qSH Waves Research Articles

Approximate reflection and transmission coefficients at an interface between two contrasting viscoelastic anisotropic half-spaces

SUMMARY We present and compare two new methods to calculate the reflection and transmission coefficients of qP, qSV and qSH waves at a flat interface between two contrasting viscoelastic anisotropic half-spaces. The first method is a new version of the real ray tracing method that extends the traditional real ray tracing method to tackle the triplications of the qSV wave. This new version is valid for computing the reflection and transmission coefficients of the qP and qSV waves for all pre- and post-critical angles of incidence, which cannot be handled by the previous approach. The second method employs the real slowness direction and Snell's law to provide approximate reflection and transmission coefficients for such wave modes. This approximate method has attractive advantages which are not only the avoidance of the searching process inherent in the real ray tracing method but also offers competitive economical solutions to the first method. Many viscoelastic VTI models are utilized to validate the two methods. The numerical results show that (1) the ray angles, ray velocities, ray Q-factors, and the magnitudes of the reflection and transmission coefficients obtained by the two methods closely agree with each other, except in the vicinity of the critical angles and (2) the phase angles of the reflected and transmitted waves exhibit complexities due to the existence of some reverse phenomena between the approximate and the real ray tracing solutions. However, such phase discrepancies do not affect the magnitudes of the reflection and transmission coefficients. Moreover, the energy ratios of the different wave modes satisfy the energy conservation criterion at the interface.

Traveltime Calculations for qP, qSV, and qSH Waves in Two‐Dimensional Tilted Transversely Isotropic Media

AbstractThis paper presents a fast sweeping method (FSM) to calculate the first‐arrival traveltimes of the qP, qSV, and qSH waves in two‐dimensional (2D) transversely isotropic media, whose symmetry axis may have an arbitrary orientation (tilted transverse isotropy [TTI]). The method discretizes the anisotropic eikonal equation with finite difference approximations on a rectangular mesh and solves the discretized system iteratively with the Gauss‐Seidel iterations along alternating sweeping orderings. At each mesh point, a highly nonlinear equation is solved to update the numerical solution until its convergence. For solving the nonlinear equation, an interval that contains the solutions is first determined and partitioned into few subintervals such that each subinterval contains one solution; then, the false position method is applied on these subintervals to compute the solutions; after that, among all possible solutions for the discretized equation, a causality condition is imposed, and the minimum solution satisfying the causality condition is chosen to update the solution. For problems with a point‐source condition, the FSM is extended for solving the anisotropic eikonal equation after a factorization technique is applied to resolve the source singularities, which yields clean first‐order accuracy. When dealing with the triplication of the qSV wave, solutions corresponding to the minimal group velocity are chosen such that continuous solutions are computed. The accuracy, efficiency, and capability of the proposed method are demonstrated with numerical experiments.

Hybrid Fast Sweeping Methods for Anisotropic Eikonal Equation in Two-Dimensional Tilted Transversely Isotropic Media

We present hybrid fast sweeping methods for computing first-arrival traveltime of the qP, qSV and qSH waves in two-dimensional tilted transversely isotropic media, based on solving the anisotropic eikonal equation. A factorization approach is applied to resolve the source singularity near the point source, which leads to a factored anisotropic eikonal equation whose solutions can be computed with high accuracy. The proposed methods solve the factored equation in a neighborhood of the point source with the size of the neighborhood independent of the mesh, and solve the original equation outside the neighborhood. The methods enjoy all the appealing features, such as efficiency, accuracy and convergence, of the usual fast sweeping method. Furthermore, the “super-convergence” property of the first-order fast sweeping method, i.e., both its numerical solution and gradient are first-order accurate, allows us to design a second-order fast sweeping method based on a linear discontinuous Galerkin formulation. As a post-processing procedure of the first-order method, the second-order method reduces the local degrees of freedom from three to one in the linear discontinuous Galerkin formulation, which implies a simple local updating formula, hence an efficient second-order scheme. Numerical experiments are presented to demonstrate the proposed methods.

Simultaneous inversion for velocity model and microseismic sources in layered anisotropic media

Microseismic monitoring is widely applied in fracturing operations, reservoir delineation and water front monitoring. Microseismic inversion generally uses an initial velocity model based on well logs and seismic data from perforation shots. This model is generally not suitable to locate microseismic sources. To reduce errors, we have developed a method that simultaneously updates the velocity model and locates events in space and time. 1-D layered anisotropy with a transversely-isotropic (TI) symmetry is assumed, where the symmetry axis can be arbitrarily oriented. The arrival times and ray paths of the qP, qSV and qSH waves are calculated with a modified multistage shortest path algorithm. Combined with the conjugate gradient method, a damped, minimum-norm, least-squares and constrained problem is solved. The numerical examples show that the proposed algorithm can be used to invert the anisotropic velocity model (elastic moduli and interface depth for each layer) and locate the microseismic sources and their onset times, simultaneously. It is shown that the algorithm is not very sensitive to random noise, which may be contained in arrival times and model anisotropy heterogeneities.

Two-Dimensional Plane Wave Reflection and Transmission in a Layered Highly Anisotropic Media under Initial Stress

ABSTRACT This theoretical work addressed an issue of the reflection and transmission of plane wave incident in a layered triclinic solids. Generalization of Snell’s law is applied and suitable boundary conditions are imposed to obtain the closed-form expressions for reflection and transmission coefficients. The reflection and transmission coefficients are computed numerically by Cramer’s method. Numerical results found to be strongly influenced by the initial stress, angle of incidence on the modulus of reflection and transmission coefficients. The slowness surfaces of qP, qSV, and qSH waves are also estimated in initially stressed triclinic medium.

2D seismic reflection tomography in strongly anisotropic media

Seismic traveltime tomography is an effective method to reconstruct underground anisotropic parameters. Currently, most anisotropic tomographic methods were developed under the assumption of weak anisotropy. The tomographic method proposed here can be implemented for imaging subsurface targets in strongly anisotropic media with a known tilted symmetry axis, since the adopted ray tracing method is suitable for anisotropic media with arbitrary degree. There are three kinds of reflection waves (qP, qSV and qSH waves) that were separately used to invert the blocky abnormal body model. The reflection traveltime tomographiy is developed here because a surface observation system is the most economical and practical way compared with crosswell and VSP. The numerical examples show that the traveltimes of qP reflection wave have inverted parameters successfully. Traveltimes of qSV reflection wave have inverted parameters successfully, with the exception of the since it is less sensitive than other parameters. Traveltimes of qSH reflection wave also have inverted parameters successfully. In addition, we find that the velocity sensitivity functions (derivatives of phase velocity with respect to elastic moduli parameters) and raypath illuminating angles have a great influence on the qualities of tomograms according to the inversion of theoretical models. Finally, the numerical examples confirm that the reflection traveltime tomography can be applied to invert strongly anisotropic models.

Reflection and refraction at the interface between distinct generally anisotropic half spaces for three-dimensional plane quasi-P waves

In this paper, reflection and refraction phenomena are studied for a three-dimensional plane quasi-P wave incident at the interface between distinct triclinic half spaces. A method has been developed to find the analytical expressions of all the phase velocities of reflected and transmitted quasi-P (qP), quasi-SV (qSV), and quasi-SH (qSH) waves. Closed form expressions for the amplitude ratios of reflection and transmission coefficients of qP, qSV, and qSH waves are also obtained. These expressions are computed numerically for a given set of data and the velocity distributions and variation of reflection and transmission coefficients with polar and azimuthal angle of incidence are shown graphically using two-dimensional single and double axes graphs. Surface plots of velocity variations of different reflected and transmitted waves are drawn to analyze the combined effect of incident polar and azimuthal angle on velocity distribution. Numerical results presented indicate that anisotropy effects on the reflection and transmission coefficients are significantly different in the three-dimensional case compared to the two-dimensional case. Certain published work has been derived as special cases to illustrate the present technique. Software in MATLAB has been designed to analyze the reflection/transmission phenomena with present geometry in simpler elastic media, such as monoclinic, transversely isotropic, isotropic, etc.

Reflection and refraction of three dimensional plane quasi-P waves at a corrugated surface between distinct triclinic elastic half spaces

The present work deals with the reflection and transmission of three-dimensional plane quasi-P (qP) waves at a corrugated surface between distinct triclinic elastic half spaces. A method has been developed to find the analytical expressions of all the three phase velocities of qP, quasi-SV (qSV) and quasi-SH (qSH) waves in three dimensions. Closed form expressions in three-dimensions for the amplitude ratios of reflection and transmission coefficients of qP, qSV and qSH waves are obtained. Graphs for reflection and transmission coefficients are drawn for different polar and azimuthal angles. These reflection and transmission coefficients are computed numerically and shown graphically for fixed values of relevant parameters. Graphical user interface software in MATLAB has been designed to generalize the result of the study, by allowing one to choose arbitrary suitable values for relevant parameters. It is found that reflection and transmission coefficients are significantly different in three dimensional case compared to the two dimensional case.

Reflection and refraction of plane quasi-P waves at a corrugated interface between distinct triclinic elastic half spaces

The reflection and refraction pattern of elastic waves at a corrugated interface between two triclinic half-spaces is discussed. The incident wave is taken to be the cause of the interface disturbance and the reflected and refracted waves are effects. This leads to the causality requirement that the reflected and refracted waves must propagate away from the interface. Closed form expressions of reflection and transmission coefficients are derived using Rayleigh’s method of approximation. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The analytical expressions of all the three phase velocities of qP, qSV and qSH waves have been derived. The variation of reflection and refraction coefficients with the angle of incidence and also with the corrugation parameter is shown. In this paper we have developed Graphical User Interface (GUI) Software in MATLAB which shows the variation of reflection and refraction coefficients with respect to incident angle and corrugation parameter. This software can be generalized to show the variation of reflection and refraction coefficients. Numerical computations are performed for a scientific model and the results obtained are shown graphically.

Reflection for three-dimensional plane waves in triclinic crystalline medium

The propagation of three-dimensional plane waves at a traction free boundary of a half-space composed of triclinic crystalline material is discussed. A method has been developed to find the analytical expressions of all the three phase velocities of quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) in three dimensions. Closed form expressions in three dimensions for the amplitude ratios of reflection coefficients of qP, qSV and qSH waves in a triclinic medium are obtained. These expressions are used for numerically studying the variation of the reflection coefficients with the angle of incidence. The graphs are drawn for different polar angle and azimuth. Numerical results presented indicate that the anisotropy affect the reflection coefficients significantly in the three dimensional case compared to the two-dimensional case.

Wave Reflection in Triclinic Crystalline Medium

In this paper, the reflection of a plane wave at a traction free boundary of a half -space composed of triclinic crystalline material is considered. It is shown that an incident plane wave generates three plane waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) waves governed by the propagation condition involving the acoustic tensor. A simple procedure is presented for the calculation of all the three phase velocities of these waves. It is demonstrated that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. A procedure is established for the calculation of the amplitude vector in terms of the phase velocity, the propagation vector, and the stiffness coefficients of the medium. Closed form solutions are obtained for the reflection coefficients of qP, qSV and qSH waves. Using the parameters of Vosges sandstone exhibiting triclinic symmetry, the graphical representations of the reflection coefficients due to an incident qP wave are given. It is observed that, in triclinic medium, the reflection coefficients are significantly different from those in an isotropic medium.

Quasi-Rayleigh Waves in Transversely Isotropic Half-Space with Inclined Axis of Symmetry

A method for determination of characteristics of quasi-Rayleigh (qR) wave in a transversely isotropic homogeneous half-space with inclined axis of symmetry is outlined. The solution is obtained as a superposition of qP, qSV and qSH waves, and surface wave velocity is determined from the boundary conditions at the free surface and at infinity, as in case of Rayleigh wave in an isotropic half-space. Though the theory is simple enough, a numerical procedure for calculation of surface wave velocity presents some difficulties. The difficulty is caused by necessity to calculate complex roots of a non-linear equation, which in turn contains functions determined as roots of non-linear equations with complex coefficients. Numerical analysis shows that roots of the equation corresponding to the boundary conditions do not exist in the whole domain of azimuths and inclinations of the symmetry axis. The domain of existence of qR wave depends on the ratio of the elastic parameters: for some strongly anisotropic models the wave cannot exist at all. For some angles of inclination qR-wave velocities deviate from those calculated on the basis of the perturbation method valid for weak anisotropy, though they have the same tendency of variation with azimuth. The phase of qR wave varies with depth unlike Rayleigh wave in an isotropic half-space. Unlike Rayleigh wave in an isotropic half-space, qR wave has three components - vertical, radial and transverse. Particle motion in horizontal plane is elliptic. Direction of the major axis of the ellipsis coincides with the direction of propagation only in azimuths 0° (180°) and 90° (270°).

Wave reflection and refraction in triclinic crystalline media

In this paper, the reflection and refraction of a plane wave at an interface between two half-spaces composed of triclinic crystalline material is considered. It is shown that due to incidence of plane wave three types of waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH), will be generated governed by the propagation condition involving the acoustic tensor. A simple procedure has been presented for the calculation of all the three phase velocities of the quasi waves. It has been established that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. Relations are established between directions of motion and propagation, respectively. The expressions for reflection and refraction coefficients of qP, qSV and qSH waves are obtained. Numerical results of reflection and refraction coefficients are presented for different types of anisotropic media and for different types of incident waves. Graphical representations have been made for incident qP waves, and for incident qSV and qSH waves numerical data are presented in tables.

The complete far–field asymptotic description of a point source acting on a transversely isotropic half–space

The elastic wavefield generated by a point source of tractions acting on the surface of a transversely isotropic half–space is studied. The symmetry axis of the solid is oriented arbitrarily with respect to the surface of the half–space. First, the integral representation of the time–harmonic Green9s tensor is given. Then the complete far–field asymptotic approximation of a quasi–longitudinal (qP) and two quasi–shear (qSH and qSV) waves is derived. The qP wave is described by the leading term of the ray series, since there is only one arrival of this wave. The qSH wave is treated similarly everywhere apart from the so–called kissing–point boundary layer, where the qSH and qSV wavefronts are tangentially close to each other. A special asymptotic formula is obtained for this case. The qSV sheet of the wave surface is allowed to have conical points and cuspidal edges. Thus, the far–field approximation of the qSV wave involves ray–asymptotic expressions while inside the geometrical regions (where either one or three qSV arrivals exist), or else boundary–layer asymptotics inside conical–point, cuspidal–edge and kissing–point boundary layers. At the end of the paper we present numerical results of the simulation of pulse propagation. A good agreement between the asymptotic and direct numerical codes is achieved but the former is orders of magnitude faster.

Analytic waveforms for a line source in a transversely isotropic medium

In order to clarify anisotropy effects on body wave propagation, exact transient solutions radiated from horizontal line sources of stress discontinuities in a transversely isotropic medium with a vertical axis of symmetry are formulated by a modified Cagniard method which gives the inverse Laplace transform in an ingenious manner. The advantage of the exact transient solutions is the clear onset of elastic displacements at the time of body wave arrivals that are defined by corresponding group velocities. The exact transient solutions display three principal effects of anisotropy: (1) noncircular propagation of wave fronts; (2) deviation of polarizations of particle motions of qP, qSV, and qSH waves from those in an isotropic medium; and (3) separation of group velocity from phase velocity, depending on the relative orientation of anisotropy and wave propagation directions.