Heterogeneous Wave Equation Research Articles

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.

Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations

A 2D wave equation with boundary damping of impedance type can be recast into an infinite-dimensional port-Hamiltonian system (pHs) with an appropriate feedback law, where the structure operator J is formally skew-symmetric. It is known that the underlying semigroup proves dissipative, even though no dissipation operator R is to be found in the pHs model. The Partitioned Finite Element Method (PFEM) introduced in Cardoso-Ribeiro et al. (2018), is structure-preserving and provides a natural way to discretize such systems. It gives rise to a non null symmetric matrix R. Moreover, since this matrix accounts for boundary damping, its rank is very low: only the basis functions at the boundary have an influence. Lastly, this matrix can be factorized out when considering the boundary condition as a feedback law for the pHs, involving the impedance parameter. Note that pHs – as open system – is used here as a tool to accurately discretize the wave equation with boundary damping as a closed system. In the worked-out numerical examples in 2D, the isotropic and homogeneous case is presented and the influence of the impedance is assessed; then, an anisotropic and heterogeneous wave equation with space-varying impedance at the boundary is investigated.

Continuous gradient and discretized layered designs for control of stress wave scattering

The scattering of stress waves in graded media is discussed in this paper using a conversion of the heterogeneous wave equation to a potential form. In this form, the scattering and inverse scattering methods of mathematical physics can be used with proper enforcement of the boundary conditions on the mechanical quantities, i.e. displacements and tractions. Two approaches for design of non-reflective media are analyzed, one for all frequencies with infinite thickness, and one with finite thickness for a target frequency range. In the latter approach, the transformed potential is set to zero, and the wave speed can have almost any desired profile, from which the profile of impedance is derived. The dispersive behavior of the finite thickness design is studied. Finally, an example is discussed in which a continuous non-reflective profile is discretized to a piece-wise constant profile and the effect of such unavoidable practical constraints on the scattering of the designed layer is shown to be minimal.

Algorithms for staggered‐grid computations for poroelastic, elastic, acoustic, and scalar wave equations

Heterogeneous wave equations are more complicated numerically than homogeneous wave equations, but are necessary for physical validity. A wide variety of numerical solutions of seismic wave equations is available, but most produce strong numerical artefacts and local instabilities where model parameters change rapidly. Accuracy and stability of heterogeneous equations is achieved through staggered‐grid formulations. A new pseudospectral staggered‐grid algorithm is developed for the poroelastic (Biot) equations. The algorithm may be reduced to handle the elastic and acoustic limits of the Biot equations. Comparisons of results from poroelastic, elastic, acoustic and scalar computations for a 2D model show that porous medium parameters may affect amplitudes significantly. The use of homogeneous wave equations for modelling of a heterogeneous medium, or of a centred rather than a staggered grid, or of simplified (e.g. acoustic) wave equations when elastic or poroelastic media are synthesized, may produce erroneous or ambiguous interpretations.

Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation

SUMMARY Artefacts and local instabilities are common in numerical solutions of seismic wave equations. Although many of these are already known, most are not well documented in an accessible form. One form of artefacts is a consequence of partial derivative operators that are non-local and are associated with the interaction between the propagating wavefield and the medium at discontinuities in material properties. When pseudo-spectral solutions are used, heterogeneous wave equations may be accurately solved in a wide dynamic range using even-based Fourier transforms on a staggered grid. Regardless of the implementation of a spatial differentiator, use of homogeneous wave equations or of standard (non-staggered) grids gives less accurate results. Synthetic examples include heterogeneous acoustic, elastic and poroelastic (Biot) equations.

On the construction of arbitrary order schemes for the many dimensional wave equation

We show that it is possible to construct arbitrary order stable schemes for the homogeneous and heterogeneous wave equation in any dimension. The construction is elementary and uses formal power series techniques. We shall also calculate exact stability limits in various cases, and apparently this limit depends only on the dimension of the space.

Book reviews

Summary. The paper presents a method of investigating seismic surface waves in (two-dimensional) layered media involving lateral heterogeneities. The method combines forward modelling by computation of synthetic seismograms for the disturbed layering and subsequent inversion of the seismogram spectra with respect to the wave modes. In terms of ‘L region’ and ‘R region', denoting the laterally homogeneous regions of the waves before and after propagation across the disturbed region, the method is described as follows. (1) Analytical definition of the wave in the L region. A time-dependent wavelet is assumed at some surface point, and the spatial displacement field corresponding to a free plane surface wave is computed from the eigen-functions of the waveguide. (2) Numerical propagation of the wave across the laterally heterogeneous region. The general heterogeneous wave equation is approximated by a finite difference time marching scheme, where the analytical wave motion enters as initial value. (3) Inversion of seismograms ‘observed’ at a number of locations in the R region. The seismogram Fourier spectra are modelled by a linear combination of modes, contaminated by random noise. Least-squares fitting between the modelled and the observed seismogram spectra yields estimates of the ‘mode participation factors'. The method is applied to Love waves of an elastic upper crustal model composed of a layer (shear velocity β=2.5 km s-1) over a homogeneous substratum (β=3.5 km s-1). Embedded in the surface layer, of 10 km thickness, is an idealized rectangular shaped sediment basin with lateral and vertical dimensions of 15 and 5 km, and a velocity of 0.6 or 0.4 of that of the layer. The initial pulse in the form of a Ricker wavelet is defined to represent a pure fundamental mode, the wavelengths of which vary between 60 and 6 km in the frequency interval 0.05-0.42 Hz ('analysis interval'), the wavelength of maximum spectral amplitude being 15 km. Observations at a distance of 60 km beyond the sediment basin yield the following results. (1) Propagation across the basin gives rise to significant scattering of the fundamental mode energy into the higher modes. The fraction of the Love wave energy transmitted is about 50 per cent for both models (slightly higher for the model of lower velocity contrast). The transmitted Love wave energy density exhibits different behaviour between the two models. For the model of lower velocity contrast, there is an increase with frequency from about 30 to 80 per cent; for the model of higher velocity contrast, there are variations between corresponding values, without a significant frequency trend. (2) Owing to the multimodal character of the Love wave transmitted, the phase velocity estimated from the horizontal phase gradient, differs from the fundamental mode velocity by amounts of up to about 10 per cent. Moreover, phase velocity as well as the seismogram spectral amplitudes are dependent upon the horizontal coordinate. This means that it is generally inadequate to interpret phase velocity (from phase differences) or attenuation (from amplitude ratios) in corresponding subsurface structures in terms of one single mode.