多层弹性半空间中的地震波（二）

一、引言 地震面波的频散性质、地震波辐射的方向性等特性已经广泛地用于地壳和上地幔结构以及震源机制的研究中,并且取得了许多有用的成果.研究地震波如何从震源辐射出来、如何在实际介质中传播和衰减的这一问题,对于利用地震波确定地壳和上地幔结构以及震源的参数,是很有必要的.关于这一问题的研究,已经作过许多工作。已往的工作中,为了分析方便,往往采用简单的地壳-上地幔模型或简单的震源模型,或两者都相     

Source function of stress waves of a spherical explosive source

Summary The source functions of the stress wave patterns at an elastic source of these waves are analysed. The comparison of the properties of the functions with the stress wave records obtained earlier showed that their parameters do not satisfy, to a greater or lesser extent, the stress wave patterns in the neighbourhood of explosive sources. For this reason a new source function (1) was defined, which fully approximates the observed stress wave patterns in gravel sandy soil. The coefficientsP ₀, , , and were experimentally determined as functions of the distance from the source, its size and the radius of the elastic source in the medium considered. The properties of source function (1) are demonstrated on an example.Paper presented at the XIIIth General Assembly of the European Seismological Commission, Braov (Romania), 28 August to 6 September, 1972.     

Reflection and refraction at an imperfectly bonded interface between poroelastic solid and cracked elastic solid

Porous solid is in contact with a cracked elastic solid at a plane interface between them. For the presence of vertically aligned microcracks, the elastic solid behaves transversely isotropic to wave propagation. The coefficients of elastic anisotropy depend on the crack density and crack porosity in the medium. A loose bonding is considered between the two solids so that a limiting case could be the welded contact. At the plane interface, the imperfection in welded bonding is represented by tangential slipping and, hence, results in the dissipation of a part of strain energy. Three types of waves propagate in an isotropic fluid-saturated porous medium, which are considered for incidence at the interface. Incidence of a wave results in three reflected waves and two refracted waves. Partition of incident energy among the reflected and refracted waves is studied for each incidence, varying from normal to grazing directions. Numerical example calculates the energy shares of reflected and refracted waves at the plane interface between water-saturated sandstone and basalt. These energy shares are computed and analyzed for different values of crack parameters as well as loose bonding parameter.     

六分量弹性波场数值模拟与分析

地震波传播过程中,质点的振动不仅包括三个独立的平移部分,还包括三个独立的旋转部分.本文基于一阶速度-应力弹性波方程,采用分裂完全匹配层(SPML)的吸收边界条件,推导了时间导数二阶精度和空间导数高阶精度的交错网格有限差分格式的弹性波速度与应力各分量计算公式,模拟了各向同性介质中均匀模型和层状模型下的六分量波场,并对二维各向同性层状模型下的三个分量地震记录做高分辨率线性拉东变换得到各自的频散能谱.数值模拟分析结果表明:(1)旋转分量的能量要比平动分量弱的多;(2)在平动分量上,面波能量强,频率低,反射P波能量较强,反射S波能量稍弱;在旋转分量上,反射P波能量很弱,S波能量强;(3)与平动分量相比,旋转分量的频散能谱效果更好,能看到基阶和完整的高阶面波,即旋转分量能反映更多的地下介质信息.     

Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.     

On the propagation of certain types of surface waves in a non-homogeneous elastic layer

Summary The propagation of a certain type of surface waves in a non-homogeneous elastic layer of finite thickness lying in welded contact with a semi-infinite homogeneous elastic medium has been investigated in this paper. The surface wave is characterised by the fact that the dilatation and the vertical displacement component are both zero. It has been proved that such disturbance can propagate only when the phase velocity of the wave is greater than the shear wave velocity and the group velocity is inversely proportional to the phase velocity.     

Hydro-elastic wave proliferation over an impermeable seabed with bottom deformation

ABSTRACT

A hydro-elastic frame has been considered to investigate the proliferation of waves over small base deformation on an infinitely extended flexible seabed. The flexible base surface is assumed as a thin elastic plate of very small thickness and it depends on the Euler–Bernoulli beam equation. For any particular frequency, there are two different modes of time-harmonic propagating wave exists rather than one mode of propagating wave along the positive horizontal direction. The waves with smaller wavenumber spread along the free-surface of the sea (say, free-surface mode) and the waves with higher wavenumber spread along the flexible base surface (say, flexural mode). A simplified perturbation approach is utilised to bring down the entire equations which govern the original boundary value problem (bvp) to a less complex bvp for the first-order velocity potential function. The first-order potential function along with the first-order reflection and transmission coefficients for both modes are calculated by a procedure based upon Fourier transform approach. A shape of sinusoidal swells flexible base surface is taken as an example to approve the scientific results. It is observed that when the train of normal incident propagating wave spreads over base distortion because of either the free-surface unsettling influence or the flexural wave movement in the sea, the reflected and transmitted energy are always feasible to be exchanged from one particular wave mode to another wave mode. Furthermore, we notice that the realistic changes in the flexural rigidity behaviour on the flexible base surface of the sea have a significant effect on the problem of water wave proliferation over small base deformation. Moreover, the energy conservation equation is derived with the help of the Green's integral theorem. The results for the values of reflection and transmission coefficients obtained for both the free-surface unsettling influence as well as flexural wave movement in the fluid are found to satisfy the energy conservation equation almost accurately.     

Spatial dispersion of surface waves at the free surface of a general anisotropic elastic medium

A new technique relates the wave velocity of the surface waves in anisotropic elastic medium to its elastic constants. Anisotropic propagation of surface waves is studied in a half-space occupied by a general anisotropic elastic solid. The phase velocity expressions of quasi-waves, in three-dimensional space, are used to derive the secular equation of surface waves. The complex secular equation is resolved, analytically, into real and imaginary parts and is then solved, numerically, for phase velocity along a given phase direction on the surface. The complete procedure is thus analogous to the one used for conventional Rayleigh waves in isotropic medium. A non-linear equation relates the ray direction of the surface waves to its phase direction on the (plane) surface of the medium. The analytical differentiation of secular equation yields the directional derivative of phase velocity. This derivative is used to calculate the wave velocity of surface waves. Spatial variations of phase velocity, wave velocity and ray direction over the free plane surface are plotted for the numerical models of crustal rocks with orthorhombic, monoclinic and triclinic anisotropies.     

Peculiarities of Rayleigh wave displacement in a transversely isotropic half space

We consider a transversely isotropic medium with vertical axis of symmetry (VTI). Rayleigh wave displacement components in a homogeneous VTI medium contain exp(±kr_jz), where z is the vertical coordinate, k is the wave number, and j?=?1, 2; r_j may be considered as depth-decay factor. In a VTI medium, r_j can be a real or imaginary as in an isotropic medium, or it can be a complex depending on the elastic parameters of the VTI medium; if complex, r₁ and r₂ are complex conjugates. In a homogeneous VTI half space, Rayleigh wave displacement is significantly modified with a phase shift when r_j changes from real to complex with variation of VTI parameters; at the transition, the displacement becomes zero when r₁?=?r₂. In a liquid layer over a VTI half space, Rayleigh waves are dispersive. Here, also Rayleigh wave displacement significantly modified with a phase shift when r_j changes from real to complex with a decrease of period. At very low period, phase velocity of Rayleigh waves becomes less than P-wave velocity in the liquid layer giving rise to Scholte waves (interface waves). The amplitudes of Scholte waves with a VTI half space are found to be significantly larger than those with an isotropic half space.     

Radiation patterns of point sources situated close to structural interfaces and to the earth's surface

The seismic wave field is considerably influenced by local structures close to the source and to the receiver. This applies to sources and receivers situated close to localized inhomogeneities, to structural interfaces, to the earth's surface, etc. In this paper we concentrate our attention mainly to the ray-theoretical radiation patterns of point sources situated close to the structural interfaces and to the earth's surface. In numerical modeling of high-frequency seismic wave fields by the ray method, the interaction of the source with the earth's surface has not usually been taken into account.The proposed procedure of the computation of the radiation patterns of point sources situated directly on structural interfaces and on the earth's surface is based on the zero-order approximation of the ray method, assuming that the length of the ray between the source and the receiver is long. The derived equations are extended to point sources located close to structural interface, to the earth's surface and to thin transition layers using the hybrid ray-reflectivity method, seeervený (1989). The thin layer need not be homogeneous; it may include an arbitrary inner layering (transition layers, laminas, etc.) The only requirement is for the layer to be thin. Roughly speaking, we require its thickness to be less than one quarter of the prevailing wavelength. The hybrid ray-reflectivity method describes well even certain non-ray effects (tunneling.S ^* waves, etc.). Explicit analytical expressions for radiation patterns for all above listed point sources are found. These expression have a local character and may be easily implemented into computer codes designed for the routine computation of ray amplitudes and synthetic ray seismograms in 2-D and 3-D, laterally varying isotropic layered and block structures by the ray method.Numerical examples of radiation patterns ofP andS waves of point sources situated close to the earth's surface and to a thin low-velocity surface layer are presented and discussed. The explosive point source (center of dilatation) and the vertical and horizontal single force point sources are considered. It has been ascertained that the radiation patterns of point sources depend drastically on the depth of the source below the surface even if the depths vary within one quarter of the prevailing wavelength.     

Simulation of full-waveform log in saturated cracked formations using Hudson's approach

We study the propagation of elastic waves that are generated in a fluid‐filled borehole surrounded by a cracked transversely isotropic medium. In the model studied the anisotropy and borehole axes coincide. To obtain the effective elastic moduli of a cracked medium we have applied Hudson's theory that enables the determination of the overall properties as a function of the crack orientation in relation to the symmetry axis of the anisotropic medium. This theory takes into account the hydrodynamic mechanism of the elastic‐wave attenuation caused by fluid filtration from the cracks into a porous matrix. We have simulated the full waveforms generated by an impulse source of finite length placed on the borehole axis. The kinematic and dynamic parameters of the compressional, shear and Stoneley waves as functions of the matrix permeability, crack orientation and porosity were studied. The modelling results demonstrated the influence of the crack‐system parameters (orientation and porosity) on the velocities and amplitudes of all wave types. The horizontally orientated cracks result in maximal decrease of the elastic‐wave parameters (velocities and amplitudes). Based on the fact that the shear‐ and Stoneley‐wave velocities in a transversely isotropic medium are determined by different shear moduli, we demonstrate the feasibility of the acoustic log to identify formations with close to horizontal crack orientations.     

Hemispheric simulation of the Asian summer monsoon

A three-level, -plane, filtered model is used to simulate the Northern Hemisphere summer monsoon. A time-averaged initial state, devoid of sub-planetary scale waves, is integrated through 30 days on a 5° latitude-longitude grid. Day 25 through day 30 integrations are then repeated on a 2.5° grid. The planetary-scale waves are forced by time-independent, spatially varying diabatic heating. Energy is extracted via internal and surface frictional processes. Orography is excluded to simplify synoptic-scale energy sources.During integration the model energy first increases, but stabilizes near day 10. Subsequent flow patterns closely resemble the hemisphere summer monsoon. Climatological features remain quasi-stationary. At 200 mb high pressure dominates the land area, large-scale troughs are found over the Atlantic and Pacific Oceans, the easterly jet forms south of Asia, and subtropical jets develop in the westerlies. At 800 mb subtropical highs dominate the oceans and the monsoon trough develops over the Asian land mass. The planetary scales at all levels develop a realistic cellular structure from the passage of transient synoptic-scale features, e.g., a baroclinic cyclone track develops near 55°N and westward propagating waves form in the easterlies.Barotropic redistribution of kinetic energy is examined over a low-latitude zonal strip using a Fourier wave-space. In contrast to higher latitudes where the zonal flow and both longer and shorter waves are fed by barotropic energy redistribution from the baroclinically unstable wavelengths, the low-latitude waves have a planetary-scale kinetic energy source. Wave numbers 1 and 2 maintain both the zonal flow and all shorter scales via barotropic transfers. Transient and standing wave processes are examined individually and in combination.Wave energy accumulates at wave numbers 7 and 8 at 200 mb and at wave number 11 in the lower troposphere. The 800-mb waves are thermally indirect and in the mean they give energy to the zonal flow. These characteristics agree with atmospheric observation. The energy source for these waves is the three wave barotropic transfer. The implications of examining barotropic processes in a Fourier wave-space, vice the more common approach of separating the flow into a mean plus a deviation are discussed.     

The discrete wave number formulation of boundary integral equations and boundary element methods: A review with applications to the simulation of seismic wave propagation in complex geological structures

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.     

Coda wave analysis for distinguishing attenuation due to isotropic scattering from attenuation due to absorption

When the quality factorQ is taken into account in attenuation studies, it is necessary to know the relative losses of wave energy due to scattering and to anelastic absorption. The coda is the most important phenomenon now known which is related to elastic scattering of seismic waves. Utilizing coda, this study presents relationships which give theQ factors of the medium around the recording station and discriminate between attenuations arising from elastic scattering (under the assumption of isotropic scattering) and those arising from anelastic absorption. This work proposes a technique for separately determining the attenuation due to isotropic scattering and that due to absorption from the observed envelope of coda waves.     

多层弹性半空间中的地震波（一）

为了了解地震震源和地球介质的性质,很有必要对地震波的辐射、传播和衰减问题作仔细的分析。作为一种近似,可以暂且忽略地球的曲率,把传播地震波的地球介质视为多层半空间。为简便起见,地震波的衰减问题另作考虑。这样,便需要研究多层、均匀、各向同性和完全弹性半空间中地震震源辐射的地震波传播问题。 用哈斯克尔（Haskell）矩阵法解多层介质中弹性波的传播问题是很方便的。如果     

The Interpretation of Gravity Changes and Crustal Deformation in Active Volcanic Areas

Simple models, like the well-known point source of dilation (Mogis source) in an elastic, homogeneous and isotropic half-space, are widely used to interpret geodetic and gravity data in active volcanic areas. This approach appears at odds with the real geology of volcanic regions, since the crust is not a homogeneous medium and magma chambers are not spheres. In this paper, we evaluate several more realistic source models that take into account the influence of self-gravitation effects, vertical discontinuities in the Earths density and elastic parameters, and non-spherical source geometries. Our results indicate that self-gravitation effects are second order over the distance and time scales normally associated with volcano monitoring. For an elastic model appropriate to Long Valley caldera, we find only minor differences between modeling the 1982–1999 caldera unrest using a point source in elastic, homogeneous half-spaces, or in elasto-gravitational, layered half-spaces. A simple experiment of matching deformation and gravity data from an ellipsoidal source using a spherical source shows that the standard approach of fitting a center of dilation to gravity and uplift data only, excluding the horizontal displacements, may yield estimates of the source parameters that are not reliable. The spherical source successfully fits the uplift and gravity changes, overestimating the depth and density of the intrusion, but is not able to fit the radial displacements.     

Reflection of waves from the boundary of a random elastic semi-infinite medium

In this paper the smooth perturbation technique is employed to investigate the problem of reflection of waves incident on the plane boundary of a semi-infinite elastic medium with randomly varying inhomogeneities. Amplitude ratios have been obtained for various types of incident and reflected waves. It has been shown that an incidentSH orSV type of wave gives rise to reflectedSH, P andSV waves, the main components beingSH andP, SV in the respective cases. The reflected amplitudes have been calculated depending upon the randomness of the medium to the square of the small quantity , where measures the deviation of the medium from homogeneity. An incidentP-type wave produces mainly aP component and also a weakSH component to the order of ². The reflected amplitudes obtainable for elastic media are also altered by terms of the same order. The direction of the reflected wave is influenced by randomness in some cases.     

Seismoelectric numerical simulation in 2D vertical transverse isotropic poroelastic medium

Seismoelectric coupling in an electric isotropic and elastic anisotropic medium is developed using a primary–secondary formulation. The anisotropy is of vertical transverse isotropic type and concerns only the poroelastic parameters. Based on our finite difference time domain algorithm, we solve the seismoelectric response to an explosive source. The seismic wavefields are computed as the primary field. The electric field is then obtained as a secondary field by solving the Poisson equation for the electric potential. To test our numerical algorithm, we compared our seismoelectric numerical results with analytical results obtained from Pride's equation. The comparison shows that the numerical solution gives a good approximation to the analytical solution. We then simulate the seismoelectric wavefields in different models. Simulated results show that four types of seismic waves are generated in anisotropic poroelastic medium. These are the fast and slow longitudinal waves and two separable transverse waves. All of these seismic waves generate coseismic electric fields in a homogenous anisotropic poroelastic medium. The tortuosity has an effect on the propagation of the slow longitudinal wave. The snapshot of the slow longitudinal wave has an oval shape when the tortuosity is anisotropic, whereas it has a circular shape when the tortuosity is isotropic. In terms of the Thomsen parameters, the radiation anisotropy of the fast longitudinal wave is more sensitive to the value of ε, while the radiation anisotropy of the transverse wave is more sensitive to the value of δ.     

On the generation of capillary-gravity waves due to two-dimensional sources

Summary With the aid of the generalized function method, a study is made of the linearized theory of transient development of capillary-gravity waves in an inviscid, incompressible and homogeneous liquid of finite and infinite depth due to an arbitrary oscillating source situated at a finite depth below the undisturbed free surface of the liquid. The initial value problem is solved by using Laplace-Fourier transforms combined with asymptotic methods. The asymptotic solution is found to consist of the steady state and the transient components which are independently modified by surface tension. The latter decays more rapidly as timet due to the presence of surface tension than in the case where surface tension is neglected. It is predicted that the principal effect of surface tension is to increase both the phase and group velocity of the waves and make the energy more readily available among the rapidly travelling progressive surface waves. In addition to the effects of surface tension on the physical properties of the wave motions, our method of solution provides an interesting illustration of the applicability of generalized functions in water wave phenomena.     

小波尺度域含气储层地震波衰减特征

黏弹性衰减因子Q的可靠估计可通过Q反褶积来提高地震资料的分辨率并有助于振幅分析. 本文从小波理论出发,结合地震波在黏弹性介质中的传播方程,推导出小波尺度域地震波能量衰减公式. 能量衰减公式具有下列性质：（1）Q值越大,能量衰减得越慢;Q值越小,能量衰减越严重;（2）尺度越小,信号中保留的能量越少;（3）对于脉冲源来说在理想的无衰减介质(即Q趋近于∞)中传播时,信号在不同尺度内的能量相同. 利用尺度能量公式,可从反射地震资料中直接估计品质因子Q（即衰减因子）,也可以提取不同尺度的能量衰减剖面作为储层描述的属性参数,用来进行岩性识别和指示气藏,与经典的谱比法相比,避免了谱比法所面临的双时窗问题以及进行谱估计的窗选择问题. 理论模型试验表明了本文方法的正确性和有效性.