Local Wavenumber Research Articles

Ultrasound shear wave phase velocity imaging using black-box system identification (BSI): a data-driven approach

Objective. Shear wave elasticity imaging (SWEI) is an established approach for diagnosing lesions in human tissue. However, the shear wavelengths used by traditional SWEI are usually not short enough, leading to inferior accuracy in small-target (<10 mm) reconstruction. To exploit short shear wavelengths (high-frequency content), this study introduces a new phase velocity (PV) estimation technique as an alternative to the conventional group velocity (GV) modality. Approach. We propose using a black-box model instead of a fully physics-based model to describe the transition process of two arbitrary shear wave signals. With this representation, local PV can be obtained via black-box system identification (BSI). For validation, two PV estimation scenarios were established: (numerical) dispersion measurements in viscoelastic media, and (real) imaging targets in a CIRS elasticity phantom. BSI was compared with a state-of-the-art PV imaging method that uses local wavenumber estimation (LWE). Main results. BSI showed excellent accuracy in the dispersion estimation for all three viscoelastic media in the simulations. In the phantom study, the two PV methods exhibited good agreement in the frequency dependence of target quantification, and could both generate a higher target reconstruction accuracy than GV. LWE images were strongly affected by noise-induced faulty estimates, whereas BSI showed no notable artifacts. Significance. This study demonstrates the advantage of the PV modality over the GV modality as the former can achieve better target visualization by increasing imaging frequency. It also implies the feasibility of data-driven modeling for soft tissue characterization.

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

In this paper, we propose self‐adaptive artificial boundary conditions (ABCs) for the two‐dimensional (2D) time‐dependent nonlinear Schrödinger equation and studied their efficiency. This equation describes the laser pulse interaction with a semiconductor layer under laser‐induced plasma generation. Because of its nonlinearity, the interaction process is very sensitive to the appearance of the optical beam spurious reflection; therefore, highly transparent ABCs are urgently required to avoid disturbing the interaction process by an unphysical reflected wave. This can be achieved by using time‐ and spatial‐dependent local wavenumbers computed near the artificial boundaries. We claim that it is necessary to account for both wavenumbers in each of the adaptive ABCs stated on both spatial coordinates. Based on computer simulation, we demonstrate an increase in the efficiency of the adaptive ABC stated on the coordinate along which the optical pulse propagates by involving both local wavenumbers. The accuracy of the developed approach is verified using a well‐known analytical solution of the linear Schrödinger equation as well as the reference problem solution, obtained by solving the problem with zero‐value boundary conditions on the extended spatial domain. For computer modeling, the finite‐difference scheme, possessing asymptotic stability, is applied together with a two‐stage iterative process for its realization. The algorithm and peculiarities of the local wavenumber computation are discussed in detail. The applicability of the developed approach is also demonstrated via the solution of both linear and nonlinear problems. The proposed method has widespread application for solving various laser physics problems governed by the Schrödinger equation.

The universal behavior of modulated stripe patterns

There are three aims of this paper. The first is to explain the reasons for behavior we had long suspected to be true but the real reasons for which we could never quite nail down. Modulated striped patterns arising from a wide class of gradient microscopic pattern forming systems display universal behavior. Their order parameter evolves according to a phase diffusion equation that derives from an averaged energy that consists of coordinate invariant combinations of the coefficients in the metric and curvature two forms of a well-defined phase surface. The evolution towards universality is asymptotic in that the pattern evolves in such a way that, over longer and longer time scales, many terms from the microscopic energy simply become negligible leaving behind canonical forms common to a wide class of microscopic pattern-forming systems. The second aim is to emphasize with some new results the key role that the Jacobian matrix of the map from physical to order parameter space plays in both two and three dimensions. In two dimensions, it is closely related to the Gaussian curvature of the phase surface. It is a conserved density whose integral over space in two dimensions or on cross-sections in three become boundary terms that measure the topological indices of point defects in two dimensions and loop defects in three. In all dimensions, the Jacobian matrix acting on the order parameter vector, the gradient of the phase, is zero when the local pattern wavenumber is close to its preferred value and this leads to the effective linearization of the phase diffusion equation. The third aim is to honor Hermann Flaschka, a close friend and scientific colleague for over fifty years, an outstanding mathematician, a true gentleman and scholar with an uncanny knack of explaining the most complicated of ideas in the simplest of ways, who passed away last year. Hermann was one of the founding editors of Physica D and served as the coordinating editor for almost twenty years.

Structurally-controlled Gold Mineralization in the Southern Zuru Schist Belt NW Nigeria: Application of Remote Sensing and Geophysical Methods

A significant number of gold occurrences are hosted by hydrothermal alteration zones in the Zuru Schist Belt, Northwestern Nigeria. The aim of this study is to identify the spatial correlation between faults, sheared, hydrothermally altered zones, and gold mineralized areas using geophysical and remote sensing data. The main structural trends of the schist belt as interpreted from the application of total gradient amplitude and local wavenumber techniques on the aeromagnetic data are in the E-W and NE-SW directions. These trends align with the locations of gold mineralization and key pathfinders such as faults in the Anka-Yauri-Iseyin shear zone. In the radiometric data, magnetic faults occurred in areas with elevated potassium enrichment within the Anka-Yauri-Iseyin shear zone and present evidence for structurally controlled hydrothermal processes. The mapped Anka-Yauri-Iseyin shear zone from the aeromagnetic data further coincides with morphological features such as ridges, valleys, and highlands, signifying that the shear zone is represented by topographical linear indicators. Importantly, the integrated approach and results presented in this work show that the Southern Zuru Schist Belt is fragmented by a NE-SW oriented AYI shear complex. Faults and hydrothermally altered zones in the AYI shear complex are new target areas for direct exploration of gold in the study area.

A Five-Point TENO Scheme with Adaptive Dissipation Based on a New Scale Sensor

In this paper, a new five-point targeted essentially non-oscillatory (TENO) scheme with adaptive dissipation is proposed. With the standard TENO weighting strategy, the cut-off parameter $C_T$ determines the nonlinear numerical dissipation of the resultant TENO scheme. Moreover, according to the dissipation-adaptive TENO5-A scheme, the choice of the cut-off parameter $C_T$ highly depends on the effective scale sensor. However, the scale sensor in TENO5-A can only roughly detect the discontinuity locations instead of evaluating the local flow wavenumber as desired. In this work, a new five-point scale sensor, which can estimate the local flow wavenumber accurately, is proposed to further improve the performance of TENO5-A. In combination with a hyperbolic tangent function, the new scale sensor is deployed to the TENO5-A framework for adapting the cut-off parameter $C_T$, i.e., the local nonlinear dissipation, according to the local flow wavenumber. Overall, sufficient numerical dissipation is generated to capture discontinuities, whereas a minimum amount of dissipation is delivered for better resolving the smooth flows. A set of benchmark cases is simulated to demonstrate the performance of the new TENO5-A scheme.

A family of spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation for compressible flows

A numerical scheme with good dissipation and dispersion properties is essential for simulations of complex compressible flows in resolving small-scale structures and capturing discontinuities. Although a sufficient numerical dissipation benefits the shock-wave capturing, the fine scales in smooth regions are dissipated either. In this work, a framework to construct arbitrarily high-order spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation is proposed, where the dispersion and dissipation properties are characterized by two free parameters respectively. The proposed scheme can automatically adjust these two free parameters to control the dispersion and dissipation according to the local flow-field properties quantified by the scale sensor. As the first step to optimize the spectral properties of the fully discrete scheme, the total dispersion and dissipation errors induced by spatio-temporal discretization are studied. Then, the dispersion property is optimized by a new integrated error function devised for fully discrete scheme. To improve the precision of prediction, the scale sensor employed to quantify the local scaled wavenumber of flow fields is optimized in wavenumber space. Furthermore, a modified dispersion-dissipation condition is developed to characterize the relationship between the total dispersion and dissipation errors. Building upon the optimized scale sensor and modified dispersion-dissipation condition, the critical-adaptive dissipation parameter is obtained, which keeps the numerical dissipation as low as possible to resolve more small-scale structures in the low-wavenumber region and introduces sufficient dissipation to capture strong discontinuities successfully. Moreover, the adaptive dispersion parameter related to the local wavenumber and Courant number is obtained to reduce the total dispersion error significantly. Meanwhile, the critical-adaptive dissipation surface and adaptive dispersion surface are constructed for different Courant number utilized in actual flow simulations, which improves the spectral properties of the proposed scheme. Finally, some benchmark cases involving strong discontinuities and multi-scale structures are employed to verify the attractive performance of the proposed scheme.

Gravity Data Imaging Using Local Wavenumber-Based Algorithm: Sustainable Development Cases Studies

A fast effective inversion algorithm is proposed herein to interpret gravitational responses caused by mineralized/ore sources (sphere, vertical and horizontal cylinders). The algorithm relies on local wavenumber and correlation imaging techniques. The correlation factor (R) between the local wavenumber of observed gravitational field and that of computed field was calculated, and the maximum Rmax was considered to correspond to the best true model (parameters). The proposed algorithm was applied to two theoretical examples, including an example contaminated with regional background and another multisource example. Besides, the proposed approach was used on three different real field cases for mining/ore investigation from Canada and Cuba. From the results obtained from the theoretical and real examples and by comparing the results with drilling and literature information, it was concluded that the method is effective, is applicable even for more than one source, is accurate, and does not necessitate any prior knowledge of the source shape.

Higher-order perturbative parabolic-equation solutions for reduced-order sound propagation modeling

The parabolic-equation (PE) approximation to the Helmholtz equation has been shown to be one of the most effective and efficient numerical methods for sound propagation modeling. In this talk, a recursive and iterative algorithm will be introduced to compute the PE solutions due to perturbations in the medium wavenumber, which is a function of frequency and sound speed. This algorithm is developed based on perturbation theory with higher-order nonlinear terms, and it can efficiently compute higher-order derivatives of the sound pressure field with respect to changes in the local wavenumber. This higher-order perturbative PE (HOPPE) method can eventually lead to model order reduction for sound propagation simulations. An example of broadband computations with nonlinear interpolations between sampled frequencies will be presented to demonstrate the application of reduced-order modeling. A sensitivity kernel with higher-order nonlinear terms will also be introduced. Applications to data assimilation modeling and sensitivity tracking will be discussed. [Work supported by the Office of Naval Research.]

Augmented reduced order models for turbulence

The authors introduce an augmented-basis method (ABM) to stabilize reduced-order models (ROMs) of turbulent incompressible flows. The method begins with standard basis functions derived from proper orthogonal decomposition (POD) of snapshot sets taken from a full-order model. These are then augmented with divergence-free projections of a subset of the nonlinear interaction terms that constitute a significant fraction of the time-derivative of the solution. The augmenting bases, which are rich in localized high wavenumber content, are better able to dissipate turbulent kinetic energy than the standard POD bases. Several examples illustrate that the ABM significantly out-performs L2-, H1- and Leray-stabilized POD ROM approaches. The ABM yields accuracy that is comparable to constraint-based stabilization approaches yet is suitable for parametric model-order reduction in which one uses the ROM to evaluate quantities of interests at parameter values that differ from those used to generate the full-order model snapshots. Several numerical experiments point to the importance of localized high wavenumber content in the generation of stable, accurate, and efficient ROMs for turbulent flows.

Aeromagnetic and Radiometric Data Interpretation to Delineate the Structural Elements and Probable Precambrian Mineralization Zones: a Case Study, Egypt

This study focused on joint interpretation of the aeromagnetic and radiometric data to allocate and describe the structural elements and mineralization zones such as uranium, gold, and sulfide in the Gebel Umm Tineidba and its surroundings, Eastern Desert, Egypt. The study region is covered by exposures of Precambrian rocks such as gneisses, migmatites, ophiolitic mélange, metavolcanics, younger granites, and Quaternary sedimentary covers. The application of the total intensity magnetic and reduced-to-pole maps, power spectrum, analytic signal, tilt-angle, and local wavenumber maps helps in recognizing magnetic sources distribution, lineament features, and mineral zones delineation. The obtained results indicate that the depths of the deeper NE–SW, NNE–SSW, and N–S trending structures extend from 400 to 1750 m and shallower ones range from 100 to 400 m. Finally, the main trends of these structures affect the probable mineralization zones.

Laser-Generated Guided Waves for Damage Detection in Metal-Lined Composite-Overwrapped Pressure Vessels.

This paper characterizes laser-generated guided waves in a metal-lined composite-overwrapped pressure vessel (COPV) to assess typical damage, including interfacial debonding and low-velocity impact damage. First, an eigenfrequency approach that avoids additional coding is utilized to theoretically analyze the dispersion characteristics of a COPV. The theoretical results show that interfacial debonding significantly alters dispersion curves, and the wavenumber of the L(0, 1) mode is sensitive to impact damage. Experimental verifications were conducted based on the full wavefield acquired using a scanning laser-ultrasonic system with a repetition rate of 1 kHz. By comparing the experimental dispersion curves with the theoretical ones, it was found that the metal-composite interface was not bonded. In addition, a local wavenumber estimation method was established to detect the impact damage by obtaining the spatial distribution of the wavenumber of the L(0, 1) mode.

Open-source software for two-dimensional Fourier processing of gridded magnetic data

Magnetic data are widely available and useful in many exploration and applied-geophysics projects. The magnetic data are usually processed, imaged, and interpreted in commercial software packages. The algorithms used in these packages are sometimes difficult to check or tune, and the code is not available for review. However, these packages often have an application programming interface (API) for people to access data and undertake their own processing and data enhancement. In many cases, these APIs use the Python programming language. In the course of developing a new method for transforming magnetic data called reduction to pole and vertical dip (RTPVD), the initial test code was written in Python. This initial code was then rewritten and incorporated into GAMS, an open-source software package capable of using a Python API to read from and then write transformed (or enhanced) data to a commercial database. In addition to RTPVD, the other enhancements GAMS can generate are the zeroth-order analytic-signal amplitude (ASA0), tilt, spatial derivatives of ASA0, the zeroth-order local wavenumber, the first-order analytic-signal amplitude, and the apparent susceptibility. These transformations require that the data be transformed to the wavenumber domain using a fast Fourier transform (FFT), operated on, and then transformed back to the space domain. The FFT and some of the preprocessing steps can be done with a number of built-in Python tools. For the preprocessing steps, some of the available Python options are fast, but they can occasionally introduce unwanted artifacts. Our open-source tool allows users to test the different options and check the intermediate steps to ensure the result is appropriate.

Bloch wavenumber identification of periodic structures using Prony’s method

Phononic crystals and metamaterials are spatially periodic waveguides that may exhibit wave attenuation bands and, therefore, can be applied as vibration attenuation devices. In this type of structure, the wave speed varies throughout the structural domain and, therefore, the local wavenumber also varies. Methods that allow the identification of representative wavenumbers are of key importance for the experimental characterization of these structures and their attenuation bands. In this context, Prony’s method can be a viable and efficient choice. In the present study, one- and two-dimensional Prony methods are, for the first time, addressed for estimating the Bloch wavenumbers of periodic structures. Bloch wavenumbers may be computed from a single unit cell by applying a periodic boundary condition. We show that the measurements used in Prony’s method must be taken at locations that are equally spaced by multiples of the spatial period of the structure. Otherwise, the spatially varying local wavenumber will cause the method to fail. The Total Least Squares Prony method is used to increase the number of observations per unit cell, as a strategy for noise reduction. This is a novel adaptation of Prony’s method for Bloch wavenumber estimation of periodic structures. A Bayesian approach is also proposed for highly noisy measurements. In the current study, we validate the numerical experiments using the theoretical wavenumber values of a periodic Euler–Bernoulli beam, a Levy-type plate periodic along one direction, and a thin square plate (Kirchhoff’s theory) with free boundary conditions periodic along two directions. The solutions for all proposed structures are obtained using the spectral element method and, for the square plate, with the finite element method. One experiment is performed for a periodic beam. Our simulation results show that the adapted Prony method, combined with the methodology to reduce the noise effect, performs very well for the considered phononic crystals.

Development of dual X-mode Doppler reflectometry system in Heliotron J

A dual X-mode Doppler reflectometry system is developed to measure the radial electric field in a stellarator/heliotron device, Heliotron J. The system is designed to have dual channels where the observation points are placed symmetrically to the equatorial plane, enabling the poloidal flow velocity measurement at two different positions in the same toroidal section, which is useful for the search for a zonal flow. In the system, an RF source generates the microwave frequency of 8.25–12.5 GHz, upconverted by an intermediate frequency of 27.5 MHz and transmitted with a coaxial cable to a transmitter located near the Heliotoron J vacuum vessel. After quadrupling the RF waves at the transmitter, the microwaves of 33–50 GHz are injected in X-mode into a plasma using a spherical focusing mirror installed inside the vacuum vessel. The local wavenumber of the probing microwaves, k ⊥, is 1.56–1.66 cm−1. The Doppler-shifted reflected wave is downconverted to a 110 MHz signal by mixing with the LO at the receiver, amplified, and then detected by an I/Q detector. In a tabletop test, we have confirmed that the phase estimated by the I/Q detector is proportionally changed as a function of the horn antenna distance. We have successfully measured the Doppler-shifted spectra of the I/Q signals and estimated the radial electric field in an electron cyclotron heated (ECH) plasma.

Application of direct source parameter imaging (direct local wave number) technique to the 2D gravity anomalies for depth determination of some geological structures

Application of direct source parameter imaging (direct local wave number) technique to the 2D gravity anomalies for depth determination of some geological structures

The effect of intrinsic instability on the surface topography of spherical 2-acetylfuran flame

The intrinsic instability of 2-acetylfuran (2-AF) expanding flame has been investigated at initial temperatures of 453–493 K, pressures of 1–3 bar, and equivalence ratios of 0.8–1.4. The linear stability theory was used to analyze the influence of the initial temperatures, initial pressures, and equivalence ratios on the 2-AF flame intrinsic instability. Moreover, an image processing technique was applied to obtain the experimental cell-wavenumber from the topography of 2-AF cellular flame to quantitatively analyze the local cell proliferation. The topography of the 2-AF expanding flame shows that the initial pressures and equivalence ratios are the two influential factors to its intrinsic instability. More specifically, the higher the initial pressures, as well as the equivalence ratios, the earlier the time of the flame becomes unstable. Furthermore, according to the comparison between the local experimental and theoretical wavenumbers, it is found that the average local experimental wavenumber falls within the unstable zone predicted by the linear stability theory. Last but not least, the cell-wavenumber distribution in the different regions on the 2-AF expanding flame surface indicates that the initial crack has a weaker shaping effect on the flame surface than the formation of small spontaneous cells.

The local wavenumber model for computation of turbulent mixing.

We present an overview of the current status in the development of a two-point spectral closure model for turbulent flows, known as the local wavenumber (LWN) model. The model is envisioned as a practical option for applications requiring multi-physics simulations in which statistical hydrodynamics quantities such as Reynolds stresses, turbulent kinetic energy, and measures of mixing such as density-correlations and mix-width evolution, need to be captured with relatively high fidelity. In this review, we present the capabilities of the LWN model since it was first formulated in the early 1990s, for computations of increasing levels of complexity ranging from homogeneous isotropic turbulence, inhomogeneous and anisotropic single-fluid turbulence, to two-species mixing driven by buoyancy forces. The review concludes with a discussion of some of the more theoretical considerations that remain in the development of this model.This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.

High-Precision Source Positions Obtained by the Combined Inversion of Different-Order Local Wavenumbers Derived from Aeromagnetic Data

The aeromagnetic survey is a common remote sensing tool for detecting iron deposits. The local wavenumber of a magnetic anomaly is used to interpret the edges or positions of the sources, and can involve first- or second-order local wavenumbers. In this paper, we derived a linear equation between the second-order local wavenumber and the source location; therefore, we propose a constraint of first and second-order local wavenumbers. Tests on synthetic data show that the source parameters, computed using a combination of equations that involved different-order local wavenumbers, are closer to the true values and show a smaller spread in estimated values. For gridded data, we proved that the different-order combination allowed us to accurately estimate the source position. When applied to the aeromagnetic data from Hebei province, China, we refined the location of most magnetic features, which we interpreted as possible iron deposits.

Evaluation of disbonds in adhesively bonded multilayer plates through local wavenumber estimation

Multilayer plates are widely used as structural components, especially in high-tech industries. However, compared to much simpler homogeneous plates, they pose unique challenges to nondestructive testing (NDT) techniques. The inspections of multilayer plates using traditional ultrasonic (US) testing modalities can be challenging and time-consuming. Faster inspection can be achieved with the use of full-field techniques based on guided elastic waves. In particular, the local wavenumber estimation (LWE) of Lamb waves has been proven effective for damage detection in various plate-like structures. However, the effectiveness of LWE depends on multiple parameters, such as the excitation frequency, the choice of a Lamb wave mode, or the inspection side, among others. This is especially significant in the case of multilayer plates, where different types of damage result in a nontrivial behavior of Lamb waves. In this work, we present a framework for the evaluation of disbonds in adhesively bonded multilayer plates through local wavenumber estimation. The efficacy of the proposed approach is demonstrated on a test sample made of three layers of aluminum with different thicknesses bonded together with an epoxy adhesive. The sample contains artificial defects of various sizes introduced at different adhesive interfaces. First, we present a theoretical analysis of the dispersion curves in the test sample to determine the optimal LWE inspection parameters for particular types of defects. Next, the approach is validated experimentally based on a set of full-wavefield datasets acquired with a scanning laser Doppler vibrometer (SLDV). We demonstrate that the proposed methodology allows obtaining clear images of defects and determining their through-thickness locations in multilayer structures. Based on the obtained results, we believe that the proposed framework can also be effectively applied to different structures and damage types.

Bathymetry Determination Based on Abundant Wavenumbers Estimated from the Local Phase Gradient of X-Band Radar Images

A phase gradient (PG)-based algorithm is proposed in this study to determine coastal bathymetry from X-band radar images. Although local wavenumbers with the same spatial resolution of the wave field can be obtained from the wave field using the PG method, only a single wavenumber result can be extracted from each location theoretically. Due to the influence of unavoidable noise on the wave field image, single wavenumber estimation often shows high uncertainty. This study combines a bandpass filter and directional pass filter to produce different nearly monocomponent wave fields from X-band radar images and then estimates more wavenumbers from these wave fields using the PG method. However, the distributions of wavenumbers in higher-frequency bins still show high variance because the strength of wave signals is weak. We confirmed that the uncertain wavenumber–frequency pairs can be improved using the Kalman filter and are more consistent with the dispersion relation curve. To decrease the influence of inaccurate wavenumbers, we also use the strength of the wave signals as the weights for the least-squares fit. Although the depth errors from shallow-water areas are still unavoidable, we can remove the inaccurate depth estimation from shallow-water areas according to the coefficients of determination of the fitting. In summary, the algorithm proposed in this study can obtain a bathymetry map with high spatial resolution. In contrast to the depth result estimated using a single wavenumber of each frequency bin, we confirm that more wavenumbers from each of the frequency bins are helpful in fitting the dispersion relation curve and obtaining a more reliable depth result.