Nyquist Wavenumber Research Articles

Turbulence Revealed by Wavelet Transform: Power Spectrum and Intermittency for the Velocity Field of the Cosmic Baryonic Fluid

We use continuous wavelet transform techniques to construct the global and environment-dependent wavelet statistics, such as energy spectrum and kurtosis, to study the fluctuation and intermittency of the turbulent motion in the cosmic fluid velocity field with the IllustrisTNG simulation data. We find that the peak scale of the energy spectrum defines a characteristic scale, which can be regarded as the integral scale of turbulence, and the Nyquist wavenumber can be regarded as the dissipation scale. With these two characteristic scales, the energy spectrum can be divided into the energy-containing range, the inertial range, and the dissipation range of turbulence. The wavelet kurtosis is an increasing function of the wavenumber k, which first grows rapidly then slowly with k, indicating that the cosmic fluid becomes increasingly intermittent with k. In the energy-containing range, the energy spectrum increases significantly from z = 2 to 1, but remains almost unchanged from z = 1 to 0. We find that both the environment-dependent spectrum and kurtosis are similar to the global ones, and the magnitude of the spectrum is smallest in the lowest-density and largest in the highest-density environment, suggesting that the cosmic fluid is more turbulent in a high-density than in a low-density environment. In the inertial range, the energy spectrum’s exponent is steeper than both the Kolmogorov and Burgers exponents, indicating more efficient energy transfer compared to Kolmogorov or Burgers turbulence.

Material Interface in the Finite-Difference Modeling: A Fundamental View

ABSTRACT By analyzing the equations of motion and constitutive relations in the wavenumber domain, we gain important insight into attributes determining the accuracy of finite-difference (FD) schemes. We present heterogeneous formulations of the equations of motion and constitutive relations for four configurations of a wavefield in an elastic isotropic medium. We Fourier-transform the entire equations to the wavenumber domain. Subsequently, we apply the band-limited inverse Fourier transform back to the space domain. We analyze consequences of spatial discretization and wavenumber band limitation. The heterogeneity of the medium and the Nyquist-wavenumber band limitation of the entire equations has important implications for an FD modeling: The grid representation of the heterogeneous medium must be limited by the Nyquist wavenumber. The wavenumber band limitation replaces spatial derivatives both in the homogeneous medium and across a material interface by continuous spatial convolutions. The latter means that the wavenumber band limitation removes discontinuities of the spatial derivatives of the particle velocity and stress at the material interface. This allows to apply proper FD operators across material interfaces. A wavenumber band-limited heterogeneous formulation of the equations of motion and constitutive relations is the general condition for a heterogeneous FD scheme.

On the accuracy and spatial sampling of finite-difference modelling in discontinuous models

Finite-difference modelling estimates the wavefield in the subsurface by solving the elastic or acoustic wave equation numerically in a discrete version of the subsurface. The derivatives in the wave equation are approximated by their finite-difference counterparts. In this paper, we investigate the accuracy of acoustic finite-difference modelling as a function of the spatial sampling rate, the frequency, the source-receiver offset and the model parameters and parametrisation. To represent discontinuities in a regular grid, we apply a tapered low-pass wavenumber filter to densely sampled compliance and density models using the Nyquist wavenumber of the desired grid as cut-off wavenumber. A long filter taper improves the finite-difference modelling accuracy but also leads to longer oscillations in the model grids. We use a 2D acoustic central-grid pseudospectral scheme and compare it to analytical solutions of the wave equation and the reflectivity method. We show that previous recommendations for the spatial sampling of four grid points per shortest wavelength in staggered-grid schemes also apply to central-grid schemes. In the case of a single thin layer, the accuracy of finite-difference modelling is dependent on the impedance contrast and the layer thickness. For accurate wide-angle reflection amplitudes of a thin layer with a strong impedance contrast, a denser grid sampling than four grid points per shortest wavelength is required. Furthermore, we demonstrate that the presented wavenumber filtering approach is better suited for the downsampling of regularly sampled compliance and density data derived from well logs than Backus averaging.

Modified wavenumber and aliasing errors of split convective forms for compressible flows

The spectral characteristics of split convective forms for compressible flows in finite difference methods are studied. It has been widely argued that the split forms are capable of reducing aliasing errors, based on the studies that consider spectral methods. However, the theoretical analysis shown here reveals that the split forms do not reduce aliasing errors in finite difference methods but rather increase aliasing errors more than the divergence form. This is because the modified wavenumber of the split forms may not become zero at the Nyquist wavenumber and is larger than that of the divergence form in the high wavenumber range. Correspondingly, this study also concludes that the superior numerical stability of kinetic energy preserving or kinetic energy and entropy preserving schemes, in which the split forms are used, is due to the enhanced preservation property of the kinetic energy and entropy and not the reduction of aliasing errors in finite difference methods. The spectral characteristics shown in the numerical tests are in good agreement with the theoretical analysis performed in this study.

Accuracy of power spectra in dissipationless cosmological simulations

AbstractWe exploit a suite of large N-body simulations (up to N = 40963) performed with abacus, of scale-free models with a range of spectral indices n, to better understand and quantify convergence of the matter power spectrum. Using self-similarity to identify converged regions, we show that the maximal wavenumber resolved at a given level of accuracy increases monotonically as a function of time. At 1 per cent level it starts at early times from a fraction of $k_\Lambda$, the Nyquist wavenumber of the initial grid, and reaches at most, if the force softening is sufficiently small, ${\sim}2{-}3 k_\Lambda$ at the very latest times we evolve to. At the $5{{\ \rm per\ cent}}$ level, accuracy extends up to wavenumbers of order $5k_\Lambda$ at late times. Expressed as a suitable function of the scale-factor, accuracy shows a very simple n-dependence, allowing a extrapolation to place conservative bounds on the accuracy of N-body simulations of non-scale-free models like LCDM. We note that deviations due to discretization in the converged range are not well modelled by shot noise, and subtracting it in fact degrades accuracy. Quantitatively our findings are broadly in line with the conservative assumptions about resolution adopted by recent studies using large cosmological simulations (e.g. Euclid Flagship) aiming to constrain the mildly non-linear regime. On the other hand, we remark that conclusions about small-scale clustering (e.g. concerning the validity of stable clustering) obtained using PS data at wavenumbers larger than a few $k_\Lambda$ may need revision in light of our convergence analysis.

Small-scale medium variations with high-order finite-difference and pseudospectral schemes

The accuracy of implementing interfaces with coarse-grid methods such as the pseudospectral method and high-order finite differences has been considered to be low. Our focus is on variations in interface locations and on inclusions that are significantly smaller than the grid step sizes. Classic implementations of these staggered-grid high-order methods are used. Band-limited versions of the Heaviside step function are used to manipulate the material-parameter grids. Interfaces can be implemented with accuracy that is one order of magnitude smaller than the step size. Small medium inclusions, diffractors, can, be three orders of magnitude smaller in area than the typical cell size and still be modeled with good accuracy. If the method used to implement an interface or to implement a small-scale inclusion is viewed as a filter, then this filter must be accurate up to the spatial Nyquist wavenumber of the simulation grid.

Variable-order optimal implicit finite-difference schemes for explicit time-marching solutions to wave equations

The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.

Accurate initial conditions for cosmological N-body simulations: minimizing truncation and discreteness errors

ABSTRACT Inaccuracies in the initial conditions for cosmological N-body simulations could easily be the largest source of systematic error in predicting the non-linear large-scale structure. From the theory side, initial conditions are usually provided by using low-order truncations of the displacement field from Lagrangian perturbation theory, with the first- and second-order approximations being the most common ones. Here, we investigate the improvement brought by using initial conditions based on third-order Lagrangian perturbation theory (3LPT). We show that with 3LPT, truncation errors are vastly suppressed, thereby opening the portal to initializing simulations accurately as late as z = 12 (for the resolution we consider). We analyse the competing effects of perturbative truncation and particle discreteness on various summary statistics. Discreteness errors are essentially decaying modes and thus get strongly amplified for earlier initialization times. We show that late starting times with 3LPT provide the most accurate configuration, which we find to coincide with the continuum fluid limit within 1 per cent for the power- and bispectrum at z = 0 up to the particle Nyquist wavenumber of our simulations (k ∼ 3h Mpc−1). In conclusion, to suppress non-fluid artefacts, we recommend initializing simulations as late as possible with 3LPT. We make our 3LPT initial condition generator publicly available.

Investigation of a Vorticity-preserving Scheme for the Euler Equations

Abstract We investigate the vorticity-preserving properties of the compressible, second-order residual-based scheme, “RBV2.” The scheme has been extensively tested on hydrodynamical problems, and has been shown to exhibit remarkably accurate results on the propagation of inviscid compressible vortices, airfoil-vortex interactions on a curvilinear mesh, vortex mergers in an astrophysical accretion disk, and the establishment of a two-dimensional inverse cascade in high-resolution turbulent simulations. Here, we demonstrate that RBV2 sustains the analytic solution for a one-dimensional shear flow. We assess the fidelity by which the algorithm maintains a skewed shear flow, and present convergence tests to quantify the magnitude of the expected numerical dispersion. We propose an adjustment to the dissipation in the algorithm that retains the vorticity-preserving qualities, and accurately incorporates external body forces, and demonstrate that it indefinitely maintains a steady-state hydrostatic equilibrium between a generic acceleration and a density gradient. We present a novel numerical assessment of vorticity preservation for discrete wavenumber, vortical modes of discrete wavenumber up to the Nyquist wavenumber. We apply this assessment to RBV2 in order to quantify the extent to which the scheme preserves vorticity for the full Euler equations. We find that RBV2 perfectly preserves vorticity for modes with symmetric wavenumbers, i.e., k x = k y , and that the error increases with asymmetry. We simulate the dynamical interaction of vortices in a protoplanetary disk to demonstrate the utility of the updated scheme for rendering astrophysical flows replete with vortices and turbulence. We conclude that RBV2 is a competitive treatment for evolving vorticity-dominated astrophysical flows, with minimal dissipation.

Analytical expressions for the reconstructed image of a homogeneous cylindrical sample exhibiting a beam hardening artifact in X-ray computed tomography

BACKGROUND:Cylindrical phantoms are often imaged by X-ray computed tomography (CT) to evaluate the extent of beam hardening (or cupping artifact) resulting from a polychromatic X-ray source.OBJECTIVE:Our goal was to derive analytical expressions for the reconstructed image of a homogeneous cylindrical phantom exhibiting a cupping artifact, to permit a quantitative comparison with experimental cupping data.METHODS:A filtered backprojection method was employed to obtain the analytical cupping profile for the phantom, assuming that the projection data could be approximated as a power series with respect to the sample penetration thickness.RESULTS:The cupping profile was obtained analytically as a series of functions by employing Ramachandran filtering with an infinite Nyquist wavenumber. The quantitative relationship between the power series of the projection and the nth moment of the linear attenuation coefficient spectrum of the phantom was also determined. Application of the obtained cupping profile to the evaluation of the practical reconstruction filters with a finite Nyquist wavenumber and to the best choice of the contrast agent was demonstrated.CONCLUSIONS:The set of exact solutions derived in this work should be applicable to the analysis of cylindrical phantom experiments intended to evaluate CT systems.

Accuracy‐constrained optimisation methods for staggered‐grid elastic wave modelling

ABSTRACTThe classical finite‐difference methods for seismic wave modelling are very accurate at low wavenumbers but suffer from inaccuracies at high wavenumbers, particularly at Nyquist wavenumber. In contrast, the optimisation finite‐difference methods reduce inaccuracies at high wavenumbers but suffer from inaccuracies at low wavenumbers, particularly at zero wavenumber when the operator length is not long and the whole range of wavenumbers is considered. Inaccuracy at zero wavenumber means that the optimisation methods only have a zeroth‐order accuracy of truncation and thus are not rigorously convergent. To guarantee the rigorous convergence of the optimisation methods, we have developed accuracy‐constrained optimisation methods. Different‐order accuracy‐constrained optimisation methods are presented. These methods not only guarantee the rigorous convergence but also reduce inaccuracies at low wavenumbers. Accuracy‐constrained optimisation methods are applied to staggered‐grid elastic wave modelling.

A Validation Study of the Compressible Rayleigh–Taylor Instability Comparing the Ares and Miranda Codes

The compressible Rayleigh–Taylor (RT) instability is studied by performing a suite of large eddy simulations (LES) using the Miranda and Ares codes. A grid convergence study is carried out for each of these computational methods, and the convergence properties of integral mixing diagnostics and late-time spectra are established. A comparison between the methods is made using the data from the highest resolution simulations in order to validate the Ares hydro scheme. We find that the integral mixing measures, which capture the global properties of the RT instability, show good agreement between the two codes at this resolution. The late-time turbulent kinetic energy and mass fraction spectra roughly follow a Kolmogorov spectrum, and drop off as k approaches the Nyquist wave number of each simulation. The spectra from the highest resolution Miranda simulation follow a Kolmogorov spectrum for longer than the corresponding spectra from the Ares simulation, and have a more abrupt drop off at high wave numbers. The growth rate is determined to be between around 0.03 and 0.05 at late times; however, it has not fully converged by the end of the simulation. Finally, we study the transition from direct numerical simulation (DNS) to LES. The highest resolution simulations become LES at around t/τ ≃ 1.5. To have a fully resolved DNS through the end of our simulations, the grid spacing must be 3.6 (3.1) times finer than our highest resolution mesh when using Miranda (Ares).

Sensitivity of near-inertial internal waves to spatial interpolations of wind stress in ocean generation circulation models

The oceanic near-inertial internal waves (NIWs) have been extensively studied using ocean general circulation models (OGCMs). Currently most OGCMs use the bilinear or bicubic interpolation to interpolate wind stress onto models’ surface grids. In this study, we examine the influences of bilinear and bicubic interpolations on the wavenumber and frequency spectra of wind stress and on the simulated NIWs in the ocean. It is demonstrated that both the bilinear and bicubic interpolations are equivalent to spatial low-pass filters with the former leading to more significant loss of wind stress variance at high wavenumbers. When coarse (e.g., 2°) wind stress is used to force OGCMs, the bilinear and bicubic interpolations significantly damp the wavenumber spectrum of wind stress at mesoscales, leading to decreased near-inertial wind stress variance. Using the bilinear (bicubic) interpolation could weaken the near-inertial wind work by ∼43% (22%) in the subtropical region (10°N–30°N) and by ∼16% (4%) at the midlatitudes (30°N–50°N).We propose a new interpolation method, i.e., the bi-sinc-function interpolation, which is able to retain all the wind stress variance within the Nyquist wavenumber. Compared to the bilinear and bicubic interpolations, the bi-sinc-function interpolation improves the simulations of NIWs and should be incorporated into OGCMs especially when coarse wind stress is used.

Gravity gradient and magnetic terrain effects for airborne applications — A practical fast Fourier transform technique

We have implemented a practical fast Fourier transform technique for fast and approximate calculation of terrain effects for airborne measurement of the gravity gradient tensor and the total magnetic field. The calculations proceed in two steps. Starting from a digital terrain model (DTM), we first calculate the fields on a plane surface lying above the highest point of the terrain in the selected area. This calculation can be made arbitrarily accurate by including a sufficiently large number of terms in Parker’s well-known Fourier transform technique. The second step involves a downward continuation of the fields to a draped surface describing the positions of the airborne measurements. The inherent instability of downward continuation through the level of the highest terrain is compensated for by low-pass filtering the calculated fields on the plane surface prior to downward continuation. We use a Gaussian filter with cutoff wavenumbers well below the Nyquist wavenumber corresponding to a wavelength equal to the distance between flight lines. Tests on synthetic data as well as on real data from a DTM from northern Sweden demonstrated that the method works well and provides a low-pass-filtered version of the true terrain effect.

A new kind of sensor array for measuring spatial coherence of surface pressure on a car's side window

Abstract The mechanical behavior of a car's side window and the resulting acoustic radiation into the cabin is mainly affected by the spatial coherence of the surface pressure exciting the glass plate. The surface pressure is a superposition of hydrodynamic and acoustic pressure whose levels differ by 2 or 3 orders of magnitude. To gain information about the coherence characteristics of the surface pressure and to separate its hydrodynamic and acoustic components, a measurement of high spatial resolution is needed. For that reason a novel pressure transducer array with a minimum distance between two adjacent measurement points of only 2 mm was developed. The pressure transducers of the array are arranged sparsely on a grid while all possible distances between the spots on the grid are covered. Due to this minimization of distance redundancy, the amount of microphones could be reduced from 1849 to 92 representing a virtual array of 43×43 measurement positions. A Nyquist wavenumber of 250 1 / m and a resolution of 11.9 1 / m using a sensor area of only 52×52 mm2 were achieved. Because of its small dimensions, this array allows for measurements at various test areas on the side window, which is a major improvement compared to former investigations. For the measurements conventional MEMS microphones are applied. It is shown that the used microphones are suitable for the requisite, even if operating in saturation. Hence, the existence of acoustic loads on the side window and the position-dependent spatial coherence of the surface pressure can be studied. Measurements using the sensor array are carried out in the anechoic wind tunnel to validate the methodology. Results of the separation between hydrodynamic and acoustic pressure as well as the identification of coherence properties are presented.

Seismic conditioning and attenuation of high-angle coherent noise in a mixed carbonate and siliciclastic setting, Campos Basin, offshore Brazil: A case study

We demonstrate the use of two industry-standard, noise-attenuating algorithms, [Formula: see text] and [Formula: see text] filtering, to attenuate coherent noise, improve event continuity, and obtain more reliable interpretation within the Pampo field, Campos Basin, offshore Brazil. We analyze 3D seismic data for noise type and associated maximum dip, and then determine optimal attenuating parameters required by the algorithms. In the crossline direction, coherent noise with dips as high as 16 to 18 ms/trace in the data masks real reflected signals. In the inline direction, the noise exhibited an approximately zero dip. The filtering process eliminated noise trains and improved event continuity, as well as obtained better well-to-seismic tie. The output wavenumber associated with noise trains is less than the Nyquist wavenumber; therefore, we conclude that the noise is not related to spatial aliasing, but to other undetermined sources. Though the noise occurs within the entire data volume, it is most severe in the vicinity of the carbonate platform. By eliminating linear noise that masquerades as geologic structures, we obtain better imaging of the Albian succession and the younger Cretaceous turbidite sequences, as well as the top of Aptian salt. The result is a better interpretation.

Numerical Method for Eulerian Vlasov Simulation Based on Multi-Moment Scheme

A new scheme referred to as the multi-moment (MM) scheme is explored to develop a more reliable Vlasov code from the viewpoint of numerical properties. The MM scheme is based on the Eulerian approach, where spatial derivatives are evaluated by interpolation functions locally constructed by not only grid values but also 0th-, 1st-, and 2nd-order moment values between grids, which largely increases numerical accuracy and resolution. Through the Fourier analyses and benchmark tests of one-dimensional (1D) and 2D transport simulations, it is found that the MM scheme exhibits significantly smaller numerical dissipation and dispersion even near the Nyquist wave-number, and as a result, the MM scheme decreases the numerical cost. The MM scheme is also applied to a 1D Vlasov-Poisson simulation and we find that the scheme captures finer scale structure in velocity space compared to the conservative form of interpolated differential operator (IDO-CF) scheme, while also maintaining good energy conservation.

Towards quantitative control on discreteness error in the non-linear regime of cosmologicalN-body simulations

The effects of discreteness arising from the use of the N-body method on the accuracy of simulations of cosmological structure formation are not currently well understood. After a discussion of how the relevant discretisation parameters introduced should be extrapolated to recover the Vlasov-Poisson limit, we study numerically, and with analytical methods we have developed recently, the central issue of how finite particle density affects the precision of results. In particular we focus on the power spectrum at wavenumbers around and above the Nyquist wavenumber, in simulations in which the force resolution is taken smaller than the initial interparticle spacing. Using simulations of identical theoretical initial conditions sampled on four different pre-initial configurations (three different Bravais lattices, and a glass) we obtain a {\it lower bound} on the real discreteness error. With the guidance of our analytical results, we establish with confidence that the measured dispersion is not contaminated either by finite box size effects or by subtle numerical effects. Our results show notably that, at wavenumbers {\it below} the Nyquist wavenumber, the dispersion increases monotonically in time throughout the simulation, while the same is true above the Nyquist wavenumber once non-linearity sets in. For normalizations typical of cosmological simulations, we find lower bounds on errors at the Nyquist wavenumber of order of a percent, and larger above this scale. The only way this error may be reduced below these levels at these scales, and indeed convergence to the physical limit firmly established, is by extrapolation, at fixed values of the other relevant parameters, to the regime in which the mean comoving interparticle distance becomes less than the force smoothing scale.

Spatial antialias filtering in the slowness-frequency domain

A rigorous, explicit spatial antialias filter is designed and applied to spatially coarsely sampled seismic data by removing all energy above the first Nyquist wavenumber, and aliased energy that is folded back across the Nyquist, in the horizontal slowness-frequency domain. The spatial filtering in the slowness-frequency domain is explicit, free from any event linearity assumption, and does not require any interpolation. The spatially aliased energy is dispersive, and present at small and large slownesses. Comparison of the output data after antialias spatial filtering, with output data after conventional antialias frequency filtering, shows that the filter removes the spatially aliased frequencies selectively at each slowness; antialias low-pass frequency filtering under- or overcorrects for spatial aliasing at all slownesses. A seismic gather can be spatially dealiased only at the expense of wavelet spectral changes; dealiasing and preservation of amplitude variations with offset are not simultaneously possible.

On the use of multicomponent streamer recordings for reconstruction of pressure wavefields in the crossline direction

Three-component measurements of particle motion would bring significant benefits to towed-marine seismic data if processed in conjunction with the pressure data. We show that particle velocity measurements can increase the effective Nyquist wavenumber by a factor of two or three, depending on how they are used. A true multicomponent streamer would enable accurate data reconstruction in the crossline direction with cable separations for which pressure-only data would be irrecoverably aliased. We also show that conventional workflows aimed at reducing these aliasing effects, such as moveout correction applied before interpolation, are compatible with multicomponent measurements. Some benefits of velocity measurements for deghosting data are well known. We outline how the new measurements might be used to address some long-standing deghosting challenges of particular interest. Specifically, we propose methods for recovering de-ghosted data between streamers and for 3D deghosting of seismic data at the streamer locations.