First-order Wave Equation Research Articles

A staggered time integrator for the linear acoustic wave equation using the Jacobi-Anger expansion

In this study, we introduce a staggered time integrator with the Fourier pseudospectral method to solve the first-order linear wave equation, which is accelerated by using the Jacobi-Anger expansion. The proposed method can reduce the computational cost by approximately half compared to the scheme whose temporal order of accuracy is extended by the Lax-Wendroff method under the equivalent modeling conditions, such as time step length, grid interval and maximum wave propagation speed. This is because the Jacobi-Anger expansion can effectively approximate the sinusoidal function, which in our case is the sine function in the wavenumber domain. Based on the wavenumber domain analysis of the proposed method, a strategy to optimally design the simulation parameters to maintain a preset level of accuracy is also introduced. According to the strategy, as the time step length increases, not only the computational cost of the simulation is reduced, but also the accuracy of the numerical solution is improved. Numerical simulations are also performed by using both homogeneous and heterogeneous models to validate the introduced strategy, which supports the practicality of the proposed method.

Computational multiscale methods for first-order wave equation using mixed CEM-GMsFEM

In this paper, we consider a pressure-velocity formulation of the heterogeneous wave equation and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible framework to construct crucial multiscale basis functions for approximating the pressure and velocity. These basis functions are constructed by solving a class of local auxiliary optimization problems over the eigenspaces that contain local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computational performance. The first-order convergence of the proposed method is proved and illustrated by several numerical tests.

Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

Generalized stiffness reduction method to remove the artificial edge-effects for seismic wave modelling in elastic anisotropic media

SUMMARYIn seismic wave modelling, the boundary reflections caused by the computational grid edges should be reduced to produce accurate simulation results. The perfectly matched layer (PML) method is one of the popular techniques to suppress such artificial reflections, because it can be easily applied to the first-order wave equation in many numerical methods. However, one issue of the PML method is that the stability condition might be violated in complex elastic anisotropic media. In these cases, the PML method will not attenuate the boundary reflections but rather introduce strong artefacts in the simulation results. To tackle this problem, we propose a generalized stiffness reduction method (GSRM) as a substitute for the PML method. We first derive the stability conditions of the PML method and analyse the suitable conditions for their application to time- and frequency-domain seismic wave modelling. Then, we develop a simple and effective numerical implementation of the GSRM to attenuate the boundary reflections and apply it to seismic wave modelling in elastic anisotropic media. We give some numerical experiments to demonstrate the feasibility and advantages of the GSRM compared to the PML method. Numerical examples show the GSRM is conceptually simpler, more computationally efficient and more straightforward in terms of numerical implementation than the PML method for seismic modelling using either first- or second-order time- and frequency-domain wave equations.

Modeling 3D elastic VTI wave propagation using an optimal k-space operator-based temporal high-accuracy staggered-grid finite-difference scheme

It is difficult to extend the present time-space-domain or temporal high-order finite-difference (FD) stencils to model anisotropic wave equations owing to anisotropy parameters and spatial derivatives are mutually coupled. Thus, high-order spatial and second-order temporal FDs are commonly used to discretize anisotropic wave equations. To improve temporal modeling accuracy, we develop an efficient temporal high-accuracy staggered-grid FD (SG-FD) scheme to solve the first-order elastic wave equations in 3D vertical transversely isotropic (VTI) media. Through combining the spatial dispersion relation of the original SG-FD stencil with the first-order k (wavenumber)-space operator, we construct a modified SG-FD dispersion relation and determine FD coefficients using least-squares (LS). We adopt the modified LS-based SG-FD scheme using k-space operator compensation to simulate 3D elastic VTI wave propagation. Dispersion analysis and numerical examples demonstrate that our optimal k-space operator-based SG-FD scheme can achieve high temporal accuracy without compromising spatial accuracy compared with the traditionally uncompensated Taylor-series expansion- and LS-based SG-FD methods. Moreover, the stability of our proposed FD scheme is superior to the conventional ones.

Application of Classical Coastal Trapped Wave Theory to High-Scattering Regions

AbstractSince the 1970s, analytical models of coastal trapped waves (CTWs) have been developed using a first-order wave equation in the long-wave limit. Formulations of this kind require assumptions of a straight coastline with similar shelf bathymetry. These assumptions prevent the models from capturing the scattering and backscattering behavior of propagating CTWs that encounter changing coastlines, bathymetry, or shelf width. CTW modes from two different analytical models, one of which includes friction and stratification, are compared with CTW observations of velocity and pressure from a study region near the Outer Banks off the North Carolina coast in the United States. The coastline in the study region is relatively straight locally but is bounded by an estuary to the north and shelf narrowing to the south, both of which induce scattering. The models suggest that the CTWs in this region are insensitive to changes in stratification, implying that observed seasonal differences in wave magnitude are due to seasonal wind forcing. Furthermore, friction is found to be important, particularly for mode-1 propagation, but higher-order modes are prevalent despite the importance of friction. There is very poor agreement between the observed and modeled free and forced CTWs because of scattering. This lack of agreement indicates that this is not a globally applicable theoretical formulation because many global coastlines violate the basic assumptions.

A rectangular staggered-grid finite-difference scheme with fourth-order temporal accuracy for pseudo-acoustic VTI modelling

ABSTRACTCurrent staggered-grid finite-difference (SFD) schemes for anisotropic wave equations can ensure only high-order spatial modelling accuracy, whereas the time discretisation accuracy is still of second-order. To improve temporal modelling accuracy, we develop a rectangular SFD scheme with fourth-order temporal accuracy to solve the first-order pseudo-acoustic wave equation in vertical transversely isotropic media. High-order temporal accuracy is achieved by applying the high-order temporal derivatives, which can be replaced with the high-order spatial derivatives to reduce the computational cost. The corresponding high-order finite-difference (FD) coefficients are generated by using Taylor-series expansion and least-squares. Dispersion and stability analyses indicate that our proposed method has higher accuracy and better stability than the conventional method. Several modelling examples confirm that our proposed scheme exhibits low temporal dispersion even for a large time step. Because of the high-order accuracy in time and space domains, our new SFD scheme allows for a larger time step and shorter operator length, which is more efficient.

Simply improved averaging for coupled oscillators and weakly nonlinear waves

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.

Second-order scalar wave field modeling with a first-order perfectly matched layer

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

Weakly nonlinear magnetic equatorial Kelvin waves in rotating spherical coordinates

ABSTRACTIn a recent paper, Marquez-Artavia, Jones and Tobias (referred to as MJT) studied wave propagation on a rotating sphere using the shallow water ideal magnetohydrodynamic (MHD) equations. These equations were linearised about a toroidal magnetic field (the Malkus field) which depends only on the latitude and is proportional to where θ is the co-latitude. These waves may play an important role in the Earth's outer core and the solar tachocline. The purpose of this paper is to extend MJT's solution for magnetic equatorial Kelvin waves to the case of weakly nonlinear waves. To do this, we use a formal asymptotic approximation when two main parameters are separately assumed to be large. The approximation recovers MJT's results in the linear case and extends those results to the weakly nonlinear case. The main result of this paper is that the zonal wave function is obtained from a first-order linear wave equation so that, unlike in the hydrodynamic case, these waves do not break or form shocks. We demonstrate a mechanism by which the presence of a magnetic field regularises the governing equations in the sense that a sufficiently large magnetic field can suppress the nonlinear terms in the Burgers equation of the hydrodynamic problem. We also find weak north–south currents, not present in the linear problem. The nonlinearities do not appear to significantly modify the linear results.

An elastic full-waveform inversion based on wave-mode separation

Multi-parameter elastic full-waveform inversion (EFWI) attempts to find high resolution model parameters that are able to match observed data exactly by minimising residuals between the observed and predicted data. However, the coupling of Vp and Vs, and the cross-talk artefacts between P- and S-wave modes increase non-uniqueness and ill-conditionedness. We propose a new EFWI method based on P- and S-wave mode separation to mitigate these problems. In this method, we derive the gradient formulas with respect to various wave modes using a P- and S-wave mode separated first-order velocity-stress wave equation, and use a step search method in subspace to calculate the corresponding step lengths. The algorithm, called wave-mode separation EFWI (SEFWI), appears to be helpful to weaken non-uniqueness and ill-conditionedness of conventional EFWI by decoupling multiple parameters. Numerical examples conducted with a synthetic dataset modelled on a simple model with anomalies reveal that SEFWI can reduce the cross-talk artefacts between P- and S-wave modes. Synthetic tests on the Marmousi2 model demonstrate that SEFWI yields better inversion results than conventional EFWI. Although the computational cost of SEFWI per iteration is 1.81 times as much as that of EFWI, the total computational cost is almost at the same level, because of its faster convergence rate.

Numerical methods for accurate description of ultrashort pulses in optical fibers

We consider a one-dimensional first-order nonlinear wave equation, the so-called forward Maxwell equation (FME), which applies to a few-cycle optical pulse propagating along a preferred direction in a nonlinear medium, e.g., ultrashort pulses in nonlinear fibers. The model is a good approximation to the standard second-order wave equation under assumption of weak nonlinearity and spatial homogeneity in the propagation direction. We compare FME to the commonly accepted generalized nonlinear Schrödinger equation, which quantifies the envelope of a quickly oscillating wave field based on the slowly varying envelope approximation. In our numerical example, we demonstrate that FME, in contrast to the envelope model, reveals new spectral lines when applied to few-cycle pulses. We analyze and compare pseudo-spectral numerical schemes employing symmetric splitting for both models. Finally, we adopt these schemes to a parallel computation and discuss scalability of the parallelization.

Some improvement strategies for the time-domain dual-sensor full waveform inversion

Full waveform inversion (FWI) is an effective method in retrieving high-resolution subsurface parameters from seismic data. In this paper, we review the dual-sensor FWI previously studied and present some improvement strategies. We first modify the low-pass Wiener filter to band-pass Wiener filter for time-domain FWI to reduce the overlapped regions. Secondly, we use spectral conjugate gradient method to update the models instead of the conventional conjugate gradient method to speed up the convergence and improve the inversion accuracy. At last, we further improve the efficient boundary storage method which is suitable for the first-order velocity-stress acoustic wave equation to reduce the storage burden. After that, we validate these new strategies for the modified Marmousi model, and the synthetic example shows the feasibility and robustness of these new strategies in terms of improving computational efficiency and alleviating large amounts of data storage.

Sedeonic Field Equations for Dyons

We discuss the theoretical description of dyons having simultaneously both electric and magnetic charges on the basis of space-time algebra of 16-component sedeons. We show that the generalized sedeonic equations for electromagnetic field of dyons can be reformulated in equivalent canonical form as the equations for redefined field potentials, field strengths and sources. The relations for energy and momentum as well as the relations for Lorentz invariants of dyonic electromagnetic field are derived. Additionally, we discuss the sedeonic second-order and first-order wave equations describing the quantum behavior of dyons in an external dyonic electromagnetic field.

Liouville term for neutrinos: flavor structure and wave interpretation

Neutrino production, absorption, transport, and flavor evolution in astrophysical environments is described by a kinetic equation Dϱ=−i[H,ϱ]+\U0001d49e[ϱ]. Its basic elements are generalized occupation numbers ϱ, matrices in flavor space, that depend on time t, space x, and momentum p. The commutator expression encodes flavor conversion in terms of a matrix H of oscillation frequencies, whereas \U0001d49e[ϱ] represents source and sink terms as well as collisions. The Liouville operator on the left hand side involves linear derivatives in t, x and p. The simplified expression D=∂t+p̂⋅∂x for ultra-relativistic neutrinos was recently questioned in that flavor-dependent velocities should appear instead of the unit vector p̂. Moreover, a new damping term was postulated as a result. We here derive the full flavor-dependent velocity structure of the Liouville term although it appears to cause only higher-order corrections. Moreover, we argue that on the scale of the neutrino oscillation length, the kinetic equation can be seen as a first-order wave equation.

Multi-scale full waveform inversion for areas with irregular surface topography in an auxiliary coordinate system

For land exploration areas with irregular surface topography, there are many challenges and problems for full waveform inversion (FWI); for example, which type of wave equation should be used to calculate high-accuracy seismic wavefields, how to deal with diffraction of irregular surface topography, what initial velocity model should be utilised to improve the inversion accuracy and how to enhance the computational efficiency of iterative FWI. Aiming at these difficulties, we first simulate the seismic waves with the first-order acoustic wave equation in an auxiliary coordinate system, which easily describes irregular surface topography. Then, we apply this wavefield simulation frame to FWI to improve inversion quality of near-surface regions with strong elevation and velocity variation. Furthermore, to enhance the robustness and computational efficiency, a time-domain multi-scale decomposition method based on the Wiener filter and an optimised encoding strategy are introduced to the proposed inversion frame, and are critical to promoting the practical application of our method. Typical numerical tests prove that the proposed method can obtain more accurate inversion results than the traditional time-domain FWI.

A comparison of the factorization approach to temporal and spatial propagation in the case of some acoustic waves

The evolution of acoustic waves can be evaluated in two ways: either as a temporal, or a spatial propagation. Propagating in space provides the considerable advantage of being able to handle dispersion and propagation across interfaces with remarkable efficiency; but propagating in time is more physical and gives correctly behaved reflections and scattering without effort. Which should be chosen in a given situation, and what compromises might have to be made? Here the natural behaviors of each choice of propagation are compared and contrasted for an ordinary second order wave equation, the time-dependent diffusion wave equation, an elastic rod wave equation, and the Stokes’/ van Wijngaarden’s equations, each case illuminating a characteristic feature of the technique. Either choice of propagation axis enables a partitioning the wave equation that gives rise to a directional factorization based on a natural ‘reference’ dispersion relation. The resulting exact coupled bidirectional equations then reduce to a single unidirectional first-order wave equation using a simple ‘slow evolution’ assumption that minimizes effect of subsequent approximations, while allowing a direct term-to-term comparison between exact and approximate theories.

Sedeonic equations of ideal fluid

In the present paper, we propose the generalized equations for an ideal fluid based on space-time algebra of sixteen-component sedeons. It is shown that the dynamics of isentropic fluid can be described by sedeonic first-order wave equation for fluid potentials. The key features of the proposed formalism are illustrated on the problem of the sound waves propagation. We consider the plane wave solution of linearized sedeonic wave equation and derive the second-order relations for the sound potential analogues to the Poynting theorem in electrodynamics. The generalization of proposed sedeonic equations for the description of viscous fluid is also discussed.

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

We introduce a space-time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer. Anal., 36, 1599-1635). Test and trial discrete functions are space-time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space-time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order $h$-convergence bounds.

High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.