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1.
In this study we apply the correspondence principle for free vibrations of a homogeneous viscoelastic solid derived by Fisher & Leitman to obtain the torsional modes of a homogeneous viscoelastic rod. We also extend the correspondence principle, showing that it may be used to find the frequencies of Love waves in a stratified viscoelastic medium. Finally, we apply the correspondence principle to four viscoelastic materials, the Kelvin-Voigt solid, the Maxwell solid, the standard linear solid, and the Achenbach-Chao solid. We show that in each of these cases some care must be used in applying the correspondence principle because of the presence of multiple solutions. We also examine measures of dissipation of the free vibrations, and we determine the conditions under which the logarithmic decrement may be approximated by the process-independent l/ Q of O'Connell & Budiansky.  相似文献   

2.
Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.  相似文献   

3.
A simple modification of the waveform inversion formula, based on the normal mode perturbation theory, is shown to lead to a formula for traveltime anomalies. The kernel which is derived can be used for traveltime inversion with automatic inclusion of finite frequency effects. Inversion for Earth structure with such kernels will lead to better resolution estimates than ray-theoretical traveltime inversion. Examples of kernels for transverse component seismograms are shown for direct S waves, ScS , Love waves and diffracted S waves. A measure of finite frequency effects is also proposed by comparing our formula with the one from ray theory. A quantity which should be 1 in the case of ray theory is computed for the finite frequency kernels and is shown to have deviations up to about 30 per cent from 1. Therefore, the use of ray theory for long-period body waves applies incorrect weight along a ray path and may introduce a small bias to an earth model.  相似文献   

4.
It is well established that the Earth's uppermost mantle is anisotropic, but there are no clear observations of anisotropy in the deeper parts of the mantle. Surface waves are well suited to observe anisotropy since they carry information about both radial and azimuthal anisotropy. Fundamental mode surface waves, for commonly used periods up to 200 s, are sensitive to structure in the first few hundred kilometres, and therefore, do not provide information on anisotropy below. Higher mode surface waves have sensitivities that extend to and beyond the transition zone, and should thus give insight about azimuthal anisotropy at greater depths. We have measured higher mode Love and Rayleigh phase velocities using a model space search approach, which provides us with consistent relative uncertainties from measurement to measurement and from mode to mode. From these phase velocity measurements, we constructed global anisotropic phase velocity maps. Prior to inversion, we determine the optimum relative weighting for anisotropy. We present global azimuthal phase velocity maps for higher mode Rayleigh waves (up to the sixth higher mode) and Love waves (up to the fifth higher mode) with corresponding average model uncertainties. The anisotropy we derive is robust within the uncertainties for all modes. Given the ray theoretical sensitivity kernels of Rayleigh and Love wave modes, the source of anisotropy is complex, but mainly located in the asthenosphere and deeper. Our models show a good correspondence with other studies for the fundamental mode, but we have been able to achieve higher resolution.  相似文献   

5.
We design a numerical algorithm for wave simulation in a borehole due to multipole sources. The stress–strain relation of the formation is based on the Kelvin–Voigt mechanical model to describe the attenuation. The modelling, which requires two anelastic parameters and twice the spatial derivatives of the lossless case, simulates 3-D waves in an axisymmetric medium by using the Fourier and Chebyshev methods to compute the spatial derivatives along the vertical and horizontal directions, respectively. Instabilities of the Chebyshev differential operator due to the implementation of the fluid–solid boundary conditions are solved with a characteristic approach, where the characteristic variables are evaluated at the source central frequency. The algorithm uses two meshes to model the fluid and the solid. The presence of the logging tool is modelled by imposing rigid boundary conditions at the inner surface of the fluid mesh. Examples illustrating the propagation of waves are presented, namely, by using monopoles, dipoles and a quadrupoles as sources in hard and soft formations. Moreover, the presence of casing and layers is considered. The modelling correctly simulates the features—traveltime and attenuation—of the wave modes observed in sonic logs, namely, the P and S body waves, the Stoneley wave, and the dispersive S waves in the case of multipole sources.  相似文献   

6.
The free oscillations of an anelastic aspherical earth   总被引：1，自引：0，他引：1  
Summary. Some century-old results, due to Rayleigh and Routh, have been adapted to investigate the normal mode eigenfrequencies and eigenfunctions of an earth with laterally variable anelasticity and to determine the transient response of such an earth to earthquakes. Using degenerate perturbation theory, the eigenfrequencies are found to first order and the associated eigenfunctions to zeroth order in the small deviations of the Earth away from a spherical perfectly elastic reference earth model. Both the eigenfrequencies and the eigenfunctions are complex and, in addition, the latter are not mutually orthogonal, reflecting the non-Hermitian character of the normal mode eigenvalue problem. The effect of laterally heterogeneous attenuation on the shape of an unresolved multiplet spectrum has been investigated in the surface-wave geometrical-optics limit. Singlet cancellation leads in that limit to the appearance of a single resonance peak whose decay rate or apparent Q −1 depends only on the average attenuation structure underlying the source—receiver great-circle path.  相似文献   

7.
This paper extends our earlier examinations of the utility of various approximations for treating the dynamics of the Earth's liquid core on time-scales of the order of 104 to 108 s. We discuss the effects of representing the response of the mantle and inner core by static (versus dynamic) Love numbers, and of invoking the subseismic approximation for treating core flow, used either only in the interior of the liquid core (SSA-1) or also at the boundaries (SSA-2). The success of each approximation (or combinations thereof) is measured by comparing the resulting surface gravity effects (computed for a given earthquake excitation), and (for the Slichter mode) the distribution of translational momentum, with reference calculations in which none of these approximations is made. We conclude that for calculations of the Slichter triplet, none of the approximations is satisfactory, i.e. a full solution (using dynamic Love numbers at elastic boundaries and no core flow approximation) is required in order to avoid spurious eigenfrequencies and to yield correct eigenfunctions (e.g. conserving translational momentum) and surface gravity. For core undertones, the use of static Love numbers at rigid boundaries is acceptable, along with SSA-1 (i.e. provided the subseismic approximation is not invoked at the core boundaries). Although the calculations presented here are for a non-rotating earth model, we argue that the principal conclusions should be applicable to the rotating Earth. Shortcomings of the subseismic approximation appear to arise because both SSA-1 and SSA-2 lower the order of the governing system of differential equations (giving rise to a singular perturbation problem), and because SSA-2 overdetermines the boundary conditions (making it impossible for solutions to satisfy all continuity requirements at core boundaries).  相似文献   

8.
Reflection coefficients for weak anisotropic media   总被引：1，自引：0，他引：1  
The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.  相似文献   

9.
Summary. Mode conversion at a continental margin between normal modes of surface waves is investigated by theoretical calculations for oblique incidence for periods longer than 15 s. It is suggested that significant conversion takes place between the various modes of Love waves in the period interval between 15 and 40 s, while there is negligible mode conversion for longer periods. The largest mode conversion involves the lowest modes. In addition the calculations have revealed a small but significant conversion between Love and Rayleigh fundamental modes around 20-s period. Reflections of Love waves are found to be significant only for the continental fundamental mode.  相似文献   

10.
Generalized spherical harmonics are used to simplify the calculation of the perturbation matrix elements (coupling coefficients) for the free oscillations of an anisotropic and laterally heterogeneous earth. In the asymptotic limit of large angular order, the local frequency pertubration which depends on the azimuth and on the location of Earth's surface is defined, and the correspondence to surface waves is established.  相似文献   

11.
Investigation of teleseismic P -wave recordings at a temporary network in the Eastern Carpathians, equipped with predominantly short-period sensors, is compared with synthetic modelling of anelastic attenuation of teleseismic waves in the upper mantle. Using the t * approach, we examine variations of amplitude decrease over frequency for teleseismic recordings in the frequency band 0.5–1.5 Hz. The results reveal a consistent pattern of increased t * values in the centre of the network, in the Vrancea region at the bend of the Carpathian Arc, although the magnitude of the observed variation in t * is much higher than expected. Synthetic t * parameter computations for the same event-receiver configurations reproduce the observed pattern in terms of relative variations. However, the amplitude of the synthetic t * values explains only 10–20 per cent of the observed variation in t *. t * is not a direct measure for anelastic attenuation but rather for a combination of anelastic and other attenuating effects such as scattering and amplitude fluctuation related to velocity inhomogeneities. If regional amplitude variations are solely attributed to anelastic attenuation, all other effects are mapped into Q . We discuss the role of anelastic attenuation and other effects in the case of the Eastern Carpathians and conclude that t * is an unsuitable parameter to characterize anelastic attenuation in the Eastern Carpathians.  相似文献   

12.
Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.  相似文献   

13.
Summary. We report the results of our continuing efforts to compute theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram — body waves and surface waves — is generated for realistic sources buried in a radially heterogeneous, anelastic, spherical earth. The results described in Paper I (Nakanishi, Schwab & Knopoff) are extended to include the summation of 11 modes; for each, the dispersion, attenuation, and excitation are computed down to a minimum period of 1 s. Examples of the theoretical seismograms, and the comparison with experimental results are presented, The results of this comparison indicate that our first application of combined body- and surface-wave generation will concern the investigation of the intrinsic anelasticity in the upper mantle. The indicated technique for such an investigation is based on body waves simply crossing the region of high attenuation a few times in passing from focus to recording station, while a guided surface wave such as Sa , experiences this anelasticity over the entire propagation path.  相似文献   

14.
Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.  相似文献   

15.
We investigate the particle orbits of long-period (about 20 s) P waves observed with the global seismic network. By analysing 84 three-component seismograms recorded at 25 stations from 60 earthquakes occurring beneath 300 km, we quantitatively evaluate the orbits by three sets of eigenvalues and eigenvectors, using a covariance matrix method. The eigenvalues for P waves recorded at stations located on continents are explained by the standard horizontal layered structure model (iasp91). On the other hand, the orbits observed at stations close to island arcs are affected not only by the horizontal layered structure but also by heterogeneity due to subducting plates, mantle diapirs and so on. On the basis of a single-scattering model for a plane P wave, we quantify the heterogeneities by an isotropic scattering coefficient g0. Fitting the theoretical eigenvalues to the observed ones, we estimate g0 for the crust and upper mantle beneath continents to be less than 0.0005 km-1, and the mean g0 for the structure beneath island arcs to be about 0.0015 to 0.003 km-1.  相似文献   

16.
Propagation characteristics of borehole flexural waves are functions of five borehole, fluid and formation parameters. Uncertainties in any of these model parameters may introduce errors in the estimate of formation shear-wave speed. Other sources of error in the estimate may be caused by deviations from the assumed circular borehole cross-section or heterogeneity in the material properties of formation in the vicinity of a borehole from the usually assumed homogeneous properties. The influence of uncertainties in model parameters on borehole flexural dispersion has been calculated from a general perturbation model based on Hamilton's principle. A sensitivity analysis of the flexural dispersion to small variations in the model parameters shows that the formation shear speed has by far the most dominant influence in a slow formation. In contrast, the flexural dispersion in a fast formation is significantly influenced by three of the five model parameters: the formation shear speed, the borehole fluid compressional speed and the borehole diameter. The frequency dependence of these sensitivity functions indicates that the inversion of flexural dispersion for formation shear speed is optimal in the range 2–4 kHz for a borehole of diameter 25.4 cm. The range of validity of the perturbation model has been estimated by comparing results of concentric annuli of different thicknesses, and shear-wave speeds different from that of the formation with those from exact numerical solutions from a modal search program. Generally, the perturbation results for altered annuli of thicknesses up to 15 cm are accurate to within 1 per cent for a shear-wave speed 10 per cent lower than that of the formation. This difference between the perturbation and exact results increases to approximately 1 to 3 per cent for a shear-wave speed 20 per cent lower than that of the formation.  相似文献   

17.
We present the results of Rayleigh wave and Love wave phase velocity tomography in the western United States using ambient seismic noise observed at over 250 broad-band stations from the EarthScope/USArray Transportable Array and regional networks. All available three-component time-series for the 12-month span between 2005 November 1 and 2006 October 31 have been cross-correlated to yield estimated empirical Rayleigh and Love wave Green's functions. The Love wave signals were observed with higher average signal-to-noise ratio (SNR) than Rayleigh wave signals and hence cannot be fully explained by the scattering of Rayleigh waves. Phase velocity dispersion curves for both Rayleigh and Love waves between 5 and 40 speriod were measured for each interstation path by applying frequency–time analysis. The average uncertainty and systematic bias of the measurements are estimated using a method based on analysing thousands of nearly linearly aligned station-triplets. We find that empirical Green's functions can be estimated accurately from the negative time derivative of the symmetric component ambient noise cross-correlation without explicit knowledge of the source distribution. The average traveltime uncertainty is less than 1 s at periods shorter than 24 s. We present Rayleigh and Love wave phase speed maps at periods of 8, 12, 16,and 20 s. The maps show clear correlations with major geological structures and qualitative agreement with previous results based on Rayleigh wave group speeds.  相似文献   

18.
On crustal corrections in surface wave tomography   总被引：1，自引：0，他引：1  
Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.  相似文献   

19.
On average, traveltimes of PKPDF for equatorial ray paths through the quasieastern hemisphere of the inner core are around 0.5 s faster than equivalent ray paths through its quasiwestern hemisphere. In these observations, the eastern hemisphere is sampled primarily by westward and the western hemisphere by eastwardpropagating waves. Noting that westward propagation is faster than eastward propagation inside a rotating earth, I estimate the expected traveltime difference from Coriolis splitting of the displacement eigenfunctions of the PKPDF equivalent modes. It turns out that Coriolis effects are too small to give rise to residuals of the required magnitude. Thus, the observations must be primarily due to velocity heterogeneities.  相似文献   

20.
Summary. Earlier efforts to generate the entire theoretical seismograms, including both body and surface waves for realistic sources buried in a radially heterogeneous anelastic, spherical earth, are extended to include the summation of 16 modes. The comparison between a real seismogram and theoretical time series, relative to different attenuation models in the upper mantle, yields information concerning the anelasticity under the Pacific Ocean.  相似文献   

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