Apparent phase velocities and fundamental-mode phase velocities of Rayleigh waves

Rayleigh wave dispersion data usually appear in the form of multimodal spectra for a layered model structure. The number of dispersion modal curves depends on the number of layers in the model. The measured dispersion velocities from the frequency–wavenumber (F–K) space, however, may not represent the true phase velocities of the fundamental-mode dispersion curve, but apparent phase velocities. The present study discusses how multimode curves are generated in the multichannel analysis of surface waves (MASW) method and the cause of the apparent velocity produced by the F–K method. Results from a field trial demonstrate that apparent phase velocities fail to reveal thin layers with low velocities. A better agreement of the inverted model with the geotechnical record is obtained by using the data points extracted from the fundamental-mode curve of the MASW spectral image.     

Simple equations guide high-frequency surface-wave investigation techniques

We discuss five useful equations related to high-frequency surface-wave techniques and their implications in practice. These equations are theoretical results from published literature regarding source selection, data-acquisition parameters, resolution of a dispersion curve image in the frequency–velocity domain, and the cut-off frequency of high modes. The first equation suggests Rayleigh waves appear in the shortest offset when a source is located on the ground surface, which supports our observations that surface impact sources are the best source for surface-wave techniques. The second and third equations, based on the layered earth model, reveal a relationship between the optimal nearest offset in Rayleigh-wave data acquisition and seismic setting—the observed maximum and minimum phase velocities, and the maximum wavelength. Comparison among data acquired with different offsets at one test site confirms the better data were acquired with the suggested optimal nearest offset. The fourth equation illustrates that resolution of a dispersion curve image at a given frequency is directly proportional to the product of a length of a geophone array and the frequency. We used real-world data to verify the fourth equation. The last equation shows that the cut-off frequency of high modes of Love waves for a two-layer model is determined by shear-wave velocities and the thickness of the top layer. We applied this equation to Rayleigh waves and multi-layer models with the average velocity and obtained encouraging results. This equation not only endows with a criterion to distinguish high modes from numerical artifacts but also provides a straightforward means to resolve the depth to the half space of a layered earth model.     

Analytical partial derivatives of the phase- and group velocities for Rayleigh waves propagating in a layer on a half-space

The dispersion relation for Rayleigh waves in a layer on a half-space is modified by introducing the quadratic wave number instead of the phase velocity. The implicit function theorem is then used to derive analytical formulae for the group velocity and for the phase- and group velocity partial derivatives with respect to the parameters of the medium. As two examples, the method is applied to the interpretation of dispersion curves for short-period Rayleigh waves observed in a sedimentary basin of the West Carpathians, and in the Moravo-Silesian region, Czech Republic.     

黏弹性介质瑞雷波有限差分模拟与特性分析

本文通过数值模拟研究了介质黏弹性对瑞雷波传播的影响.模拟采用结合了交错Adams-Bashforth时间积分法、应力镜像法和多轴完美匹配层的标准交错网格高阶有限差分方案.通过模拟结果和理论结果对比,测试了方法的精度,验证了结果的正确性.在均匀半空间模型中,分别从波场快照、波形曲线及频散能量图三个角度,对黏弹性介质瑞雷波衰减和频散特性进行了详细分析.两层速度递增模型被用于进一步分析瑞雷波在黏弹性层状介质中的特性.结果表明：由于介质的黏弹性,瑞雷波振幅发生衰减,高频成分比低频成分衰减更剧烈,衰减程度随偏移距增大而增强;瑞雷波相速度发生频散,且随频率增大而增大,频散能量的分辨率有所降低;黏弹性波动方程中的参考频率,不会影响瑞雷波振幅衰减和相速度频散的程度,但决定了黏弹性和弹性介质瑞雷波相速度相等的频率位置.本研究有助于人们更好地理解地球介质中瑞雷波的行为,并为瑞雷波勘探的应用和研究提供了科学和有价值的参考.     

Transmission and reflection of Rayleigh waves at a thin low velocity vertical layer: A laboratory and theoretical study

A three-dimensional ultrasonic modelling was used to study the Rayleigh waves reflected from a thin low velocity vertical layer and those transmitted through the same zone. The reflection and transmission coefficients of the Rayleigh waves have been calculated by an approximate method based on the Green function technique. Nonrigid contact boundary conditions were used for the low velocity layer modelling. A rather good correspondence between the theoretical and experimental absolute values of the reflection and the transmission coefficients was obtained. Some discrepancies between experimental and theoretical results, in particular with regard to phase shifts, may be explained by a certain inadequacy of the theoretical model to the low velocity layer investigated by the ultrasonic modelling.     

Analysis of Short-Period Rayleigh Waves Recorded in the Bohemian Massif Area During Celebration 2000 Experiment

The aim of this paper is to show the application of short-period surface waves recorded during deep seismic sounding experiment for constraining shallow velocity structure of the crust. Phase velocity of fundamental mode Rayleigh waves, observed along the CELEBRATION 2000 experiment profile CEL09, were obtained by a p-ω method and has been subsequently inverted for one-dimensional shear velocity models for the top 2 km. Multiple filter technique applied to one shot gather was used to carry out a joint inversion of phase and group velocity data and to provide γ_R data to be used for Q_β inversion. Validity of obtained V_S and Q_β models was confirmed by the reflectivity method. Noticeably, no clear dispersive wawes were observed in the Tepla-Barrandian Unit. Quasi-2D model based on the individual 1D V_S models is well correlated with the surface geology. Lower V_S are observed in the Saxothuringian Zone in comparison to the Moldanubian Zone. In the vicinity of the Central Bohemian and Moldanubian Plutons, the near-surface V_S values are relatively low, but below 1 km depth, they are higher than in surrounding areas. We interpret it as the result of the weathering and cracks within the granitoid rocks.     

Quasi-Rayleigh Waves in Transversely Isotropic Half-Space with Inclined Axis of Symmetry

A method for determination of characteristics of quasi-Rayleigh (qR) wave in a transversely isotropic homogeneous half-space with inclined axis of symmetry is outlined. The solution is obtained as a superposition of qP, qSV and qSH waves, and surface wave velocity is determined from the boundary conditions at the free surface and at infinity, as in case of Rayleigh wave in an isotropic half-space. Though the theory is simple enough, a numerical procedure for calculation of surface wave velocity presents some difficulties. The difficulty is caused by necessity to calculate complex roots of a non-linear equation, which in turn contains functions determined as roots of non-linear equations with complex coefficients. Numerical analysis shows that roots of the equation corresponding to the boundary conditions do not exist in the whole domain of azimuths and inclinations of the symmetry axis. The domain of existence of qR wave depends on the ratio of the elastic parameters: for some strongly anisotropic models the wave cannot exist at all. For some angles of inclination qR-wave velocities deviate from those calculated on the basis of the perturbation method valid for weak anisotropy, though they have the same tendency of variation with azimuth. The phase of qR wave varies with depth unlike Rayleigh wave in an isotropic half-space. Unlike Rayleigh wave in an isotropic half-space, qR wave has three components - vertical, radial and transverse. Particle motion in horizontal plane is elliptic. Direction of the major axis of the ellipsis coincides with the direction of propagation only in azimuths 0° (180°) and 90° (270°).     

Techniques for mode separation in Rayleigh wave testing

The spectral analysis of surface wave (SASW) developed in the early eighties has opened the way to the use of surface waves for the definition of shear wave velocity profiles in soil deposits or pavement structures without the need of any borings or intrusion. The SASW testing procedure was designed to minimize the contribution of higher modes and thus assumes that the Rayleigh waves which propagate at the surface belong only to the fundamental mode. Several studies have however demonstrated that, in some conditions, higher Rayleigh modes can contribute significantly to the dispersion curve. Different tests configurations exist today to deal with Rayleigh mode problem by the use of an array of receivers. In spite of that, the SASW configuration remains attractive due to the limited number of receivers, as well as, the Rayleigh modes contributing in SASW records configuration can be identified by multiple-filter technique and isolated using time-variable filters. The proposed techniques are first validated by simulated records and then applied to SASW records obtained in the field. The study confirms that higher modes can participate and even dominate in SASW records. An important contribution of higher Rayleigh modes can also exist, even if the shear wave velocity increases regularly with depth. The higher Rayleigh modes can significantly affect the accuracy of the shear wave velocity profile if they are not properly identified and separated. A multi-mode inversion process is shown to be important to have an accurate soil characterization.     

Rayleigh-wave group velocity distribution in the Antarctic region

We determined 2D group velocity distribution of Rayleigh waves at periods of 20-150 s in the Antarctic region using a tomographic inversion technique. The data are recorded by both permanent networks and temporary arrays. In East Antarctica the velocities are high at periods of 90-150 s, suggesting that the root of East Antarctica is very deep. The velocities in West Antarctica are low at all periods, which may be related to the volcanic activity and the West Antarctic Rift System. Low velocity anomalies appear at periods of 40-140 s along the Southeastern Indian Ridge and the western part of the Pacific Antarctic Ridge. The velocities are only slightly low around the Atlantic Indian Ridge, Southwestern Indian Ridge, and the eastern part of the Pacific Antarctic Ridge, where the spreading rates are small. Around two hotspots, the Mount Erebus and Balleny Islands, the velocity is low at periods of 50-150 s.     

基阶与高阶瑞利波联合反演研究

研究了六层层状介质模型瑞利波基阶和高模式波相速度对横波速度、深度的敏感性,结果表明：基阶波较高模式波对7 m以内浅部地层的横波速度更敏感,敏感性频带在10～25 Hz范围内,峰值频带集中在18 Hz左右;高模式波较基阶波对深部地层的横波速度更敏感,敏感性频带宽,峰值分散.基阶波对浅层的敏感性和高模式波穿透深度更深的特点为近地表岩土层二维横波速度结构的联合反演提供了前提条件.利用阻尼最小二乘SVD（Singular Value Decomposition）算法联合基阶与高模式波对理论模型和实例数据进行横波速度反演,反演结果表明联合反演增强了反演的稳定性,提高了反演的精度.     

利用地脉动台阵观测推算Love波频散曲线

土层剪切波速结构是工程场地条件的重要的基本参数，许多研究者关注利用从地脉动台阵中推得的Rayleigh波或Love波的相速度频散曲线反演实际工程场地剪切波速度结构。探讨从地脉动台阵观测的记录中提取Love波相速度的频散曲线的方法，从厦门某工程场地的地脉动台阵观测三分量记录中提取了Love波相速度的频散曲线，将提取的结果与根据实际场地资料构成的水平成层介质模型计算的Love波相速度的频散曲线进行了对比，为下一步的反演奠定了基础。     

云南地区地壳中上部横波速度结构研究

根据云南地区的基阶瑞利波相速度频散资料，用面波层析成像方法反演得到该区域中上地壳S波速度结构. 给出了研究区域内在4个深度上的S波速度水平分布图像和沿100.5°E、24°N、25°N、26°N及27°N的S波速度-深度剖面图. 结果表明：在小江断裂与红河断裂围成的川滇菱形块体内，26～30km深度处的速度明显低于周边地区，其南段从地表到15km深度均为明显的低速区域. 云南地区的强震（M＞6.0）震中位置与S波速度分布图像具有明显的相关性，主要分布于高速与低速的过渡区域.     

Simultaneous measurement and inversion of surface wave dispersion and attenuation curves

Surface wave tests are non-invasive seismic techniques that have traditionally been used to determine the shear wave velocity (i.e. shear modulus) profile of soil deposits and pavement systems. Recently, Rix et al. [J. Geotech. Geoenviron. Engng 126 (2000) 472] developed a procedure to obtain near-surface values of material damping ratio from measurements of the spatial attenuation of Rayleigh waves. To date, however, the shear wave velocity and shear damping ratio profiles have been determined separately. This practice neglects the coupling between surface wave phase velocity and attenuation that arises from material dispersion in dissipative media. This paper presents a procedure to measure and invert surface wave dispersion and attenuation data simultaneously and, thus, account for the close coupling between the two quantities. The methodology also introduces consistency between phase velocity and attenuation measurements by using the same experimental configuration for both. The new approach has been applied at a site in Memphis, TN and the results obtained are compared with independent measurements.     

Seismic tomography of Yunnan region using short-period surface wave phase velocity

Introduction The Yunnan region is located on the east margin of the collision zone between the Indian and Eurasian plates; it belongs to the south section of the N-S Seismic Belt of China and is the junc-ture of the Yangtze metaplatform, Songpan-Garz?fold system, Sanjiang fold system and South China fold system. This region has complex tectonic movements, crisscross faults and frequently occurring strong earthquakes, and hence it is one of the regions with the strongest earthquake ac-tivi…     

Determining Q of near-surface materials from Rayleigh waves

High-frequency (≥2 Hz) Rayleigh wave phase velocities can be inverted to shear (S)-wave velocities for a layered earth model up to 30 m below the ground surface in many settings. Given S-wave velocity (V_S), compressional (P)-wave velocity (V_P), and Rayleigh wave phase velocities, it is feasible to solve for P-wave quality factor Q_P and S-wave quality factor Q_S in a layered earth model by inverting Rayleigh wave attenuation coefficients. Model results demonstrate the plausibility of inverting Q_S from Rayleigh wave attenuation coefficients. Contributions to the Rayleigh wave attenuation coefficients from Q_P cannot be ignored when Vs/V_P reaches 0.45, which is not uncommon in near-surface settings. It is possible to invert Q_P from Rayleigh wave attenuation coefficients in some geological setting, a concept that differs from the common perception that Rayleigh wave attenuation coefficients are always far less sensitive to Q_P than to Q_S. Sixty-channel surface wave data were acquired in an Arizona desert. For a 10-layer model with a thickness of over 20 m, the data were first inverted to obtain S-wave velocities by the multichannel analysis of surface waves (MASW) method and then quality factors were determined by inverting attenuation coefficients.     

利用多重滤波方法提取面波频散曲线

多重滤波方法已被证明在分析面波的频散特性时是一种有效而快速的方法。本文用单个台站的单次地震事件记录,应用MFT(Multiple Filter Technique)方法提取了内华达地区的面波频散曲线并分析了其特征,认为其具有大陆基振型瑞利波频散曲线的典型特征。还利用广义反透射系数法快速计算了瑞利面波频散曲线,尤其是基阶瑞利波相速度及群速度频散曲线;并且在由振型叠加方法合成的理论地震图的基础上,通过比较理论频散曲线和MFT方法得到的频散曲线,研究了多重滤波的可靠性。     

蒙古中南部地区面波相速度层析成像

本文采用小波变换频时分析技术提取了1893条蒙古中南部地区双台间基阶Rayleigh波相速度频散曲线,通过对提取到的相速度频散进行二维反演,首次重构了蒙古高原中南部10~80s周期内分辨率为0.5°×0.5°的相速度分布图.结果表明,短周期相速度频散(10~20s)明显受地表地形控制,表现为杭爱—肯特山盆地为相对高速异常,而南部的戈壁带表现为相对低速异常;在30s到60s周期内,南部的戈壁带和北部的杭爱—肯特山盆地都显示出稳定的高速异常,而中部戈壁带则表现为低速异常.该低速异常区与新生代火山岩出露位置的一致性,暗示该低速异常可能与新生代火山活动有关.穿过蒙古高原中部的两条主要断裂带与研究区内的地震波速度分布具有很好的一致性,可能暗示两条断裂带一直延伸到整个岩石圈.此外,蒙古中南部地区的壳幔速度低于全球平均大陆值,且其相速度频散曲线与遭受破坏的克拉通,尤其大陆裂谷很相似,可能暗示着研究区具有薄的、活跃的岩石圈构造.     

Crustal and upper mantle structures of the brazilian coast

The Rayleigh wave phase and group velocities in the period range of 24–39 sec, obtained from two earthquakes which occurred in northeastern brazil and which were recorded by the Brazilian seismological station RDJ (Rio de Janeiro), have been used to study crustal and upper mantle structures of the Brazilian coastal region. Three crustal and upper mantle models have been tried out to explain crustal and upper mantle structures of the region. The upper crust has not been resolved, due basically to the narrow period range of the phase and group velocities data. The phase velocity inversions have exhibited good resolutions for both lower crust and upper mantle, with shear wave velocities characteristic of these regions. The group velocity data inversions for these models have showed good results only for the lower crust. The shear wave velocities of the lower crust (3.86 and 3.89 km/sec), obtained with phase velocity inversions, are similar to that (=3.89 km/sec) found byHwang (1985) to the eastern South American region, while group velocity inversions have presented shear velocity (=3.75 km/sec) similar to that (=3.78 km/sec) found byLazcano (1972) to the Brazilian shield. It was not possible to define sharply the crust-mantle transition, but an analysis of the phase and group velocity inversions results has indicated that the total thickness of the crust should be between 30 and 39 km. The crustal and upper mantle model, obtained with phase velocity inversion, can be used as a preliminary model for the Brazilian coast.     

Approximation to Cutoffs of Higher Modes of Rayleigh Waves for a Layered Earth Model

A cutoff defines the long-period termination of a Rayleigh-wave higher mode and, therefore is a key characteristic of higher mode energy relationship to several material properties of the subsurface. Cutoffs have been used to estimate the shear-wave velocity of an underlying half space of a layered earth model. In this study, we describe a method that replaces the multilayer earth model with a single surface layer overlying the half-space model, accomplished by harmonic averaging of velocities and arithmetic averaging of densities. Using numerical comparisons with theoretical models validates the single-layer approximation. Accuracy of this single-layer approximation is best defined by values of the calculated error in the frequency and phase velocity estimate at a cutoff. Our proposed method is intuitively explained using ray theory. Numerical results indicate that a cutoffs frequency is controlled by the averaged elastic properties within the passing depth of Rayleigh waves and the shear-wave velocity of the underlying half space.     

Modifications of the dispersion relations for surface waves in a layer on a half-space

Summary Dispersion relations for Love and Rayleigh waves in a layer on a half-space are modified by introducing quadratic slownesses instead of velocities. The advantages of this approach are demonstrated on analytical formulae for computing the group velocity.