Scattered Body Waves Research Articles

Spurious Rayleigh-wave apparent anisotropy near major structural boundaries: a numerical and theoretical investigation

SUMMARY The recent developments in array-based surface-wave tomography have made it possible to directly measure apparent phase velocities through wave front tracking. While directionally dependent measurements have been used to infer intrinsic $2\psi $ azimuthal anisotropy (with a 180° periodicity), a few studies have also demonstrated strong but spurious $1\psi $ azimuthal anisotropy (360° periodicity) near major structure boundaries particularly for long period surface waves. In such observations, Rayleigh waves propagating in the direction perpendicular to the boundary from the slow to the fast side persistently show a higher apparent velocity compared to waves propagating in the opposite direction. In this study, we conduct numerical and theoretical investigations to explore the effect of scattering on the apparent Rayleigh-wave phase velocity measurement. Using 2-D spectral-element numerical wavefield simulations, we first reproduce the observation that waves propagating in opposite directions show different apparent phase velocities when passing through a major velocity contrast. Based on mode coupling theory and the locked mode approximation, we then investigate the effect of the scattered fundamental-mode Rayleigh wave and body waves interfering with the incident Rayleigh wave separately. We show that scattered fundamental-mode Rayleigh waves, while dominating the scattered wavefield, mostly cause short wavelength apparent phase velocity variations that could only be studied if the station spacing is less than about one tenth of the surface wave wavelength. Scattered body waves, on the other hand, cause longer wavelength velocity variations that correspond to the existing real data observations. Because of the sensitivity of the $1\psi $ apparent anisotropy to velocity contrasts, incorporating such measurements in surface wave tomography could improve the resolution and sharpen the structural boundaries of the inverted model.

Shake to the Beat: Exploring the Seismic Signals and Stadium Response of Concerts and Music Fans

Abstract Large music festivals and stadium concerts are known to produce unique vibration signals that resemble harmonic tremor, particularly at frequencies around 1–10 Hz. This study investigates the seismic signals of a Taylor Swift concert performed on 5 August 2023 (UTC) as part of a series at SoFi Stadium in Inglewood, California, with an audience of ∼70,000. Signals were recorded on regional seismic network stations located within ∼9 km of the stadium, as well as on strong-motion sensors placed near and inside the stadium prior to the concert series. We automatically identified the seismic signals from spectrograms using a Hough transform approach and characterized their start times, durations, frequency content, particle motions, radiated energy, and equivalent magnitudes. These characteristics allowed us to associate the signals with individual songs and explore the nature of the seismic source. The signal frequencies matched the song beat rates well, whereas the signal and song durations were less similar. Radiated energy was determined to be a more physically relevant measure of strength than magnitude, given the tremor-like nature of the signals. The structural response of the stadium showed nearly equal shaking intensities in the vertical and horizontal directions at frequencies that match the seismic signals recorded outside the stadium. In addition, we conducted a brief experiment to further evaluate whether the harmonic tremor signals could be generated by the speaker system and instruments, audience motions, or something else. All evidence considered, we interpret the signal source as primarily crowd motion in response to the music. The particle motions of the strongest harmonics are consistent with Rayleigh waves influenced by scattered body waves and likely reflect how the crowd is moving. Results from three other musical performances at SoFi in summer 2023 were similar, although differences in the signals may relate to the musical genre and variations in audience motions.

Distributed Extension Across the Ethiopian Rift and Plateau Illuminated by Joint Inversion of Surface Waves and Scattered Body Waves

AbstractThe East African Rift System provides a rare location in which to observe a wide scope of rifting states. Well‐defined active narrow rifting in the Main Ethiopian Rift (MER) transitions to incipient extension and eventually pre‐rifted lithosphere through the northwestern flank of the Ethiopian Plateau (EP). Although the MER is well studied, the off‐axis region has received less attention. We develop Rayleigh wave phase velocity maps, Ps receiver functions, and H‐κ stack surfaces, and jointly invert these data using a trans‐dimensional, hierarchical Bayesian inversion algorithm to create shear velocity profiles across the MER and EP. All shear velocities observed are slower than the PREM global average, a reflection of the elevated temperatures that persist from plume impingement. In the EP, we find a shallow mantle slow shear velocity lineament parallel to the MER axis, amidst otherwise faster shear velocities. The crust is shallow in the MER, and also in the northwestern‐most EP flank. Thicker crust found elsewhere throughout the plateau is caused by crustal underplating and flood basalt emplacement. Shear velocities more reduced than the already low regional average, in concert with surficial volcanic features, geodetic observations, and slow P‐ and S‐wave anomalies, support off‐axis extension in the Ethiopian plateau, requiring reevaluation of the localization of continental breakup in the narrow MER.

Time–frequency windowing in multiparameter elastic FWI of shallow seismic wavefield

SUMMARYFull-waveform inversion of shallow seismic wavefields is a promising method to infer multiparameter models of elastic material properties (S-wave velocity, P-wave velocity and mass density) of the shallow subsurface with high resolution. Previous studies used either the refracted Pwaves to reconstructed models of P-wave velocity or the high-amplitude Rayleigh waves to infer the S-wave velocity structure. In this work, we propose a combination of both wavefields using continuous time–frequency windowing. We start with the contribution of refracted P waves and gradually increase the time window to account for scattered body waves, higher mode Rayleigh waves and finally the fundamental Rayleigh wave mode. The opening of the time window is combined with opening the frequency bandwidth of input signals to avoid cycle skipping. Synthetic reconstruction tests revealed that the reconstruction of P-wave velocity model and mass density can be improved. The S-wave velocity reconstruction is still accurate and robust and is slightly benefitted by time–frequency windowing. In a field data application, we observed that time–frequency windowing improves the consistency of multiparameter models. The inferred models are in good agreement with independent geophysical information obtained from ground-penetrating radar and full-waveform inversion of SH waves.

Matrix Approach of Seismic Imaging: Application to the Erebus Volcano, Antarctica

AbstractMultiple scattering of seismic waves is often seen as a nightmare for conventional migration techniques that generally rely on a ballistic or a single‐scattering assumption. In heterogeneous areas such as volcanoes, the multiple‐scattering contribution limits the imaging‐depth to one scattering mean free path, the mean distance between two successive scattering events for body waves. In this Letter, we propose a matrix approach of passive seismic imaging that pushes back this fundamental limit by making an efficient use of scattered body waves drowned into a noisy seismic coda. As a proof of concept, the case of the Erebus volcano in Antarctica is considered. The Green's functions between a set of geophones placed on top of the volcano are first retrieved by the cross correlation of coda waves induced by multiple icequakes. This set of impulse responses forms a reflection matrix. By combining a matrix discrimination of singly scattered waves with iterative time reversal, we are able to push back the multiple scattering limit beyond 10 scattering mean free paths. The matrix approach reveals the internal structure of the Erebus volcano: A chimney‐shaped structure at shallow depths, a magma reservoir at 2,500 m and several cavities at sea level and below it. The matrix approach paves the way toward a greatly improved monitoring of volcanic structures at depth. Beyond this specific case, the matrix approach of seismic imaging can generally be applied to all scales and areas where multiple scattering events undergone by body waves prevent in‐depth imaging of the Earth's crust.

An adaptive Bayesian inversion for upper-mantle structure using surface waves and scattered body waves

SummaryWe present a methodology for 1-D imaging of upper-mantle structure using a Bayesian approach that incorporates a novel combination of seismic data types and an adaptive parametrization based on piecewise discontinuous splines. Our inversion algorithm lays the groundwork for improved seismic velocity models of the lithosphere and asthenosphere by harnessing the recent expansion of large seismic arrays and computational power alongside sophisticated data analysis. Careful processing of P- and S-wave arrivals isolates converted phases generated at velocity gradients between the mid-crust and 300 km depth. This data is allied with ambient noise and earthquake Rayleigh wave phase velocities to obtain detailed V S and V P velocity models. Synthetic tests demonstrate that converted phases are necessary to accurately constrain velocity gradients, and S–p phases are particularly important for resolving mantle structure, while surface waves are necessary for capturing absolute velocities. We apply the method to several stations in the northwest and north-central United States, finding that the imaged structure improves upon existing models by sharpening the vertical resolution of absolute velocity profiles, offering robust uncertainty estimates, and revealing mid-lithospheric velocity gradients indicative of thermochemical cratonic layering. This flexible method holds promise for increasingly detailed understanding of the upper mantle.

Transdimensional inversion of scattered body waves for 1D S-wave velocity structure – Application to the Tengchong volcanic area, Southwestern China

Inversion of receiver functions is commonly used to recover the S-wave velocity structure beneath seismic stations. Traditional approaches are based on deconvolved waveforms, where the horizontal component of P-wave seismograms is deconvolved by the vertical component. Deconvolution of noisy seismograms is a numerically unstable process that needs to be stabilized by regularization parameters. This biases noise statistics, making it difficult to estimate uncertainties in observed receiver functions for Bayesian inference. This study proposes a method to directly invert observed radial waveforms and to better account for data noise in a Bayesian formulation. We illustrate its feasibility with two synthetic tests having different types of noises added to seismograms. Then, a real site application is performed to obtain the 1-D S-wave velocity structure beneath a seismic station located in the Tengchong volcanic area, Southwestern China. Surface wave dispersion measurements spanning periods from 8 to 65 s are jointly inverted with P waveforms. The results show a complex S-wave velocity structure, as two low velocity zones are observed in the crust and uppermost mantle, suggesting the existence of magma chambers, or zones of partial melt. The upper magma chambers may be the heart source that cause the thermal activity on the surface.

Depth sensitivity of seismic coda waves to velocity perturbations in an elastic heterogeneous medium

SUMMARY Numerous monitoring applications make use of seismic coda waves to evaluate velocity changes in the Earth. This raises the question of the spatial sensitivity of coda wave-based measurements. Here, we investigate the depth sensitivity of coda waves to local velocity perturbations using 2-D numerical wavefield simulations. We calculate the impulse response at the surface before and after a slight perturbation of the velocity within a thin layer at depth is introduced. We perform a parametric analysis of the observed apparent relative velocity changes, e obs , versus the depth of the thin perturbed layer. Through the analysis of the decay of e obs , we can discriminate two different regimes: one for a shallow perturbation and the other for a deep perturbation. We interpret the first regime as the footprint of the 1-D depth sensitivity of the fundamental surface wave mode. To interpret the second regime, we need to model the sensitivity of the multiply scattered body waves in the bulk. We show that the depth sensitivity of coda waves can be modelled as a combination of bulk wave sensitivity and surface wave sensitivity. The transition between these two regimes is governed by mode conversions due to scattering. We indicate the importance of surface waves for the sensitivity of coda waves at shallow depths and at early times in the coda. At later times, bulk waves clearly dominate the depth sensitivity and offer the possibility of monitoring changes at depths below the sensitivity of the surface waves. Based on the transition between the two regimes, we can discriminate a change that occurs at the surface from a change that occurs at depth. This is illustrated for shallow depth perturbations through an example from lunar data.

Synthesis of coda wave envelopes in randomly inhomogeneous elastic media in a half-space: single scattering model including Rayleigh waves

SUMMARY Seismograms of local earthquakes consist of not only direct waves but also coda waves originating from scattering due to lithospheric inhomogeneities. As a mathematical basis for interpreting these seismograms, we present a synthesis of coda wave envelope propagating through randomly inhomogeneous medium in a half-space by considering elastic conversion scattering modes of body-to-body waves, Rayleigh-to-body waves, body-to-Rayleigh waves and Rayleigh-to-Rayleigh waves. We first evaluate scattering amplitudes of above scattering modes by elastic inhomogeneity localized in a block by applying the Born approximation to the elastic wave equation in the frequency domain. Scattering amplitudes of all conversion scattering modes are represented by the Fourier spectrum of the elastic inhomogeneity multiplied by the basic scattering pattern which determines the non-isotropic scattering pattern in the long-wavelength limit. Then, considering an ensemble of random media, we derive scattering coefficients which represent the scattering power per unit volume as ensemble averages. By summing up the scattered energy from distributed scattering blocks in a half-space for a point shear dislocation source with the omega-square model, we synthesize the coda-wave envelope for each scattering mode. Numerical evaluation of three-component MS envelopes for a shallow source shows strong frequency dependence. Though scattered body waves are dominant in high frequencies, Rayleigh-to-Rayleigh scattered waves with smaller decay with increasing lapse time become dominant for low frequencies.

Scattered Surface Wave Energy in the Seismic Coda

One of the many important contributions that Aki has made to seismology pertains to the origin of coda waves (Aki, 1969; Aki and Chouet, 1975). In this paper, I revisit Aki's original idea of the role of scattered surface waves in the seismic coda. Based on the radiative transfer theory, I developed a new set of scattered wave energy equations by including scattered surface waves and body wave to surface wave scattering conversions. The work is an extended study of Zeng et al. (1991), Zeng (1993) and Sato (1994a) on multiple isotropic-scattering, and may shed new insight into the seismic coda wave interpretation. The scattering equations are solved numerically by first discretizing the model at regular grids and then solving the linear integral equations iteratively. The results show that scattered wave energy can be well approximated by body-wave to body wave scattering at earlier arrival times and short distances. At long distances from the source, scattered surface waves dominate scattered body waves at surface stations. Since surface waves are 2-D propagating waves, their scattered energies should in theory follow a common decay curve. The observed common decay trends on seismic coda of local earthquake recordings particular at long lapse times suggest that perhaps later seismic codas are dominated by scattered surface waves. When efficient body wave to surface wave conversion mechanisms are present in the shallow crustal layers, such as soft sediment layers, the scattered surface waves dominate the seismic coda at even early arrival times for shallow sources and at later arrival times for deeper events.

Simulated envelopes of non-isotropically scattered body waves as compared to observed ones: another manifestation of fractal heterogeneity

Envelopes of scalar waves are simulated at various distances from an instant point source embedded in a random uniformly scattering medium by means of direct Monte-Carlo modelling of wave-energy transport. Three types of scattering radiation pattern ('indicatrix') are studied, for media specified by (1) a Gaussian autocorrelation function of inhomogeneities, (2) a power-law ('fractal', k−α) inhomogeneity spectrum and (3) the mix of case (1) and the isotropic indicatrix (very small + large inhomogeneities). We look for a model that can qualitatively reproduce the two most characteristic features of real S-wave envelopes of near earthquakes, namely (1) the broadening of the ‘direct’ wave group with distance and (2) the monotonously decaying shape of the coda envelope that does not deviate strongly from that expected in the isotropic scattering case. Both properties are observed for any band over a wide frequency range (1–40 Hz). The well-studied isotropic scattering model realistically predicts the appearance of codas but fails to predict pulse broadening. The model of large-scale inhomogeneity realistically predicts the mode of pulse broadening but fails to predict codas. We have found that, for a particular frequency band, within each class of inhomogeneity studied, both requirements can be qualitatively satisfied by a certain choice of parameters. In the Gaussian-ACF case, however, this match can be obtained only for a narrow frequency range. In contrast, the fractal case (with a value of exponent a of about 3.5–4) reproduces qualitatively the observed wide-band behaviour, and we consider it a reasonable representation of the gross properties of the earth medium.

Complexities in high‐frequency seismic waveforms due to three‐dimensional structure in the New Madrid Seismic Zone

Effects of three‐dimensional structure on 2 to 10 Hz seismic signals in the New Madrid Seismic Zone (NMSZ) are investigated using waveforms from similar microearthquakes that occurred near Ridgely, Tennessee. Slowness power spectral (SPS) analysis is used to study the plane wave composition of P, S, and coda waves. The observed SPS at four stations suggest that the early S coda is composed mainly of wavelets leaving the source with slownesses of the direct S wave. At greater lapse times, coda is increasingly affected by random scattering. SPS observed on the vertical components and/or near the S wave nodes are more comlplex due to practical limitations of the algorithm. P coda and early S coda at a fifth station, located southwest of the junction, show the most complex SPS. Analysis of that complexity, together with waveform synthetics and travel time analysis, strongly suggests that scattering/multipathing due to source zone velocity heterogeneities occurs near the path, probably due to dilatancy occurring southwest of the Ridgely junction, or due to internal fault zone complexities. Coda Q, when averaged over three components, show no station variations. We infer that (1) coda at 1 Hz is mainly caused by scattering in the sedimentary layer where Q is frequency independent, (2) at higher frequencies the coda contains an increasing amount of scattered body waves, and (3) the source radiation pattern and path variable scattering/multipathing effects in early coda do not affect coda Q.

Seismic Wavefields in Multi-Layered Media Calculated by Hybrid Combination of Boundary Element Method and Thin Layer Finite Element Method. The Case of Two-Dimensional SH-Wavefields.

Two-dimensional SH-wavefields in laterally heterogeneous multi-layered media are calculated by using a hybrid method in which the boundary element method (BEM) is combined with the thin layer finite element method (TLFEM) by matching the boundary conditions on vertical boundaries. The traction operators defined on vertical boundaries are used to match the boundary conditions. The traction operators can be approximately calculated from eigenvalues and eigenfunctions for the multi-layered media, which are obtained by using the TLFEM. This method is efficient to calculate scattering of surface waves in laterally heterogeneous multi-layered media. Scattering of Love waves in multi-layered slope structure is investigated. For the case in which a source is in the shallower side, Love waves mainly propagate in the forward direction. The dispersion of Love waves shows an averaged structure of both sides. For the case in which a source is in the deeper side, the amplitude of scattered waves is large, and scattered body waves and back-scattered Love waves exist.

Rayleigh-wave scattering at various wedge corners: Investigation in the wider range of wedge angles

Abstract A mathematically simple procedure was developed that secures a great increase in the range of wedge angles for which numerical computation of the reflection and transmission coefficients of Rayleigh waves in a wedge-shaped medium can be performed. Whereas in previous research the range of angles in such a medium for which computation was possible was treated as necessarily lying between 72° and 108°, the present procedure extends that range to between 36° and 180°. Furthermore, by completely separating the poles located on the original integral path, it becomes possible to perform numerical calculations on the original path itself. The numerical results thus obtained were found to coincide almost perfectly with experimental ones. It is pointed out that in cases of wedge angles exceeding 180°, the present procedure allows a greater extension of the computable range than was previously possible. Moreover, the features of scattered body waves inside wedges are discussed over the wide range of wedge angles.

Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake

Several models which describe the intensity of single and multiply scattered waves radiated by an instant point source into a medium with uniformly dispersed scatterers and uniform absorption are reviewed. The model of multiple low-angle scattering (MLAS) is then considered and we show that in the realistic case of dominantly forward scattering one can employ the model to determine the mean free path l from pulse broadening of scattered waves. This fast (∝ r 2 ) pulse broadening leads to r −2 amplitude decay and can explain known fast-amplitude decay of short-range magnitude calibration curves and of peak acceleration attenuation laws. Since analytical theory is lacking for an important case of scattered body waves at source distances around r = l , we employ the previously developed technique of Monte-Carlo simulation of a scattered wavefield to obtain a set of theoretical formulae and master curves. These enable us to estimate mean free path l in two independent ways: from intensity ratio of direct and scattered waves ( l A ) and from pulse duration or retardation ( l T ). In both cases, the estimates must depend only weakly on errors of Q determination. We applied the developed theory to the interpretation of records of earthquakes near Kamchatka recorded by frequency selecting (‘ChISS’) stations. For Shipunsky (SPN) station in the 1.5–6.0 Hz frequency range the estimates are l A ≈ 150 km, l T ≈ 110 km. In the 6–25 Hz range, l A is decreasing, roughly as ƒ −0.65 . We could expect that improved theoretical coda shapes will resemble the observed ones, leading to accurate intrinsic Q estimates. This is not the case however, and our Q estimates depend in fact on the choice of lapse time window. This indicates that uniform medium models are insufficient for interpretation. We could demonstrate directly the depth dependence of l based on l T estimates.

Rayleigh wave scattering at wedge corners with major wedge angles

Theoretical calculations were performed on reflected and transmitted Rayleigh waves and scattered body waves, in the case where a two-dimensional Rayleigh wave is incident to a wedge-shaped medium having a wedge angle between 250° and 290° and arbitrary value of Poisson ratio. The reflection and transmission coefficients of Rayleigh waves were also experimentally measured in cases of wedges with 190° to 330° wedge angles. The method of theoretical analysis and the techniques of experiment are based on those developed in our preceding research (W-1. W-2 and W-3). Compared with the results where the wedge angle is smaller than 180° (W-1 and W-2), all features show consistent variation with wedge angle.

The scattering of Rayleigh waves at a variously inclined discontinuity.

A theoretical analysis is performed on the scattering of Rayleigh waves at a variously inclined discontinuity, where a two-dimensional sinusoidal Rayleigh wave is incident. Reflection and transmission coefficients are numerically derived. The calculable extent of the inclination angle of the interface ranges from 74° to 106° to the free surface, though it is reduced to a certain degree, according to the assumption of elasticities in both sides of the discontinuity. Within this range the variations of the coefficients are discussed under various assumptions of elastic constants in the media. The amplitude ratios of reflection and transmission do not greatly change with the inclination of the interface, but the phase shift of reflection varies linearly, and hence, by the use of the latter it is possible to determine the actual inclination of the interface from the observed Fourier spectra of incident and reflected Rayleigh waves. Scattered body waves, especially S waves, from an interface show rather complicated distributions, depending on the properties of the media.

SCATTERING OF RAYLEIGH WAVES IN AN ELASTIC THREE-QUARTER SPACE

This study investigates the generation of waves, due to the incidence of Rayleigh waves upon the corner of an elastic three-quarter space. Incident Rayleigh waves travel along one surface of the elastic three-quarter space. Integral equations are derived by use of the Fourier transform technique, and are solved by deforming the integration paths into paths along which the integrands vary smoothly in magnitude. Expressions for the energy flux of Rayleigh waves along two free surfaces and for scattered body waves, were obtained. Partition of energy flux and directivity of the scattered P and S waves are discussed. The agreement between our theory and the experiment made by another investigator is considerably good. Substantial amounts of energy of incident Rayleigh waves are scattered away in the form of S waves, in the same direction as incident Rayleigh waves. Another interesting and important feature is the fact that, just around the corner of the elastic medium, the dilatational and distortional parts of waves are very large in magnitude. In other words, a kind of trapping of these waves occurs around the corner. These two kinds of waves are out of phase and the resultant waves are small in magnitude, due to the result of cancelling.

Coda wave excitation due to nonisotropic scattering and nonspherical source radiation

Under the assumption that seismic coda waves may be modeled as a superposition of single scattered body waves due to scatterers randomly distributed in the three‐dimensional earth, we study how nonisotropic scattering and nonspherical source radiation patterns affect the coda wave excitation. The energy density of coda waves is represented by an integral over scattering angle or radiation angle. Properties of each integral kernel are analytically studied in detail. The energy density of coda waves is factored into 2 parts: a geometrical spreading factor and a function of the ratio of the lapse time measured from the earthquake origin time to the travel time of direct wave. The equations predict that the influence of nonisotropic scattering and that of nonspherical source radiation will be significant, particularly on the early part of coda waves.

Surface waves in structures with locally irregular boundaries

abstract The transmitted field due to surface waves incident on a local irregularity of a plane-layered medium has been studied. A perturbation method to the first order and the Born approximation can be used if the variations in the thickness of the layers are sufficiently smooth and the wavelengths are long when compared to the size of the irregularities. The spectrum of the perturbed part of the displacement field at the surface is a sum over the surface-wave modes for the regular medium, with an additional term involving the scattered body waves. Numerical computations have been performed for structures composed of a layer overlying a half-space. The contribution of the various modes to the transmitted Love or Rayleigh fields has been studied for several structures. A general method has been obtained to analyze the effect of a complex structure as the superposition of the fields due to simpler ones. When the layer thickness is kept unchanged, the incident mode is not perturbed to the first order. Synthetic seismograms, computed at stations sufficiently close to the irregular region, show how the perturbation of the signal depends on distance. A comparison has been made for Love waves with a finite element method. Both methods give very similar results when the stations are not too close to the irregularities so that the body-wave contribution is negligible. The local phase velocity shows departures from the curves for a regular model.