic algorithm and time-distance analysis of the reflected signal, semi-automatic detection could be achieved by searching the best fitting hyperbola at every potential location in the binary image, and matching them with the reflected signal of targets according to several carefully chosen matching criteria. The impact of waveform distortion is also considered in the matching process to ensure a good adaptability of this method to the GPR data. Precision and efficiency of this method are analyzed lastly by four measured data of different antenna frequencies and targets, the average misjudgment and the undetected rate are 9.08% and 11.98% respectively. Meanwhile, the same GPR data is detected by four antenna central frequencies to find that the missed rate of semi-automatic detection decreases with the increase of the antenna frequency, when the misjudgment rate remains stable.

Non-destructive evaluation of isotropic plate structures by means of mode filtering in the frequency-wavenumber domain

Non-Destructive Testing (NDT) techniques are prevalent in the aerospace, green energy and automotive industries. These techniques, including ultrasound, despite their use, still have limitations surrounding speed and resolution. This work presents the development of a band-pass mode filtering technique in the frequency wavenumber domain for the purpose of damage detection in isotropic materials. Data was captured in the temporal and spatial domain using a 3D Scanning Laser Doppler Vibrometer (SLDV) with piezoelectric transducers exciting the structure with a variety of steady state signals ranging in frequency from 75 kHz to 400 kHz and a frequency modulated signal from 75 kHz to 400 kHz. A thickness map was created based on the frequency and wavelength of the A0 Lamb wave mode. The technique was demonstrated on two aluminium specimens with dimensions of 400 mm by 400 mm with a thickness range of 0.5 mm to 8 mm with distinct geometric features. Through using multi-frequency excitation combined with mode based filters an estimation of thickness was achieved with a mean percentage thickness error of 15%. Circular thickness reductions with a diameter of 10 mm were clearly identified at the maximum plate thickness of 8 mm. This technique was shown to perform better than wavenumber filtering by allowing data from multi frequency steady state excitation to be combined into a single resulting thickness map. This improvement was shown to be particularly important at greater thicknesses.

Knudsen number effects on the nonlinear acoustic spectral energy cascade.

We present a numerical investigation of the effects of gas rarefaction on the energy dynamics of resonating planar nonlinear acoustic waves. The problem setup is a gas-filled, adiabatic tube, excited from one end by a piston oscillating at the fundamental resonant frequency of the tube and closed at the other end; nonlinear wave steepening occurs until a limit cycle is reached, resulting in shock formation for sufficiently high densities. The Knudsen number, defined here as the ratio of the characteristic molecular collision timescale to the resonance period, is varied in the range Kn=10^{-1}-10^{-5}, from rarefied to dense regime, by changing the base density of the gas. The working fluid is Argon. A numerical solution of the Boltzmann equation, closed with the Bhatnagar-Gross-Krook model, is used to simulate cases for Kn≥0.01. The fully compressible one-dimensional Navier-Stokes equations are used for Kn<0.01 with adaptive mesh refinement to resolve the resonating weak shocks, reaching wave Mach numbers up to 1.01. Nonlinear wave steepening and shock formation are associated with spectral broadening of the acoustic energy in the wavenumber-frequency domain; the latter is defined based on the exact energy corollary for second-order nonlinear acoustics derived by Gupta and Scalo [Phys. Rev. E 98, 033117 (2018)2470-004510.1103/PhysRevE.98.033117], representing the Lyapunov function of the system. At the limit cycle, the acoustic energy spectra exhibit an equilibrium energy cascade with a -2 slope in the inertial range, also observed in freely decaying nonlinear acoustic waves by the same authors. In the present system, energy is introduced externally via a piston at low wavenumbers or frequencies and balanced by thermoviscous dissipation at high wavenumbers or frequencies, responsible for the base temperature increase in the system. The thermoviscous dissipation rate is shown to scale as Kn^{2} for fixed Reynolds number based on the maximum velocity amplitude, i.e., increasing with the degree of flow rarefaction; consistently, the smallest length scale of the steepened waves at the limit cycle, corresponding to the thickness of the shock (when present) also increases with Kn. For a given fixed piston velocity amplitude, the bandwidth of the inertial range of the spectral energy cascade decreases with increasing Knudsen numbers, resulting in a reduced resonant response of the system. By exploiting dimensionless scaling laws borrowed by Kolmogorov's theory of hydrodynamic turbulence, it is shown that an inertial range for spectral energy transfer can be expected for acoustic Reynolds numbers Re_{U_{max}}>100, based on the maximum acoustic velocity amplitude in the domain.

Image reconstruction algorithm based on frequency-wavenumber decoupling for three-dimensional MIMO-SAR imaging.

In this paper, a frequency-wavenumber decoupling algorithm with high-efficiency and high-precise for three-dimensional (3-D) multiple-input-multiple-output synthetic aperture radar (MIMO-SAR) imaging is proposed. Based on one-dimensional (1-D) MIMO array combined with synthetic aperture scan along another dimension, MIMO-SAR imaging scheme allows the number of array elements to be greatly reduced compared with the two-dimensional (2-D) MIMO arrays. By multi-dimensional Fourier transforming and Method of Stationary Phase (MSP), analytical expression of the object function in the frequency-wavenumber domain was derived. By further expanding the range Fourier transform factor to its Taylor series form, the range compression can be realized by a simple fast Fourier transform (FFT) without multi-dimensional interpolation. After that, a decoupling factor was multiplied to compensate for the cross-range and range coupling in frequency domain. Finally, 2-D IFFT is carried out after rearrangement in the MIMO spatial frequency to get a fully focused 3-D image. Simulation and experimental results demonstrated that the algorithm can obtain the same high-precision images as back projection (BP) algorithm, and has the same high efficiency as range migration algorithm (RMA) while avoiding cumbersome multi-dimensional interpolation. A bistatic prototype imaging system in 0.1 THz band was designed for the proof-of-principle experiments. The 3-D reconstruction results of different targets were presented to verify the theoretical results and effectiveness of the proposed algorithm for MIMO-SAR imaging.

Seismogram Synthesis for Multilayered Heterogeneous Media with Irregular Interfaces by the Global Generalized Reflection–Transmission Matrices Method

ABSTRACTWe present a semianalytical numerical method to simulate SH-wave propagation in heterogeneous multiple layers with smooth property variations. Wave propagation in such piecewise heterogeneous media will dominate the reflection and transmission at interfaces, whereas the scattered waves by volume heterogeneities inside each geological layer are superimposed on the boundary waves to cause fluctuations in amplitude and phase. The incident, boundary-scattering, and volume-scattering waves are accurately superposed through the generalized Lippmann–Schwinger integral (GLSI) equation. The multilayered heterogeneous media with irregular interfaces are sliced into horizontal and heterogeneous slabs, with each discretized into a series of volume elements. The traditional generalized reflection–transmission matrices (GRTMs) method is extended to solve the GLSI equation for each heterogeneous slab in the frequency–wavenumber domain by incorporating the boundary–volume integral equation numerical techniques. The global GRTM is obtained by integrating all the slab’s GRTMs in terms of the boundary condition across adjacent slabs. The wavefield at an arbitrary depth can be computed in a recursive relation from a source term. The accuracy of the method is tested by comparing it with other full-wave solutions to the SH problem. To show the applicability of the method, we calculate synthetic seismograms to demonstrate the significant impact of volume heterogeneities on seismic responses in layered heterogeneous media.

Lamb Waves and Adaptive Beamforming for Aberration Correction in Medical Ultrasound Imaging.

Phase aberration in transcranial ultrasound imaging (TUI) caused by the human skull leads to an inaccurate image reconstruction. In this article, we present a novel method for estimating the speed of sound and an adaptive beamforming technique for phase aberration correction in a flat polyvinylchloride (PVC) slab as a model for the human skull. First, the speed of sound of the PVC slab is found by extracting the overlapping quasi-longitudinal wave velocities of symmetrical Lamb waves in the frequency-wavenumber domain. Then, the thickness of the plate is determined by the echoes from its front and back side. Next, an adaptive beamforming method is developed, utilizing the measured sound speed map of the imaging medium. Finally, to minimize reverberation artifacts caused by strong scatterers (i.e., needles), a dual probe setup is proposed. In this setup, we image the medium from two opposite directions, and the final image can be the minimum intensity projection of the inherently co-registered images of the opposed probes. Our results confirm that the Lamb wave method estimates the longitudinal speed of the slab with an error of 3.5% and is independent of its shear wave speed. Benefiting from the acquired sound speed map, our adaptive beamformer reduces (in real time) a mislocation error of 3.1, caused by an 8 mm slab, to 0.1 mm. Finally, the dual probe configuration shows 7 dB improvement in removing reverberation artifacts of the needle, at the cost of only 2.4-dB contrast loss. The proposed image formation method can be used, e.g., to monitor deep brain stimulation procedures and localization of the electrode(s) deep inside the brain from two temporal bones on the sides of the human skull.