Wave Processes in Inhomogeneous Nonlinear-Dissipative Media

Several problems for nonlinear localized wave processes are analysed. Wave solutions of the kink type, concentrated in the vicinity of some curve l, are considered. Both the solution and the curve l are unknown. It is shown that the determination of l may be realized without knowledge of the solution. For one class of problems, a variational principle for finding l, similar to the Fermat principle, is obtained. Types of waves are found which exist due to the inhomogeneity of the medium or due to the initial front curvature only.     

Asymptotic Ray Theory for Seismic Surface Waves in Laterally Inhomogeneous Media

A recurrence procedure is outlined for constructing asymptotic series for surface wave field in a half-space with weak lateral heterogeneity. Both horizontal variations of the elastic parameters and of the wave field are assumed small on the distances comparable with the wavelength. This is equivalent to the condition that the frequency is large. The Surface Wave Asymptotic Ray Theory (SWART) is an analog of the asymptotic ray theory (ART) for body waves. However the case of surface waves presents additional difficulty: the rate of amplitude variation is different in vertical and horizontal directions. In vertical direction it is proportional to the large parameter . To overcome this difficulty the transformation equalizing vertical and horizontal coordinated is suggested, Z = z. In the coordinates x,y,Z the wave field is represented as an asymptotic series in inverse powers of . The amplitudes of successive terms of the series are determined from a recurrent system of equations. Attention is paid to similarity and difference of the procedures for constructing the ray series in SWART and ART. Applications of SWART to interpretation of seismological observations are discussed.     

Hybrid Method for Love Wave Dispersion in Vertically Inhomogeneous Media

—Love wave dispersion in a vertically inhomogeneous multilayered medium is studied by a combination of analytical and numerical methods for arbitrary variation of rigidity and density with depth. The problem is reduced to a boundary value problem for a differential equation and solved numerically. The method compares favourably with other methods in use. Simple particular cases are considered and interesting results are exhibited graphically.     

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, evanescent or inhomogeneous waves can be studied and modelled using the notion of complex rays. The wavefront or eikonal equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates. In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays. Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismograms. For anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.     

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, “evanescent” or “inhomogeneous” waves can be studied and modelled using the notion of “complex rays”. The wavefront or “eikonal” equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates.In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays.Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismogramsFor anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.     

P-wave Tomography in Inhomogeneous Orthorhombic Media

— A P-wave tomographic method for 3-D complex media (3-D distribution of elastic parameters and curved interfaces) with orthorhombic symmetry is presented in this paper. The technique uses an iterative linear approach to the nonlinear travel-time inversion problem. The hypothesis of orthorhombic anisotropy and 3-D inhomogeneity increases the set of parameters describing the model dramatically compared to the isotropic case. Assuming a Factorized Anisotropic Inhomogeneous (FAI) medium and weak anisotropy, we solve the forward problem by a perturbation approach. We use a finite element approach in which the FAI medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, analytical expressions for rays and travel times, valid to first-order, are given for P waves in orthorhombic inhomogeneous media. More complex media can be modeled by introducing interfaces separating FAI media with different elastic properties. Simple formulae are given for the Fréchet derivatives of the travel time with respect to the elastic parameters and the interface parameters. In the weak anisotropy hypothesis the P-wave travel times are sensitive only to a subset of the orthorhombic parameters: the six P-wave elastic parameters and the three Euler angles defining the orientation of the mirror planes of symmetry. The P-wave travel times are inverted by minimizing in terms of least-squares the misfit between the observed and calculated travel times. The solution is approached using a Singular Value Decomposition (SVD). The stability of the inversion is ensured by making use of suitable a priori information and/or by applying regularization. The technique is applied to two synthetic data sets, simulating simple Vertical Seismic Profile (VSP) experiments. The examples demonstrate the necessity of good 3-D ray coverage when considering complex anisotropic symmetry.     

Hermite-Gaussian Beams in Inhomogeneous Elastic Media

Hermite-Gaussian beams in a 3D elastic inhomogeneous medium are obtained as high-frequency asymptotic solutions of equations of motion concentrated in a vicinity of P- and S-wave rays. Equations of motion are transformed into the parabolic equation (Schroedinger equation) in this case. Explicit expressions for a complete set of linearly independent solutions of a parabolic equation are derived. The method of creation and anihilation operators known from quantum mechanics are used in the derivation.     

Inhomogeneous Harmonic Plane Waves in Viscoelastic Anisotropic Media

In viscoelastic media, the slowness vector p of plane waves is complex-valued, p = P + iA. The real-valued vectors P and A are usually called the propagation and the attenuation vector, repectively. For P and A nonparallel, the plane wave is called inhomogeneousThree basic approaches to the determination of the slowness vector of an inhomogeneous plane wave propagating in a homogeneous viscoelastic anisotropic medium are discussed. They differ in the specification of the mathematical form of the slowness vector p. We speak of directional specification, componental specification and mixed specification of the slowness vector. Individual specifications lead to the eigenvalue problems for 3 × 3 or 6 × 6 complex-valued matrices.In the directional specification of the slowness vector, the real-valued unit vectors N and M in the direction of P and A are assumed to be known. This has been the most common specification of the slowness vector used in the seismological literature. In the componental specification, the real-valued unit vectors N and M are not known in advance. Instead, the complex-valued vactorial component p of slowness vector p into an arbitrary plane with unit normal n is assumed to be known. Finally, the mixed specification is a special case of the componental specification with p purely imaginary. In the mixed specification, plane represents the plane of constant phase, so that N = ±n. Consequently, unit vector N is known, similarly as in the directional specification. Instead of unit vector M, however, the vectorial component d of the attenuation vector in the plane of constant phase is known.The simplest, most straightforward and transparent algorithms to determine the phase velocities and slowness vectors of inhomogeneous plane waves propagating in viscoelastic anisotropic media are obtained, if the mixed specification of the slowness vector is used. These algorithms are based on the solution of a conventional eigenvalue problem for 6 × 6 complex-valued matrices. The derived equations are quite general and universal. They can be used both for homogeneous and inhomogeneous plane waves, propagating in elastic or viscoelastic, isotropic or anisotropic media. Contrary to the mixed specififcation, the directional specification can hardly be used to determine the slowness vector of inhomogeneous plane waves propagating in viscoelastic anisotropic media. Although the procedure is based on 3 × 3 complex-valued matrices, it yields a cumbersome system of two coupled equations.     

Computation of Ray Amplitudes in Inhomogeneous Media with Curved

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1, 1993 to January 3, 2001 (cycles 11–305) was used for investigating the long-term variations of the geoidal geopotential W ₀ and the geopotential scale factor R ₀ = GM÷W ₀ (GM is the adopted geocentric gravitational constant). The mean values over the whole period covered are W ₀ = (62 636 856.161 ± 0.002) m²s^-2, R ₀ = (6 363 672.5448 ± 0.0002) m. The actual accuracy is limited by the altimeter calibration error (2–3 cm) and it is conservatively estimated to be about ± 0.5 m²s^-2 (± 5 cm). The differences between the yearly mean sea surface (MSS) levels came out as follows: 1993–1994: –(1.2 ± 0.7) mm, 1994–1995: (0.5 ± 0.7) mm, 1995–1996: (0.5 ± 0.7) mm, 1996–1997: (0.1 ± 0.7) mm, 1997–1998: –(0.5 ± 0.7) mm, 1998–1999: (0.0 ± 0.7) mm and 1999–2000: (0.6 ± 0.7) mm. The corresponding rate of change in the MSS level (or R ₀) during the whole period of 1993–2000 is (0.02 ± 0.07) mm÷y. The value W ₀ was found to be quite stable, it depends only on the adopted GM, and the volume enclosed by surface W = W ₀. W ₀ can also uniquely define the reference (geoidal) surface that is required for a number of applications, including World Height System and General Relativity in precise time keeping and time definitions, that is why W ₀ is considered to be suitable for adoption as a primary astrogeodetic parameter. Furthermore, W ₀ provides a scale parameter for the Earth that is independent of the tidal reference system. After adopting a value for W ₀, the semi-major axis a of the Earth's general ellipsoid can easily be derived. However, an a priori condition should be posed first. Two conditions have been examined, namely an ellipsoid with the corresponding geopotential which fits best W ₀ in the least squares sense and an ellipsoid which has the global geopotential average equal to W ₀. It is demonstrated that both a-values are practically equal to the value obtained by the Pizzetti's theory of the level ellipsoid: a = (6 378 136.7 ± 0.05) m.     

A Wave-Front Curvature Approach to Computing Ray Amplitudes in Inhomogeneous Media with Curved Interfaces

A system of three ordinary non-linear first order differential equations is proposed for the computation of the geometrical spreading of the wave front of a seismic body wave in a three-dimensional medium. The variables of the system are the parameters which provide a second order approximation of the wave front.     

SH Wave Propagation in Viscoelastic Media

When comparing solutions for the propagation of SH waves in plane parallel layered elastic and viscoelastic (anelastic) media, one of the first things that becomes apparent is that in the elastic case the location of the saddle points required to obtain a high frequency approximation are located on the real p axis. This is true of the branch points also. In a viscoelastic medium this is not typical. The saddle point corresponding to an arrival lies in the first quadrant of the complex p-plane as do the branch points. Additionally, in the elastic case the saddle point and branch points lie on a straight line drawn through the origin (the positive real axis in the complex p-plane), while in the viscoelastic case this is generally not the case and the saddle point and branch points lie in such a manner as to indicate the degree of their complex values.In this paper simple SH reflected and transmitted particle displacement arrivals due to a point torque source at the surface in a viscoelastic medium composed of a layer over a half space will be considered. The path of steepest descent defining the saddle point in the first quadrant will be parameterized in terms of a real variable and the high frequency solutions and intermediate analytic results obtained will be used to formulate more specific constraints and observations regarding saddle point location relative to branch point locations in the complex p-plane.As saddle point determination for an arrival is, in general, the solution of a non-linear equation in two unknowns (the real and imaginary parts of the complex saddle point p ₀), which must be solved numerically, the use of analytical methods for investigating this problem type is somewhat limited.Numerical experimentation using well documented solution methods, such as Newton's method, was undertaken and some observations were made. Although fairly basic, they did provide for the design of algorithms for the computation of synthetic traces that displayed more efficient convergence and accuracy than those previously employed. This was the primary motivation for this work and the results from the SH problem may be used with minimal modifications to address the more complicated subject of coupled P-SV wave propagation in viscoelastic media.Another reason for revisiting a problem that has received some attention in the literature was to approach it in a fairly comprehensive manner so that a number of specific observations may be made regarding the location of the saddle point in the complex p-plane and to incorporate these into computer software. These have been found to result in more efficient algorithms for the SH wave propagation and a significant enhancement of the comparable software in the P-SV problem.     

基于波动方程的微地震震源位置、震源时间及VTI介质各向异性参数联合反演

对于非常规油气开发,水力压裂监控的效果取决于对微地震事件的分析、解释.准确的微地震震源位置是关乎施工成败的重要因素.微地震震源位置的准确性与多个参数相关,其不仅依赖于微地震事件的激发时间,同时也依赖于储层介质参数信息,因此进行微地震震源位置、震源时间、储层介质参数的联合反演尤为重要.页岩气储层通常表现出较强的各向异性,...     

应力诱导各向异性介质中地震波的传播

主要讨论了应力变化如何影响各向异性介质中波速度的问题。推导了一般各向异性介质在初始应力下的Christoffel方程,得到介质中3种波的相速度和初始应力的关系表达式;通过实验数据验证了单轴应力能够诱导各向异性,当施加单轴应力时,速度在沿应力的方向增加最大,在垂直应力的方向增加最小,实验结果与理论推导一致;用Christoffel方程的数值解模拟在3种对称情况下的弹性各向异性介质中初始应力对波速度的影响。数值结果表明:初始应力对各向异性介质中波传播速度的影响,随着各向异性强度的增加而增大,而且速度越慢,影响越大。     

含方形凹陷半无限非均匀介质波动问题FDM模拟

采用规则网格有限差分方法对二维平面弹性波动方程进行差分离散,得到相应的弹性波动方程的有限差分方程,再将弹性波动方程的差分格式与吸收边界、自由边界的离散形式结合形成弹性波动方程有限差分方程解决问题的主体,将其应用于含方形凹陷半无限非均匀介质的模型中进行数值模拟,得到此离散化模型中不同时刻不同节点的位移值。针对具体算例,运用上述方法结合科学计算软件MATLAB和结果后处理软件DIFEM ISOLINE PLOTER得到不同时刻的水平方向位移等值线图与接收器测量点处的合成位移记录,讨论非均匀介质、吸收边界、方形凹陷等对波动特性的影响。     

波动方程多次波压制技术的进展

多次波问题是海洋地震勘探中最突出的问题之一，如何有效地压制多次波是数据处理中的一个关键问题，多次波压制技术分为两大类：基于有效波和多次波之间差异的滤波方法和基于波动方程的多次波预测减去法。针对有代表性的波动方程方法－－基于反馈模型的迭代反演多次波压制技术做了较深入的讨论，目前在解决多次波问题时，基于波动理论的方法应得到重视。     

用四面体单元模型研究三维不均匀介质中的地震波走时

利用四面体单元模型讨论了地震波在三维横向不均匀介质中的传播路径和走时的计算方法 ,给出震源所在四面体的确定方法。以华北地区大地构造模型为例 ,计算了直达波走时曲线 ,其结果较为满意     

SH波在表面多层介质中传播的精确模拟

针对地震横波在地表低速层内的振幅放大效应问题，提出了一种模拟SH波在地表层状介质中传播的递推算法，并用它模拟了新西兰Alfredton盆地A10场址的SH波地震动响应特性。这个方法适用于具线性吸收性质的粘弹性介质。由于方法不受介质层厚薄制约，层厚可以无限薄化，实践上可以用许多薄层逼近的办法来模拟纵向上任意变化的连续介质。通过求取不同频率不同波数平面简谐波解并按实际问题的加权迭加可求解具特定波形和传播方向组合的任意SH波场。此方法在计算上具有解析解特有的精确性，稳定性和方便性，特别适用于模拟薄层介层，次波长现象及需要进行大量而又精确模拟计算的情形。     

瞬态弹性波散射的变换域积分方程法

针对瞬态弹性波散射的问题,从弹性动力学问题的积分表示定理出发,采用Laplace变换的方法,得到了变换域内均质体位移场的积分方程表示;在此基础上推导了适合瞬态弹性波对异质体散射求解的变换域位移场积分方程。     

频率域波动方程的双参数识别方法

利用反射波地震记录恢复波动方程的两个系数以研究地球内部构造，是当前应用地球物理研究的热点之一，本文在波动方程反演方法的理论基础上，导出了频率域识别介质的密度波速两个重要参数的新算法，从而把当前的波动方程单参数反演双参数反演方法，通过理论模型及实际测井资料的反演试算对比，均藜得良好效果，在算法中还省去了Ｇｒｅｅｎ函数的繁杂计算，节省了计算量，增加实用性。