Numerical modeling of Gaussian beam propagation and diffraction in inhomogeneous media based on the complex eikonal equation

Gaussian beam is an important complex geometrical optical technology for modeling seismic wave propagation and diffraction in the subsurface with complex geological structure. Current methods for Gaussian beam modeling rely on the dynamic ray tracing and the evanescent wave tracking. However, the dynamic ray tracing method is based on the paraxial ray approximation and the evanescent wave tracking method cannot describe strongly evanescent fields. This leads to inaccuracy of the computed wave fields in the region with a strong inhomogeneous medium. To address this problem, we compute Gaussian beam wave fields using the complex phase by directly solving the complex eikonal equation. In this method, the fast marching method, which is widely used for phase calculation, is combined with Gauss–Newton optimization algorithm to obtain the complex phase at the regular grid points. The main theoretical challenge in combination of this method with Gaussian beam modeling is to address the irregular boundary near the curved central ray. To cope with this challenge, we present the non-uniform finite difference operator and a modified fast marching method. The numerical results confirm the proposed approach.     

A combined study of macroseismic data and focal mechanisms applied to the West-Bohemian earthquake,Czechoslovakia

The main shock of the West-Bohemian earthquake swarm, Czechoslovakia, (magnitudem=4.5, depthh=10 km) exhibits an irregular areal distribution of macroseismic intensities 6° to 7° MSK-64. Four lobes of the 6° isoseismal are found and the maximum observed intensity is located at a distance of 8 km from the instrumentally determined epicentre. This distribution can be explained by the energy flux of the directS wave generated by a circular source, the hypocentral location and focal mechanism of which are taken from independent instrumental studies. The theoretical intensity, which is assumed to be logarithmically proportional to the integrated squared ground-motion velocity (i.e.,I=const+log v ² (t)dt), fits the observed intensity with an overall root-mean-square error less than 0.5°. It is important that the present intensity data can also be equally well explained by the isotropic source. The fit was attained by means of a horizontally layered model though large fault zones and an extended sedimentary basin suggest a significant lateral heterogeneity of the epicentral region. The results encourage a broader application of the simple modelling technique used.     

Effects of causal absorption on seismic body waves

Summary The effects of the causal absorption on seismic wave fields are discussed. Several computer programs for the evaluation of seismic wave fields have been modified to include the causal absorption. The programs are based mostly on matrix and ray methods. The main attention is paid to ray methods. A simple plane wave algorithm is described, which can be used along the ray to compute elementary seismograms in dissipative media. For a certain class of signals, commonly used in seismology, its application does not require any convolution or frequency-domain computations, and is only slightly slower than the evaluation of elementary seismograms in non-dissipative media. The derived algorithm even describes the velocity dispersion related to absorption and the decrease of the prevailing frequency of the signal well. Numerical examples are presented, both exact and approximate.     

A new approximation of the velocity-depth distribution and its application to the computation of seismic wave fields

Summary A new approximation of the velocity-depth distribution in a vertically inhomogeneous medium is suggested. This approximation guarantees the continuity of velocity and of its first and second derivatives and does not generate false low-velocity zones. It is very suitable for the computations of seismic wave fields in vertically inhomogeneous media by ray methods and its modifications, as it removes many false anomalies from the travel-time and amplitude-distance curves of seismic body waves. The ray integrals can be evaluated in a closed form; the resulting formulae for rays, travel times and geometrical spreading are very simple. They do not contain any transcendental functions (such asln (x) orsin ^–1, (x)) like other approximations; only the evaluation of one square root and of certain simple arithmetic expressions for each layer is required. From a computational point of view, the evaluation of ray integrals and of geometrical spreading is only slightly slower than for a system of homogeneous parallel layers and even faster than for a piece-wise linear approximation.     

Estimating the elastic parameters of anisotropic media using a joint inversion of P-wave and SV-wave traveltime error

A method is presented to estimate the elastic parameters and thickness of media that are locally laterally homogeneous using P‐wave and vertically polarized shear‐wave (SV‐wave) data. This method is a ‘layer‐stripping’ technique, and it uses many aspects of common focal point (CFP) technology. For each layer, a focusing operator is computed using a model of the elastic parameters with which a CFP gather can be constructed using the seismic data. Assuming local homogeneity, the resulting differential time shifts (DTSs) represent error in the model due to anisotropy and error in thickness. In the (τ?p) domain, DTSs are traveltimes Δτ that connect error in layer thickness z, vertical slowness q, and ray parameter p. Series expansion is used to linearize Δτ with respect to error in the elastic parameters and thickness, and least‐squares inversion is used to update the model. For stability, joint inversion of P and SV data is employed and, as pure SV data are relatively rare, the use of mode‐converted (PSV) data to represent SV in the joint inversion is proposed. Analytic and synthetic examples are used to demonstrate the utility and practicality of this inversion.     

地震波场的复射线展开

利用复射线理论将惠更斯原理推广到复空间,发展出一种简便的波场计算方法.本文以均匀柱面波在平界面上的反射场为例,用复射线展开法计算了时谐地震波及瞬变地震波在部分反射、全反射和临界角过渡区的场分布,并与平面波谱积分结果进行了比较.结果表明,用复射线展开法可以自动平滑过渡区反射场的奇异性,因而该方法可直接处理焦点、焦散线和全反射过渡区等复杂的波场分析与综合问题,并可简化数值计算过程.     

Anisotropic radiation modelling of macroseismic intensities for estimation of the attenuation structure of the upper crust in Greece

A method is suggested for the analysis of macroseismic intensity data in order to accurately determine an average attenuation structure of the upper part of the crust in an area. The method is based on a model which assumes that the observed intensities depend on source properties (radiation pattern, size, focal depth), geometrical spreading and anelastic attenuation. The method is applied to 13,008 intensity values, observed in corresponding sites of Greece and grouped (in 4228 groups), according to their spatial clustering in order to diminish observational errors and site effects. An average intensity attenuation coefficient,c=–0.0039±0.0016, corresponding to a quality factor, Q=350±140, is determined for the upper 20 km of the crust in this area. This value is relatively low, in good agreement with the relatively high heat flow and high seismic activity of this area. A byproduct of the present study is the determination, for each earthquake, of a macroseismic focal depth and of a macroseismic size, which is strongly correlatted with both the earthquake's magnitude and its seismic moment determined by independent methods.     

Shape Analysis of Isoseismals Based on Empirical and Synthetic Data

—?We present an attempt to compare modeled ground-motion acceleration fields with macroseismic observations. Two techniques for the representation of the observed intensities by isoseismals, a smoothing technique and one which visualizes the local uncertainty of an isoseismal, are tested with synthetic and observed data. We show how noise in the data and irregularities in the distribution of observation sites affect the resolution of the isoseismal's shape. In addition to “standard” elongated shapes, we identify cross-like patterns in the macroseismic observations for two Italian earthquakes of strike-slip type; similar patterns are displayed by the theoretical peak acceleration fields calculated assuming the point source models given in the literature.     

The 23 April 1909 Benavente earthquake (Portugal): macroseismic field revision

The 23 April 1909 earthquake, with epicentre near Benavente (Portugal), was the largest crustal earthquake in the Iberian Peninsula during the twentieth century (M _w = 6.0). Due to its importance, several studies were developed soon after its occurrence, in Portugal and in Spain. A perusal of the different studies on the macroseismic field of this earthquake showed some discrepancies, in particular on the abnormal patterns of the isoseismal curves in Spain. Besides, a complete list of intensity data points for the event is unavailable at present. Seismic moment, focal mechanism and other earthquake parameters obtained from the instrumental records have been recently reviewed and recalculated. Revision of the macroseismic field of this earthquake poses a unique opportunity to study macroseismic propagation and local effects in central Iberian Peninsula. For this reasons, a search to collect new macroseismic data for this earthquake has been carried out, and a re-evaluation of the whole set has been performed and it is presented here. Special attention is paid to the observed low attenuation of the macroseismic effects, heterogeneous propagation and the distortion introduced by local amplifications. Results of this study indicate, in general, an overestimation of the intensity degrees previously assigned to this earthquake in Spain; also it illustrates how difficult it is to assign an intensity degree to a large town, where local effects play an important role, and confirms the low attenuation of seismic propagation inside the Iberian Peninsula from west and southwest to east and northeast.     

Strong motion records and synthetic isoseismals of the Pyrgos,Peloponnisos, Southern Greece,Earthquake Sequence of March 26, 1993

In this work, several seismological observations are presented in order to explain characteristic features of the earthquake sequence which occurred in March 1993 in southwestern Greece, very close to the city of Pyrgos.Fault plane solutions of the largest fore- and aftershocks and the main shock, as well as the directions at which the maximum ground accelerations were recorded suggest that this earthquake sequence has been developed by rupturing three distinct focal planes with different focal mechanisms. The first focal plane, located in the off-shore area, strikes NW-SE, dips SE and includes most of the foreshock activity. The foreshock activity migrated to the northeastern part of the city of Pyrgos and took place on planes with a predominant direction NE-SW. The main shock ofM _x =5.5 occurred in a focal plane located between the two above-mentioned areas. Strong motion records of significant shocks of the sequence show peak acceleration values on components consistent with the relevant fault plane solutions.Furthermore, the observed macroseismic field has been compared with synthetic isoseismals computed by using a certain velocity model and the focal mechanism parameters of the main shock.     

汶川8.0级地震仪器烈度与宏观烈度对比分析

利用2008年汶川M8.0地震获得的强震动记录数据,根据《仪器地震烈度计算暂行规程》计算得到各台站处的仪器地震烈度值,分析仪器地震烈度与宏观地震烈度的对应关系,研究该仪器烈度计算方法的适用性。结果表明,利用该算法所得的仪器烈度值与宏观烈度完全吻合的比率为47.5%,偏差±1度以内的比率为89.1%,说明二者对应情况较为理想,仪器烈度可在一定程度上客观反映实际的震害情况;在各宏观烈度区内仪器烈度值虽然具有一定的离散性,但其均值与宏观烈度区值的偏差相对较小,均控制在±0.3度以内。另外,文中还绘制了汶川地震仪器烈度分布图,虽然与宏观烈度在整体分布上具有一定的对应关系,但受多种因素的影响,仪器烈度分布与宏观烈度分布不可能完全一致。仪器烈度与宏观烈度的概念和属性有所差异,发挥的作用也不尽相同,不应混淆和相互替代。     

含衰减地层微地震震源机制反演及其反演分辨率

微地震震源机制的反演对于非常规油气开发具有至关重要的作用.微地震信号主频高、能量小,容易受地层吸收衰减作用的影响使其波形发生畸变,本文提出了一种考虑地层吸收衰减作用的微震源机制反演方法,并利用费雷谢偏导矩阵的SVD分解(特征值分解)方法,分析研究了地层的吸收衰减因子的变化对于微地震震源机制反演分辨率的影响,根据理论计算给出了不同地震数据对各种震源机制反演的适用条件.理论计算证明,采用直达P波和S波数据联合反演震源的T值,单独利用直达P波反演震源的k值,可以有效降低地层吸收衰减作用对反演结果的影响.     

2014年阿拉善块体西缘两次中强地震震源机制解

2014年2月12日新疆于田发生了MS7.3地震,之后在阿拉善块体西缘相继发生了蒙古4.9级、新疆哈密5.1级地震。利用甘肃"十五"数字地震台网的波形资料,采用CAP方法反演这两次地震的震源机制解。结果显示蒙古4.9级地震为走滑型,而新疆哈密5.1级属于逆冲型,表明青藏块体与阿拉善块体之间的作用增强引发了这两次地震活动。     

Back‐projection stacking of P‐ and S‐waves to determine location and focal mechanism of microseismic events recorded by a surface array

We present an automatic method of processing microseismic data acquired at the surface by a star‐like array. The back‐projection approach allows successive determination of the hypocenter position of each event and of its focal mechanisms. One‐component vertical geophone groups and three‐component accelerometers are employed to monitor both P‐ and S‐waves. Hypocenter coordinates are determined in a grid by back‐projection stacking of the short‐time‐average‐to‐long‐time‐average ratio of absolute amplitudes at vertical components and polarization norm derived from horizontal components of the P‐ and S‐waves, respectively. To make the location process more efficient, calculation is started with a coarse grid and zoomed to the optimum hypocenter using an oct‐tree algorithm. The focal mechanism is then determined by stacking the vertical component seismograms corrected for the theoretical P‐wave polarity of the focal mechanism. The mechanism is resolved in the coordinate space of strike, dip, and rake angles. The method is tested on 34 selected events of a dataset of hydraulic fracture monitoring of a shale gas play in North America. It was found that, by including S‐waves, the vertical accuracy of locations improved by a factor of two and is equal to approximately the horizontal location error. A twofold enhancement of horizontal location accuracy is achieved if a denser array of geophone groups is used instead of the sparse array of three‐component seismometers. The determined focal mechanisms are similar to those obtained by other methods applied to the same dataset.     

河北省强震区内外深部S波速度结构特征研究

利用短周期记录的S波记录反演了沙城老震区及其邻区共3个台站下方的剪切波速度结构，并进行了对比分析。结合以前得到的邢台、唐山震区及其附近5个台站的速度结构，发现强震区具有几个大致相同的构造特征，即：（1）地壳厚度低于周围台站；（2）S波低速层速度低于周围地区，高低速层差异大；（3）具有较低的上地幔S波速度；（4）强震往往发生在低速层上面的高速层内或高低速层交界面上；（5）易震层往往是两个低速层夹持的高速层等，并对这种结构易于发震的机理给出了分析。总之强震的发生是与低速层紧密相磁的，而剪切波对于反映低速体构造非常敏感，因此S波速度结构研究对于揭示强震的深部构造背景，深入研究震源机制都是很有意义的。而且，强震我所共同具有的这种特征为我们今后强震危险区的划分和强震地点的预测提供了较为科学的依据。     

陕西地区小微震震源机制研究

小微震事件的震源机制是区域应力场及诸多地球动力学研究的基础资料。陕西地区为多个地震带的交汇区,近些年积累了丰富的小微震波形资料。运用新近发展的适用于求解小微震震源机制的广义极性振幅技术(GPAT),结合陕西2015地壳速度模型,求解陕西测震台网2011年4月至2015年12月间记录的121次ML1.5~3.5事件的震源机制。反演结果表明:(1)以上事件的震源机制大部分为走滑及正断类型,其比例占64.5%;逆断型机制占22.3%。(2)反演震源机制得到的震源深度与定位深度具有良好的一致性;矩震级与近震震级间存在差别,且这种差别随事件的变小而增大。(3)对比渭河断陷带相关研究成果,验证了该区域震源机制以正断型为主,具有拉张应力状态。     

Determination of focal depth from macroseismic data

Summary Authors give a diagram and procedure based onKövesligethy's formula for determining focal depth from macroseismic data. The method has the advantages that a) it supplies all the constants characterising, the macroseismal area, b) the personal effects are reduced, c) only one diagram is necessary.     

The mechanism of small mining tremors from amplitude inversion

A method of inverting seismic wave amplitudes from mining tremors is proposed to reveal their mechanism. A simple shear-implosive model of a mechanism is proposed. The shapes of theoretical and observed radiation patterns are compared in the least squares norm and the parameters describing the shear-slip together with the ratio of implosive and shear dislocations are optimized. The efficiency of the method, even in the case of a small number of stations, is demonstrated by applying it to weak mining tremors from two mine regions in Czechoslovakia.The weak mining tremors are found to be of a predominantly shear mechanism with a clear coincidence of nodal planes to local tectonics, which could imply their tectonic nature.     

Maslov面波理论地震图

研究了横向非均匀介质中的Ｍａｓｌｏｖ面波渐近理论，在横向弱非均匀介质的假设下，介质的纵向非均匀性反应在局部本征函数中，以局部本征函数近似真本征函数是射线理论的直接推论．由此，三维结构下的面波计算退化为准二维问题．由于本文方法属于慢度法，面波的频散使得在一般情况下得不到与体波ＷＫＢＪ方法相似的褶积结果；在震源函数为高斯波包的假设下，得出了与二维体波Ｍａｓｌｏｖ理论图形式上完全相同的褶积结果．还讨论了吸收介质中的面波波包理论图计算，最后结果与二维体波吸收介质中的结果相似．