Calculation of Synthetic Seismograms with Gaussian Beams

— In this paper, an overview of the calculation of synthetic seismograms using the Gaussian beam method is presented accompanied by some representative applications and new extensions of the method. Since caustics are a frequent occurrence in seismic wave propagation, modifications to ray theory are often necessary. In the Gaussian beam method, a summation of paraxial Gaussian beams is used to describe the propagation of high-frequency wave fields in smoothly varying inhomogeneous media. Since the beam components are always nonsingular, the method provides stable results over a range of beam parameters. The method has been shown, however, to perform better for some problems when different combinations of beam parameters are used. Nonetheless, with a better understanding of the method as well as new extensions, the summation of Gaussian beams will continue to be a useful tool for the modeling of high-frequency seismic waves in heterogeneous media.     

A tale of two beams: an elementary overview of Gaussian beams and Bessel beams

An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that are ideally diffraction-free and do not diverge when they propagate. They can be described by plane waves with normal vectors along a cone with a fixed angle from the beam propagation direction. X-waves are an example of pulsed beams that propagate in an undistorted fashion. For realizable localized beam solutions, Bessel beams must ultimately be windowed by an aperture, and for a Gaussian tapered window function this results in Bessel-Gauss beams. Bessel-Gauss beams can also be realized by a combination of Gaussian beams propagating along a cone with a fixed opening angle. Depending on the beam parameters, Bessel-Gauss beams can be used to describe a range of beams solutions with Gaussian beams and Bessel beams as end-members. Both Gaussian beams, as well as limited diffraction beams, can be used as building blocks for the modeling and synthesis of other types of wave fields. In seismology and geophysics, limited diffraction beams have the potential of providing improved controllability of the beam solutions and a large depth of focus in the subsurface for seismic imaging.     

2D共炮时间域高斯波束偏移

针对传统射线方法在奇异区成像精度不高,而2D频率域高斯波束叠前深度偏移需要计算成像点处每个频率的格林函数,影响计算效率的问题,本文通过使用复走时代替实走时,改变频率域下成像公式的积分顺序,给出了在时间域下进行高斯波束偏移的方法和计算公式.本文使用复杂数值模型验证了2D时间域高斯波束叠前偏移方法的正确性,并同传统射线偏移成像结果做了对比.对比结果表明时间域高斯波束偏移在成像精度上优于传统射线偏移.     

Optimization of the shape of Gaussian beams

The applicability and accuracy of the Gaussian beam method depend on the proper choice of the shape of beams. Gaussian beams become inaccurate solutions of the elastodynamic equation if the velocity field changes considerably within the beam width. We present a procedure of determining the optimum initial shape of Gaussian beams based on minimizing the average squared widths of Gaussian beams and smoothing the distribution of the optimum parameters of Gaussian beams on the Hamiltonian hypersurface in the phase-space. The original method of smoothing represents an essential part of the algorithm, which is designed particularly for the optimization of the shape of Gaussian beams for Gaussian beam or packet migrations.     

Imaging Offsets in the Moho: Synthetic Tests using Gaussian Beams with Teleseismic Waves

We carry out a sequence of numerical tests to understand conditions under which rapid changes in crustal thickness can be reliably imaged by teleseismic body waves. Using the finite-difference method over a 2-D grid, we compute synthetic seismograms resulting from a planar P-wavefield incident below the grid. We then image the Moho using a migration scheme based on the Gaussian beam representation of the wavefield. The use of Gaussian beams for the downward propagation of the wavefield is particularly advantageous in certain geologically critical cases such as overthrusting of continental lithosphere, resulting in the juxtaposition of high-velocity mantle material over crustal rocks. In contrast to ray-based methods, Gaussian beam migration requires no special treatment to handle such heterogeneities. Our results suggest that with adequate station spacing and signal-to-noise ratios, offsets of the Moho, on the order of 10 km in height, can be reliably imaged beneath thickened crust at depths of about 50 km. Furthermore, even sharp corners and edges are faithfully imaged when precise values of seismic wave speeds are available. Our tests also demonstrate that flexibility in choices of different types of seismic phases is important, because any single phase has trade-offs in issues such as spatial resolution, array aperture, and amplitude of signals.     

基于高斯束与高斯波包的Gabor框架散射波模拟方法

在给出真实模型和相应光滑背景模型的情况下,如何计算扰动模型(散射体)产生的散射波场是一个有实际意义的正演问题.在Gabor变换域描述散射体,且入射波场为短时宽带信号时,散射波场可以在频率域用高斯束或时间域用高斯波包描述.相对于波动方程方法,高斯束和高斯波包的计算效率更高;背景模型光滑时,高斯束和高斯波包方法的精度也接近波动方程方法.文中导出了声波假设下应用高斯束和高斯波包计算散射波的方法.测试分析了高斯波包的计算精度.给出了一般散射体的散射波模拟策略.同时针对一个理论模型完成了本文方法计算散射波的实验,实验结果表明高斯波包散射波计算方法是有效可行的.     

声波介质一次散射波场高斯束Born正演

Born正演是一种常用的地震波场正演模拟方法,也是线性化地震反演的理论基础.在实际应用时,Born正演通常结合常规的地震射线方法进行实现.为了克服常规地震射线方法的弊端,并且保证地震波场的模拟精度和计算效率,本文提出了一种基于高斯束的一阶散射波场Born正演方法.该方法分为两个环节:首先,我们利用高斯束的走时和振幅信息将地下散射点处的反射率映射为地表束中心位置处的局部平面波;然后,我们利用逆倾斜叠加将局部平面波转化为接收点处的时空域散射波场.在具体的实施过程中,我们提出一种以wavelet-bank方式实现的局部平面波合成方法,同现有的算法相比,可以在保持计算精度的同时,大大减少计算时间;此外,我们还利用最速下降法优化了高斯束的迭代循环过程,进一步提高了Born正演的计算效率.两个模型的应用效果证明,本文所提出的高斯束Born正演方法可以精确、高效的实现声波介质一次散射波场的正演模拟,为三维大规模地震波场的正演问题提供了一种切实可行的实现方案.     

True amplitude imaging by inverse generalized Radon transform based on Gaussian beam decomposition of the acoustic Green's function

True amplitude migration is one of the most important procedures of seismic data processing. As a rule it is based on the decomposition of the velocity model of the medium into a known macrovelocity component and its sharp local perturbations to be determined. Under this decomposition the wavefield can be considered as the superposition of an incident and reflected/scattered waves. The single scattering approximation introduces the linear integral operator that connects the sharp local perturbations of the macrovelocity model with the multishot/multioffset data formed from reflected/scattered waves. We develop the pseudoinverse of this operator using the Gaussian beam based decomposition of acoustic Green's functions. The computation of this pseudoinverse operator is done pointwise by shooting Gaussian beams from the target area towards the acquisition system. The numerical implementation of the pseudoinverse operator was applied to the synthetic data Sigsbee2A. The results obtained demonstrate the high quality of the true amplitude images computed both in the smooth part of the model and under the salt body.     

The Relation between Gaussian Beams and Maslov Asymptotic Theory

The application of Maslov asymptotic theory in a general 3-D mixed subspace of 6-D complex phase space is proposed to obtain the integral superpositions of Gaussian packets and beams. The ray method and the superposition of plane waves (Maslov method of Chapman and Drumond [7]) are special limiting cases of the above mentioned approach. The same high-frequency asymptotic expansion formulae for seismic body waves were derived previously in [8] using the Gaussian beam method.     

True amplitude elastic Gaussian beam imaging of multicomponent walkaway vertical seismic profiling data

We develop the true‐amplitude prestack migration of multicomponent data based on the use of elastic Gaussian beams for walkaway vertical seismic profile (VSP) acquisition systems. It consists in a weighted summation of multishot data with specific weights, computed by tracing elastic Gaussian beams from each imaging point of the target area towards the sources and receivers. Each pair of beams may be connected with either a pair of P‐rays (PP‐image) or the P‐ray towards sources and the S‐ray to receivers (PS‐image) and is uniquely determined by dip (the angle of the bisector between the rays and the vertical direction) and opening (the angle between the rays) angles. Shooting from the bottom towards the acquisition system helps to avoid well‐known troubles, in particular multipathing for the imaging conditions in complex velocity models. The ability to fix the dip angle and implement summation over opening angles leads to the so‐called selective images that contain mostly interfaces with desired slopes. On the other hand, a set of images computed for a range of opening angles by summation over all available dip angles is used as input of an AVO‐like inversion procedure for the recovery of elastic parameters. The feasibility of this imaging procedure is verified by synthetic data for 2D realistic elastic models.     

基于Born散射理论的二维黏声介质高斯波束正演

Born散射理论可以通过省略高阶项实现针对一次散射波场的模拟.在这一理论的基础上,本文提出了一种针对二维黏声介质的一次散射波场高斯束Born正演方法.在该方法中,格林函数通过一系列不同初射方向的高斯波束累加获得,可以计算多至走时波场,保证了正演算法的计算精度.同时为了提高计算效率,正演方法使用了wavelet-bank...     

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.     

弹性各向同性介质一次散射波场高斯束Born正演

本文提出了一种弹性波一次散射波场的正演方法——弹性波高斯束Born正演.该方法以线性散射理论为基础,通过Born近似建立起地下散射点处不同波型的反射率同弹性波主分量波场之间的数据映射关系,利用高斯束所包含的走时、振幅和极性信息进行不同波型的局部平面波的合成,进而通过逆倾斜叠加将所合成的局部平面波转化为时空域的多分量地震记录.该方法不但保持了射线类方法高效的优点,还具备了处理多次走时波场的能力,从而保证了复杂构造的波场模拟的精度.文中两个数值模型的应用效果表明,本文所提出的弹性波高斯束Born正演算法具有近似于波动方程有限差分法的波场模拟精度以及高得多的计算效率.     

水与流体饱和孔隙介质界面上非镜面反射声场的实验研究

用光学法和发射-接收法研究了水与流体饱和孔隙介质界面上声波的非镜面反射声场．测量了入射角θ_i为16°－80°范围内的反射系数R，记录了θ_i在Rayleigh波临界角θ_c附近±4°范围内两束反射波的反射角、波束位移△、波形和频谱．实验发现：(1)在θ_c附近±4°范围内，观察到θ_i两束反射波A和B，其中波束B沿界面有波束位移△，实验测出△随θ_i增大而减小，并非在θ_i=θ_c时最大．(2)反射角等于θ_i的波束的频谱很接近于入射波的频谱，反射角等于θ_c的波束的频谱相对于入射波有频谱的负偏移．(3)波束A的相位与发射换能器直达波的相位一致，波束B的相位与波束A相反     

Gabor‐frame‐based Gaussian packet migration

We present a Gaussian packet migration method based on Gabor frame decomposition and asymptotic propagation of Gaussian packets. A Gaussian packet has both Gaussian‐shaped time–frequency localization and space–direction localization. Its evolution can be obtained by ray tracing and dynamic ray tracing. In this paper, we first briefly review the concept of Gaussian packets. After discussing how initial parameters affect the shape of a Gaussian packet, we then propose two Gabor‐frame‐based Gaussian packet decomposition methods that can sparsely and accurately represent seismic data. One method is the dreamlet–Gaussian packet method. Dreamlets are physical wavelets defined on an observation plane and can represent seismic data efficiently in the local time–frequency space–wavenumber domain. After decomposition, dreamlet coefficients can be easily converted to the corresponding Gaussian packet coefficients. The other method is the Gabor‐frame Gaussian beam method. In this method, a local slant stack, which is widely used in Gaussian beam migration, is combined with the Gabor frame decomposition to obtain uniform sampled horizontal slowness for each local frequency. Based on these decomposition methods, we derive a poststack depth migration method through the summation of the backpropagated Gaussian packets and the application of the imaging condition. To demonstrate the Gaussian packet evolution and migration/imaging in complex models, we show several numerical examples. We first use the evolution of a single Gaussian packet in media with different complexities to show the accuracy of Gaussian packet propagation. Then we test the point source responses in smoothed varying velocity models to show the accuracy of Gaussian packet summation. Finally, using poststack synthetic data sets of a four‐layer model and the two‐dimensional SEG/EAGE model, we demonstrate the validity and accuracy of the migration method. Compared with the more accurate but more time‐consuming one‐way wave‐equation‐based migration, such as beamlet migration, the Gaussian packet method proposed in this paper can correctly image the major structures of the complex model, especially in subsalt areas, with much higher efficiency. This shows the application potential of Gaussian packet migration in complicated areas.     

Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media

Explicit equations for the spatial derivatives and perturbation derivatives of amplitude in both isotropic and anisotropic media are derived. The spatial and perturbation derivatives of the logarithm of amplitude can be calculated by numerical quadratures along the rays. The spatial derivatives of amplitude may be useful in calculating the higher-order terms in the ray series, in calculating the higher-order amplitude coefficients of Gaussian beams, in estimating the accuracy of zero-order approximations of both the ray method and Gaussian beams, in estimating the accuracy of the paraxial approximation of individual Gaussian beams, or in estimating the accuracy of the asymptotic summation of paraxial Gaussian beams. The perturbation derivatives of amplitude may be useful in perturbation expansions from elastic to viscoelastic media and in estimating the accuracy of the common-ray approximations of the amplitude in the coupling ray theory.     

二维弹性多波时空域高斯束偏移方法

随着我国勘探开发难度逐步增大,勘探目标开始向裂缝油气藏、岩性油气藏等复杂探区转移,研究高精度、适应性强的多波多分量深度偏移算法在后续的地震解释、属性分析及储层预测中具有重要意义.针对多波多分量地震数据,本文提出了一种二维弹性波时空域高斯束偏移方法.时空域高斯束沿中心射线传播时能够面向成像目标描述局部波场,且对振幅和频率可调制的Gabor基函数有天然的适应性,因而将基于Gabor分解的子波重构方法应用于震源波场构建,从而得到任意点源函数产生的时空域高斯束波场.该方法由于直接在时间域进行计算,可以避开频率域中出现的假频和边缘截断效应等问题.基于各向同性弹性波动方程的Kirchhoff-Helmholtz积分解,利用矢量时空域高斯束传播算子构建格林函数和格林位移张量,并结合上行射线追踪策略,实现了检波点波场的反向延拓.针对矢量波成像问题,本文借鉴弹性波逆时偏移方法从矢量延拓波场中分离出纯纵波分量和纯横波分量,进而采用修改后的内积成像条件产生具有明确物理意义的PP、PS成像结果,避免了转换波成像的极性反转问题.最后利用简单两层模型和不含盐体构造的部分Sigsbee2a模型的成像结果,并将其与应用近似纵横波成像条件、标量和矢量势成像条件的偏移剖面进行对比,验证了本文方法的正确性和有效性.     

VTI介质角度域转换波高斯束偏移成像方法研究

转换波偏移可以利用纵横波波场信息,得到高分辨率的成像结果,从而为油藏描述提供高质量的地震资料.目前的研究主要是利用纵波波场信息进行偏移成像,然而,传统的纵波方法在复杂探区成像时具有一定的局限性.为此,本文在各向异性介质声波射线追踪算法的基础上,推导出各向异性介质转换波射线追踪方程,发展了一种转换波射线追踪算法;并将研究的追踪算法应用到偏移成像中,提出了一种各向异性VTI介质角度域转换波高斯束偏移成像方法.通过各向异性VTI介质断块模型和复杂构造模型试算,说明了本文方法的正确性和有效性.模型试算的结果表明,在考虑地下各向异性时,本文研究的方法具有更好的成像效果,提取的角道集结果可以为偏移速度分析提供依据.     

Fracture detection by Gaussian beam imaging of seismic data and image spectrum analysis

Localization of fractured areas is of primary interest in the study of oil and gas geology in carbonate environments. Hydrocarbon reservoirs in these environments are embedded within an impenetrable rock matrix but possess a rich system of various microheterogeneities, i.e., cavities, cracks, and fractures. Cavities accumulate oil, but its flow is governed by a system of fractures. A distinctive feature of wave propagation in such media is the excitation of the scattered/diffracted waves by the microheterogeneities. This scattering could be a reliable attribute for characterization of the fine structure of reservoirs, but it has extremely low energy and any standard data processing renders them practically invisible in comparison with images produced by specular reflections. Therefore, any attempts to use these waves for image congestion of microheterogeneities should first have a preliminary separation of the scattering and specular reflections. In this paper, the approach to performing this separation is based on the asymmetric summation. It is implemented by double focusing of Gaussian beams. To do this, the special weights are computed by propagating Gaussian beams from the target area towards the acquisition system separately for sources and receivers. The different mutual positioning of beams in each pair introduces a variety of selective images that are destined to represent some selected singular primitives of the target objects such as fractures, cavities, and edges. In this way, one can construct various wave images of a target reservoir, particularly in scattered/diffracted waves. Additional removal of remnants of specular reflections is done by means of spectral analysis of the scattered/diffracted waves' images to recognize and cancel extended lineaments. Numerical experiments with Sigsbee 2A synthetic seismic data and some typical structures of the Yurubcheno‐Tokhomskoye oil field in East Siberia are presented and discussed.     

基于分灾模式的钢筋混凝土梁抗弯加固性能评估方法

针对当前钢筋混凝土梁抗弯加固性能评估法未考虑不同建筑结构的抗灾差异,所求得的各项承载力参数不够客观,从而造成评估结果与实际检测结果差异大,评估精度低的问题,提出基于分灾模式的钢筋混凝土梁抗弯加固性能评估方法。计算梁底初始拉应变、梁正截面极限抗弯承载力和钢筋混凝土梁正截面荷载挠度,然后设计分灾模式的钢筋混凝土梁抗弯加固,通过分析载荷与压应变、载荷与挠度以及剪力与位移变化,得到钢筋混凝土梁抗弯加固后各项承载力参数,输入各项梁承载力参数到ANSYS通用程序,利用该程序结合钢筋混凝土梁抗弯加固性能进行评估。通过进行仿真实验,结果表明加入分灾元件的钢筋混凝土梁抗弯加固性能有明显提高。将本文方法评估结果与实验实际评估结果对比可知,提出的评估方法与实际检测结果基本一致,评估精度准确性较高。