Frequency-wavenumber Domain Research Articles

A New Inversion Method to Obtain Upper-Ocean Current-Depth Profiles Using X-Band Observations of Deep-Water Waves

AbstractA new method for estimating current-depth profiles from observations of wavenumber-dependent Doppler shifts of the overlying ocean wave field is presented. Consecutive scans of marine X-band backscatter provide wave field measurements in the time–space domain that transform into the directional wavenumber–frequency domain via a 3D fast Fourier transform (FFT). Subtracting the linear dispersion shell yields Doppler shift observations in the form of (kx, ky, Δω) triplets. A constrained linear regression technique is used to extract the wavenumber-dependent effective velocities, which represent a weighted depth average of the Eulerian currents (Stewart and Joy). This new method estimates these Eulerian currents from the effective velocities via the inversion of the integral relationship, which was first derived by Stewart and Joy. To test the effectiveness of the method, the inverted current profiles are compared to concurrent ADCP measurements. The inversion method is found to successfully predict current behavior, with a depth-average root-mean-square (RMS) error less than 0.1 m s−1 for wind speeds greater than 5 m s−1 and a broad wave spectrum. The ability of the inversion process to capture the vertical structure of the currents is assessed using a time-average RMS error during these favorable conditions. The time-averaged RMS error is found to be less than 0.1 m s−1 for depths shallower than 20 m, approximately twice the depth of existing methods of estimating current shear from wave field measurements.

A mixed space-time and wavenumber-frequency domain procedure for modelling ground vibration from surface railway tracks

This paper presents a methodology for studying ground vibration in which the railway track is modelled in the space-time domain using the finite element method (FEM) and, for faster computation, discretisation of the ground using either FEM or the boundary element method (BEM) is avoided by modelling it in the wavenumber-frequency domain. The railway track is coupled to the ground through a series of rectangular strips located at the surface of the ground; their vertical interaction is described by a frequency-dependent dynamic stiffness matrix whose elements are represented by discrete lumped parameter models. The effectiveness of this approach is assessed firstly through frequency domain analysis using as excitation a stationary harmonic load applied on the rail. The interaction forces at the ballast/ground interface are calculated using the FE track model in the space-time domain, transformed to the wavenumber domain, and used as input to the ground model for calculating vibration in the free field. Additionally, time domain simulations are also performed with the inclusion of nonlinear track parameters. Results are presented for the coupled track/ground model in terms of time histories and frequency spectra for the track vibration, interaction forces and free-field ground vibration. For the linear track model, the results from the mixed formulation are in excellent agreement with those from a semi-analytical model formulated in the wavenumber-frequency domain, particularly in the vicinity of the loading point. The accuracy of the mixed formulation away from the excitation point depends strongly on the inclusion of through-ground coupling in the lumped parameter model, which has been found to be necessary for both track dynamics and ground vibration predictions.

Vibration of infinite Timoshenko beam on Pasternak foundation under vehicular load

The vibration of beams on foundations under a vehicular load has attracted wide attention for decades. The problem has numerous applications in several fields such as highway structures. However, most of analytical or semi-analytical studies simplify the vehicular load as a concentrated point or distributed line load with the constant or harmonically varying amplitude, and neglect the presence of the vehicle and the road irregularity. This article carries out an analytical study of vibration on an infinite Pasternak-supported Timoshenko beam under vehicular load which is generated by the passage of a quarter car on a road with harmonic surface irregularity. The governing equations of motion are derived based on Hamilton’s principle and Timoshenko beam theory and then are solved in the frequency–wavenumber domain with a moving coordinate system. The analytical solutions are expressed in a general form of Cauchy’s residue theorem. The results are validated by the case of an Euler–Bernoulli beam on a Winkler foundation, which is a special case of the current system and has an explicit form of solution. Finally, a numerical example is employed to investigate the influence of properties of the beam (the radius of gyration and the shear rigidity) and the foundation (the shear viscosity, rocking, and normal stiffness) on the deflected shape, maximum displacement, critical frequency, and critical velocity of the system.

Wave modulation of the extratropical tropopause inversion layer

Abstract. This study aims to quantify how much of the observed strength and variability in the zonal-mean extratropical tropopause inversion layer (TIL) comes from the modulation of the temperature field and its gradients around the tropopause by planetary- and synoptic-scale waves. By analyzing high-resolution observations, it also puts other TIL enhancing mechanisms into context.Using gridded Global Positioning System radio occultation (GPS-RO) temperature profiles from the COSMIC mission (2007–2013), we are able to extract the extratropical wave signal by a simplified wavenumber–frequency domain filtering method and quantify the resulting TIL enhancement. By subtracting the extratropical wave signal, we show how much of the TIL is associated with other processes, at mid- and high latitudes, for both hemispheres and all seasons.The transient and reversible modulation by planetary- and synoptic-scale waves is almost entirely responsible for the TIL in midlatitudes. This means that wave-mean flow interactions, inertia–gravity waves and the residual circulation are of minor importance for the strength and variability in the midlatitude TIL.At polar regions, the extratropical wave modulation is dominant for the TIL strength as well, but there is also a clear fingerprint from sudden stratospheric warmings (SSWs) and final warmings in both hemispheres. Therefore, polar vortex breakups are partially responsible for the observed polar TIL strength in winter (if SSWs occur) and spring. Also, part of the polar summer TIL strength cannot be explained by extratropical wave modulation.We suggest that our wave modulation mechanism integrates several TIL enhancing mechanisms proposed in previous literature while robustly disclosing the overall outcome of the different processes involved. By analyzing observations only, our study identifies which mechanisms dominate the extratropical TIL strength and their relative contribution. It remains to be determined, however, which roles the different planetary- and synoptic-scale wave types play within the total extratropical wave modulation of the TIL, as well as what causes the observed amplification of extratropical waves near the tropopause.

Fundamental solutions of a multi-layered transversely isotropic saturated half-space subjected to moving point forces and pore pressure

The steady-state dynamic response of a multi-layered transversely isotropic (TI) saturated half-space due to point forces and pore pressure moving with a constant speed is investigated in this paper. To solve this problem, the dynamic stiffness method combined with the inverse Fourier transform is employed. First, the governing equations in terms of the displacement components and pore fluid pressure are solved in the transformed domain by employing the Fourier transform. Next, the exact three-dimensional (3D) dynamic stiffness matrices for the TI saturated layer, as well as the TI saturated half-space, are constructed, and the global dynamic matrix of the problem is formulated by assembling the dynamic matrices of the discrete layers and the underlying half-space. Finally, solutions in the frequency-wavenumber domain of the displacement, pore pressure and stress are obtained through the dynamic stiffness method. The result in the time-space domain is recovered by the Fourier synthesis of the frequency responses which, in turn, are obtained by numerical integration over on one horizontal wavenumber. The accuracy of the developed formulations is confirmed by comparison with existing solutions for an isotropic and saturated medium that is a special case of the more general problem addressed. Numerical results for both low and high source velocities are presented, and the effects of moving speed, material anisotropy, permeability, surface drainage condition and TI saturated layer on the dynamic response are analyzed. It is observed that the dynamic responses reach their peak values when the source velocity is equal to or approaches the phase velocities of SH-, qP1-, qP2- and qSV- in the horizontal direction and the phase velocity of qRayleigh waves. Material anisotropy is very important for the accurate assessment of the dynamic response due to the moving point forces and pore pressure in a TI saturated medium.

Semi-analytical approach to modelling the dynamic behaviour of soil excited by embedded foundations

The underlying soil has a significant effect on the dynamic behaviour of structures. The paper proposes a semi-analytical approach based on a Green’s function solution in frequency–wavenumber domain. The procedure allows calculating the dynamic stiffness for points on the soil surface as well as for points inside the soil body. Different cases of soil stratification can be considered, with soil layers with different properties overlying a half-space of soil or bedrock. In this paper, the soil is coupled with piles and surface foundations. The effects of different foundation modelling configurations are analysed. It is determined how simplification of the numerical model affects the overall dynamic behaviour.

Ultrasonic zero-lag cross correlation based on frequency-wavenumber domain analysis for internal delamination detection

This paper proposes a zero-lag cross correlation (ZLCC) method based on frequency-wavenumber (f-k) domain analysis for internal delamination detection in composite structures. The proposed ZLCC method is able to identify and localise hidden delamination inside composite layers without relying on baseline data accumulated from the pristine condition of a target structure, making it possible to minimise false alarms caused by environmental effects. The ZLCC method is theoretically developed by investigating the physical relationship between ZLCC and ultrasonic wave interaction with delamination. Then, it is numerically validated using a 2D finite element model. The finite element analysis results reveal that internal delamination is successfully localised and highlighted by ZLCC.

Dynamic Analysis of an Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

Nonstationary random vibration analysis of an infinitely long beam resting on a Kelvin foundation subjected to moving random loads is studied in this paper. Based on the pseudo excitation method (PEM) combined with the Fourier transform (FT), a closed-form solution of the power spectral responses of the nonstationary random vibration of the system is derived in the frequency-wavenumber domain. On the numerical integration scheme a fast Fourier transform is developed for moving load problems through a parameter substitution, which is found to be superior to Simpson’s rule. The results obtained by using the PEM-FT method are verified using Monte Carlo method and good agreement between these two sets of results is achieved. Special attention is paid to investigation of the effects of the moving load velocity, a few key system parameters, and coherence of loads on the random vibration responses. The relationship between the critical speed and resonance is also explored.

A 2.5D spectral approach to represent acoustic and elastic waveguides interaction on thin slab structures

In this paper, we propose a spectral element method (SEM) to study guided waves in coupled problems involving thin-walled structures and fluid-acoustic cavities. The numerical method is based on the subdomain decomposition of the fluid-structure system. Two spectral elements are developed to represent the fluid and the structure. A plate element based on a mixed Reissner-Mindlin and Kirchhoff-Love formulation is proposed to represent the thin-walled structure. This element uses C0 approximation functions to overcome the difficulties to formulate elements with arbitrary order from C1 functions. The proposed element uses a substitute transverse shear strain field resulting free shear locking. The fluid element is derived from the Helmholtz equation.These elements use Lagrange polynomials as shape functions at the Legendre-Gauss-Lobatto (LGL) points. The analysis is carried out by a two-and-a-half dimension (2.5D) approach in the wavenumber-frequency domain. The guided wave in a fluid cavity with a flexible side is analysed.

Vibrations of Flexible Strip on Viscoelastic Halfspace

The dynamic response of rigid and flexible foundations on the soil has been subject of extensive study in the past decades. A hybrid method using a combined finite element method (FEM) and boundary element method (BEM) is the most common method used for solving this problem. The objective of this paper is to present an effective frequency domain method to obtain the dynamic response of a flexible strip foundation resting on a viscoelastic halfspace. The foundation is treated with the spectral element method (SEM), while the soil is modelled using the integral transform method (ITM). Both SEM and ITM are based on the analytical solution of the Lame-equations in the frequency domain and therefore are suitable for combining. The solution is obtained in the transformed space-frequency or wavenumber-frequency domain using the Fourier transformation. The study is performed as a 2D plane-strain analysis, assuming that the foundation cross-section behaves as an Euler-Bernoulli beam and that there is no sliding between the foundation and the soil, nor discontinuities in terms of the displacement field. The vertical displacements field of the foundation is described by a set of modal functions corresponding to free vibration mode shapes of a SEM Euler-Bernoulli beam element. The coupling between the foundation and the soil is achieved using the modal soil impedance functions, which are determined by using the ITM. The displacements of the coupled foundation-soil system are solved by the modal superposition method. The accuracy of the proposed method is assessed by comparing the obtained results with the results obtained by a commercial software package SASSI2000. The comparison shows that the presented method is accurate and less costly in terms of computational effort, especially in the high frequency range. The presented method can be easily extended to provide the solution of the response of a flexible strip on a layered halfspace due to a horizontal and vertical excitation.

Ultrasonic zero-lag cross correlation based on frequency-wavenumber domain analysis for internal delamination detection

Ultrasonic zero-lag cross correlation based on frequency-wavenumber domain analysis for internal delamination detection

Passive ranging technique using waveguide invariant in shallow water with thermocline

Waveguide invariant is widely used in underwater target passive ranging. In shallow water with uniform sound speed profile, the value of waveguide invariant is approximately a constant, while in shallow water with thermocline, it varies in a wide range. The waveguide invariant distributions and striations in these two conditions are analyzed respectively. On the basis of wavenumber difference between reflected modes and refracted modes, a wavenumber-frequency domain filtering technique is proposed to separate the two groups of modes. The required relationship between array element space, total array length and target azimuth for effective application is discussed. Finally, the simulation results indicate that in shallow water with a thermocline, refracted modes can be effectively filtered out using the wavenumber-frequency domain filtering technique and the target's range is estimated accurately.

Quantification of Shear Wave Scattering From Far-Surface Defects via Ultrasonic Wavefield Measurements.

Nondestructive evaluation methods rely on prior knowledge of the expected interaction of ultrasonic waves with defects to inform detection and characterization decisions. Wavefield imaging, which refers to the measurement of signals originating from a spatially fixed source on a 2-D rectilinear grid, can be applied to visualize the effect of a subsurface scatterer on surface-measured wave motion. Here, obliquely incident shear waves are directed at the far surface of a plate containing a through-hole using the well-known angle-beam ultrasonic inspection method. A laser vibrometer and laboratory scanner are used to record the resulting out-of-plane motion on the plate surface in the vicinity of the through-hole both before and after a far-surface corner notch is introduced and subsequently enlarged. Waves scattered from the notch are isolated from the incident and hole-scattered waves via baseline subtraction of wavefields. The scattered wavefields are then filtered in the frequency-wavenumber domain to separate Rayleigh, shear, and longitudinal contributions to the scattered wavefield. The filtered wavefields are interpolated in space to obtain 2-D radial wavefield slices originating at the base of the notch. Each radial slice is analyzed to quantify scattering as a function of observation direction, resulting in Rayleigh, shear, and longitudinal scattering profiles for each notch size. The results are compared for four different notch sizes and two transducer orientations.

Spectrally formulated one-dimensional element for analysis of wave propagation in pretwisted anisotropic strips

A spectrally formulated one-dimensional element is developed to study the wave propagation in pretwisted anisotropic strips. The element is based on linear sectional analysis of pretwisted anisotropic strips with small pretwist. Firstly, the governing equations of the strip are obtained through dimensional reduction of laminated shell theory to a linear one dimensional (1-D) theory using the variational asymptotic method (VAM). Next, an exact dynamic stiffness matrix is derived in the frequency-wavenumber domain using spectral finite element (SFE) method. In SFE formulation the mass distribution is modeled exactly and as a result a single element is sufficient to capture the exact frequency response of a regular structure. For numerical validation of the proposed model, the natural frequencies of the strip are compared to the modal behavior of pretwisted composite strips available in literature. The model is used to predict wave responses due to modulated sinusoidal input. The different wave modes present, namely axial, flexural, and torsional modes are seen and their velocities are compared to that obtained from the dispersion plot. This proposed modeling strategy is a first step towards the development of a comprehensive model to do structural health monitoring of structures idealized as pretwisted strips.

Static and dynamic behaviours of isolated or unisolated ballast tracks using a fast wavenumber domain method

The dynamics of unisolated and isolated ballast tracks have been analysed by multi-beam models for the track and by a layered half-space model for the soil. The solution is calculated in frequency–wavenumber domain and transformed back to space domain by a wavenumber integral. This is a faster method compared to other detailed track–soil interaction methods and almost as fast as the widely used Winkler soil method, especially if the compliances of the soil have been stored for repeated use. Frequency-dependent compliances and force transfer functions have been calculated for a variety of track and soil parameters. The ballast has a clear influence on the high-frequency behaviour, whereas the soil is dominating the low-frequency behaviour of the track. A layering of the soil may cause a moderate track–soil resonance, whereas more pronounced vehicle–track resonances occur with elastic track elements like rail pads, sleeper pads and ballast mats. Above these resonant frequencies, a reduction in the excitation forces follows as a consequence. The track deformation along the track has been analysed for the most interesting track systems. The track deformation is strongly influenced by the resonances due to layering or elastic elements. The attenuation of amplitudes and the velocity of the track–soil waves change considerably around the resonant frequencies. The track deformation due to complete trains have been calculated for different continuous and Winkler soils and compared with the measurement of a train passage showing a good agreement for the continuous soil and clear deviations for the Winkler soil model.

Comparative analysis on spatiotemporal pore pressure evolution under surface–wave perturbations

The near surface spatiotemporal pore pressure fluctuations in a porous half space induced by seismic surface–wave perturbations in different poroelastic regimes are analytically compared. The results for the pore pressure in the space–time domain are derived by inverse transforms of the fundamental solutions of dynamic poroelasticity in the frequency–wavenumber domain. Different poroelastic regimes depend on the different low frequency approximation for Biot's multi wave mode physical behaviors. In low seismic frequency band, whereas the fast wave mode inference can generate surface wave along the interface, fluid pressure is also significantly perturbated as in solid phase, but behaves as a slow P2 mode reflected by the R surface wave. It is also shown that the amount of induced pore pressure fluctuations are different by P2 diffusivity (dispersion–difusivity regime, DD and non–dispersion–diffusivity regime, NDD) and P2 wave (full–wave regime, FW). The strong solid and fluid coupling on the interface generates the transferred RP2 mode. Therefore, the decoupled viscoleastic approximation (VA) does not represent these near surface fluid characteristics. The results can be applied to predict near surface pore pressure responses under seismic perturbations, or as a benchmark for other numerical approaches for pressure solutions in a dynamic poroelastic half space, and also favor a potential interpretation for near surface soil liquefaction failure.

Signal apparition applied to towed marine simultaneous sources – a case study on synthesized real data from the Viking Graben

Encoding sources using random dithers for simultaneous source acquisition offers significant scope for enhanced towed marine operations. However, so far it has proven to be challenging to reduce the decoding residual noise from these techniques to a level that is widely accepted by the industry. The emerging technology of signal apparition offers a new approach to overcome this challenge. In contrast to established thinking, the signal apparition concept capitalizes on periodic rather than random shot-to-shot modulation functions to facilitate shot separation. The approach allows us to efficiently populate the available data space in, for instance, the frequency wavenumber domain with energy from different simultaneous sources. The introduction of periodic modulation functions in seismic acquisition produces an effect where parts of the energy of one or more sources are shifted to different empty parts within the frequency wavenumber (f-k) domain. This so-called apparated energy can then be used to perfectly predict at low frequencies the remaining part of the signal in the regions of the f-k domain where the wavefields from the different sources overlap and to deterministically separate the sources. On a synthesized real data set from the Viking Graben in the North Sea, the suitability of this concept to marine seismic simultaneous sources is demonstrated. An operational scenario for a single vessel is designed where the source arrays are excited simultaneously. Having two closely spaced simultaneous sources is a challenging case for shot separation but, if successful, such a cost-effective single-vessel configuration offers highly significant productivity gains. The wavefield decoding results analysed pre-stack and post-stack are very encouraging and indicate that the signal apparition concept indeed has the potential to deliver the high quality source decoding required for many seismic applications.

High-productivity seabed time-lapse seismic data acquisition using simultaneous sources enabled by seismic apparition: A synthetic-data study

Seismic apparition is a recent signal-processing advance that trades signal interference and aliasing between different spatial (and temporal) dimensions. In particular, an important application of seismic apparition is for simultaneous-source separation to better exploit the available data space in the frequency-wavenumber (f-k) domain with energy from different simultaneous sources. The introduction of periodic modulation functions in seismic acquisition produces an effect where parts of the energy of one or more sources are partially shifted to different empty parts within the f-k domain. This so-called apparated energy then can be used to perfectly predict (at low frequencies) the remaining part of the signal in the regions of the f-k domain where the wavefields from the different sources overlap and to deterministically separate the sources at predetermined (repeated) positions — a prerequisite for 4D seismic processing. At higher frequencies, the apparition separation is dealiased using directional information, taking full advantage of the perfect separation at lower frequencies to achieve the required low-error separation. To apparate seismic energy to different portions within the f-k domain, we introduce a periodic modulation function, consisting for instance of a small time delay/advance or an amplitude scaling to every second shot. Our simultaneous-source approach is thus opposite to established industry practices of maximizing randomness using dithered sources. A deterministic simultaneous-source-separation approach in which shot points and simultaneous-source patterns are repeated accurately is a major advantage over industry-standard stochastic simultaneous-source-acquisition approaches, particularly for 4D applications. We demonstrate the suitability of the seismic-apparition simultaneous-source technique to time-lapse seismic, having a permanent-reservoir-monitoring context in mind, by investigating two operational scenarios of single-vessel acquisition where two sources are excited simultaneously. Single-vessel simultaneous-source acquisition represents the most challenging case for shot separation in general and for time-lapse seismic applications in particular. The quantitative and qualitative wavefield decomposition results analyzed pre- and poststack are highly encouraging on synthetic data and warrant further testing of seismic-apparition technology in a real 4D seismic test case.

Imaging of local porosity/voids using a fully non-contact air-coupled transducer and laser Doppler vibrometer system

This study exploits the feasibility of imaging zones of local porosity/voids simulated by introducing microspheres during layup of a unidirectional carbon fiber–reinforced polymer composite panel. A fully non-contact hybrid system primarily composed of an air-coupled transducer and a laser Doppler vibrometer was used for imaging the local porosity/void zones from the guided wave response. To improve image resolution, several preprocessing techniques are performed. The wavefield reconstructed from the laser Doppler vibrometer measurements was first “denoised” using a one-dimensional wavelet transform in the time domain followed by a two-dimensional wavelet transform in the spatial domain. From the total wavefield, the much weaker backscattered waves were separated from the stronger incident wave by frequency–wavenumber domain filtering. In order to further enhance the signal-to-noise ratio and sharpen the image, the attenuation of incident wave propagation to the damage site was compensated through two proposed weight functions. Finally, a zero-lag cross-correlation was performed for imaging the zone where the compensated incident and backscattered waves were in phase. This improved imaging condition, the “denoised” weighted zero-lag cross-correlation, was proposed and tested for defect imaging in the composite panel with eight intentionally introduced zones of high porosity/voids of varying diameters (1.59–6.35 mm) and depths (0.36–1.08 mm). As expected, the sensitivity of the non-contact air-coupled transducer/laser Doppler vibrometer hybrid system was limited by the wavelength of the excitation signal. The system incorporated with the denoised weighted zero-lag cross-correlation imaging condition for guided wave interrogation gave similar image quality in comparison with that by the immersion C-scan.

Baroclinic turbulence on the polar β-plane in the rotating tank: Down to submesoscale

The energy spectra of the baroclinic turbulence are examined in finely resolved laboratory flows. A wide range of wavenumbers and frequencies is accessible, including those dynamically similar to the oceanic meso- and submesoscales. Oceanographically relevant phenomena observed in the experiments include Rossby waves, alternating zonal jets, baroclinically unstable coastal currents as well as submesoscale filaments and eddies. Independent spectral decomposition methodologies (Fourier and Fourier–Bessel) applied in Cartesian and polar coordinates respectively provide a complementary framework for representing and interpreting the measured flows. Evolution of energy spectra in the wavenumber domain demonstrates that energy concentrates in zonal modes. The spectra in the frequency-wavenumber domain reveal a significant role of the linear dynamics in the form of the Rossby waves and baroclinic instability modes. The spectral analyses are extended to obtain energy fluxes between lengthscales and reveal an inverse cascade at larger (meso-) scales and a direct cascade at smaller (submeso-) scales.