共查询到20条相似文献，搜索用时 125 毫秒
1.
B.L.N.Kennett 《Geophysical Journal International》1998,133(1):159-174
A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal field: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in different directions in the horizontal plane. The influence of 3-D structure is included by allowing coupling between different modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial differential equations for the weight functions for different modal orders.
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation. 相似文献
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation. 相似文献
2.
Matthias Zillmer Boris Kashtan & Dirk Gajewski 《Geophysical Journal International》1998,132(3):643-653
Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails. 相似文献
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails. 相似文献
3.
Summary. The paper gives the results of a study of the anisotropy of seismic wave velocities within the Ashkhabad test field in Central Asia. The anisotropy was studied by analysing variations in the values of apparent velocities of first arrivals for epicentral distances ranging from 30 to 130 km and by analysing the delays (Δ ts1 -s2 ) between the arrival times of shear waves with different polarizations.
The velocities of P -waves vary with azimuth from 5.3 to 6.27 km s-1 and the velocities of S -waves vary from 3.15 to 3.5 km s-1 .
The delay times Δ tS1 - S2 depend on the direction of the propagation. The character of the variation of the propagation velocity of the longitudinal wave, the presence of two differently polarized shear waves S 1 and S 2 propagating at different velocities, and the character of the distribution of Δ tS1 - S2 on the stereogram suggest that the symmetry of the anisotropic medium is close to hexagonal with a nearly horizontal symmetry axis coinciding with the direction of maximal velocity. The azimuth of the symmetry axis of the medium is 140° and coincides with the direction of geological faults. 相似文献
The velocities of P -waves vary with azimuth from 5.3 to 6.27 km s
The delay times Δ t
4.
M. D. Sharma 《Geophysical Journal International》2008,174(3):971-977
Out of the four waves in an anisotropic poroelastic medium, two are termed as quasi-transverse waves. The prefix 'quasi' refers to their polarizations being nearly, but not exactly, perpendicular to direction of propagation. In this composite medium, unlike perfectly elastic medium, the propagation of a longitudinal wave along a phase direction may not be accompanied by transverse waves. The existence of a transverse wave in anisotropic poroelastic media is ensured by the two equations restricting the choice of elastic coefficients of porous aggregate as well as fluid–solid coupling. Necessary and sufficient conditions for the existence of transverse waves along the coordinate axes and in the coordinate planes for general anisotropy are discussed. The discussion is extended to the case of orthotropic materials and existence for few specific phase directions is also explored. The conditions for the transverse waves decided on the basis of their apparent polarizations, that is, particle motion being perpendicular to ray direction, are also discussed. For a particular numerical model, the existence of these apparent transverse waves is solved numerically for phase directions in coordinate planes. For general directions of phase propagation, the existence of these transverse waves is checked graphically for the chosen numerical model. 相似文献
5.
Summary. A connection is established between the group velocity of Rayleigh waves, the spectral amplitudes of surface waves generated by a source, and the resonance of vertically travelling P waves. It implies that a minimum in a group velocity curve is reflected in the spectral amplitudes as a maximum. That this is so, appears to have been first noticed by Longuet-Higgins in a study of microseisms. Also when a sharp impedance contrast occurs in a plane-layered model of the crust, the group velocity minimum in the fundamental mode occurs close to a period equal to four times the travel time of P -waves from the surface to the interface. More than one such contrast gives rise in general to more than one minimum. Similar relations hold for the higher modes. 相似文献
6.
Summary The coherence of atmospheric acoustic-gravity waves has been measured in the period range 10–100 s at the Large Aperture Microbarograph Array in south-eastern Montana. The acoustic-gravity waves observed were signals generated by presumed nuclear explosions. The decrease of coherence with increasing distance between pairs of microbarographs is less rapid in the direction of wave propagation than transverse to it. Variation of direction of arrival over a small range of azimuth (±5°) explains the spatial behaviour of coherence in the direction normal to the wave propagation; variation of phase velocity of ±10 ms-1 explains the behaviour along the direction of wave propagation. Both effects may be due to inhomogeneities in the atmosphere; the velocity variation may be due to the presence in the signal of several normal modes of acoustic- gravity waves, each travelling at a slightly different phase velocity in the range 300–330 ms-1 . 相似文献
7.
Summary. The symmetry relations between the reflection and transmission coefficients for plane elastic waves incident upon an arbitrary horizontally stratified medium are derived by a novel approach. Previous results, particularly for a single interface, are obtained as special cases of this treatment.
In addition, for perfectly elastic media, projection operators for travelling and evanescent waves are introduced and used to derive a number of new relationships between the reflection and transmission coefficients. 相似文献
In addition, for perfectly elastic media, projection operators for travelling and evanescent waves are introduced and used to derive a number of new relationships between the reflection and transmission coefficients. 相似文献
8.
Summary. Lg -wave observations at NORSAR from a sequence of explosions straddling the Central Graben in the North Sea, show that the phase is almost extinguished by passing through this structure. For models of the graben structure, based on seismic refraction experiments, with crustal thinning beneath a zone of thickened sediments, numerical modelling of Lg propagation shows very poor transmission. The inclusion of attenuation simulating the effects of scattering from faults and other complex structures further diminishes the transmittivity. The theoretical effect of a crustal pinch is to redistribute S energy over a wide range of groups velocities, a pattern that fits well with the observations.
Three components recordings at NORSAR of explosive charges fired in water covering a wide range of azimuths, show significant transverse components to the seismograms. The transverse energy builds up with travel time and for Lg is of the same order as the vertical component. These results are consistent with the theoretical prediction of progressive net transfer of energy from Rayleigh to Love modes as the wavetrain propagates through a three-dimensionally heterogeneous medium. 相似文献
Three components recordings at NORSAR of explosive charges fired in water covering a wide range of azimuths, show significant transverse components to the seismograms. The transverse energy builds up with travel time and for Lg is of the same order as the vertical component. These results are consistent with the theoretical prediction of progressive net transfer of energy from Rayleigh to Love modes as the wavetrain propagates through a three-dimensionally heterogeneous medium. 相似文献
9.
Summary. Using a single scattering approximation, we derive equations for the scattering attenuation coefficients of P- and S -body waves. We discuss our results in the light of some recent energy renormalization approaches to seismic wave scattering. Practical methods for calculating the scattering attenuation coefficients for various earth models are emphasized. The conversions of P - to S -waves and S- to P -waves are included in the theory. The earth models are assumed to be randomly inhomogeneous, with their properties known only through their average wavenumber power spectra. We approximate the power spectra with piecewise constant functions, each segment of which contributes to the net, frequency-dependent, scattering attenuation coefficient. The smallest and largest wavenumbers of a segment can be plotted along with the wavevectors of the incident and scattered waves on a wavenumber diagram. This diagram gives a geometric interpretation for the frequency behaviour associated with each spectral segment, including a 'transition' peak that is due entirely to the wavenumber limits of the segment. For regions of the earth where the inhomogeneity spectra are concentrated in a band of wavenumbers, it should be possible to observed such a peak in the apparent attenuation of seismic waves. We give both the frequency and distance limits on the accuracy of the theoretical results. 相似文献
10.
Toshiro Tanimoto 《Geophysical Journal International》1984,78(3):641-660
Summary. Normal mode theory, extended to the slightly laterally heterogeneous earth by the first-order Born approximation, is applied to the waveform inversion of mantle Love wave (200–500 s) for the Earth's lateral heterogeneity at l = 2 and a spherically symmétric anelasticity ( Q μ ) structure. The data are from the Global Digital Seismograph Network (GDSN). The l =2 pattern is very similar to the results of other studies that used either different méthods, such as phase velocity measurements and multiplet location measurements, or a different data set, such as mantle Rayleigh waves from different instruments. The results are carefully analysed for variance reduction and are most naturally explained by heterogeneity in the upper 420 km. Because of the poor resolution of the data set for the deep interior, however, a fairly large heterogeneity in the transition zones, of the order of up to 3.5 per cent in shear wave velocity, is allowed. It is noteworthy that Love waves of this period range cannot constrain the structure below 420 km and thus any model presented by similar studies below this depth are likely to be constrained by Rayleigh waves (spheroidal modes) only.
The calculated modal Q values for the obtained Qμ model fall within the error bars of the observations. The result demonstrates the discrepancy of Rayleigh wave Q and Love wave Q and indicates that care must be taken when both Rayleigh and Love wave data, including amplitude information, are inverted simultaneously.
Anomalous amplitude inversions of G2 and G3, for example, are observed for some source-receiver pairs. This is due to multipathing effects. One example near the epicentral region, which is modelled by the obtained l = 2 heterogeneity, is shown. 相似文献
The calculated modal Q values for the obtained Q
Anomalous amplitude inversions of G2 and G3, for example, are observed for some source-receiver pairs. This is due to multipathing effects. One example near the epicentral region, which is modelled by the obtained l = 2 heterogeneity, is shown. 相似文献
11.
Summary. The seismic structure has been measured to a depth of about 3 km along a 30 km seismic profile in east central Ireland. This profile is unusual in that it is the S -wave velocity—depth structure that has been measured to a degree of precision more normally associated with P -wave results. One reason for this is that the sources used were quarry blasts which generated strong S -waves and short-period surface waves but rather weak P -waves.
The results show a layer of Carboniferous limestone with shear velocity 2.65 km−1 s overlying a layer with a velocity of 3.06 km s−1 . This second layer was interpreted as Lower Palaeozoic strata (Silurian/Ordovician) since this velocity was evident in an inlier seen at the surface at the northern end of the line. A third refraction horizon, shear velocity 3.45 km s−1 and displaying a basinal structure, was also recognized. This may be Cambrian or Precambrian basement. 相似文献
The results show a layer of Carboniferous limestone with shear velocity 2.65 km
12.
Summary. The propagation of a pulsed elastic wave in the following geometry is considered. An elastic half-space has a surface layer of a different material and the layer furthermore contains a bounded 3-D inhomogeneity. The exciting source is an explosion, modelled as an isotropic pressure point source with Gaussian behaviour in time.
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves. 相似文献
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves. 相似文献
13.
Pekka R. Saastamoinen 《Geophysical Journal International》1984,79(1):269-284
Summary. A technique based on ray asymptotics has been developed to propagate complex spectra of elastic normal mode surface waves in a waveguide with material and geometrical properties varying smoothly in the lateral directions. In the technique, the original problem defined in the unstretched coordinates has been transformed into an eiconal equation as well as into a certain number of transport equations defined in stretched coordinates.
The solution of the eiconal equation is equal to the solution of the eigenproblem of the eiconal operator A0 . Due to the self-adjointness of A0 , in each of the relevant local inner product spaces, LIPS, the solution of the eigenproblem, A0 ψ= v ψ results in the set { v t } of real local eigenvalues and in the orthonormal system {ψt } of local eigenvectors.
As the Hamiltonian function of an initial value problem, each eigenvalues gives birth to a bicharacteristic curve as well as to the related ray. The introduction of the rays induces connections between the vertical cross-sections of the waveguide.
Finally, for each asymptotic order j , the LIPS-valued transport equations are reduced to a set of matricial propagation equations in the local spectral amplitude vectors, LSAVs. Consequently, a knowledge of the initial conditions at a vertical cross-section makes it possible to propagate the LSAVs along the rays of the relevant modes. However, to complete the propagation one needs, in addition to the initial values, information about certain additional quantities, non-diagonal terms of order j , diagonal terms of orders lower than j and the auxiliary boundary terms of orders from 1 to j . The treatment has been completed by the propagation of the modal phases along the relevant rays. 相似文献
The solution of the eiconal equation is equal to the solution of the eigenproblem of the eiconal operator A
As the Hamiltonian function of an initial value problem, each eigenvalues gives birth to a bicharacteristic curve as well as to the related ray. The introduction of the rays induces connections between the vertical cross-sections of the waveguide.
Finally, for each asymptotic order j , the LIPS-valued transport equations are reduced to a set of matricial propagation equations in the local spectral amplitude vectors, LSAVs. Consequently, a knowledge of the initial conditions at a vertical cross-section makes it possible to propagate the LSAVs along the rays of the relevant modes. However, to complete the propagation one needs, in addition to the initial values, information about certain additional quantities, non-diagonal terms of order j , diagonal terms of orders lower than j and the auxiliary boundary terms of orders from 1 to j . The treatment has been completed by the propagation of the modal phases along the relevant rays. 相似文献
14.
15.
Robert I. Odom 《Geophysical Journal International》1986,86(2):425-453
Summary. A coupled mode theory is used to examine surface wave propagation in a laterally inhomogeneous acoustic waveguide. The theory is developed from the equations of motion for the pressure and velocity fields. The presence of lateral inhomogeneities in the form of varying layer thickness causes coupling among the discrete modes of the waveguide and radiation to the continuum. Expressions for the coupling coefficients among all mode types including coupling to the continuum spectrum are derived. The coupling coefficients are proportional to the horizontal derivative of the function describing the interface between layers of constant material properties but varying thickness. The coupled mode equations are solved in approximation for the case of a sinusoidal boundary and a sloping boundary. The results for radiation losses due to interaction with the irregular boundary of the waveguide are presented in analytical form, which clearly show the primary physical effects on the wavefield of the interaction. The far field amplitude of the scattered modes, excited by the interaction of some incident signal with a weak boundary irregularity, is modulated by the spatial Fourier transform of the irregularity. 相似文献
16.
A review of the effects of anisotropic layering on the propagation of seismic waves 总被引：10，自引：0，他引：10
Stuart Crampin 《Geophysical Journal International》1977,49(1):9-27
Summary. This paper reviews recent work, much of it unpublished, 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. 相似文献
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. 相似文献
17.
Summary. Based on accurately located 23 very shallow earthquakes ( h = 1–14 km) in northern and central Greece by portable networks of seismic stations and by the joint epicentre method, the travel times of the Pn -waves from the foci of these earthquakes to the sites of 54 permanent stations in the Balkan region have been determined. The travel times of Pn -waves in the central and eastern part of the area (eastern Greece, south-eastern Yugoslavia, the Aegean Sea, Bulgaria, southern Romania, western Turkey) fit a straight line very well with the Pn velocity equal to 7.9 ± 0.1 km s-1 . On the contrary, the travel times of Pn -waves to stations in the western part of the area (Albania, western Greece) do not fit this curve because the Pn -waves travelling to these stations are delayed by more than 1 s due to the thicker crust under the Dinarides–Hellenides mountain range. Time delays for Pn -waves have been calculated for each permanent station in the Balkan area with respect to the mean travel-time curve of these waves in the central and eastern part of the area. Corrections of the travel times for these delays contribute very much to the improvement of the accuracy in the location of the shallow earthquakes in the Aegean and surrounding area. 相似文献
18.
Reciprocity theorems for one-way wavefields 总被引：1，自引：0，他引：1
Acoustic reciprocity theorems have proved their usefulness in the study of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and are thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation into a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theorems" along the same lines as the usual derivation of the 'two-way reciprocity theorems'. However, for the one-way reciprocity theorems it is not directly obvious that the 'contrast term' vanishes when the medium parameters in the two different states are identical. By introducing a modal expansion of the Helraholtz operator, its square root can be derived, which appears to have a symmetric kernel. This symmetry property appears to be sufficient to let the contrast term vanish in the above-mentioned situation.
The one-way reciprocity theorem of the convolution type is exact, whereas the one-way reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators. 相似文献
The one-way reciprocity theorem of the convolution type is exact, whereas the one-way reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators. 相似文献
19.
20.
Roger D. Borcherdt 《Geophysical Journal International》1982,70(3):621-638
Summary. The reflection and refraction of general (homogeneous or inhomo-geneous) plane P and type-I S ( SV ) body waves incident on plane boundaries are considered for general linear viscoelastic solids. Reflection—refraction laws, physical characteristics of the waves, and the nature of critical angles are examined in detail at welded boundaries and a free surface. General visco-elasticity with no low-loss approximations predicts that contrasts in intrinsic absorption at boundaries give rise to inhomogeneous reflected and refracted waves with elliptical particle motions, velocities and maximum attenuations that vary with frequency and angle of incidence, energy propagation at speeds and directions different from phase propagation, phase propagation that in general is parallel to the boundary for at most one angle of incidence, and reflection—transmission coefficients dependent on energy flow due to wave interaction. None of these physical characteristics are predicted for waves incident on boundaries that respond instantaneously. 相似文献