Coda waves: A review

The analysis and interpretation of coda waves have received increasing attention since the early seventies. In the past few years interest in this subject has spread worldwide, and the study of high-frequency seismic coda waves has become a very important seismological topic. As a conclusion of the studies accomplished in this time, coda waves are considered the result of scattering processes caused by heterogeneities acting on seismic waves.P andS waves play a particularly important role in this interaction. The process introduces an attenuation which, added to the intrinsic absorption, gives the observed apparent attenuation. Therefore, coda waves constitute a thumbprint left by the heterogeneities on the seismograms. Coda waves offer decisive information about the mechanism of how scattering and attenuation take place. This review describes coda waves in detail, and summarizes the work done in this subject to 1986. The relation between coda waves and attenuation in the context of research on seismic scattering problems is stressed. Particular attention has been given to the application of coda waves to estimate source and medium parameters. The state-of-the-art of the temporal variations of coda wave shape, and the possible use of these variations as an earthquake precursor also are presented. Care has been taken to introduce the statistical models used to deal with the heterogeneities responsible for scattering.     

Probing mid-mantle heterogeneity using PKP coda waves

We investigate the utility of PKP coda waves for studying weak scattering from small-scale heterogeneity in the mid-mantle. Coda waves are potentially a useful probe of heterogeneity in the mid-mantle because they are not preferentially scattered near the CMB, as PKP precursors are, but are sensitive to scattering at all depths. PKP coda waves have not been used for this purpose historically because of interference with other late-arriving energy due to near-surface resonance and scattering. Any study of deep mantle scattering using coda waves requires the removal of near-surface effects from the data. We have analyzed 3624 recordings of PKP precursors and coda made by stations in the Incorporated Research Institutions for Seismology (IRIS) Global Seismographic Network (GSN). To study the range and time dependence of the scattered waves, we binned and stacked envelopes of the recordings. We have considered precursors that arrive within a 20 s window before PKP and coda waves in a 60 s window after PKP. The PKP scattered waves increase in amplitude rapidly with range as predicted by scattering theory. At ranges below 125°, we predict and observe essentially no scattered energy preceding PKP. Coda amplitudes at these ranges are independent of range and provide an estimate of energy due to near-surface effects that we can expect at all ranges. We use the average coda amplitude at ranges from 120 to 125° to correct coda amplitudes at other ranges. PKP coda waves show a strong dependence on time and range and are clearly influenced by scattering in the lower mantle. PKP coda waves, however, do not provide a tighter constraint on the vertical distribution of mantle heterogeneity than is provided by precursors. This is due, in part, to relatively large scatter in coda amplitudes as revealed by a resampling analysis. Modeling using Rayleigh–Born scattering theory and an exponential autocorrelation function shows that PKP coda amplitudes are not highly sensitive to the vertical distribution of heterogeneity in the mantle. To illustrate this we consider single-scattering in two extreme models of mantle heterogeneity. One allows heterogeneity just at the CMB; the other includes heterogeneity throughout the mantle. The amplitudes of precursors are tightly constrained by our stack and support our earlier conclusion that small-scale heterogeneity is uniformly distributed throughout the lower mantle. The best-fit model includes 8 km scale length heterogeneity with an rms velocity contrast throughout the mantle of 1%.     

Phase and Correlation in `Random' Seismic Fields and the Reconstruction of the Green Function

We first present a summary of recent results on coda interpretation. We emphasize the observation of the stabilization of P to S energy ratio indicating the modal equipartition of the wavefield. This property clearly shows that the coda waves are in the regime of multiple scattering. Numerical solutions of the elastic radiative transfer equation are used to illustrate the evolution of the wave-field towards P-to-S energy stabilization, and asymptotically to complete isotropy. The energy properties of the coda have been widely studied but the phase properties have often been neglected. The recently observed coherent backscattering enhancement, an expression of the so-called `weak localization', demonstrates that interference effects still persist for multiple diffracted waves. Another manifestation of the persistence of the phase is the possibility to reconstruct the Green function between two stations by averaging the cross correlation of coda waves produced by distant earthquakes and recorded at those two stations. This reconstruction is directly related to the properties of reciprocity and time reversal of any wavefield. Using broadband seismic coda waves, we show that the dominant phases of the Green function in the band 2 s–10 s, namely fundamental mode Rayleigh and Love waves, are reconstructed. We analyze the time symmetry of the cross correlation and show how the level of symmetry evolves with the isotropization of the diffuse field with lapse time. Similarly we investigate the correlation in continuous ambient noise records. Whereas the randomness of the coda results from multiple scattering by randomly distributed scatterers, we assume that the seismic noise is random mostly because of the distribution of sources at the surface of the Earth. Surface waves can be extracted from long time series. The dispersion curves of Rayleigh waves are deduced from the correlations. On paths where measurements from earthquake data are also available, we show that they are in good agreement with those deduced from noise correlation. The measurement of velocities from correlation of noise along paths crossing different crustal structures opens the way for a `passive imaging' of the Earth's structure.     

Poroelastic finite-difference modeling for ultrasonic waves in digital porous cores

Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with small-scale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference (FD) method of Biot’s poroelastic equations, with an arbitrary even-order (2L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer (CPML) absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultrasonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thicknesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Q _sc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.     

Scattering and dissipation of seismic waves in the presence of nonlinearity

The processes of seismic wave scattering and dissipation are examined in a medium that, in addition to being inhomogeneous and anelastic, is nonlinear and seismically active (seismic emission). In such a medium, there is a complex interrelation between the effects of nonlinearity, scattering and dissipation. Thus, nonlinear interactions between the various components (primary, scattered and induced) of a developing wavefield cause nonlinear (or wave-on-wave) scattering and, by transferring part of the wave energy to the high-frequency region, contribute to its scattering and dissipation. On the other hand, whereas dissipation opposes the accumulation of nonlinear effects by reducing the wave amplitudes, scattering assists it by increasing the propagation distance (and hence the interaction time).Estimates based on results of field experiments involving vibrators indicate that, as a rule, scattering on inhomogeneities is much stronger than nonlinear scattering, and that nonlinear effects may often dominate dissipative ones.The nonlinearity of the transmitting medium explains observedQ-value anomalies, and its seismic activity explains the existence ofP coda and the temporal changes in codaQ.     

地震多级散射波正演模拟方法

地震波在穿越地下散射体群时会产生多级散射波,分析其地震响应特征,可推断散射体的分布情况和性质。本文从二维标量波动方程出发,结合地震散射理论和波恩近似理论,推导了多级散射波方程。在此基础上,采用高阶有限差分法对双点散射体模型和复杂散射体模型进行数值模拟,分析了多级散射波的传播规律和波场特征,并通过抽取多级散射记录和各级散射记录的单道记录与参考单道记录的对比,验证了本文推导散射波方程的准确性。      

上月壳中的散射引起月震尾波的数值模拟研究

在阿波罗月震记录中普遍存在着强烈持久的尾波信号,这样的波形特征无法用均匀分层月球模型解释.一个普遍被接受的解释是月震尾波由月球浅层结构对月震波的散射引起.我们采用基于交错网格的伪谱和有限差分混合方法模拟研究非均匀上月壳对月震波的散射效应,在此基础上解释月震尾波的形成机制,并估计出上月壳速度扰动的强度.我们发现,在均匀分层模型基础上,进一步考虑上月壳中的非均匀结构对月震波的散射效应,能有效地解释月震信号中强烈持久的尾波.我们认为月震尾波可能是由上月壳中的低波速、低衰减和散射这三个因素的共同作用所引起.采用不同的扰动标准差模拟上月壳的非均匀性,并比较模拟波形与真实月震图的相似程度,我们发现上月壳中速度扰动的标准差应该在3%到5%之间,很可能接近于3%.     

利用超声尾波观测1.5 m长岩石断层黏滑实验的波速变化

在实验室内利用超声尾波观测大尺度(1.5 m)岩石断层的黏滑过程.利用基于尾波干涉的观测方法,我们获得了高达10~(-6)的相对波速变化的观测精度,这相当于~10 kPa的应力变化.利用高精度的测量,我们获取两种不同加载速率下(1μm·s~(-1),10μm·s~(-1))黏滑过程三个阶段(恢复、加载和滑动)基于波速变化的特征量.我们更进一步获取了断层失稳阶段波速变化的时空演化过程.最后讨论了该观测方法需要改进的地方.以上研究结果表明作为一种对现有实验观测手段的有益补充,利用超声尾波观测实验室大尺度岩石断层的动力学过程是可行的.     

High-frequency interference waves in low strain dynamic testing of X-section concrete piles

Stress waves propagate along vertical, radial and circumferential directions when a non-uniformly distributed load is applied at one end of a three-dimensional shaft. As a result, the receiving signals are usually mixed with undesired interference components, often featuring as high-frequency fluctuations. Previous studies have revealed that sectional geometry(shape and size) greatly affects the high-frequency interference. In this study, low strain dynamic testing on full-scale X-section concrete is conducted in order to investigate the influences of high-frequency interference on velocity responses at the pile head. Emphasis is placed on the frequency and peak value of interference waves at various receiving points. Additionally, the effects of the geometrical, and mechanical properties of the pile shaft on high-frequency interference are elaborated on through the three-dimensional finite element method. The results show that the measured wave is obscured by interference waves superposed by two types of high-frequency components. The modulus and cross-sectional area are contributing factors to the frequency and peak value of the interference waves. On the other hand, the position with the least interference is determined, to some extent, by the accurate shape of the X-section.     

Comparison of crustal and upper mantle heterogeneity in different time periods: Indonesian subduction zone to northern Australia

Earthquake events from the Indonesian subduction zone recorded in northern Australia show a long and high-frequency coda associated with both P and S waves. Regional events recorded by Warramunga array in northern Australia can separate out wave propagation through the mantle by focusing on the coherent signal across the medium-aperture array. Most of the incoherent wave components result from structures in the vicinity of the array with small-scale lengths of 1–2 km or smaller. The coherent phases with relatively rapid changes in waveforms are associated with the scattering of seismic waves by crustal and mantle heterogeneity, but in some case can be related to structural effects near the source. As the depth of the source increases, the coherent portion of the seismic wavefield tends to become much simpler, which suggests that the heterogeneity tends to weaken at depth with larger-scale length. We compare the coherent signal features of earthquakes from the Indonesian subduction zone that have occurred in recent years with those in the early 1980s, first studied by Kennett (Phys Earth Planet Inter 47: 319–332, 1987). The general characteristics of the coherent signal variation with depth in recent years are the same as those observed in 1980s, but the variations are larger. This change suggests a stringer variation in heterogeneity with depth than before, which may bear important information about the dynamic processes and evolution of the crust and upper mantle.     

龙马溪组页岩数字岩芯超声响应数值模拟及散射特征分析

超高频（几百兆赫兹）超声数值模拟微米至纳米尺度的龙马溪组页岩数字岩芯及其强非均质性严重挑战数值模拟算法的精度和数值稳定性.本文利用图像阈值分割算法将龙马溪组页岩数字岩芯主要成分分解为石英类、黏土类、黄铁矿及孔隙四种类型,假定液相（油）均匀分布在整个介质模型中,根据岩芯的孔隙度、渗透率和各类矿物的岩石物理参数,建立了精细的非均质双相介质模型.采用基于Biot双相介质方程的不分裂卷积完全匹配层与高精度旋转交错网格有限差分方法精确模拟超声波在页岩岩芯中传播的散射衰减.通过精确控制匹配层吸收边界数来模拟边界反射量及其对尾波的干涉强度,并与超声实验尾波散射Q值进行比较,估算超声实验中的边界反射量及其对尾波的干涉强度.对不同超声子波主频的数值模拟试验,结合L/a-ka散射态式图分析表明：本文采用的龙马溪组页岩数字岩芯的非均质强度与600 MHz波长尺度相当,产生的散射衰减达到最大.开展页岩岩芯超声波散射数值模拟研究,据此评估页岩岩芯的非均质性,为页岩储层声学非均质预测提供依据.     

超声实验中谱比法衰减的散射与本征吸收特性

实际地球介质的弹性波衰减,无论是对于地球内部结构反演、含油气储层描述,还是对于地震岩石物理本身的基础研究而言,都十分重要.通过岩石物理实验测量岩石的弹性波衰减是相关研究的基本手段,而谱比法是一种最常用的测量方法.利用谱比法测量的弹性波衰减,简称谱比法衰减,由介质的本征吸收特性和非均质散射特性共同决定.然而,在谱比法理论中,没有将这两种衰减特性的作用区分对待.因此,实验室测量的谱比法衰减中是否含有散射特性,其散射特性是否可以忽略,常不明确.本文基于对岩石物理实验中样品波形特征的基本认识,通过能流理论描述了弹性波的本征吸收和非均质散射过程,以及对应的尾波形成机理.文章还应用能流理论测量岩石的弹性波衰减,即能流法衰减.无论是通过理论推导,还是通过实验验证,谱比法衰减与能流法衰减都显示出良好的一致性.这两种直达波衰减的一致性暗示了谱比法衰减中存在着两种衰减特性.文中还对比分析了谱比法衰减与喷射流本征衰减的实验结果,该结果同样展示了谱比法衰减中可能存在着散射特性.因此,通过能流理论,可以利用样品波形的尾波特征定性判断谱比法衰减中可能存在的散射特性.即异常明显的尾波振幅可能意味着强烈的直达波非均质散射,谱比法衰减中的散射特性因而不可忽略.     

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.     

EXPERIMENTAL STUDY ON THE CHANGES OF ULTRASONIC CODA WAVE AND ACOUSTIC EMISSION DURING ROCK LOADING AND DEFORMATION

The coda wave propagation path has received extensive attention as it is more sensitive to small changes in the medium than the direct wave. During the process of loading, the wave velocity, medium or source changes may cause the coda wave to change. The physical mechanism of change in the ultrasonic coda wave varies during different deformation stages. Meanwhile, there exist local damages in the rock sample during the deformation, and it will be accompanied by acoustic emission. Combining the ultrasonic coda wave and acoustic emission is beneficial to characterize the coda wave characteristics and damage degree of the sample at different deformation stages. In this paper, three kinds of rocks, including granodiorite, marble and sandstone with the sizes of 50mm×50mm×150mm, are used to carry out observations of ultrasonic coda wave and acoustic emission during the whole process of loading so as to study characteristics of the coda wave at different deformation stages. The major results are given below: 1)There is a good correspondence between the coda wave variation and the acoustic emission evolution process. When the acoustic emission frequency increases, the coda wave changes accordingly. In particular, the coda wave changes in the early stages of increased acoustic emission frequency, which indicates that the early damage information of rock can be obtained by analysis of the coda wave. 2)The physical mechanism of the coda wave change is different in different deformation stages. At the initial stage of loading, there are obvious scatterer changes in the coda wave change; then, in the linear elastic deformation stage, the wave velocity change is dominating; in the late-stage of loading, the scatterer change increases and coexists with the wave velocity change, the scatterer change effect is related with the rock micro-fracture degree, the rock will locally be damaged before rupturing, and the role of the scatterer will be enhanced. 3)With the increase of loading, the amplitude of increase of the wave velocity generally decreases gradually, which is basically consistent with the understanding obtained through the direct wave. The interference of acoustic emission can be eliminated because of the Kaiser effect when analyzing the coda wave. The consistency of the wave velocity change and stress loading and unloading is further verified. 4)The micro-fracture generated during rock deformation will change the physical mechanism of the coda wave change, and the scatterer effect will be significantly enhanced. At the same time, the acoustic emission waveform will cause interference to the ultrasonic coda wave. This means that attention needs to be paid when analyzing rock damage using only coda wave data. In short, the ultrasonic coda wave and acoustic emission can reflect the damage inside the rock, and the change mechanism of the coda wave in different deformation stages is different. The joint observation of the two can play a mutual verification role, which is conducive to improving the reliability of the observation results.     

双相介质中地震波衰减的物理机制

High-frequency seismic attenuation is conventionally attributed to anelastic absorption. In this paper, I present three studies on high-frequency seismic attenuation and propose that the physical mechanism results from the interference of elastic microscopic multiple scattering waves. First, I propose a new theory on wave propagation in a two-phase medium which is based on the concept that the basic unit for wave propagation is a nano- mass point. As a result of the elasticity variations of pore fluid and rock framework, micro multiple scattering waves would emerge at the wavelength of the seismic waves passing through the two-phase medium and their interference and overlap would generate high- frequency seismic attenuation. Second, I present a study of the frequency response of seismic transmitted waves by modeling thin-layers with thicknesses no larger than pore diameters. Results indicate that high-frequency seismic waves attenuate slightly in a near-surface water zone but decay significantly in a near-surface gas zone. Third, I analyze the seismic attenuation characteristics in near-surface water and gas zones using dual-well shots in the Songliao Basin, and demonstrate that the high-frequency seismic waves attenuate slightly in water zones but in gas zones the 160-1600 Hz propagating waves decay significantly. The seismic attenuation characteristics from field observations coincide with the modeling results. Conclusions drawn from these studies theoretically support seismic attenuation recovery.     

单次与多次散射对地方震尾波的作用

在Aki(1969)的单次反向散射理论和高龙生(1983)的多次散射理论基础上,提出了震源距不为零的单次与多次散射模型。在此模型下,研究了二维无限介质空间中,由统计上均匀分布的各向同性散射体引起的单次与多次散射对地方震尾波功率谱的作用。推导出了二维情况下单次散射和多次散射尾波功率谱的解析表达式,并在二维情况下对不同震源距的单次散射与多次散射对尾波功率谱的作用进行了比较。      

On the generation and decay of the long-period coda energy of large earthquakes

Seismic coda wave is the tail portion of the earthquake record after main arrivals. Studies on the coda usually focus on high-frequency data within several hours after regional events and attribute them to the scattering effect of the heterogeneities inside the earth. Here, we use records of seven large earthquakes at globally distributed seismic stations to examine the decay of long-period (100 s to 300 s) coda in the time window of 10,000 s to 140,000 s after the origin time and fit it with a statistical model. The geometric spreading effect in the estimated initial energy and a location-independent equivalent attenuation coefficient indicate that the long-period coda energy is less affected by the heterogeneity-induced scattering effect than that of shorter-period coda. The coda energy can reach the earth's inner core and can be explained by a 1D earth model, making it more effective for constraining the global attenuation model. It also has the potential to determine the magnitudes of large earthquakes and to explore the interior of planetary bodies.     

地震波的散射与盆地的场地效应

根据射线理论对二维盆地型试场地与Ｐ波波列特征的关系，波列中震相的性质及其产生的原因等进行了研究，对决定震相振幅大小的散射系数，几何扩散和波阻抗３个因子等的研究表明，因界面弯曲引起的广角反射和聚焦分别Ｐ波波阵中出现大能量团的第一和第二位的重要因素，在层内纵，横波波速比和层间波速比大范围变动的条件下，对界面散射系数的计算表明，介质内界面上Ｐ到Ｓ的反射转换波是指示界面两侧速度反差强弱以及入射波所在介质层