Space-time ray method for seismic wave fields

Summary The space-time ray method can be applied to the evaluation and continuation (extrapolation) of the complete seismic wave field in laterally inhomogeneous media with curved interfaces. The wave field propagates along certain space-time curves, called space-time rays. Their space projections correspond to standard rays. Examples of possible applications of the space-time ray method, where the standard ray method fails, are as follows: a) The propagation of seismic waves in slightly dissipative media, b) The computation of seismic wave fields generated by seismic sources with direction-dependent source-time variations. c) Downward continuation of the seismic wave field (actual seismograms) measured at the Earth's surface.     

起伏层状介质中曲线网格有限差分法与射线法波场模拟对比研究

针对处理起伏地表(或含地下不规则波阻抗界面)条件下发展起来的地震波场数值模拟算法的模拟结果与解析解(大多数情形下无法得到)无法进行对比,且其有效性和正确性难以验证的情况,本文提出了一种可以相互验证波场数值模拟结果与射线追踪数值模拟结果的正确性和有效性的佐证方法,验证了参考射线追踪法.其中,波场数值模拟中采用曲线网格DRP/opt MacCormack有限差分法,射线追踪模拟则采用分区多步三角网格最短路径算法.通过系统对比上述两种方法得到的波场快照、单炮地震记录,以及合成理论地震图的结果显示,本方法相互作证了两种方法所得结果的正确性和有效性.双层和三层起伏层状模型的对比分析结果表明,这种方法不但可以加深理解地震波在复杂介质中的传播规律,同时射线法的引入为清晰识别和标定地震波场数值模拟中各种不同震相提供了一种便捷的途径.      

TIME-DOMAIN COMPUTATION OF NON-NORMAL INCIDENCE WAVEFIELDS IN PLANE LAYERED MEDIA1

A new time-domain method is introduced for the calculation of theoretical seismograms which include frequency dependent effects like absorption. To incorporate these effects the reflection and transmission coefficients become convolutionary operators. The method is based on the communication theory approach and is applicable to non-normal incidence plane waves in flat layered elastic media. Wave propagation is simulated by tracking the wave amplitudes through a storage vector inside the computer memory representing a Goupillaud earth model discretized by equal vertical transit times. Arbitrary numbers of sources and receivers can be placed at arbitrary depth positions, while the computational effort is independent of that number. Therefore, the computation of a whole plane-wave vertical seismic profile is possible with no extra effort compared to the computation of the surface seismogram. The new method can be used as an aid to the interpretation of plane-wave decomposed reflection data where the whole synthetic vertical seismic profile readily gives the interpreter the correct depth position of reflection events.     

Application of the edge wave superposition method

Numerical examples of high-frequency synthetic seismograms of body waves in a 2-D layered medium with complex interfaces (faults, wedges, curvilinear, corrugated) are presented. The wave field modeling algorithm combines the possibilities of the ray method and the edge wave superposition method. This approach preserves all advantages of the ray method and eliminates restrictions related to diffraction by boundary edges and to caustic effects in singular regions. The method does not require two-point ray tracing (source-to-receiver), and the position of the source, as well as the type of source, and the position of receivers can be chosen arbitrarily. The memory and the time required for synthetic seismogram computation are similar to ray synthetic seismograms. The computation of the volume of the medium (the Fresnel volume or Fresnel zones), which gives the essential contribution to the wave field, is included in the modeling program package. In the case of complicated irregular interface (or a layered medium with a regular ray field at the last interface), the method displays a high accuracy of wave field computation. Otherwise, the method can be considered a modification of the ray method with regularization by the superposition of edge waves.     

地震波在垂向不均匀介质中传播问题的某些探讨

本文计算了含有高速夹层介质中首波的理论地震图。通过分析得到,当高速夹层薄到一定程度时,就会产生干涉型首波,从而从一个侧面证明了射线理论的局限性。通过对地震波反射—折射系数能量守恒关系的分析,探讨了反射—折射系数大于1的可能性。最后,介绍了一种计算垂向不均匀介质中拉梅问题理论地震图的数值方法——有限差分法。     

A STUDY OF THE SEISMIC SIGNATURES OF SEDIMENTATION MODELS USING SYNTHETIC SEISMOGRAMS*

In modern exploration for hydrocarbons there is a great emphasis on the location of stratigraphic traps and estimation of lithologic information like sand-shale ratios from seismic data. In order to investigate the possibilities of success in this endeavour we have studied the synthetic seismograms for wave form and spectral characteristic for four basic sedimentation models: (I) interbedded sand-shale model representing the sediments of generally fluviatile origin, (2) interbedded coal-shale model representing deltaic deposits, (3) sedimentary models representing transgression and regression of shore lines, and (4) a basal sand model. The results have shown that for the first two models a change in the sand-shale or coal-shale ratio results in a characteristically different seismogram. The nature of the seismogram, however, is also strongly dependent on how the sand-shale or coal shale layers are arranged to ultimately give the same number of total layers, thus implying the same coal-shale or sand-shale ratios. The transgression, regression, and basal sand models also produce characteristically different seismic signatures. The spectra of these seismograms show attendant characteristic changes. However, it seems that in the case of real data which are disturbed by noise and the effects of overlying layers these characteristic features may not always be distinguishable.     

Synthetic seismograms for finite sources in spherically symmetric Earth using normal-mode summation

Normal-mode summation is the most rapidly used method in calculating synthetic seismograms. However, normal-mode summation is mostly applied to point sources. For earthquakes triggered by faults extending for as long as several 100 km, the seismic waves are usually simulated by point source summation. In this paper, we attempt to follow a different route, i.e., directly calculate the excitation of each mode, and use normal-mode summation to obtain the seismogram. Furthermore, we assume the finite source to be a ‘‘line source' and numerically calculate the transverse component of synthetic seismograms for vertical strike-slip faults. Finally, we analyze the features in the Love waves excited by finite faults.     

Point source radiation in inhomogeneous anisotropic structures

The ray formulae for the radiation from point sources in unbounded inhomogeneous isotropic as well as anisotropic media consist of two factors. The first one depends fully on the type and orientation of the source and on the parameters of the medium at the source. We call this factor the directivity function. The second factor depends on the parameters of the medium surrounding the source and this factor is the well-known geometrical spreading. The displacement vector and the radiation pattern defined as a modulus of the amplitude of the displacement vector measured on a unit sphere around the source are both proportional to the ratio of the directivity function and the geometrical spreading.For several reasons it is desirable to separate the two mentioned factors. For example, there are methods in exploration seismics, which separate the effects of the geometrical spreading from the observed wave field (so-called true amplitude concept) and thus require the proposed separation. The separation also has an important impact on computer time savings in modeling seismic wave fields generated by point sources by the ray method. For a given position in a given model, it is sufficient to calculate the geometrical spreading only once. A multitude of various types of point sources with a different orientation can then be calculated at negligible additional cost.In numerical examples we show the effects of anisotropy on the geometrical spreading, the directivity and the radiation pattern. Ray synthetic seismograms due to a point source positioned in an anisotropic medium are also presented and compared with seismograms for an isotropic medium.     

PHASE AND GROUP TIME SECTIONS AND POSSIBILITIES FOR THEIR USE IN SEISMIC INTERPRETATION OF COMPLEX MEDIA*

Interpretation problems are discussed for a new class of models for complex seismic media, called heterogeneous models formed by inclusions (HMI). Examples of such models in geology are destructive deformation zones, tectonic raptures, complex folds, magmatogene formations, fronts of metamorphism and of phase transition, etc., which are of importance in interpretation of seismic data. The wavefields in such media have a complicated interferential character and should be considered as complex wave groups characterized by their phase and group properties. To study the phase and group characteristics of such wavefields, a method of construction and comparative analysis of so-called phase and group sections is introduced. This method is based on a transformation of the wavefield (seismogram) into a normalized seismogram (cos of the phase) and a perigram (a low cut version of the trace envelope). The group sections obtained on the perigrams represent zones of energy concentration and give stable estimates of the average characteristics of model structure. The phase sections are obtained on the normalized seismograms and represent primarily, the inner structure of the model. The method was applied to both synthetic and field data. The results of the combined analysis of the phase and group sections show that in many cases there are significant differences between them. On the basis of this analysis, several types of seismic objects may be distinguished which can serve as a basis for seismic interpretation.     

A cloud-based synthetic seismogram generator implemented using Windows Azure

Synthetic seismograms generated by solving the seismic wave equation using numerical methods are being widely used in seismology. For fully three-dimensional seismic structure models, the generation of these synthetic seismograms may require large amount of computing resources. Conventional high-performance computer clusters may not provide a cost-effective solution to this type of applications. The newly emerging cloud-computing platform provides an alternative solution. In this paper, we describe our implementation of a synthetic seismogram generator based on the reciprocity principle using the Windows Azure cloud application framework. Our preliminary experiment shows that our cloud-based synthetic seismogram generator provides a cost-effective and numerically efficient approach for computing synthetic seismograms based on the reciprocity principle.     

华北地区地震地面危害分布的定量估算

根据1500年以来的地震活动,用确定性方法对华北地区地震地面危害分布图进行了定量计算。震源和研究区域均用网格单元表示,网格点的间距取为0.2°。计算中主要的输入参量为该地区的地震目录、震源机制、地震活动水平、深部结构模型等。各个单元上模拟的理论地震图考虑到了来自不同地点的震源和传播路径的影响。计算获得了华北地区最大地面运动和设计地面加速度(DGA)的分布图。最大DGA值位于北京以东,达0.75g。该结果可对未来华北地区地震危害的分布特征研究和该地区的防震减灾工作提供依据     

Validation of 3D synthetic seismograms based on the ray-Born approximation

The first-order Born approximation is a weak scattering perturbation method which is a powerful tool. The combination of the Born approximation and the ray theory enables to extend the applicability of the ray theory in terms of the required smoothness of the model and ensures faster computations than with, e.g., the finite difference method. We are motivated to describe and explain the effects of the numerical discretization of the Born integral on the resulting seismograms. We focus on forward modelling and study the cases in which perturbation from the background model contains the interface. We restrict ourselves to isotropic models that contain two homogeneous layers. We compare the 2D and 3D ray-based Bornapproximation seismograms with the ray-theory seismograms. The Born seismograms are computed using a grid of finite extent. We anticipate that the computational grid should contain an appropriate number of gridpoints, otherwise the seismogram would be inaccurate. We also anticipate that the limited size of the computational grid can cause problems. We demonstrate numerically that an incorrect grid can produce significant errors in the amplitude of the wave, or it can shift the seismogram in time. Moreover, the grid boundaries work as interfaces, where spurious waves can be generated. We also attempt to explain these phenomena theoretically. We give and test the options of removing the spurious waves. We show that it is possible to compute the Born approximation in a sparser grid, if we use elastic parameters averaged from some dense grid.     

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.     

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 NONITERATIVE PROCEDURE FOR INVERTING PLANE-WAVE REFLECTION DATA AT SEVERAL ANGLES OF INCIDENCE USING THE RICCATI EQUATION*

Various exact methods of inverting the complete waveform of vertical seismic reflection data to produce acoustic impedance profiles have been suggested. These inverse methods generally remain valid for nonvertical, plane-wave data, provided total reflection does not occur. Thus, in principle, the “seismogram” at each ray parameter in a slant stack can be interpreted separately. Rather than invert each plane-wave seismogram separately, they can all be interpreted simultaneously and an “average” model thus obtained. Inversion for both the velocity and the density also becomes possible when two or more plane-wave seismograms are simultaneously inverted. The theory for a noniterative inversion method, based on the time-domain Riccati equation, is discussed. Numerical examples of inversions using this technique on synthetic data demonstrate its numerical stability and the advantage of simultaneous inversion of several seismograms to reduce the effect of noise in the data and increase the stability of the inversion process.     

火山地震的理论地震图计算方法

本文首先分析了不同类型火山地震的波形,提出了相应的震源模型.然后发展了计算层状介质理论地震图的部分分离变量—有限差分方法,其最重要的环节是引用了吸收边界条件,使计算工作得以简化.这种方法特别适用于火山地震的理论地震图计算.最后,本文给出了计算实例.     

PS转换波界面二次源法射线追踪

界面二次源法是最近提出的一种最小走时射线追踪方法,尤其适合层状介质中走时和射线路径的计算.该方法相对于传统的最小走时树方法（如Moser法）,仅在物性界面上设置二次源,射线路径的方向只在层界面处发生改变,该方法最大程度地消除了射线路径的锯齿状现象,同时也避免了低变速区的射线路径多值现象,因此,它具有更高的追踪精度和效率.本文采用界面二次源法在各向同性介质中实现了PS转换波射线追踪,理论模型的计算证实了界面二次源法追踪PS转换波的准确性和高效性,同时该方法在各向异性介质中也很好地追踪出分离的PSV波和PSH波,因此该方法有利于横波分裂在地震勘探中的研究和应用     

Teleseismic receiver function using stacking and smoothing of multi seismic-records at a single station

Receiver functions (RFs) obtained using teleseismic wave records at a seismic station and synthetic seismograms indicate that RF with a single teleseismic wave record is related to the selection of record section and to the calculating parameters of the RF. The scatter noise contained in the seismogram also affects the quality of RF. A new method for calculating receiver function, stacking and smoothing multi-seismic records in a single station, is presented in this paper. The RF results using some records and some synthetic seismograms with different noises indicate prominent mantle discontinuity and thus prove that the method is effective and satisfied.     

ON THE POSSIBILITY OF SEISMIC EXPLORATION USING SURFACE TORQUE SOURCE AND RELATED TOPICS*

In this study we derive expressions for particle displacement or particle velocity anywhere inside a stratified earth and at its surface due to horizontal torque source located in the top layer. Equivalently, invoking Green's function reciprocity theorem, the solution applies also to the case of a surface or subsurface source when the resulting displacement or velocity is measured within the top layer. In order to evaluate the closed-form analytical solution economically and accurately it is advisable to introduce inelastic attenuation. Causal inelastic attenuation also lends the necessary realism to the computed seismic trace. To provide proof that the analytical solution is indeed correct and applicable to the multilayer case, a thick uniform overburden was assumed to consist of many thin layers. The correctness of the computed particle velocity response can be very simply verified by inspection. The computed response can also serve as a check on other less accurate methods of producing synthetic seismograms, such as the techniques of finite differences, finite elements, and various sophisticated ray-tracing techniques. It is not difficult to construct horizontal surface torque source. It appears that such source is well suited for seismic exploration in areas with a high-velocity surface layer. A realistic source function is analyzed in detail and normalized displacement response evaluated at different incidence angles in the near and the far fields. In an effort to distinguish the features of an SH torque seismogram from a pressure seismogram two models with identical layerings and layer parameters have been set up. As expected the torque seismogram is very different from the compressional seismogram. One desirable feature of a torque seismogram is the fast decay of multiples. Exact synthetic seismograms have many uses; some of them, such as the study of complex interference phenomena, phase change at wide angle reflection, channeling effects, dispersion (geometrical and material), absolute gain, and inelastic attenuation, can be carried out accurately and effortlessly. They can also be used to improve basic processing techniques such as deconvolution and velocity analysis. The numerical evaluation of the analytical solution of the wave equation as described in this paper has a long history. Most of the work leading to this paper was carried out by one of us (M. J. K.) in the years 1957 to 1968 at the Geophysical Research Corporation. However, the full testing of the various computer codes was carried out only very recently at the Phillips Petroleum Company.     

Theoretical study on the influence of CMB topography on the core reflection ScS

Computing synthetic seismograms for media with localized heterogeneous regions can be performed using hybrid methods. Here, a combination of a finite-difference (FD) technique and a frequency-wavenumber (ω − k) filtering is applied to model wave reflection at different kinds of core-mantle boundary (CMB) topography. The FD method is only applied in the neighbourhood of the CMB, while the ω − k filter is used to continue the reflected wavefield to the Earth's surface. Synthetic SH-seismograms for ScS with a dominant frequency of 0.5 Hz are computed at epicentral distances from 44° to 69°. The topography varies in amplitude (maximum amplitude of 1.0–2.7 km) and in its wavenumber spectrum; it is either monochromatic (wavelengths from 55 to 270 km) or statistical (coloured noise). The seismograms for a CMB with topography are compared with those for a plane CMB. We observe that monochromatic topography with short wavelengths (less than 100 km) results in amplitude reduction and shorter travel times than in the case of a plane CMB, but no variations with epicentral distance appear, whereas greater wavelengths exhibit amplitude variations with distance as well as travel time residuals, which both correlate with the CMB topography. Statistical models show amplitude variations with epicentral distance, while the travel time residuals are very small (less than 0.1 s). All synthetics illustrate that wavefront healing occurs along the ray path from the CMB to the Earth's surface. While the seismograms at the CMB exhibit strong fluctuations, the fluctuations at the surface are smoothed and reduced. This demonstrates that it is necessary to use wave theoretical methods for computing synthetic seismograms for complicated structures at greater depth. It also follows that travel times are less sensitive to the structure than the amplitudes.