Efficient Simulation of Elastic Waves Propagation in Anisotropic Media

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Primary attention is given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the Lebedev scheme approximates a system of equations, which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts.” Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of the initial system on a sufficiently smooth solution. The results obtained and the knowledge of the complete set of characteristics are important for constructing reflectionless boundary conditions during approximation of point sources, etc.

Dynamic stiffness of three-dimensional anisotropic multi-layered media based on the continued-fraction method

Wavefront construction methods allow for efficient simulation of seismic wave propagation in anisotropic media using ray theory. This approach models an individual wavefront as a mesh that is adaptively refined as the wave travels through an earth model. The algorithm, implemented here for quasi-compressional waves, provides an effective tool for modeling and analyzing a large vertical seismic-profile data set from the Gulf of Mexico. Seismic velocities in this region generally increase linearly with depth, but simulations show that an isotropic vertical-gradient model cannot accurately reproduce traveltimes of first arrivals. However, a simple velocity-analysis inversion based on the wavefront-construction method allows a straightforward estimation of improved isotropic and transversely isotropic models that significantly reduce the misfit between computed traveltimes and the observations. The goal of this analysis is to produce a simple model suitable for use as a starting point for migrations and for assessing the importance of anisotropy in the field site. In fact, the final anisotropic model is much more consistent with background geologic information than the best isotropic model, which also has a larger rms traveltime error. The results strongly support the presence of measurable anisotropy.

Since a good many of the papers published regularly in Radio Science deal with wave problems in magnetoplasmas, especially those of the ionospheric variety, the readership has constant access to information and techniques pertaining to electromagnetic wave propagation in anisotropic media. Nevertheless, it is appropriate at times to draw attention to work in a given subject area by a concentrated grouping of papers dealing with various facets. The present spedal issue on Electromagnetic Wave Propagation in Anisotropic Media should be regarded in this light.

Modified Christoffel equations are derived for three-dimensional wave propagation in a general anisotropic medium under initial stress. The three roots of a cubic equation define the phase velocities of three quasi-waves in the medium. Analytical expressions are used to calculate the directional derivatives of phase velocities. These derivatives are, further, used to calculate the group velocities and ray directions of the three quasi-waves in a pre-stressed anisotropic medium. Effect of initial stress on wave propagation is observed through the deviations in phase velocity, group velocity and ray direction for each of the quasi-waves. The variations of these deviations with the phase direction are plotted for a numerical model of general anisotropic medium with triclinic/ monoclinic/orthorhombic symmetry

Anisotropy is widely-used nowadays in seismic processing as it may characterize fractured reservoirs, pre-stressed state of formations etc. However, numerical simulation of seismic wave propagation in anisotropic media requires the use of advanced, computationally-intense algorithms. Meanwhile anisotropic formations take relatively small part, several percents, of the model. In this situation it is reasonable to develop a finite-difference algorithm coupling the anisotropy-oriented scheme with efficient standard staggered grid, or Virieux, scheme. So, that the computationally-intense algorithm is used only inside the anisotropic formations, while the Virieux scheme is used in the major isotropic part of the model. The coupled scheme may be almost as efficient as Virieux scheme but providing full detailed knowledge about wave propagation in anisotropic formations.

Seismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

Numerical simulation of wave propagation in anisotropic media

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common‐midpoint (CMP) spreads is usually well described by the normal‐moveout (NMO) velocity defined in the zero‐offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry.For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero‐offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix‐type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi‐axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions.Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero‐offset) ray. Remarkably, the expressions for geometrical spreading along the zero‐offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi‐azimuth, multi‐offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip‐moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces.The high accuracy of our NMO expressions is illustrated by comparison with ray‐traced reflection traveltimes in piecewise‐homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P‐wave moveout can be used to find the depth‐dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.

The mathematical description of phenomena related to wave propagation in anisotropic media is considerably more complicated than that for isotropic media. This complexity stems from many physical properties distinguishing the anisotropic from isotropic medium. Therefore, calculating the angles of reflection and transmission for a ray impining on a boundary between two anisotropic media become more complicated in mathematical implication. Snell law in isotropic medium has not been applicable in anisotropic medium, which must be modified. This article presents a new method of calculating the angles of reflection and transmission on a boundary between two anisotropic media, called as the modified Snell law.

This paper deals with radiation and low frequency scattering of electromagnetic waves in a general anisotropic homogeneous medium. The explicit exact form of the four spectral Green's dyads is obtained and asymptotic expressions for the radiation field due to arbitrary sources are derived, which show that the associated modes and the eigenmodes determine radiation along a singular optic axis. Meanwhile, the analytical solution of low frequency scattering by a small body immersed into an anisotropic medium is developed.

We propose a domain decomposition method based on a combination of the finite element and finite difference methods for numerical simulation of elastic wave propagation. It is not limited by model geometries and easy to be applied to seismic wave modelling with surface topography. It is also convenient to be implemented by parallel computational programs. The results of numerical experiments validate the accuracy of the algorithm.

Structure damage modelling for the simulation of elastic wave propagation is essential for simulation-based testing of structural health monitoring (SHM) systems. Reflection, transmission and other complex phenomena related to elastic wave interaction with structure defects need to be reproduced by the simulation method to replace physical testing by numerical one and aid the design process of SHM system. Authors review commonly used methods for the simulation of elastic wave propagation pointing out their advantages and disadvantages. Subsequently methods for crack and delamination modelling are presented. Proposed models are used for numerical testing of designed SHM system. Simulation results are compared with experimental ones, showing very good agreement. Finally, structure damage models are used for testing delamination localization algorithm.

Studying the anisotropy of seismic wave velocity can provide a theoretical basis for understanding the characteristics of seismic wave propagation in anisotropic media, such as forward modeling of seismic wavefield and simultaneous inversion of anisotropic parameters. It is an important work to calculate the group velocities of three kinds of seismic waves (qP, qS1, and qS2) and the analytical expressions of the partial derivatives of the group velocities with respect to 21 elastic parameters in general anisotropic media. Therefore, a new relationship between the phase and group velocity in the general anisotropic medium is introduced first, and then the analytical expressions of the group velocity of three kinds of waves are deduced. Then, the analytical formulas of partial derivatives of the group (or phase) velocity with respect to 21 elastic parameters are derived. Finally, the distribution of partial derivatives of group slowness with respect to 21 elastic parameters with varied ray angles is analyzed and discussed. The final analytical expressions of group velocity and its partial derivatives are relatively concise and can deal with the singular points and multivalued problems of qS waves. By drawing comparison to the numerical solution of the eigenvector method, the accuracy of the analytical formula of group velocity and its partial derivatives with respect to elastic parameters are verified. The numerical simulation results indicate that the algorithm can accurately obtain the partial derivatives of group slowness with respect to 21 elastic parameters and would provide a theoretical basis for seismic ray tracing and traveltime inversion in general anisotropic media.

A split-step finite-difference time-domain (FDTD) method is presented for 3-D Maxwell's equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. Stability analysis of the proposed split-step FDTD method using the Fourier method is presented. The eigenvalues of the Fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the split-step FDTD method in general anisotropic media. Numerical results are presented to further validate the accuracy and stability of the proposed split-step FDTD method in general anisotropic media.