The scattering of SH waves by a finite crack with a superposition-based diffraction technique

The problem of diffraction of cylindrical and plane horizontally polarized shear waves (SH waves) by a finite crack embedded in a plane bidimensional elastic full-space is revisited. Particularly, we construct an approximate solution by the addition of independent diffracted terms. In our method the derivation of the fundamental case of a semi-infinite crack obtained as a degenerate case of a generalized wedge is first considered. This result is then used as a building block to compute the diffraction of the main incident waves. The interaction between the opposite edges of the crack is later considered in terms of a series, one term at a time until a desired tolerance is reached. Moreover, we propose a procedure to determine the number of required interactions as a function of frequency. The solution derived with the superposition technique is shown to be effective at low and high frequencies and as shown by comparisons with a direct boundary element method software, highly accurate solutions are obtained after retaining just a few terms of the infinite series.     

Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid‐characteristic method

This paper introduces a novel method of modelling acoustic and elastic wave propagation in inhomogeneous media with sharp variations of physical properties based on the recently developed grid‐characteristic method which considers different types of waves generated in inhomogeneous linear‐elastic media (e.g., longitudinal, transverse, Stoneley, Rayleigh, scattered PP‐, SS‐waves, and converted PS‐ and SP‐waves). In the framework of this method, the problem of solving acoustic or elastic wave equations is reduced to the interpolation of the solutions, determined at earlier time, thus avoiding a direct solution of the large systems of linear equations required by the FD or FE methods. We apply the grid‐characteristic method to compare wave phenomena computed using the acoustic and elastic wave equations in geological medium containing a hydrocarbon reservoir or a fracture zone. The results of this study demonstrate that the developed algorithm can be used as an effective technique for modelling wave phenomena in the models containing hydrocarbon reservoir and/or the fracture zones, which are important targets of seismic exploration.     

Propagating Stationary Surface Potential Waves in a Deep Ideal Fluid

A new exact solution of the problem for propagating stationary potential wave of an arbitrary amplitude in a deep ideal homogeneous fluid was constructed. Calculated wavy surface is represented by transcendental Lambert’s complex functions. For a physical interpretation of the results real linear combinations of the solutions were formed. The range of the wave steepness values, in which the real sum of constructed comprehensive solutions describes waves with smooth crests, is defined. In the limiting case of waves with small but finite amplitude as well as infinitesimal amplitude, the real combinations of the solutions are transferred in classical nonlinear and linear asymptotic Stokes expressions. Another real combination of constructed complex solutions describing waves with cusped crests do not fall within the range of conditions for the existence of stationary waves.     

Weighted residual method for diffraction of plane P-waves in a 2-D elastic half-space III: on an almost circular arbitrary-shaped alluvial valley

Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV- scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by Lee and Liu. This paper uses their technique of defining these stress-free scattered waves, which Brandow and Lee previously used to solve the problem of the scattering and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape, to the study of the scattering and diffraction of these in-plane waves on an almost circular arbitrary-shaped alluvial valley.     

局域双曲线Radon变换压制散射波研究与应用

一般而言,由于地下非均质体的存在所产生的二次波源,由它再生成新的波场,叫散射波场。目前,将散射波作为有效波来成像,已开始在溶洞和裂缝等特殊地质体的识别中得到应用。但对于野外资料采集来说,地表复杂地区,如戈壁、砾石区和山前带,大量存在的散射波却是干扰波,它们的存在会严重影响资料的品质,而其研究与实际应用国内外还很少。因此,通过正演模拟,分析散射波的基本特征,在此基础上研究散射波的去噪方法显得十分必要。本文从地震波运动学时距关系出发,研究了反射波和散射波的几何特征;然后用有限差分正演,模拟了散射波场,用理论模型研究并测试了局域双曲线Radon变换散射波去噪新方法。对于实际炮集资料,分析了F-K滤波方法压制散射噪声的局限,采用局域双曲Radon变换有效地去除了炮集中存在的散射噪声,取得了较好的应用效果。     

Scattering and diffraction of plane P-waves in a 2-D elastic half-space Ⅱ:shallow arbitrary shaped canyon

Scattering and Diffraction of elastic in-plane P- and SV- waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH waves on the same elastic canyon that is semi-circular in shape on the half-space surface is the first such problem that was solved by analytic closed form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because, the in-plane P- and SV- scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattered and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape.     

THE PERFECTLY REFLECTING WEDGE USED AS A CONTROL MODEL IN SEISMIC DIFFRACTION MODELLING*

The numerical modelling of seismic diffraction, e.g., at faults and other discontinuities, generally requires the use of fast approximate methods. The geophysicist responsible for the development of such numerical methods has a real need of exact solutions to certain ideal geometries to check the accuracy of his calculations. One such exact solution, which is available, is the acoustic wave solution to the perfectly reflecting wedge. The solution is three-dimensional and the source is an explosive point source. This model is ideal for seismic diffraction; the solution has the advantage of being exact, truly three-dimensional and of being in the convenient form of the temporal and spatial impulse response. More complicated sources which are extended in either space or time can, therefore, be modelled exactly by numerical integration. This paper presents some examples of the use of the perfectly reflecting wedge as a control model for an asymptotic high frequency diffraction modelling method. This control model has revealed that certain survey and wedge configurations can yield significant disagreement with, e.g., the Kirchhoff approximation. Such configurations could occur during VSP modelling when the survey lies in the near field or in the shadow zone of a high contrast fault. This control model has also been instructive in demonstrating why the high frequency, asymptotic, approximation is generally very good and has indicated a possible improvement to the Kirchhoff approximation for edge diffraction.     

Diffraction of plane SV waves by a cavity in poroelastic half-space

This paper presents an indirect boundary integration equation method for diffraction of plane SV waves by a 2-D cavity in a poroelastic half-space.The Green’s functions of compressive and shear wave sources are derived based on Biot’s theory. The scattered waves are constructed using fi ctitious wave sources close to the boundary of the cavity, and their magnitudes are determined by the boundary conditions. Verifi cation of the accuracy is performed by: (1) checking the satisfaction extent of the boundary c...     

Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel

A closed-form analytic solution of two-dimensional scattering and diffraction of plane SH waves by a semicylindrical hill with a semi-cylindrical concentric tunnel inside an elastic half-space is presented using the cylindrical wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations. Fourier expansion theorem with the form of complex exponential function and cosine function is used. Numerical solutions are obtained by truncation of the infinite equations. The accuracy of the presented numerical results is carefully verified.     

Scattering and reflection of SH waves around a slope on an elastic wedged space

The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating.     

Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space (Ⅰ): Formulation

This paper presents an indirect boundary integration equation method for diffraction of plane P waves by a two-dimensional canyon of arbitrary shape in poroelastic half-space. The Green's functions of compressional and shear wave sources in poroelastic half-space are derived based on Biot's theory. The scattered waves are constructed using the fictitious wave sources close to the boundary of the canyon, and magnitude of the fictitious wave sources are determined by the boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, the comparison between the degenerated solutions of single-phased half-space and the well-known solutions, and the numerical stability of the method.     

Weighted residual method for diffraction of plane P-waves in a 2-D elastic half-space Ⅲ: on an almost circular arbitrary-shaped alluvial valley

Scattering and diffraction of elastic in-plane P-and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong motion seismologists for over forty years. The case of out-ofplane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P-and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P-and SV-scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by Lee and Liu. This paper uses their technique of defining these stress-free scattered waves, which Brandow and Lee previously used to solve the problem of the scattering and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape, to the study of the scattering and diffraction of these in-plane waves on an almost circular arbitrary-shaped alluvial valley.     

Marine seismic wavefield measurement to remove sea-surface multiples

We propose a new method for removing sea-surface multiples from marine seismic reflection data in which, in essence, the reflection response of the earth, referred to a plane just above the sea-floor, is computed as the ratio of the plane-wave components of the upgoing wave and the downgoing wave. Using source measurements of the wavefield made during data acquisition, three problems associated with earlier work are solved: (i) the method accommodates source arrays, rather than point sources; (ii) the incident field is removed without simultaneously removing part of the scattered field; and (iii) the minimum-energy criterion to find a wavelet is eliminated. Pressure measurements are made in a horizontal plane in the water. The source can be a conventional array of airguns, but must have both in-line and cross-line symmetry, and its wavefield must be measured and be repeatable from shot to shot. The problem is formulated for multiple shots in a two-dimensional configuration for each receiver, and for multiple receivers in a two-dimensional configuration for each shot. The scattered field is obtained from the measurements by subtracting the incident field, known from measurements at the source. The scattered field response to a single incident plane wave at a single receiver is obtained by transforming the common-receiver gather to the frequency–wavenumber domain, and a single component of this response is obtained by Fourier transforming over all receiver coordinates. Each scattered field component is separated into an upgoing wave and a downgoing wave using the zero-pressure condition at the water-surface. The upgoing wave may then be expressed as a reflection coefficient multiplied by the incident downgoing wave plus a sum of scattered downgoing plane waves, each multiplied by the corresponding reflection coefficient. Keeping the upgoing scattered wave fixed, and using all possible incident plane waves for a given frequency, yields a set of linear simultaneous equations for the reflection coefficients which are solved for each plane wave and for each frequency. To create the shot records that would have been measured if the sea-surface had been absent, each reflection coefficient is multiplied by complex amplitude and phase factors, for source and receiver terms, before the five-dimensional Fourier transformation back to the space–time domain.     

Diffraction of a compressional wave by a rigid barrier in a liquid half-space

Summary The problem of diffraction of compressional waves by a rigid barrier of finite height fixed in a liquid half space has been studied. Wiener-Hopf technique forms the basis of the methods used to solve the problem. Exact solutions have been obtained in terms of Fourier integrals whose evaluation along an appropriate contour gives the transmitted, reflected and diffracted waves. The diffracted waves decay rapidly away from the barrier.     

P波入射时含圆柱形空腔板弹性波散射与动应力集中

采用波函数展开法给出了板的横截面处孔洞在平面P波入射下动应力集中问题的解。本文使用大圆弧假定法来满足板中的自由表面的边界条件。数值结果表明,上表面和孔洞间的距离与下表面和孔洞间的距离都对结果有影响。如果孔洞半径相对板的厚度很小,解近似等于孔洞全空间解。所给出的解是半解析解。因此能被用来验证诸如BEM,FEM,FDM等数值法。     

Wave propagation in semi-infinite media with topographical irregularities using Decoupled Equations Method

In this paper, a novel semi-analytical method, called Decoupled Equations Method (DEM), is presented for modeling of elastic wave propagation in the semi-infinite two-dimensional (2D) media which are involved surface topography. In the DEM, only the boundaries of the problem are discretized by specific subparametric elements, in which special shape functions as well as higher-order Chebyshev mapping functions are implemented. For the shape functions, Kronecker Delta property is satisfied for displacement function. Moreover, the first derivatives of displacement function with respect to the tangential coordinates on the boundaries are assigned to zero at any given node. Employing the weighted residual method and using Clenshaw–Curtis numerical integration, coefficient matrices of the system of equations are transformed into diagonal ones, which leads to a set of decoupled partial differential equations. To evaluate the accuracy of the DEM in the solution of scattering problem of plane waves, cylindrical topographical features of arbitrary shapes are solved. The obtained results present excellent agreement with the analytical solutions and the results from other numerical methods.     

饱和土沉积谷场地对平面SV波的散射问题的解析解

把波函数展开方法用于饱和多孔介质中波的传播的研究中,给出了不同土层界面条件（透水条件和不透水条件）下具有饱和土沉积层的圆弧形沉积河谷场地对平面SV波散射问题的解析解. 其中沉积谷软土场地用饱和多孔介质的Biot动力学理论模拟,半空间场地用单相介质弹性动力理论模拟. 对于入射角大于临界入射角时,产生的面波的波函数用有限Fourier级数展开,这种方法适用于较大的入射波频率范围,这是现存的数值方法所不能比拟的一大优点. 文中算例分析了入射波频率和入射角对地震地面运动的影响.     

Analytical representation of Green’s functions of the Biot equations for elastic waves in a homogeneous two-phase medium

The system of Biot vector equations in the frequency space includes two elliptic-type vector partial differential equations with unknown displacement vectors in the solid and liquid phases. Considering the Biot equations, alongside with Pride’s equations, the key approaches to the theoretical study of the elastic waves in the two-phase fluid-saturated media, the author suggests an analytical solution for the inhomogeneous Biot equations in the frequency space, which is reduced to finding its fundamental solution (Green’s function). The solution of this problem consists of solutions for two systems of Biot equations. In the first system, only the first equation is inhomogeneous, while in the second system, only the second equation is inhomogeneous and, as it is shown, its right-hand side is exclusively a potential function. The fundamental solution of the full system of inhomogeneous Biot equations (in which both equations are inhomogeneous) is represented in the form of Green’s matrix-tensor, for the scalar elements of which the analytical relations are presented. The obtained formulas describing the elastic displacements of both the solid and liquid phases reflect three wave types, namely, compressional waves of the first and the second kind (the fast and the slow waves, respectively) and shear waves. Similar terms (those describing the same type of the elastic waves in the solid and liquid phases) in the expressions for Green’s functions are linked with each other through the coefficient that links the components of the displacement vectors of the solid and liquid phases corresponding to the given wave type.

     

Transient Scattering of Elastic Waves by Dipping Layers of Arbitrary Shape Part 1: Antiplane Strain Model

Scattering of elastic waves by two dimensional multilayered dipping sediments of arbitrary shape embedded in an elastic half-sapce is investigated by using a bondary method. The displancement field is evaluated throughout the elastic media for both steady state and transient incident SH waves. The unknown scattered field is expressed in terms of wave functions which satisfy the equation of motion, traction-free boundary condition and appropariate radiation conditions. The transient response is constructed from the steady state solution by using the fast Fourier transform technique. The numerical results presented demonstrate that scattering of waves by subsurface irregularities may cause locally very large amplification of surface ground motion. The motion can be affected greatly by the scattered surface waves in the sediments. The results clearly indicate that the surface ground motion depends upon a number of parameters present in the problem, such as frequency and the angle of incidence of the incoming wave, impedance contrast between the layers and location of the observation point.     

地震初至波走时的有限差分计算

本文提出了用有限差分解程函方程求取地震初至波走时的快速、精确方法。算法考虑了首波，散射波开采新的延拓方法。在任意复杂的速度结构中能得到精确的结果。本方法对叠前偏移、层析成像是非常适宜的。