Wave Fields In Inhomogeneous Media Research Articles

Reconstruction of a 2D seismic wavefield by seismic gradiometry

We reconstructed a 2D seismic wavefield and obtained its propagation properties by using the seismic gradiometry method together with dense observations of the Hi-net seismograph network in Japan. The seismic gradiometry method estimates the wave amplitude and its spatial derivative coefficients at any location from a discrete station record by using a Taylor series approximation. From the spatial derivatives in horizontal directions, the properties of a propagating wave packet, including the arrival direction, slowness, geometrical spreading, and radiation pattern can be obtained. In addition, by using spatial derivatives together with free-surface boundary conditions, the 2D vector elastic wavefield can be decomposed into divergence and rotation components. First, as a feasibility test, we performed an analysis with a synthetic seismogram dataset computed by a numerical simulation for a realistic 3D medium and the actual Hi-net station layout. We confirmed that the wave amplitude and its spatial derivatives were very well-reproduced for period bands longer than 25 s. Applications to a real large earthquake showed that the amplitude and phase of the wavefield were well reconstructed, along with slowness vector. The slowness of the reconstructed wavefield showed a clear contrast between body and surface waves and regional non-great-circle-path wave propagation, possibly owing to scattering. Slowness vectors together with divergence and rotation decomposition are expected to be useful for determining constituents of observed wavefields in inhomogeneous media.

Differential form and numerical implementation of Biot’s poroelasticity equations with squirt dissipation

The squirt-flow wave attenuation mechanism is implemented in Biot’s theory of poroelasticity in the form of differential equations. All the stiffnesses involved in the stress-strain relation become complex and frequency dependent, which can exactly be expressed in terms of kernels based on the Zener mechanical model. In the time domain, this approach implies time convolutions, which are circumvented by introducing memory variables. The differential equations are consistent with Gassmann’s and Mavko-Jizba equations at low and high frequencies, respectively. All the coefficients in the poro-viscoelastic differential equations have a clear physical meaning and can be obtained or estimated from independent measurements. The key additional parameters are the dry-rock bulk modulus at a confining pressure where all the compliant pores are closed, i.e., a hypothetical rock without the soft porosity, the grain-contact aspect ratio and the compliant porosity. We recasted the wave equation in the particle-velocity/stress formulation and solved it by using a time-splitting technique and the Fourier pseudospectral method to compute the spatial derivatives. The algorithm can be used to obtain synthetic wave fields in inhomogeneous media.

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

Simulations of acoustic wavefields in inhomogeneous media are always performed on finite numerical domains. If contrasts actually extend over the domain boundaries of the numerical volume, unwanted, non-physical reflections from the boundaries will occur. One technique to suppress these reflections is to attenuate them in a locally reflectionless absorbing boundary layer enclosing the spatial computational domain, a perfectly matched layer (PML). This technique is commonly applied in time-domain simulation methods like finite element methods or finite-difference time-domain, but has not been applied to the integral equation method. In this paper, a PML formulation for the three-dimensional frequency-domain integral-equation-based acoustic scattering problem is derived. Three-dimensional acoustic scattering configurations are used to test the PML formulation. The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs themselves are virtually reflectionless. In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions.

Simulation of the vector wave field of a low-frequency sound source in the ocean: Some important features

An approach to the simulation of low frequency vector wave fields in stratified media (mainly in the ocean) is considered. The approach is characterized by an improved stability with respect to dividing the medium into many layers of arbitrary thickness. The model for the sound field of a point source is based on an integral representation of two-dimensional, cylindrically symmetric vector wave fields in inhomogeneous media, so that the contributions of all types of waves are included automatically. The model medium is subdivided into N horizontally homogeneous layers for which 4(N−1) equations are formulated to satisfy the boundary conditions between adjacent layers. The method of the generalized Schmidt matrix is used to obtain the coefficients of the equations; these coefficients are substituted into the expressions (of the Fourier-Bessel integral type) for the local parameters of the field. The latter are calculated according to the numerical procedure, and the results are used to model the distributions of the acoustic pressure and the horizontal and vertical components of the particle velocity in liquid and elastic media. The instability of the calculation procedure may result in a disagreement between the model and the exact solution. However, the disagreement is shown to occur mainly in models containing excessively thick layers. A way for improving the stability of the numerical model is suggested. The simulation results are compared with the exact analytical solution for the simplest example and with the results obtained according to the commonly used generalized matrix procedure (the benchmark problem). The examples of the practical application of the model for investigating more complex seismoacoustic wave fields in the ocean are presented.

Symmetries and modes of wave fields in inhomogeneous media

In this paper the 2D Helmholtz equation with space-dependent dielectric permittivity and/or magnetic permeability is considered. The structure of the determining equations for the symmetry generator is analysed. It allows one to find many nontrivial spatial profiles of dielectric functions for which one can still separate variables in both the Helmholtz equation and corresponding Hamilton-Jacobi equation. For the case of the Dirichlet problem of the half-plane in an inhomogeneous medium the diffractive modes of the field are characterized. Several specific examples are provided.

The parameter perturbation method on elastic wave equation in inhomogeneous medium

In this paper, the medium parameters of the elastic wave equation in inhomogeneous medium are rewritten by introducing the referential variables and the perturbational variables, and the wave equation whose sources are the medium parameter perturbational term in homogeneous medium is obtained. By using the Green function theory, the integral equation of the perturbational parameters is obtained. Then the displacement field in homogeneous medium is considered the result of the first iteration, and the displacement field is solved by this integral equation. When the perturbations of medium parameters are about 50 percent, this method can solve the displacement field effectively. From the analysis of the numerical results, the characteristics of wave field in inhomogeneous medium are obtained. The results conform with the local principles of wave function in inhomogeneous medium.

Gaussian beam synthetic seismograms

This presentation is a tutorial on the Gaussian beam method used for the asymptotic synthesis of seismic and acoustic wave fields in inhomogeneous media. The method is based on the superposition of beam solutions, each of which is an approximate solution of the wave equation along particular rays. Smoothness conditions on the medium are required for the approximate propagation of beam solutions. Within a smoothly varying medium, various choices of the beam parameters can be used. Specifying broad planar beams at the source would result in a plane‐wave decomposition of the visible spectrum. The standard ray method would result by specifying narrow planar beams at the receiver. Another choice is to specify the minimum integral beam width along each ray. There are several advantages of the Gaussian beam method, including finite amplitudes at caustics, smoothing of endpoint errors, and the reduction of amplitude variability resulting from model parameterization. Several numerical examples will be given in 2‐D and 3‐D, and comparisons will be shown with other numerical methods.

Paraxial ray approximations in the computation of seismic wavefields in inhomogeneous media

Summary. Several important applications of the paraxial ray approximation (PRA) to numerical modelling of high-frequency seismic body wavefields are discussed. The PRA can be used to evaluate the displacement vector not only directly on the ray, as in the standard ray method. but also approximately in the vicinity of this ray. The PRA also offers simple ways of approximate evaluation of paraxial rays, situated in the vicinity of the central ray, and of two-point ray tracing. A very important application of the PRA consists in a simple, fast and effective Computation of body-wave synthetic seismograms in general, 3-D, laterally inhomogeneous, layered structures. Examples of synthetic seismograms for 3-D structures, computed using the PRA, are presented.

The two‐dimensional Gaussian beam synthetic method: Testing and application

The Gaussian beam method of Červený et al. (1982) is an asymptotic method for the computation of wave fields in inhomogeneous media. The method consists of tracing rays and then solving the wave equation in “ray‐centered coordinates.” The parabolic approximation is applied to find the asymptotic local solution in the neighborhod of each ray. The approximate global solution for a given source is then constructed by a superposition of Gaussian beams along nearby rays. The Gaussian beam method is tested in a two‐dimensional inhomogeneous medium using two approaches. One is the application of the reciprocal theorem for Green's functions in an arbitrarily heterogeneous medium. The discrepancy between synthetic seismograms for reciprocal cases is considered as a measure of the error. The other approach is to apply Gaussian beam synthesis to cases for which solutions are known by other approximate methods. This includes the soft basin problem that has been studied by finite difference, finite element, discrete wavenumber, and glorified optics. We found that the results of these tests were in general satisfactory. We have used the Gaussian beam method for two applications. First, the method is used to study volcanic earthquakes at Mount Saint Helens. The observed large differences in amplitude and arrival time between a station inside the crater and stations on the flanks can be explained by the combined effects of an anomalous velocity structure and a shallow focal depth. The method is also applied to scattering of teleseismic P waves by a lithosphere with randomly fluctuating velocities.

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.

Computation of wave fields in inhomogeneous media -- Gaussian beam approach

An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.