Stratigraphic filtering of seismic waves: The influence of mode conversion multiples onP-wave andS-wave phase and attenuation

We demonstrate how multiples, generated at the interfaces of plane parallel beds, modify the propagation characteristics of an originally coherent seismic wave. For waves propagating at an angle to the bedding plane we find that theSV andP-waves couple so that neither is a pure mode. TheSH-wave, while modified in its propagation characteristics by multiples, remains a pure mode. The coupling ofSV-multiples into the quasi-P-mode appears weaker than the coupling ofP-wave multiples into the quasi-SV mode; at least this is so for the two simple cases of (a) density fluctuations only and (b) correlatedV _p andV _s fluctuations which conserve Poisson's ratio.We also find that the coupling is sensitive to both the angle of propagation and frequency. In addition there is a cut-off angle forP-wave multiples influencing the quasi-SV mode. Propagation angles larger than the cut-off permit theP-multiples to modify the phase of the quasi-SV mode, but not its effective attenuation. No such cut-off effect is found for SV-multiples influencing the quasi-P mode, whose angle-dependent and frequency-dependent phase distortion and effective attenuation are influenced both byP-wave multiples andSV-multiples.In view of the mathematical complexity of the expressions describing the phase, and effective attenuation of modes when allowance is made forP-andS-wave multiples, we strongly advocate numerical coding of the major mathematical formulae. By so doing a systematic study can be undertaken of the frequency and offset dependence of seismic waves as a function of seismic source input and power spectral behavior of the fluctuations in density and elastic constants of beds. It is our opinion that the full mathematical expressions are too involved to permit an analytic, systematic investigation to be given of the phase and attenuation of seismic waves with any degree of sophistication or generality.     

高频面波方法的若干新进展

面波多道分析方法(MASW)通过分析高频瑞雷波确定浅地表剪切波速度.在过去的20年中,由于该方法具有非侵入性、无损、高效及价格低的特点,越来越受到浅地表地球物理和地质工程学界的重视,视为未来最有希望的技术之一.这篇综述论文将介绍中国地质大学(武汉)浅地表地球物理团队近年来在研究高频面波的传播理论和应用中取得的部分成果.非几何波是一种仅存在于浅地表介质,尤其是未固结的沉积物中的独特的地震波.它的存在对快速而准确地获得表层S波速度有一定价值.我们的研究表明非几何波是一种具有频散特性的泄漏波.泄漏波的存在可能导致将其误认为瑞雷波的基阶或高阶能量,从而造成模式误判.这种模式误判会导致错误的反演结果.我们通过求取高基阶分离后的瑞雷波格林函数证明虚震源法瑞雷波勘探的可行性.这个结果将极大地降低野外瑞雷波勘探成本.勒夫波多道分析方法(MALW)中未知参数比瑞雷波的少,这使得勒夫波的频散曲线比瑞雷波的简单.因此,勒夫波反演更稳定,非唯一性更低.勒夫波数据生成的能量图像通常比瑞雷波的清晰,并具有更高的分辨率,从而可以更容易地拾取精确的勒夫波的相速度.利用雅克比矩阵分析波长与探测深度的关系表明对相同波长的基阶模式而言,瑞雷波的探测深度是勒夫波的1.3~1.4倍;而两种波的相同波长的高阶模式波的探测深度相同.我们也尝试了时间域勒夫波反演.按照勒夫波分辨率将地球模型剖分成了不同尺寸的块体,利用反卷积消除了地震子波对勒夫波波形的影响,通过更新每个块体的S波速度来拟合勒夫波波形,从而获得地下S波速度模型.该方法不基于水平层状模型假设,适用于任意二维介质模型.     

Sources of microseismicP waves

Summary Origin ofP waves detected earlier in microseisms of very quiet locations in the USSR is discussed in detail. It appears that the most pronounced sources ofP waves are tropical cyclones over the Pacific. The amplitude of the force in the source for a medium power typhoon is found to be of the order of 10¹⁶ dynes. The effective source area is estimated as 10⁴–10⁵ km² approximately. The shape of the amplitude spectrum ofP wave corrected for the absorption in the mantle does not contradict with the standing wave theory of microseisms generation. Results of observations at various epicentral distances give strong evidences of the predominant attenuation of the fundamental Rayleigh mode as compared with higher Rayleigh modes andP waves in the frequency band of 0.3–0.15 cps.     

The dynamic response of a fluid‐filled borehole under a normal point load on the free surface

The dynamic response of a semi‐infinite fluid‐filled borehole embedded in an elastic half‐space under a concentrated normal surface load is analysed in the long‐wavelength limit. The solution of the problem is obtained with integral transforms in the form of a double integral with respect to the slowness and frequency. The partial P‐ and SV‐wave responses are further transformed to path integrals along Cagniard paths in the complex slowness plane. Unlike the traditional Cagniard‐de Hoop technique based on the Laplace transform of time dependence, this paper is based on the Fourier transform. The tube‐wave response is presented as a causal integral over a slowness range. The resultant representation in the time‐domain is suitable for the numerical evaluation of the complete response in the fluid‐filled borehole, especially at large distances. Asymptotic analysis of seismic phases arising in the borehole is performed on the basis of the obtained solution. The complete asymptotic wavefield consists in P‐ and SV‐waves, the Rayleigh wave and the low‐frequency Stoneley (tube) wave. Pressure synthetics obtained by the use of the asymptotic formulas are shown to be in good agreement with straightforward calculations.     

两种不同势函数下半空间饱和多孔介质中Rayleigh波求解比较

分别对"考虑两种压缩波和幅值比例系数"和"考虑一种压缩波(P1或P2波)但不考虑幅值比例系数"两种不同势函数下的半空间饱和多孔介质中Rayleigh波求解进行详细推导,理论分析表明"考虑两种压缩波和幅值比例系数"下Rayleigh波求解推导更为严密,与饱和多孔介质中存在两种压缩波的事实相一致。在研究半空间饱和多孔介质中Rayleigh波时应采用"考虑两种压缩波和幅值比例系数"的势函数。     

An attenuating structure beneath the Aso Caldera determined from the propagation of seismic waves

The attenuation of amplitude is seen in seismic waves which pass through the central region of the Aso caldera, in Kyushu, Japan. It is also recognized from spectral analysis of seismic waves that the higher frequencies of the P-wave are reduced in the waves which pass through the central region of the caldera. It is shown that the relative attenuation increases remarkably for the frequency range of 5 to 10 Hz. The specific attenuation factor Q of the P-wave train is about 100. From the surface projection of the ray paths with low Q values through the Aso caldera to each station, the attenuating region is located beneath the center of the caldera, extending to the north of the central cones. In conjunction with the low Q value of the P-wave and the decreases of S-wave amplitudes, the relative P-wave residual times have comparatively large values for seismic waves passing through the central region beneath the caldera. In order to attempt to provide additional information on the depth configuration of the attenuating material, the ray paths of P-wave's first arrivals are located in three-dimensional space. It indicates that the low-velocity material is located beneath the center of the caldera at depths of about 6 to 9 km. However, lowvelocity anomalies above the depth of 6 km and below the depth of 15 km were not able to be detected, because most of the available seismic ray paths had crossed the caldera at depths of about 6 to 15 km. Furthermore, the relative residual times have numerous errors resulting from incorrect hypocenter locations, origin times, inhomogeneities in the structure and uncertainty of the velocity structure. At shallow depths in the Aso caldera, refraction or reflection studies are required for an accurate estimate of the structure and more detailed properties of the attenuating material.     

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.     

Effects of Impedance Contrast and Soil Thickness on Basin-Transduced Rayleigh Waves and Associated Differential Ground Motion

This paper presents the effects of impedance contrast (IC) across the basin edge, velocity contrast between the basin and underlying bedrock, Poisson’s ratio and soil thickness on the characteristics of basin-transduced Rayleigh (BTR) waves and associated differential ground motion (DGM). Analysis of simulated results for a two-dimensional (2D) basin revealed complex mode transformation of Rayleigh waves after entering the basin. Excellent correlation of frequencies corresponding to different spectral ratio peaks in ellipticity curves of BTR waves and spectral amplification peaks was obtained. However, such correlation was not observed between values of peaks in ellipticity curves and spectral amplification at the corresponding frequencies. An increase of spectral amplification with IC was obtained. The largest spectral amplification was more than twice the IC in the horizontal component and more than the IC in the vertical component in the case of large and same impedance contrast for P- and S-waves. It was concluded that the frequency corresponding to the largest spectral amplification was greater than the fundamental frequency of soil by around 14% and 44% in the vertical and horizontal components, respectively. Spectral amplification of the vertical component was negligible when soil thickness was less than around 15–20 times the S-wave wavelength in the basin. The largest values of peak ground displacement (PGD) and peak differential ground motion (PDGM) were obtained very near the basin edge, and their values with offset from the edge were strongly dependent on the IC across the basin edge, Poisson’s ratio, velocity contrast between the basin and underlying bedrock (dispersion), damping and soil thickness. The obtained value of PDGM for a span of 50 m in the horizontal and vertical components due to the BTR wave was of the order of 0.75 × 10^?3 and 1.32 × 10^?3 for unit amplitude (1.0 cm) in the horizontal component of the Rayleigh wave at rock very near the basin edge.     

Regional Seismic Wavefield Computation on a 3-D Heterogeneous Earth Model by Means of Coupled Traveling Wave Synthesis

-- I present a new algorithm for calculating seismic wave propagation through a three-dimensional heterogeneous medium using the framework of mode coupling theory originally developed to perform very low frequency (f < ~0.01т.05 Hz) seismic wavefield computation. It is a Greens function approach for multiple scattering within a defined volume and employs a truncated traveling wave basis set using the locked mode approximation. Interactions between incident and scattered wavefields are prescribed by mode coupling theory and account for the coupling among surface waves, body waves, and evanescent waves. The described algorithm is, in principle, applicable to global and regional wave propagation problems, but I focus on higher frequency (typically f S ~0.25 Hz) applications at regional and local distances where the locked mode approximation is best utilized and which involve wavefields strongly shaped by propagation through a highly heterogeneous crust. Synthetic examples are shown for P-SV-wave propagation through a semi-ellipsoidal basin and SH-wave propagation through a fault zone.     

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.     

Source parameters for the January 1975 Brawley — Imperial valley earthquake swarm

The source parameters, moment, stress drop and source dimension are estimated for 61 events from the January 1975 Brawley earthquake swarm. Earthquakes studied range in local magnitude from 1.0 to 4.7. Stress drops range from 1 to 636 bars and increase with source depth. It is estimated that the sedimentary structure of the Imperial Valley amplifies shear waves by a factor of 2 to 3 in addition to the free surface amplification of 2. Estimates of moment from 10 sec surface waves are 4 to 6 times larger than the moment estimated from the relatively flat part of the local body wave spectrum at 1 sec. This may be due to after-slip on the fault, a long thin fault, or partial stress drop. It is shown that the experimentally determined ratio of stress drop to apparent stress should be approximately 4.0 when spectrum integration is used to obtainS-wave energy and theP-wave energy is 1/3 theS-wave energy.     

Q-structure beneath the Tibetan Plateau from the inversion of Love- and Rayleigh-wave attenuation data

The fundamental mode Love and Rayleigh waves generated by ten earthquakes and recorded across the Tibet Plateau, at QUE, LAH, NDI, NIL, KBL, SHL, CHG, SNG and HKG are analysed. Love- and Rayleigh-wave attenuation coefficients are obtained at time periods of 5–120 s using the spectral amplitudes of these waves for 23 different paths. Love wave attenuation coefficient varies from 0.0021 km^?1, at a period of 10 s, to 0.0002 km^?1 at a period of 90 s, attaining two maxima at time periods of 10 and 115 s, and two minima at time periods of 25 and 90 s. The Rayleigh-wave attenuation coefficient also shows a similar trend. The very low value for the dissipation factor, Q_β, obtained in this study suggests high dissipation across the Tibetan paths. Backus-Gilbert inversion theory is applied to these surface wave attenuation data to obtain average Q_β^?1 models for the crust and uppermost mantle beneath the Tibetan Plateau. Independent inversion of Love- and Rayleigh-wave attenuation data shows very high attenuation at a depth of ～50–120 km ($\text{Q}{}_{\mathrm{\beta }}\text{? 10}$). The simultaneous inversion of the Love and Rayleigh wave data yields a model which includes alternating regions of high and low Q_β^?1 values. This model also shows a zone of high attenuating material at a depth of ～40–120 km. The very high inferred attenuation at a depth of ～40–120 km supports the hypothesis that the Tibetan Plateau was formed by horizontal compression, and that thickening occurred after the collision of the Indian and Eurasian plates.     

Effects of source and cavity depths on wave fields in homogeneous half spaces

In homogeneous half spaces excited by small buried, spherically symmetric P-wave sources, Rayleigh waves (R-waves) could be generated. The effects of the source depth on the induced wave pattern and propagation behavior of R-waves are analyzed using the thin layer method. When a cavity is present in a homogeneous half space, R-waves could be formed in the scattered wave field. It is found that the energy of R-waves in the incident (direct) surface wave-field is related to the ratio of the source depth to the wavelength of R-waves; R-waves have relatively strong energy when the ratio is less than 1; the buried source induced R-waves approximately travel at the velocity of the planar R-waves in the range of the offset beyond about one wavelength; the energy of R-waves in the back-scattered surface wave-field depends on the ratio of the depth of cavity to the wavelength of R-waves; and for the case of the cavity presented at depths less than one wavelength, R-waves can be clearly observed in the back-scattered wave field. The results are helpful for selecting the source depth and the frequency component in seismic surveys and interpreting both the incident wave and the scattered wave patterns.     

利用CAP方法和瑞利面波振幅谱联合反演宁强5.3级地震震源深度

2018年9月12日19时6分,陕西省汉中市宁强县发生5.3级地震,不同机构给出的震源深度结果相差较大。为进一步确定宁强5.3级地震的震源深度,基于区域速度模型,首先利用CAP方法反演得到该地震的震源机制解,然后采用瑞利面波振幅谱和CAP深度误差函数联合反演,进一步测定了此次地震的矩心深度。结果显示:CAP方法得到的陕西宁强5.3级地震矩心深度约为12km,瑞利面波振幅谱测定的矩心深度为13km,结合引入的误差函数联合反演,最终确定陕西宁强5.3级地震的矩心深度为13km左右,表明此次地震仍属于发生于上地壳的地震。     

Site-specific response spectra from the combination of microzonation with probabilistic seismic hazard assessment—An example for the Cologne (Germany) area

Seismic noise was measured at some 20 sites in the Cologne area (Germany) aligned nearly perpendicular to a graben structure. The H/V spectral noise ratio for each site was used to derive realistic S-wave velocity profiles down to the bedrock by means of a genetic algorithm inversion. Numerical simulations were performed for different combinations of source and propagation path parameters: focal depth, epicentral distance, attenuation and fault mechanism. Synthetic seismograms were produced and converted to Fourier and response spectra. Finally, the site-specific values from response spectral ratios, with their uncertainties, were used to modify attenuation functions entering the logic-tree algorithm of the probabilistic seismic hazard assessment (PSHA). The site-specific response spectra show the significance of taking into account the local S-wave velocity structure in PSHA.     

P- and S-Wave Site Response of the Seismic Network RESNOM Determined from Earthquakes of Northern Baja California, Mexico

—We determined the response to P- and S-wave incidence of the permanent stations of the seismic network of Baja California (RESNOM) using two independent methods. We selected 65 events with magnitudes between 2.2 and 4.8 and hypocentral distances ranging between 5 and 330 km. The site response of the ten stations analyzed was first estimated using average spectral ratios between the horizontal and the vertical components of motion (H/V ratios). As a second approach we performed a simultaneous inversion for source and site. In order to invert the spectral records to determine the site response, we made an independent estimate of the attenuation for two different source-station path regions. Then we corrected the spectral records for the attenuation effect before we made the inversion. Although the average H/V ratio of many sites is inside the error bars of the site response estimated with the spectral inversion, the spectral inversion tends to give higher values. For the S wave some sites show similar frequency of predominant peak when comparing the responses obtained with both methods. In contrast, for the P waves the H/V ratios disagree with the results of the inversion. In general, the site response of the stations is strongly frequency dependent for both P and S waves. We also found that the natural frequency of resonance of the sites is near 0.5 Hz for P and near 0.8 Hz for the S waves.     

The 1986 Kermadec earthquake and its relation to plate segmentation

To evaluate the tectonic significance of the October 20, 1986 Kermadec earthquake (M _w =7.7), we performed a comprehensive analysis of source parameters using surface waves, body waves, and relocated aftershocks. Amplitude and phase spectra from up to 93 Rayleigh waves were inverted for centroid time, depth, and moment tensor in a two-step algorithm. In some of the inversions, the time function was parameterized to include information from the body-wave time function. The resulting source parameters were stable with respect to variations in the velocity and attenuation models assumed, the parameterization of the time function, and the set of Rayleigh waves included. The surface wave focal mechanism derived (=275°, =61°, =156°) is an oblique-compressional mechanism that is not easy to interpret in terms of subduction tectonics. A seismic moment of 4.5×10²⁰ N-m, a centroid depth of 45±5 km, and a centroid time of 13±3 s were obtained. Directivity was not resolvable from the surface waves. The short source duration is in significant contrast to many large earthquakes.We performed a simultaneous inversion ofP andSH body waves for focal mechanism and time function. The focal mechanism agreed roughly with the surface wave mechanism. Multiple focal mechanisms remain a possibility, but could not be resolved. The body waves indicate a short duration of slip (15 to 20 s), with secondary moment release 60s later. Seismically radiated energy was computed from the body-wave source spectrum. The stress drop computed from the seismic energy is about 30 bars. Sixty aftershocks that occurred within three months of the mainshock were relocated using the method of Joint Hypocentral Determination (JHD). Most of the aftershocks have underthrusting focal mechanisms and appear to represent triggered slip on the main thrust interface. The depth, relatively high stress drop, short duration of slip, and paucity of true aftershocks are consistent with intraplate faulting within the downgoing plate. Although it is not clear on which nodal plane slip occurred, several factors favor the roughly E-W trending plane. The event occurred near a major segmentation in the downgoing plate at depth, near a bend in the trench, and near a right-lateral offset of the volcanic are by 80 km along an E-W direction. Also, all events in the region from 1977 to 1991 with CMT focal mechanisms similar to that of the Mainshock occurred near the mainshock epicenter, rather than forming an elongate zone parallel to the trench as did the aftershock activity. We interpret this event as part of the process of segmentation or tearing of the subducting slab. This segmentation appears to be related to the subduction of the Louisville Ridge, which may act as an obstacle to subduction through its buoyancy.     

Rayleigh波在浅圆凹陷地形附近的散射:高频解答

利用波函数展开法给出了Rayleigh波在浅圆凹陷地形附近散射的一个高频解析解，并分析了入射频率、凹陷地形宽度和深度等因素对波散射的影响。数值结果表明，由于Rayleigh波幅值随深度而衰减，凹陷地形表面位移幅值整体上较小，且随着凹陷地形深度的增加而减小；由于Rayleigh波幅值还随频率而衰减，随着入射频率的升高，凹陷地形表面位移幅值逐渐减小；由于凹陷地形的屏障作用，在人射波的近端，地表位移分布变得相对复杂，地表位移峰值出现在左角点附近，而在入射波远端，地表位移分布相对简单，地表位移峰值出现在距凹陷地形较远的地方。     

Analytical expressions for the borehole seismic signatures excited by an explosive source buried in an elastic half‐space

Borehole guided waves that are excited by explosive sources outside of the borehole are important for interpreting borehole seismic surveys and for rock property inversion workflows. Borehole seismograms are typically modelled using numerical methods of wave propagation. In order to benchmark such numerical algorithms and partially to interpret the results of modelling, an analytical methodology is presented here to compute synthetic seismograms. The specific setup is a wavefield emanating from a monopole point source embedded within a homogeneous elastic medium that interacts with a fluid‐filled borehole and a free surface. The methodology assumes that the wavelength of the seismic signal is much larger than the borehole radius. In this paper, it is supposed that there is no poroelastic coupling between the formation and the borehole. The total wavefield solution consists of P, PP, and PS body waves; the surface Rayleigh wave; and the low‐frequency guided Stoneley wave (often referred as the tube wave) within the borehole. In its turn, the tube wave consists of the partial responses generated by the incident P‐wave and the reflected PP and PS body waves at the borehole mouth and by the Rayleigh wave, as well as the Stoneley wave eigenmode. The Mach tube wave, which is a conic tube wave, additionally appears in the Mach cone in a slow formation with the tube‐wave velocity greater than the shear one. The conditions of appearance of the Mach wave in a slow formation are formulated. It is shown that the amplitude of the Mach tube wave strongly depends on Poisson's ratio of the slow surrounding formation. The amplitude of the Mach tube wave exponentially decreases when the source depth grows for weakly compressible elastic media with Poisson's ratio close to 0.5 (i.e., saturated clays and saturated clay soils). Asymptotic expressions are also provided to compute the wavefield amplitudes for different combinations of source depth and source‐well offset. These expressions allow an approximate solution of the wavefield to be computed much faster (within several seconds) than directly computing the implicit integrals arising from the analytical formulation.     

Scattering ofP andS waves by a spherically symmetric inclusion

Scattering of an arbitrary elastic wave incident upon a spherically symmetric inclusion is considered and solutions are developed in terms of the spherical vector system of Petrashen, which produces results in terms of displacements rather than displacement potentials and in a form suitable for accurate numerical computations. Analytical expressions for canonical scattering coefficients are obtained for both the cases of incidentP waves and incidentS waves. Calculations of energy flux in the scattered waves lead to elastic optical theorems for bothP andS waves, which relate the scattering cross sections to the amplitude of the scattered fields in the forward direction. The properties of the solutions for a homogeneous elastic sphere, a sphere filled by fluid, and a spherical cavity are illustrated with scattering cross sections that demonstrate important differences between these types of obstacles. A general result is that the frequency dependence of the scattering is defined by the wavelength of the scattered wave rather than the wavelength of the incident wave. This is consistent with the finding that the intensity of thePS scattering is generally much stronger than theSP scattering. When averaged over all scattering angles, the mean intensity of thePS converted waves is2V _p ² /V _s ⁴ times the mean intensity of theSP converted waves, and this ratio is independent of frequency. The exact solutions reduce to simple and easily used expressions in the case of the low frequency (Rayleigh) approximation and the low contrast (Rayleigh-Born) approximation. The case of energy absorbing inclusions can also be obtained by assigning complex values to the elastic parameters, which leads to the result that an increase in attenuation within the inclusion causes an increased scattering cross section with a marked preference for scatteredS waves. The complete generality of the results is demonstrated by showing waves scattered by the earth's core in the time domain, an example of high-frequency scattering that reveals a very complex relationship between geometrical arrivals and diffracted waves.