Effective-mode analysis of elastic waves

SUMMARYTo describe the energy transport in the seismic coda, we introduce a system of radiative transfer equations for coupled surface and body waves in a scalar approximation. Our model is based on the Helmholtz equation in a half-space geometry with mixed boundary conditions. In this model, Green’s function can be represented as a sum of body waves and surface waves, which mimics the situation on Earth. In a first step, we study the single-scattering problem for point-like objects in the Born approximation. Using the assumption that the phase of body waves is randomized by surface reflection or by interaction with the scatterers, we show that it becomes possible to define, in the usual manner, the cross-sections for surface-to-body and body-to-surface scattering. Adopting the independent scattering approximation, we then define the scattering mean free paths of body and surface waves including the coupling between the two types of waves. Using a phenomenological approach, we then derive a set of coupled transport equations satisfied by the specific energy density of surface and body waves in a medium containing a homogeneous distribution of point scatterers. In our model, the scattering mean free path of body waves is depth dependent as a consequence of the body-to-surface coupling. We demonstrate that an equipartition between surface and body waves is established at long lapse-time, with a ratio which is predicted by usual mode counting arguments. We derive a diffusion approximation from the set of transport equations and show that the diffusivity is both anisotropic and depth dependent. The physical origin of the two properties is discussed. Finally, we present Monte Carlo solutions of the transport equations which illustrate the convergence towards equipartition at long lapse-time as well as the importance of the coupling between surface and body waves in the generation of coda waves.

The objective of this paper is to provide a unified treatment of elastic and electromagnetic (EM) wave propagation in horizontally layered media for which the parameters in the partial differential equations are piece‐wise continuous functions of only one spatial variable. By applying a combination of Fourier, Laplace, and Bessel transforms to the partial differential equations describing the elastic or EM wave propagation I obtain a system of 2n linear ordinary differential equations. The [Formula: see text] coefficient matrix is partitioned into [Formula: see text] submatrices. By a proper choice of variables, the diagonal submatrices are zero and the off‐diagonal submatrices are symmetric. All the results in the paper are derived from the symmetry properties of this general equation. In the appendices it is shown that three‐dimensional elastic waves, cylindrical P‐SV waves, acoustic waves, and electromagnetic waves in isotropic layered media can all be represented by an equation with the same properties. The symmetry properties of the system matrix are used to derive simplified equations for computing the propagator matrix for a stack of inhomogeneous layers. The wave field is also decomposed into upgoing and downgoing waves by an eigenvector decomposition which is much simplified compared with the general case of a full [Formula: see text] system matrix. This wave field decomposition is a suitable starting point for deriving one‐way wave equations and WKB‐approximations of different order. These approximations have potential applications in generalized migration schemes. Two propagation invariants are derived using the symmetry properties of the system matrix. One of these is only valid for lossless media and corresponds to the conservation of energy. For a stack of inhomogeneous layers, transmission and reflection matrices for upward and downward propagation are defined. Using the two propagation invariants, a number of symmetry properties of the reflection and transmission matrices are derived. The relationship between the reflection and transmission matrices and the propagator matrix is given. Computation of the reflection and transmission matrices for two inhomogeneous layers is done by Redheffer’s star product. This composition rule has been derived for P‐SV waves by Kennett. The inverse of the star product, apparently unknown in seismology, is also given. This is a rule which may be used to remove the effect of an inhomogeneous layer at the top or bottom of a stack of layers. Such layer stripping techniques have possible applications in general inversion schemes. It is also shown that the reflection and transmission matrices of an inhomogeneous medium can be found by solving a matrix Riccati equation. For a stack of inhomogeneous layers bounded above by a free surface, modified reflection and transmission matrices are defined. Using the two propagation invariants, a number of symmetry properties of the modified reflection and transmission matrices are derived. For lossless media a generalized Kunetz equation is given. The modified reflection and transmission matrices are expressed in terms of the partitioned submatrices of the propagator matrix and in terms of the usual reflection and transmission matrices. I also derive the response of a buried point source for a layered medium bounded by a free surface and a homogeneous half‐space, and for a layered medium bounded by two homogeneous half‐spaces. More general sources can be treated by superposition. In an appendix, I use multidimensional Fourier transforms to derive the decomposition of a spherical wave into plane waves (the Weyl integral) and into cylindrical waves (the Sommerfeld integral). Both these decompositions include inhomogeneous or evanescent waves. The Whittaker integral represents a decomposition into traveling waves only, but an explicit form of the Whittaker integral for spherical waves does not appear to be known. The decomposition into upgoing and downgoing waves breaks down for horizontally traveling waves such as channel waves and surface waves. The reflection and transmission matrices do not exist in this case. This fact is used to derive dispersion relationships for channel waves and surface waves by requiring that certain determinants shall be zero.

Festschrift to mark the sixtieth birthday of Professor Jens Lothe

Summary. This paper reviews recent work, much of it unpub~shed, on the effects of anisotropy on seismic waves, and lays the theoretical background for some of the other papers in this number of the Geophysical Journal. The propagation of both body and surface waves in anisotropic media is fundamentally different from their propagation in isotropic media, although the differences in behaviour may be comparatively subtle and difficult to observe. One of the most diagnostic of these anomalies, which has been observed on. some surface-wave trains, and should be evident in body-wave arrivals, is generalized, three-dimensional polarization, where the Rayleigh motion is coupled to the Love, and the P and SV motion is coupled to the SH. This coupling introduces polarization anomalies which may be used to investigate anisotropy within the Earth. 1 Introduction A material displaying velocity anisotropy must have its effective elastic constants arranged in some form of crystalline symmetry. The behaviour of both body and surface waves in such anisotropic structures differs from that in isotropic structures, and the variation of velocity with direction is only one of the anomalies which may occur, where we use anomaly to mean differences in behaviour from that expected in isotropic material. Within an anisotropic material three body waves propagate in any direction, having different and varying velocity, and different and varying polarization. Away from directions of crystal symmetry there may be anomalous phases, body and surface waves will have anomalous polarizations, and energy propagation of body and surface waves will not be parallel to the propagation vector. It appears intuitively that many of the anomalies can be attributed to the subtle interplay of the three varying body waves, making the variations of these anomalies difficult to predict. Similarly, smd differences in the structure, such as the thickness of the layer, can make radical changes in the anomalous behaviour. In this paper, we shall describe the type of phenomena to be expected in seismic waves from the presence of a layer of anisotropy within the Earth. A more complete treatment of the mathematics for the general problem of a plane layered structure containing a layer of anisotropy can be found in Keith (1975) for body waves, and Crampin (1970) and Taylor & Crampin (1977) for surface waves.

SUMMARYSeismic body wave traveltime tomography and surface wave dispersion tomography have been used widely to characterize earthquakes and to study the subsurface structure of the Earth. Since these types of problem are often significantly non-linear and have non-unique solutions, Markov chain Monte Carlo methods have been used to find probabilistic solutions. Body and surface wave data are usually inverted separately to produce independent velocity models. However, body wave tomography is generally sensitive to structure around the subvolume in which earthquakes occur and produces limited resolution in the shallower Earth, whereas surface wave tomography is often sensitive to shallower structure. To better estimate subsurface properties, we therefore jointly invert for the seismic velocity structure and earthquake locations using body and surface wave data simultaneously. We apply the new joint inversion method to a mining site in the United Kingdom at which induced seismicity occurred and was recorded on a small local network of stations, and where ambient noise recordings are available from the same stations. The ambient noise is processed to obtain inter-receiver surface wave dispersion measurements which are inverted jointly with body wave arrival times from local earthquakes. The results show that by using both types of data, the earthquake source parameters and the velocity structure can be better constrained than in independent inversions. To further understand and interpret the results, we conduct synthetic tests to compare the results from body wave inversion and joint inversion. The results show that trade-offs between source parameters and velocities appear to bias results if only body wave data are used, but this issue is largely resolved by using the joint inversion method. Thus the use of ambient seismic noise and our fully non-linear inversion provides a valuable, improved method to image the subsurface velocity and seismicity.

Full waveform tomography for near-surface applications has garnered increased attention in recent years due to increasing computational capabilities and rapid developments in the continued search for hydrocarbon sources. Full waveform tomography attempts to solve an inversion problem whereby the entirety of the seismic waveforms measured at a site are matched with waveforms acquired from numerical simulations of wave propagation in a subsurface model. One of the strengths of the method is the ability to incorporate all types of waves into the inversion. With careful placement of measurement locations, combined recordings of body waves and surface waves can be exploited to improve the inversion results. One near-surface engineering application where a combined body and surface wave full waveform inversion (FWI) technique can potentially improve resolution capabilities is for evaluation of liquefaction triggering. Liquefaction of granular soils refers to the loss of shear strength caused by rapid dynamic loading as encountered in earthquake events. The evaluation of liquefaction triggering typically involves site characterization at pre-determined locations with standard penetration tests (SPT) and cone penetration tests (CPT) to estimate the resistance of the subsurface soils against liquefaction. Geophysical measurements from either downhole, crosshole, or surface-wave testing methods can be used to augment this SPT/CPT information. Since boreholes or CPT soundings are often already present at a site being investigated for liquefaction hazards, a receiver array could be simultaneously deployed at the surface and within the subsurface to capture both surface and body waves generated by a single source. This study uses numerical simulations to examine how the accuracy and resolution of FWI can be improved with combined body and surface wave measurements within the context of characterizing liquefaction triggering. The results demonstrated that the spatial extent of liquefaction is better estimated using the combined full waveform approach when compared to full waveform of only surface measurements and when compared to interpolation between localized in-situ measurements.

<p>Different phases of seismic waves generated by earthquakes carry considerable information about the subsurface structures as they propagate within the earth. Depending on the scope and objective of an investigation, various types of seismic phases are studied. Studying surface waves image shallow and large-scale subsurface features, while body waves provide high-resolution images at higher depths, which is otherwise impossible to be resolved by surface waves. The most challenging aspect of studying body waves is extracting low-amplitude P and S phases predominantly masked by high amplitude and low attenuation surface waves overlapping in time and frequency. Although body waves generally contain higher frequencies than surface waves, the overlapping frequency spectrum of body and surface waves limits the application of elementary signal processing methods such as conventional filtering. Advanced signal processing tools are required to work around this problem. Recently the Sparsity-Promoting Time-Frequency Filtering (SP-TFF) method was developed as a signal processing tool for discriminating between different phases of seismic waves based on their high-resolution polarization information in the Time-Frequency (TF)-domain (Mohammadigheymasi et al., 2022). The SP-TFF extracts different phases of seismic waves by incorporating this information and utilizing a combination of amplitude, directivity, and rectilinearity filters. This study implements SP-TFF by properly defining a filter combination set for specific extraction of body waves masked by high-amplitude surface waves. Synthetic and real data examinations for the source mechanism of the  M<sub>w</sub>=7.5 earthquake that occurred in November 2021 in Northern Peru and recorded by 58 stations of the United States National Seismic Network (USNSN) is conducted. The results show the remarkable performance of SP-TFF extracting P and SV phases on the vertical and radial components and SH phase on the transverse component masked by high amplitude Rayleigh and Love waves, respectively. A range of S/N levels is tested, indicating the algorithm’s robustness at different noise levels. This research contributes to the FCT-funded SHAZAM (Ref. PTDC/CTA-GEO/31475/2017) and IDL (Ref. FCT/UIDB/50019/2020) projects. It also uses computational resources provided by C4G (Collaboratory for Geosciences) (Ref. PINFRA/22151/2016).</p><p>REFERENCE<br>Mohammadigheymasi, H., P. Crocker, M. Fathi, E. Almeida, G. Silveira, A. Gholami, and M. Schimmel, 2022, Sparsity-promoting approach to polarization analysis of seismic signals in the time-frequency domain: IEEE Transactions on Geoscience and Remote Sensing, 1–1.</p>

SUMMARY Analysis of long recordings of ambient seismic noise has shown to be effective for estimation of seismic responses between points located on the surface. This includes both the ballistic and the coda part of the waveforms. Passive image interferometry is used to analyse perturbations in the reconstructed coda, to detect and locate changes in the medium. This method has been shown to be effective in monitoring variations in seismic velocity produced by a wide range of phenomena. However, localization of the sources of these changes is still an open problem for a 3-D half-space, given the difficulties of integrating body and surface waves within the same framework. In this study, we approach this problem by developing the sensitivity kernels of a scalar model that integrates the body and surface scalar waves. First, we establish a parallel between the penetration depth of the surface waves for the elastic and scalar cases, which equips the latter with a natural scaling with frequency that is otherwise not included in the model. Next, using a variational approach, we quantify how a velocity perturbation in the medium affects the propagation velocity of the surface waves. Based on these results, we extend the sensitivity theory to include the body and surface waves as modes of propagation and detection, as restricted to a 1-D depth-dependent perturbation description, for simplicity. The obtained kernel can be expressed as the sum of a surface and a body waves sensitivity kernels, which are inter-dependent through a set of traveltime distributions. These distributions are estimated with Monte Carlo simulations based on the radiative transfer equations of the system, with the source and the receiver located in the same position at the surface. The sensitivity at depth is in good agreement with previous results based on full wavefield elastic simulations in 3-D inhomogeneous half-space. The temporal evolution of the body and surface waves sensitivity is quantified, as well as the contribution of all the possible modes of propagation and detection to each of these sensitivities. We show how the position of the source affects the sensitivity between the two types of waves. We find that the efficacy of energy conversion from surface to body waves is controlled by the ratio between the surface wave penetration depth and the mean free path, a feature that has not been reported in previous studies. This means that configurations that share this ratio have the same sensitivity as long as all the spatial and temporal variables (e.g. elapsed time, depth) are non-dimensionalized with the mean free path and the mean free time, respectively.

A surface wave method is often used to map shear wave velocity variation of soil with depth. However, measured seismic wave field usually contains such unfavorable waves as higher modes of surface waves, body waves, and ambient noise. These waves can significantly influence results of the analysis if not properly handled. Since the conventional surface wave method of spectral analysis of surface waves (SASW) is based on the two-receiver acquisition and processing scheme, these complications are usually not effectively accounted and the field procedure tends to be labor intensive. In an attempt to increase confidence in the interpreted Vs profile as a result of the ambiguity in the analyzed dispersion characteristics, multichannel method is used in this research to characterize a test site on the campus of University of Tennessee, Knoxville that is seismically active. The multichannel analysis of surface waves (MASW) method originated from the traditional seismic exploration approach that employs multiple (twelve or more) receivers placed along a linear survey line. Main advantage is its capability of recognizing different types of seismic waves based on wave propagation characteristics such as velocity and attenuation. The MASW method utilizes this capability to discriminate the fundamental-mode Rayleigh wave against all other types of surface and body waves generated not only from the seismic source but also from the ambient site conditions. Dispersive characteristics of seismic waves are imaged from an objective 2-D wavefield transformation. Since the multichannel pattern-recognition ability can tolerate certain extent of adverse influence from the near-field effects as well as from the noise waves, data acquisition procedure is also a simple task, being insensitive to such field factors as seismic source, receiver spacing, and distance from source. In this way, the MASW method can produce a 2-D cross section map of Vs distribution within soil in accurate and efficient manner. The present paper indicates results from MASW survey at a site along the Tennessee river with relatively shallow bed rock. Both SASW and MASW techniques will prove to be important tools for evaluating liquefaction potential for future geophysical and geotechnical engineering community and this paper presents important aspects of MASW technique and its effectiveness in geophysical site characterization.

Elastic waves produced by an impact were recorded at the surface of a solid 12.0 inch diameter steel sphere coated with a 0.3 inch copper layer. Conventional modeling techniques employing both compressional and shear piezoelectric transducers were used to record elastic waves for one millisecond at various points around the great circle of the sphere. Body, PL, and surface waves were observed. Density, layer thickness, compressional and shear-wave velocities were measured so that accurate surface-wave dispersion curves could be computed. Surface-wave dispersion was measured as well as computed. Measured PL mode dispersion compared favorably with theoretical computations. In addition, dispersion curves for Rayleigh, Stoneley, and Love modes were computed. Measured surface-wave dispersion showed Rayleigh and Love modes were observed but not Stoneley modes. Measured dispersion compared favorably with theoretical computations. The curvature correction applied to dispersion calculations in a flat space has been estimated to correct dispersion values at long-wave lengths to about one per cent of correct dispersion in a spherical model. Measured dispersion compared with such flat space dispersion corrected for curvature proved accurate within one per cent at long wave lengths. Two sets of surface waves were observed. One set was associated with body waves radiating outward from impact. The other set was associated with body waves reflecting at the pole opposite impact. For each set of surface waves, measured dispersion compared favorably with computed dispersion.

We derive a radiative transfer equation that accounts for coupling from surface waves to body waves and the other way around. The model is the acoustic wave equation in a two-dimensional waveguide with reflecting boundary. The waveguide has a thin, weakly randomly heterogeneous layer near the top surface, and a thick homogeneous layer beneath it. There are two types of modes that propagate along the axis of the waveguide: those that are almost trapped in the thin layer, and thus model surface waves, and those that penetrate deep in the waveguide, and thus model body waves. The remaining modes are evanescent waves. We introduce a mathematical theory of mode coupling induced by scattering in the thin layer, and derive a radiative transfer equation which quantifies the mean mode power exchange. We study the solution of this equation in the asymptotic limit of infinite width of the waveguide. The main result is a quantification of the rate of convergence of the mean mode powers toward equipartition.

ABSTRACTThree sites of the Italian Strong Motion Network (RAN) have been selected for detailed S‐wave profiling, using both borehole and surface wave seismic methods. At these sites, the presence of stiffness contrasts within the soil column is found to influence the surface wave propagation profoundly. Advanced aspects in surface wave inversion such as resolution, accuracy and higher‐mode interpretation must be properly taken into account to obtain realistic results from the surface wave dispersion observations. The possibility of mode misidentification and the loss of resolution with depth in surface wave interpretation are explored using synthetic modelling together with active and passive seismic data sets. With high stiffness contrasts, the possibility of mode jumps and higher mode dominance over specific frequency ranges is very probable. This is true also for normally dispersive sites, where the shear velocity increases with depth, though higher mode dominance is recognized as more common in the case of a shear‐wave velocity inversion within the soil column and the sensitivity of the dispersion curves with respect to those layers beneath the low‐velocity zone may be significantly reduced. Pitfalls in the inversion resulting from mode misidentification can be avoided by investigating the effective phase velocity distribution, using active data sets and full waveform seismic modelling. When an unambiguous modal identification is achieved, the results obtained by surface wave inversion are very satisfactorily consistent with borehole data.

Effects due to aliasing on surface-wave extraction and suppression in frequency-velocity domain

Seismic moment tensors are routinely determined for global earthquakes using waveform inversions, and the resulting double‐couple and non‐double‐couple characteristics are extensively used in tectonic analyses. For shallow events, several moment tensor components are intrinsically poorly resolved, especially for surface wave inversions, which can lead to artificial non‐double‐couple components or can obscure actual non‐double‐couple components in the source representations. The relatively greater instability of surface wave inversions can cause differences between inversions based on only body waves and joint inversions of body and surface waves. The Harvard CMT catalog exhibits systematic variations in non‐double‐couple components with seismic moment, which can be attributed to use of only body waves in inversions for small events and combined body and surface waves for large events. Systematic behavior of non‐double‐couple components with regional strain environment is significantly less dependent on seismic moment when solutions influenced by less stable surface waves are omitted.

The frequency-dependent properties of Rayleigh-type surface waves can be utilized for imaging and characterizing the shallow subsurface. Most surface-wave analysis relies on the accurate calculation of phase velocities for the horizontally traveling fundamental-mode Rayleigh wave acquired by stepping out a pair of receivers at intervals based on calculated ground roll wavelengths. Interference by coherent source-generated noise inhibits the reliability of shear-wave velocities determined through inversion of the whole wave field. Among these nonplanar, nonfundamental-mode Rayleigh waves (noise) are body waves, scattered and nonsource-generated surface waves, and higher-mode surface waves. The degree to which each of these types of noise contaminates the dispersion curve and, ultimately, the inverted shear-wave velocity profile is dependent on frequency as well as distance from the source. Multichannel recording permits effective identification and isolation of noise according to distinctive trace-to-trace coherency in arrival time and amplitude. An added advantage is the speed and redundancy of the measurement process. Decomposition of a multichannel record into a time variable-frequency format, similar to an uncorrelated Vibroseis record, permits analysis and display of each frequency component in a unique and continuous format. Coherent noise contamination can then be examined and its effects appraised in both frequency and offset space. Separation of frequency components permits real-time maximization of the S/N ratio during acquisition and subsequent processing steps. Linear separation of each ground roll frequency component allows calculation of phase velocities by simply measuring the linear slope of each frequency component. Breaks in coherent surface-wave arrivals, observable on the decomposed record, can be compensated for during acquisition and processing. Multichannel recording permits single-measurement surveying of a broad depth range, high levels of redundancy with a single field configuration, and the ability to adjust the offset, effectively reducing random or nonlinear noise introduced during recording. A multichannel shot gather decomposed into a swept-frequency record allows the fast generation of an accurate dispersion curve. The accuracy of dispersion curves determined using this method is proven through field comparisons of the inverted shear-wave velocity (v s ) profile with a downhole v s profile.