Ray tracing based on Fermat's principle in irregular grids1

A new method to trace rays in irregular grids based on Fermat's principle of minimum time is introduced. Besides the usual transmitted and reflected waves, refracted, diffracted and converted waves can also be simulated. The proposed algorithm is fast and stable, and respects the reciprocity principle between source and receiver better than procedures adopting the shooting method. It is particularly suited to form part of a traveltime inversion procedure. The use of irregular grids allows adaptation of the earth discretization to the available acquisition geometry and ray distribution, to obtain more stable and reliable tomographic images.     

Diffraction imaging by uniform asymptotic theory and double exponential fitting

Seismic diffracted waves carry valuable information for identifying geological discontinuities. Unfortunately, the diffraction energy is generally too weak, and standard seismic processing is biased to imaging reflection. In this paper, we present a dynamic diffraction imaging method with the aim of enhancing diffraction and increasing the signal‐to‐noise ratio. The correlation between diffraction amplitudes and their traveltimes generally exists in two forms, with one form based on the Kirchhoff integral formulation, and the other on the uniform asymptotic theory. However, the former will encounter singularities at geometrical shadow boundaries, and the latter requires the computation of a Fresnel integral. Therefore, neither of these methods is appropriate for practical applications. Noting the special form of the Fresnel integral, we propose a least‐squares fitting method based on double exponential functions to study the amplitude function of diffracted waves. The simple form of the fitting function has no singularities and can accelerate the calculation of diffraction amplitude weakening coefficients. By considering both the fitting weakening function and the polarity reversal property of the diffracted waves, we modify the conventional Kirchhoff imaging conditions and formulate a diffraction imaging formula. The mechanism of the proposed diffraction imaging procedure is based on the edge diffractor, instead of the idealized point diffractor. The polarity reversal property can eliminate the background of strong reflection and enhance the diffraction by same‐phase summation. Moreover,the fitting weakening function of diffraction amplitudes behaves like an inherent window to optimize the diffraction imaging aperture by its decaying trend. Synthetic and field data examples reveal that the proposed diffraction imaging method can meet the requirement of high‐resolution imaging, with the edge diffraction fully reinforced and the strong reflection mostly eliminated.     

Application of diffracted wave analysis to time‐lapse seismic data for CO2 leakage detection

Time‐lapse seismic analysis is utilized in CO₂ geosequestration to verify the CO₂ containment within a reservoir. A major risk associated with geosequestration is a possible leakage of CO₂ from the storage formation into overlaying formations. To mitigate this risk, the deployment of carbon capture and storage projects requires fast and reliable detection of relatively small volumes of CO₂ outside the storage formation. To do this, it is necessary to predict typical seepage scenarios and improve subsurface seepage detection methods. In this work we present a technique for CO₂ monitoring based on the detection of diffracted waves in time‐lapse seismic data. In the case of CO₂ seepage, the migrating plume might form small secondary accumulations that would produce diffracted, rather than reflected waves. From time‐lapse data analysis, we are able to separate the diffracted waves from the predominant reflections in order to image the small CO₂ plumes. To explore possibilities to detect relatively small amounts of CO₂, we performed synthetic time‐lapse seismic modelling based on the Cooperative Research Centre for Greenhouse Gas Technologies (CO2CRC) Otway project data. The detection method is based on defining the CO₂ location by measuring the coherency of the signal along diffraction offset‐traveltime curves. The technique is applied to a time‐lapse stacked section using a stacking velocity to construct offset‐traveltime curves. Given the amount of noise found in the surface seismic data, the predicted minimum detectable amount of CO₂ is 1000–2000 tonnes. This method was also applied to real data obtained from a time‐lapse seismic physical model. The use of diffractions rather than reflections for monitoring small amounts of CO₂ can enhance the capability of subsurface monitoring in CO₂ geosequestration projects.     

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.     

断层衍射波及其应用ht

本文从衍射波的物理定义出发,简化了Trorey提出的断层上Kirchhoff-Helmholtz衍射方程的解法;计算了不同深度、不同测线方向下的断层衍射波理论地震图;计算了衍射波振幅谱与相位谱,给出了利用衍射波求断层位置的公式;得出了一些新的结论,通过人工地震测深与地震勘探的实例,对断层衍射波的特性有了更明确的认识。 本文的结果表明:断层衍射波发生在地球介质剧烈变化处;衍射点两侧的衍射波走时曲线呈双曲线状;波初动清晰且半周期小;其优势频率振幅谱与反射波的相同,在衍射波与反射波走时曲线相切处附近,记录图中出现衍射波最大振幅,且波反相;视断点与真实断点一般不重合。上述特点可能为判定衍射波并确定断层位置提供判据。      

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.     

Tutorial on the numerical modeling of edge diffracted waves by the ray method

Summary The concept of diffracted rays introduced by Keller (1962) opened the way for a ray approach in the numerical modeling of the diffraction phenomena frequently seen in the seismic field records. Unlike the computation of travel times, which has never caused any difficulties being governed by the same eikonal equation as are the rays of ordinary seismic body waves, the evaluation of the diffracted ray amplitude has been another story. In this paper we give a tutorial account of a highly promising technique for the computation of the ray amplitude of seismic waves diffracted by linear edges on seismic interfaces originally presented by Klem-Musatov (1980, 1995). We demonstrate the adequacy of this technique on a series of SH synthetic traces containing diffracted arrivals, some of them diffracted more than once. Our computer program is based on a successful combination of Klem-Musatov's approach with the zero order approximation of the Asymptotic Ray Theory (Červeny and Hron, 1980). We showed in the paper that both techniqúes are ideally suited for such a combination, since the ray amplitude evaluation in each of them is based on the same transport equation. In our tutorial review we present all formulae which are needed for a practical implementation of the method and provide their physical interpretation, wherver possible, by using the numerical examples presented in the paper.     

Scattering and diffraction of earthquake motions in irregular,elastic layers,II: Rayleigh and body P and SV waves

We present our study of the wave propagation in an irregularly layered, elastic wave-guide excited by incoming Rayleigh surface waves and P and SV body waves. Our aim is to show examples of applying a method that will make it possible to analyze the distribution and amplification of displacements, rotations, curvatures, strains, and stresses on or below the ground surface during passage of strong earthquake ground motion. We employ the weighted-residuals method, which makes it possible to calculate the scattered and diffracted waves, and then we illustrate the amplification of motions in the vicinity of inhomogeneity.     

用地震反射波全部时間值求有效速度

本文讨论了用地震反射波的全部时间值求有效速度的方法,得出了四种计算有效速度的公式。经过实际运用,认为有一定的效果,同时可用来计算断层面引起的绕射波的有效速度,对识别绕射波提供了一些线索。本文仅作为地震勘探工作交流实际经验而提出。     

Scattering and diffraction of earthquake motions in irregular elastic layers,I: Love and SH waves

We describe the wave propagation through an irregularly layered, elastic medium for incoming body (SH) and surface (Love) waves. As a result of irregular geometry, each layer generates additional waves by scattering and diffraction. These additional waves modify the input motions and locally may lead to larger motions and concentrations of stresses, strains, and rotations on or below the ground surface. For engineering design, and in particular for analyses of soil-structure interaction, it is important to understand the nature and the consequences of such motions.The scattering and diffraction of Love and SH waves by irregular layers will be investigated by the weighted-residuals method. The scattered and diffracted mode shapes and spectral amplification characteristics at different frequencies will be examined and discussed.     

智利地震震相特征分析

对智利地震在温州地震台的记录特征进行分析,对分析方法做阐述.其结论是:可清晰记录智利地震衍射波PDIF震相;PDIF震相波形“孤立”,起始平缓,周期较大;利用PDIF、PKP、SKKS、SS等明显震相的到时进行大震速报,可提高大震速报的速度和质量.     

Revisiting the Basin-edge Effect at Kobe During the 1995 Hyogo-Ken Nanbu Earthquake

The basin edge effect, i.e., the interference of the direct S wave with the surface wave diffracted off the basin edge has been invoked by many authors to explain the damage distribution during the January 17, 1995 Hyogo-Ken Nanbu (Kobe) earthquake. Here we present the results of numerical experiments obtained with the spectral element method in 2-D geometry. Our results confirm that the amplification of horizontal motion close to the basin edge can be twice as large as the one measured in the center of the basin. This additional amplification is shown to depend strongly on the edge geometry and on frequency, due to physical dispersion of diffracted surface waves. In particular we obtain maximal amplification around 3 Hz, at frequencies critical for buildings.     

折射波法探测断层的模拟试验——记录图的特征

用折射波法对断层进行勘探的实际工作中,不仅观察到上、下盘的正常折射波及棱上的绕射波,而且还可以看到因绕射、透过所产生的一些次生异常波.这些波郁同时在记录上出现,并构成一幅复杂的波形图.如果不能一一地识别它们,并掌握它们的运动学及动力学特征,就无法正确地识别及解释断层.以往一些文章多半讨论板状介质边缘上的绕射现象,而未涉及上、下盘同时存在时,折射法记录上可能出现的次生绕射、折射波. 本文为了解决这一问题,利用了超声波脉冲地震模拟仪进行了实验,对不同大小断距情况下的折射异常波及次生波进行了研究,回答了有关断层勘探中必须阐明的一些问题.     

Detection of waves converted from P to SV in the mantle

A method is described for the detection of P to SV converted waves in the long-period P coda. The procedure involves axis rotation, transformation of records to a standard form and stacking of processed records from events of various epicentral distances. When applied to NORSAR records, the procedure detected converted phases corresponding to the boundaries in the 410–440- and 640–690-km depth ranges.     

DIFFRACTION BY A WEDGE IN AN ACOUSTIC CONSTANT DENSITY MEDIUM1

A new method is presented for solving the 2D problem of diffraction of a plane wave by a wedge of arbitrary angle in a purely acoustic, constant-density medium with different constant compressional wave speeds inside and outside the wedge. The diffraction problem is formulated as integral equations, and a wavenumber–frequency representation of the scattered field is obtained. With the aid of the Cagniard–de Hoop method, exact analytical expressions in the space–time domain are obtained for the different wave constituents, i.e. geometric optical scattered waves and edge diffracted waves including head waves. These expressions can be computed to any degree of accuracy within reasonable computation times on a computer, and the semi-analytical method of solution presented thus constitutes a means of constructing reference solutions for wedge configurations. Such highly accurate reference solutions are of importance for verification of results that include diffraction phenomena modelled by general numerical approximate methods, e.g. finite differences, finite elements and spectral methods. Examples of such applications of the method of solution are given.     

Least square approach to simulate wave propagation in irregular profiles using the indirect boundary element method

In earthquake engineering and seismology it is of interest to know the surface motion at a given site due to the incoming and scattered seismic waves by surface geology. This can be formulated in terms of diffraction of elastic waves and then the indirect boundary element method (IBEM) for dynamic elasticity is used. It is based on the explicit construction of diffracted waves at the boundaries from which they radiate. This provides the analyst with insight on the physics of diffraction. The IBEM has been applied to study the amplification of elastic waves in irregular soil profiles. From the strong or weak satisfaction of boundary conditions and a simple analytical discretization scheme a linear system of equations for the boundary sources is obtained. Here, we explore the use of a weak discretization strategy with more collocation points than force densities. The least squares enforcement of boundary conditions leads to a system with reduced number of unknowns. This approach naturally allows one to use both coarser and finer boundary discretizations for smooth and rapidly varying profiles, respectively. A well studied semicircular canyon under incident P or SV in-plane waves is used to calibrate this method. Several benefits are obtained using mixed meshing that leads to the least squares condensation of the IBEM.     

A comparison of the reflectivity method and full wave theory as applied to SH diffraction around the core

Summary To derive the structure of the Lower Mantle, observations of P and S waves at epicentral distances beyond 90 degrees have been used. The arrival times and amplitude decay with distance proved to be a source of information to derive the structure near the core-mantle interface. These observations were compared with results of theoretical calculations based on full wave theory [1, 2, 5, 6]. Another important theoretical method is the reflectivity method [4]. Differences between the two methods have a direct bearing on the interpretation of the observations and therefore need to be investigated.In the two methods the wave propagation involved in diffraction is simulated in different ways. In this paper the importance of approximations involved in each of the methods will be discussed. For this purpose the calculation of the frequency-dependent decay with distance of diffracted SH waves has been chosen. The approximations are considered to be important both in P and SH diffraction. The advantage of studying SH waves is the simpler and more straightforward formulation.     

NON-LINEAR ESTIMATION OF REFLECTION COEFFICIENTS FROM SEISMIC DATA*

For years, reflection coefficients have been the main aim of traditional deconvolution methods for their significant informational content. A method to estimate seismic reflection coefficients has been derived by searching for their amplitude and their time positions without any other limitating assumption. The input data have to satisfy certain quality constraints like amplitude and almost zero phase noise—ghosts, reverberations, long period multiples, and diffracted waves should be rejected by traditional processing. The proposed algorithm minimizes a functional of the difference between the spectra of trace and reflectivity in the frequency domain. The estimation of reflection coefficients together with the consistent “wavelet’ is reached iteratively with a multidimensional Newton-Raphson technique. The residual error trace shows the behavior of the process. Several advantages are then obtainable from these reflection coefficients, like conversion to interval velocities with an optimum calibration either to the well logs or to the velocity analysis curves. The procedure can be applied for detailed stratigraphic interpretations or to improve the resolution of a conventional velocity analysis.     

Scholte waves on fluid-solid interfaces by means of an integral formulation

The present work shows the propagation of Scholte interface waves at the boundary of a fluid in contact with an elastic solid, for a broad range of solid materials. It has been demonstrated that by an analysis of diffracted waves in a fluid it is possible to infer the mechanical properties of the elastic solid medium, specifically, its propagation velocities. For this purpose, the diffracted wave field of pressures and displacements, due to an initial wave of pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion. The source in the fluid is represented by a Hankel’s function of second kind and zero order. The solution to this wave propagation problem is obtained by means of the Indirect Boundary Element Method, which is equivalent to the well-known Somigliana representation theorem. The validation of the results is carried out by using the Discrete Wave Number Method and the Spectral Element Method. Firstly, we show spectra of pressures that illustrate the behavior of the fluid for each solid material considered, then, we apply the Fast Fourier Transform to show results in time domain. Snapshots to exemplify the emergence of Scholte’s waves are also included.     

Mie elastic scattering generated by cracks

Some numerical experiments are designed to elucidate geometrical details of fractures by studying scattering of elastic waves. The response in the time domain does not seem to provide much information. In the frequency domain, geometrical features of diffractors stamp a signature on reflected and diffracted wave fields. Results show that radiation patterns in the Mie regime are a promising device to infer some aspects such as shape and connectivity of the fractures. A combination of the direct and the indirect boundary element methods is used to compute numerical results. The numerical method is validated with analytic solutions for canonical problems.