Algorithm for calculating synthetic seismograms in a layered half-space with application of matrix impedance

A new algorithm is proposed for calculating the complete synthetic seismograms from a point source in the form of the sum of a single force and a dipole with an arbitrary seismic moment tensor in a plane layered medium composed of homogenous elastic isotropic layers. Following the idea of (Alekseev and Mikhailenko, 1978), an artificial cylindrical boundary is introduced, on which the boundary conditions are specified. For this modified problem, the exact solution (in terms of the displacements and stresses on the horizontal plane areal element) in the frequency domain is derived and substantiated. The unknown depth-dependent coefficients form the motion-stress vector, whose components satisfy the known system of ordinary differential equations. This system is solved by the method that involves the matrix impedance and propagator for the vector of motion, as previously suggested by the author in (Pavlov, 2009). In relation to the initial problem, the reflections from the artificial boundary are noise, which, to a certain degree, can be suppressed by selecting a long enough distance to this boundary and owing to the presence of a purely imaginary addition to the frequency. The algorithm is not constrained by the thickness of the layers, is applicable for any frequency range, and is suitable for computing the static offset.     

The algorithms for calculating synthetic seismograms from a dipole source using the derivatives of Green’s function

The problem of calculating complete synthetic seismograms from a point dipole with an arbitrary seismic moment tensor in a plane parallel medium composed of homogeneous elastic isotropic layers is considered. It is established that the solutions of the system of ordinary differential equations for the motion–stress vector have a reciprocity property, which allows obtaining a compact formula for the derivative of the motion vector with respect to the source depth. The reciprocity theorem for Green’s functions with respect to the interchange of the source and receiver is obtained for a medium with cylindrical boundary. The differentiation of Green’s functions with respect to the coordinates of the source leads to the same calculation formulas as the algorithm developed in the previous work (Pavlov, 2013). A new algorithm appears when the derivatives with respect to the horizontal coordinates of the source is replaced by the derivatives with respect to the horizontal coordinates of the receiver (with the minus sign). This algorithm is more transparent, compact, and economic than the previous one. It requires calculating the wavenumbers associated with Bessel function’s roots of order 0 and order 1, whereas the previous algorithm additionally requires the second order roots.     

Propagator-matrix Technique for the Viscoelastic Response of a Multi-layered Sphere to Surface Toroidal Traction

We derive the matrix propagator for the response of a multi-layered, Maxwell viscoelastic sphere to a surface toroidal force in analytical form. Several numerical methods for 2-D and 3-D glacial-isostatic adjustment (GIA) modelling have been applied recently. The present analytical approach may be used as an aid for testing more general algorithms of GIA. The propagator-matrix solution has been tested by comparing it with the analytical solution of two-layer and three-layer spheres, with full agreement obtained.     

LEVINSON INVERSION OF EARTHQUAKE GEOMETRY SH-TRANSMISSION SEISMOGRAMS IN THE PRESENCE OF NOISE*

The Kunetz-Claerbout equation for the acoustic transmission problem in a layered medium in its original form establishes the relation between the transmission and the reflec tion response for P-waves in an horizontally layered medium and with vertical incidence. It states that the reflection seismogram due to an impulsive source at the surface, is one side of the autocorrelation of the seismogram due to an impulsive source at depth and a surface receiver. By adapting Claerbout's formulation to the transmission of SH-waves, the Kunetz-Claerbout equation also holds for reflection and transmission coefficients dependent on the incident angle. Thus, earthquake geometry SH-transmission seismograms can be used to caculate corresponding pseudoreflection seismograms which, in turn, can be inverted for the impedance structure using the Levinson algorithm. If the average incidence angle is known, a geometrical correction on the resulting impedance model can improve the resolution of layer thicknesses. In contrast to the inversion of reflection seismograms, the Levinson algorithm is shown to yield stable results for the inversion of transmission seismograms even in the presence of additive noise. This noise stabilization is inherent to the Kunetz-Claerbout equation. Results of inverted SH-wave microearthquake seismograms from the Swabian Jura, SW Germany, seismic zone obtained at recording site Hausen im Tal have been compared with sonic-log data from nearby exploration drilling at Trochtelfingen. The agreement of the main structural elements is fair to a depth of several hundred metres.     

Inversion of seismograms to determine simultaneously the moment tensor components and source time function for a point source buried in a horizontally layered medium

Summary A method of determining simultaneously the moment tensor and source time function in the point source approximation is presented. For trial values of the moment tensor components and of the source time function, parametrized by the sum of overlapping triangles delayed in time, theoretical seismograms can be synthetized and compared with the recorded ones. The iterative procedure determines the adjustment of source parameters until a good correlation of both synthetic and observed records is reached. The Green functions in a horizontally stratified medium are constructed with the use of a modal summation method.The limits of applicability of the algorithm are illustrated by the inversion of four synthetic seismograms constructed for two horizontally stratified models of the structure in Friuli area, Italy. The records constructed for the same structural model as for which the Green functions were computed can be inverted even in the high-frequency range. In the opposite case, when the records and Green functions used corresponded to different structural models, a good correlation of the input records with the final synthetics was obtained for low - pass filtered data only.Additional tests performed with input seismograms contaminated with random noise yielded good resolution of the moment tensor and the duration of the source time function even for a high noise to signal ratio.     

Time-harmonic response of transversely isotropic multilayered half-space in a cylindrical coordinate system

With the aid of the analytical layer-element method, a comprehensive analytical derivation of the response of transversely isotropic multilayered half-space subjected to time-harmonic excitations is presented in a cylindrical coordinate system. Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the Fourier expansion, Hankel and Laplace integral transform, analytical layer-elements for a finite layer and a half-space are derived. Considering the continuity conditions on adjacent layers׳ interfaces and the boundary conditions, the global stiffness matrix equations for multilayered half-space are assembled and solved. Finally, some numerical examples are given to make a comparison with the existing solution and to demonstrate the influence of parameters on the dynamic response of the medium.     

THE EXACT SEISMIC RESPONSE OF AN OCEAN AND A N-LAYER CONFIGURATION*

The space-time acoustic wave motion generated by an impulsive monopole source is calculated with the aid of the Cagniard-de Hoop technique. Two configurations with plane interfaces are discussed: an air/fluid/solid configuration with the source and the receiver located in the fluid layer; and a stack of n fluid layers between two acoustic half-spaces where the source and the receiver are located in the upper half-space. Synthetic seismograms are generated for the pressure of the reflected wavefield, using the source signature of an airgun.     

弱各向异性介质弹性波的准各向同性近似正演模拟

准各向同性(QI)近似可用于弱各向异性介质的正演模拟.本文通过运用QI方法的零阶和一阶近似,计算了VTI介质模型的地震记录.得出的地震记录与标准各向同性射线理论(IRT)和基于伪谱法的三维地震正演模拟得出的地震记录作了比较,可以认为是精确的合成地震记录.     

风化半圆形河谷对柱面SH波的散射解析解

风化河谷地震效应对建在此类场地上工程结构(如大坝、桥梁)的动力稳定性具有潜在的威胁,为揭示河谷风化层对地面运动的影响,利用波函数展开法推导风化半圆形河谷对线源柱面SH波散射问题的解析解。计算不同震源位置条件下风化半圆形河谷的地表位移反应和地面运动放大因子,分析地面运动的幅值和形态,发现随着震源距离的增大,在河谷附近地震动的放大和衰减的交替更频繁。     

层状各向同性多孔介质中井间地震的数值计算

依据Ｂｉｏｔ理论，提出计算理论地震图的相应传播矩阵算法，并针对井间地震震源和检波器的不同配置进行数值计算，结果表明，本文算法高频可达１０００Ｈ２以上．文中算例说明利用震源和检波器相对于储油层的不同位置，可从反射震相记录波形上的差异，判定储油层存在与否．     

二维弹性波的有限元模拟及其初步实践

本文讨论了二维介质中弹性波有限元模拟的一种方法;导出了二维无界空间中集中力点源的理论初动表达式和位移波形表达式,并和相似情况下的有限元结果进行了比较;对用有限元方法算得的几个中小模型的节点位移进行了分析。从所能鉴别的震相的到时、初动符号分布、波形、瑞利波质点运动轨迹等证据看,在得到比较真实的全波理论地震图方面,有限元方法具有很大的潜力。     

DOWNHOLE SYNTHETIC SEISMIC PROFILES IN ELASTIC MEDIA1

A multichannel lattice filter structure is utilized to represent seismic waves propagating in adjacent layers in an elastic medium. Using this model, an explicit time-domain solution for arbitrary source and receiver locations is obtained as an ARMA (AutoRegressive and Moving-Average) process. The lattice and ARMA structures have given rise to an effective algorithm for the calculation of offset/downhole synthetic seismograms. A large range of recently developed offset/downhole seismic survey geometries, such as the ‘Yo-Yo’ arrangement, can thus be simulated. In addition, the explicit solutions for upgoing and downgoing waves provide new insight into the properties of general downhole seismic signals, including wave-mode conversion effects and multiple reflections. Furthermore, offset/downhole seismograms generated by a line source (i.e. 2D point source) can also be constructed by superposition of plane waves with different incidence angles. Synthetic seismograms generated using a different source-receiver arrangement indicate that the properties especially associated with offset/downhole seismic signals can be predicted by this modelling method. These properties include arrival times, amplitude attenuation and wave-mode conversion effects. Finally, utilizing this numerical modelling method to a real downhole survey with Yo-Yo geometry may lead to a proper data acquisition and processing procedure, and improves the interpretation confidence of the field section.     

Theoretical seismogram constraints on some crustal velocity models in the central United States

Summary Earthquake investigations in the central United States require a realistic crustal model for accurate location of seismic events, for the determination of earthquake source parameters, and for theoretical ground motion prediction for an assumed earthquake source. A number of crustal models have been proposed which were based on first arrival refraction data obtained from artificial and natural seismic sources.The reflectivity method for generating realistic theoretical seismograms is used together with recently acquired refraction survey data to distinguish among several of the proposed crustal models. The data used in this study preclude a definitive statement about the nature of the lower crust in the region. However, a crust with two or more layers in the upper 20 km immediately below the basement is preferred. The absence of prominent near yertical reflections suggests that the layers may be separated by rapid velocity gradients rather than by sharp discontinuities.     

用传播矩阵法研究层状正交各向异性地层中多分量感应测井响应

本文采用传播矩阵技术研究并建立了层状正交各向异性地层中多分量感应测井响应的有效算法.首先通过Fourier变换将频率空间域中的Maxwell方程组求解问题转化为频率波数域中关于电磁场水平分量常微分方程组的定解问题.利用该方程组系数矩阵的本征值和归一化本征向量将电磁场分解成上行波和下行波模式的组合,推导出均匀正交各向异性介质中由任意方向磁偶极子产生的电磁波模式解析表达式;在此基础上,利用叠加原理和边界条件研究了电磁波在层状正交各向异性地层中的反射和透射,给出各个界面上的广义反射系数和不同地层中电磁波振幅的递推公式,进而得到电磁波模式的解析解.为了有效确定频率空间域中的电磁场,采用二维Patterson自适应求积算法结合有限连分式展开技术计算傅氏逆变换.最后通过数值模拟结果证明了该算法的有效性,考察了不同各向异性系数、不同井眼倾角以及仪器长度和工作频率变化等情况下的多分量感应测井响应特征.     

层状介质中地震面波频散函数和体波广义反射系数的计算

本文在B、P、C坐标中给出了弹性波在横向均匀介质中的传播矩阵,并将其表示为五个形式简单的矩阵乘积,其中有四个矩阵是与频率无关的。我们用传播矩阵的分离方式,将Abo-Zena（1979）算法大量简化。用本文的算法计算综合地震图时,计算量要比文献[2,7,8]的算法少一半以上。 文中还给出了地震面波频散函数和体波广义反射系数的快速计算步骤。     

基于理论地震图方法的波导结构对 地下核爆炸Lg波传播特性研究

基于频率-波数域算法的理论地震波形图方法, 可以数值模拟频率达到10 Hz、 震中距达1000 km的区域理论地震波形图. 该算法适用于计算大量分层地壳结构中激发的导波Lg波. 本文在前苏联东哈萨克斯坦地下核试验场至我国乌鲁木齐台站间的地球介质速度模型中, 引入速度梯度结构、 速度扰动分布的薄叠加层结构、 降低Q值结构以及速度扰动与Q值变化的综合结构来模拟实际地壳波导结构, 较好地模拟出东哈萨克斯坦地下核爆炸地震在乌鲁木齐台站记录的宽频带地震波形图, 模拟出完整的Lg波序列, 该序列符合Lg波能量分布特征, 且能够解释Lg波波尾的特征. 结果表明, Lg波的形状和峰值结构均依赖于地壳的不同波导结构.      

基于频率-慢度域叠前地震记录的合成与并行方法研究

正演模拟是叠前弹性波反演的基础.采用慢度法计算层状介质的叠前地震记录,分别对频率和慢度进行积分变换得到时-空域的地震道集,并对在慢度积分过程中产生的计算噪音提出了解决方案.为得到高精度合成地震记录,需将地层细分,但地层层数很多时,计算量较大;而对地层粗分虽然会大大加快运算速度,但合成记录会丢失很多信息,文中给出了地层的划分原则.该方法能够计算出包括转换波和多次反射在内的全地震响应.但在提高合成记录精度的同时,也导致计算量增大、计算效率降低,因此,本文对基于慢度法全波场模拟进行了并行算法设计,采用计算域分割、工作池并行技术,建立了慢度法全波场正演模拟的并行算法,使得弹性波正演问题求解更加高效,为充分利用叠前地震资料进行叠前反演提供了研究基础.     

使用地面单台三分向地震记录资料评估场地效应方法的研究

利用唐山强震观测台阵提供的井下和地面数字测震资料，研究了用单台地震记录评估场地效应的方法。利用井下基岩地震记录作参考台，用地面和井下的记录求取经验传递函数。同时应用Nakamura单台评估方法求解Nakamura谱比率。通过对比，证实了对场地效应的评估可以使用地面单台测震资料进行，不必参考基岩台的资料。因此，该方法是评估重要工程场地效应的一种简便可行的方法。使用该方法可以降低地震小区划的成本。     

局部角度域波传播步进算法研究

本文从非均匀介质中波动方程出发,提出了基于一般标架的相空间（局部角度域）波传播的步进算法. 该方法在构造单程波的步进算法时,在选择标架或正交基等方面有更大的自由度. 我们以不随频率及深度变化且具有变尺度特性的Gabor_Daubechies紧标架为例,给出了单程波传输算子的具体形式及相应的波场步进算法;详细讨论了基于Gabor_Daubechies标架的传输算子的高频渐近展开问题,得出了在高频、小传输步长条件下传输算子的近似解析表达式,并给出使用条件. 通过模型算例,比较了精确传输算子与高频近似传输算子用于非均匀介质中波传播的结果,说明在一定条件下由两者得出的波场几乎是相同的.     

Thermoelastic deformation of a transversely isotropic and layered half-space by surface loads and internal sources

A general method is developed for the study of transient thermoelastic deformation in a transversely isotropic and layered half-space by surface loads and internal sources. A Laplace transform is first applied to the field quantities; Cartesian and cylindrical systems of vector functions are then introduced for reducing the basic equations to three sets of simultaneous linear differential equations. General solutions are obtained from these sets, and propagator matrices from the solutions by a partitioned matrix method.

Source functions for a variety of sources are derived in the Cartesian and cylindrical systems, and the Laplace transformed expressions of the field variables at the surface presented explicitly in the two systems in terms of a layer matrix. The effect of gravity is included by multiplying simply an effect matrix resulting from the modification of continuity conditions at the surface and the layer interfaces.

It should be noted that the present analytical method has great advantages over either the classical thin plate approach or the finite element method, and that the present result can be reduced directly to the solutions of the corresponding isotropic case.