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1.
We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A. 相似文献
2.
Summary. The study of the asymptotic behaviour of eigenfrequencies of torsional modes of the Earth is of interest in the problem of the existence of discontinuities in the Earth's interior and the determination of their depth and scale, since the solotone effect, which is a persistent oscillatory cohponent in the asymptotic overtone structure, is very sensitive to the presence of discontinuities. The asymptotic behaviour of torsional eigenfrequencies of the second order differential equation for the Earth's free oscillations can be compared with that obtained from eigenfrequencies evaluated from synthetic SH-wave seismograms by Brune's phase correlation method, using various earth models. The solotone effect that appears in the former for discontinuous models can be explained in terms of multiple reflections fram the discontinuities, and can be reconstructed from synthetic SH -wave pulses arising from these reflections. Its properties vary systematically with the depth and the scale of discontinuities and can be superposed for several discontinuities. 相似文献
3.
In contrast to previous work, which treats the Earth's lateral heterogeneity as an infinitesimal perturbation to a spherically symmetrical starting model, we conduct iterative linearized waveform inversion for the Earth's laterally heterogeneous structure. We use the Direct Solution Method (DSM) (Geller et al. 1990a) to calculate synthetic seismograms and their partial derivatives for a laterally heterogeneous earth model. We invert surface-wave data from the IDA and GEOSCOPE networks. We expand the lateral heterogeneity of rigidity in spherical harmonics up to angular order number 8 and use three parameters to specify the depth dependence of each harmonic, giving us a total of 240 unknowns. The short-wavelength lateral heterogeneity (s = 4, 6 and 8) in the deeper part of the upper mantle obtained by this study differs significantly from M84A. The relative improvement in the variance reduction as compared with model M84A is about 20 per cent for the IDA data and more than 100 per cent for the GEOSCOPE data. 相似文献
4.
Barbara Romanowicz 《Geophysical Journal International》1987,90(1):75-100
Summary. Starting with the first-order formulation of quasi-degenerate splitting theory for the normal modes of a laterally heterogeneous earth, we have obtained an asymptotic expression for the coupling terms corresponding to neighbouring multiplets along the same dispersion branch as the mode considered, valid to order 1/ℓ, where ℓ is the angular order of this mode (ℓ≥ 1).
We show that, to order zero, these coupling terms introduce a small shift in epicentral distance into the expression for the long period seismogram obtained by normal mode summation. This shift depends on the difference between the great circle and the minor arc averages of the local frequency. the coupling terms thus permit us to reconcile results obtained by normal-mode summation and by a propagating wave approach, as far as the dependence on structure of the phase of surface waves is concerned.
To order 1/ℓ, the coupling terms result in a perturbation in the amplitude of the mode considered, which depends on spatial derivatives of the local frequency and thus on the structure in the vicinity of the source station great circle path. We show that this term is equivalent to that which is found using ray perturbation methods for propagating surface waves. We compare and discuss the assumptions underlying both approaches and illustrate, by an example, the potential of the asymptotic normal-mode formulation for improved modelling of lateral heterogeneity in the earth. 相似文献
We show that, to order zero, these coupling terms introduce a small shift in epicentral distance into the expression for the long period seismogram obtained by normal mode summation. This shift depends on the difference between the great circle and the minor arc averages of the local frequency. the coupling terms thus permit us to reconcile results obtained by normal-mode summation and by a propagating wave approach, as far as the dependence on structure of the phase of surface waves is concerned.
To order 1/ℓ, the coupling terms result in a perturbation in the amplitude of the mode considered, which depends on spatial derivatives of the local frequency and thus on the structure in the vicinity of the source station great circle path. We show that this term is equivalent to that which is found using ray perturbation methods for propagating surface waves. We compare and discuss the assumptions underlying both approaches and illustrate, by an example, the potential of the asymptotic normal-mode formulation for improved modelling of lateral heterogeneity in the earth. 相似文献
5.
Summary. Synthetic seismograms based upon first-order perturbation theory are analysed to test the validity of assumptions which form the basis of current velocity inversion procedures. It is found that the lowest order geometrical optics approximation, namely that measured normal mode eigen-frequencies reflect the average structure underlying the source–receiver great circle path, becomes less valid near nodes in the source radiation pattern and near the surface wave foci at the source and its antipode. These failures are a consequence of singlet interference within an isolated normal mode multiplet. The technique of determing frequency by fitting a single resonance peak to a multiplet yields results which agree well with the first-order theory for slow and fast paths where excitation is dominated by one pair of singlets but on intermediate paths where singlet interference is more of a problem, agreement is not as good. Inversion of small data sets is particularly sensitive to frequency fluctuations near radiation nodes, while larger sets are influenced more by antipodal deviations from geometrical optics. The latter leads to inversions which fail to recover the short wavelength structure of the starting model. Basing inversions directly upon first-order theory shows promise of improving recovery of short wavelengths. 相似文献
6.
Stephen L. Gillett 《Geophysical Journal International》1986,84(3):641-646
Summary. In cases where directional data, such as palaeomagnetic directions, lie nearly along a great circle, a good approximation to the maximum likelihood estimate of the intermediate concentration parameter k 2 in the Bingham probability distribution is given by: 2( t 2 / N ) – 1 = I 1 (1/2 k 2 )/ I 0 (1/2 k 2 ), where t 2 is the intermediate eigenvalue, N is the number of samples, and the Ii are the appropriate modified Bessel functions of the first kind. This estimate, the asymptotic limit as the smallest eigenvalue t 1 → 0, corresponds to restricting all points to lie on a great circle. The limit is also useful as an endpoint for interpolation, especially since numerical calculation in this region is difficult. 相似文献
7.
R. S. Anderssen 《Geophysical Journal International》1977,50(2):303-309
Summary. This paper pursues theoretically and comments on results obtained by Wang, Gettrust & Cleary from a model study of the asymptotic overtone structure of torsional eigenfrequencies of the Earth. It shows that a number of qualitative results, which they derive about the dependence of the asymptotic overtone structure on the position and nature of discontinuities in density and shear velocity, can be derived quantitatively. 相似文献
8.
F. A. Dahlen 《Geophysical Journal International》1979,59(1):19-42
Summary. Exact spectra of a normal mode multiplet n Sl or n Tl on a non-rotating ellipsoidal earth are composed of l + 1 lines. Asymptotically, in the limit of large l , such spectra appear, however, as a single line or peak broadened by attenuation alone. The location within the multiplet of this single peak depends only upon the angle of inclination of the source-receiver great circle path to the axis of ellipsoidal symmetry. The appearance of a single peak is produced by the cancellation of nearby singlets. To assess the utility of the single peak approximation, exact and asymptotic synthetic spectra have been compared directly for a number of multiplets and for a variety of source-receiver configurations. Except in the immediate vicinity of the source and its antipode, the approximation appears to be satisfactory on an ellipsoidal earth down to about l ∼ 10. Additional studies will be required to determine the limits of validity on a laterally heterogeneous earth. 相似文献
9.
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
10.
The free oscillations of an anelastic aspherical earth 总被引：1，自引：0，他引：1
F. A. Dahlen 《Geophysical Journal International》1981,66(1):1-22
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
11.
12.
We investigate the effect of laterally varying earth structure on centroid moment tensor inversions using fundamental mode mantle waves. Theoretical seismograms are calculated using a full formulation of surface wave ray theory. Calculations are made using a variety of global tomographic earth models. Results are compared with those obtained using the so-called great-circle approximation, which assumes that phase corrections are given in terms of mean phase slowness along the great circle, and which neglects amplitude effects of heterogeneity. Synthetic tests suggest that even source parameters which fit the data very well may have large errors due to incomplete knowledge of lateral heterogeneity. The method is applied to 31 shallow, large earthquakes. For a given earthquake, the focal mechanisms calculated using different earth models and different forward modelling techniques can significantly vary. We provide a range of selected solutions based on the fit to the data, rather than one single solution. Difficulties in constraining the dip-slip components of the seismic moment tensor often produce overestimates of seismic moment, leading to near vertical dip-slip mechanisms. This happens more commonly for earth models not fitting the data well, confirming that more accurate modelling of lateral heterogeneity can help to constrain the dip-slip components of the seismic moment tensor. 相似文献
13.
C. Wang 《Geophysical Journal International》1979,56(1):199-209
Summary. In examining the effect of discontinuities in the Earth's interior on free oscillations, McNabb, Anderssen & Lapwood derived an equation for the asymptotic behaviour of torsional overtone eigenfrequencies of a discontinuous earth model, the constants in their equation being explicitly determined only for the case of one internal discontinuity. Since Brune's phase correlation method for the evaluation of eigenfrequencies from body-wave data implies a ray-mode duality only for continuous earth models, it is desirable to justify the McNabb et al. formulation from the point of view of ray theory.
By a novel method of ray analysis, Wang, Cleary & Anderssen showed that, for earth models with a single discontinuity between the Earth's surface and the core—mantle boundary, the McNabb et al. formulation can be derived from an adaptation of Brune's method to multiply reflected SH body waves recorded at small epicentral distances. In this paper, the technique of Wang et al. is extended to derive the McNabb et al. formulation (with constants explicitly determined) for the general case of earth models with N discontinuities. This establishes a basis for a ray-mode duality for discontinuous earth models. 相似文献
By a novel method of ray analysis, Wang, Cleary & Anderssen showed that, for earth models with a single discontinuity between the Earth's surface and the core—mantle boundary, the McNabb et al. formulation can be derived from an adaptation of Brune's method to multiply reflected SH body waves recorded at small epicentral distances. In this paper, the technique of Wang et al. is extended to derive the McNabb et al. formulation (with constants explicitly determined) for the general case of earth models with N discontinuities. This establishes a basis for a ray-mode duality for discontinuous earth models. 相似文献
14.
Po-Yu Shen 《Geophysical Journal International》1976,46(2):307-318
A rotating incompressible fluid bounded by two concentric spherical rigid surfaces can exhibit purely toroidal free oscillations. The eigenfrequencies are fractions of the angular frequency of rotation. If the bounding surfaces are slightly ellipsoidal, secondary spheroidal fields become existent, and in general, a free mode splits into a doublet with one of which exists only when the inner bounding surface is present.
For the real earth, the compressibility of the outer core, the elasticity of the solid earth, and the self-gravitation of the entire earth modify the toroidal core oscillations. The present treatment gives explicitly the effects of these parameters on the eigenfrequencies. 相似文献
For the real earth, the compressibility of the outer core, the elasticity of the solid earth, and the self-gravitation of the entire earth modify the toroidal core oscillations. The present treatment gives explicitly the effects of these parameters on the eigenfrequencies. 相似文献
15.
Annie Souriau 《Geophysical Journal International》1985,80(3):783-790
Summary. Synthetic great circle Rayleigh wave phase velocities are generated from a 3-D upper mantle model for two great circle distributions: a uniform random distribution and an experimental distribution arising from a real set of sources and receivers. Given a tectonic regionalization, the retrieval of the pure path velocities from these great circle data sets leads to a strongly biased solution when the uniform distribution is used, whereas the results from the experimental distribution are closer to the true values. An analysis of the data importances and the information distribution in the data indicates that the great circles introduce coupling between tectonic regions; thus the degeneracy of the solution inherent in great circle data cannot be completely overcome by introducing a tectonic regionalization. It is shown that a satisfactory solution is obtained by selecting data as pure as possible, i.e. data that over-sample one tectonic region without oversampling the others. This condition is better fulfilled by the experimental distribution than by the uniform distribution. 相似文献
16.
3-D linearized scattering of surface waves and a formalism for surface wave holography 总被引：3，自引：1，他引：2
Roel Snieder 《Geophysical Journal International》1986,84(3):581-605
Summary. Scattering of surface waves by lateral heterogeneities is analysed in the Born approximation. It is assumed that the background medium is either laterally homogeneous, or smoothly varying in the horizontal direction. A dyadic representation of the Green's function simplifies the theory tremendously. Several examples of the theory are presented. The scattering and mode conversion coefficients are shown for scattering of surface waves by the root of an Alpine-like crustal structure. Furthermore a 'great circle theorem'in a plane geometry is derived. A new proof of Snell's law is given for surface wave scattering by a quarter-space. It is shown how a stationary phase approximation can be used to simplify the Fourier synthesis of the scattered wave in the time domain. Finally a procedure is suggested to do 'surface wave holography'. 相似文献
17.
18.
Elastic scattering from a continuous and laterally unbounded heterogeneous layer has been formulated using the Born approximation. A general solution of the scattered wave equation for the above-stated medium has been given in terms of a Fourier integral over plane waves. Far-field asymptotic expressions for weak elastic scattering by a finite, continuous and inhomogeneous layer have been presented which agree with earlier results. For perturbations of the two elastic parameters and the density having the same form of spatial variation, the spectrum of plane waves scattered from a heterogeneous layer is expressed as a product of an 'elastic scattering factor'and a 'distribution factor'. As in earlier results for small-scale heterogeneity, the scattering pattern depends on various combinations of perturbations of elastic parameters and density. In order to show the general characteristics of the elastic wave scattering, some scattering patterns have been given. 相似文献
19.
EXPERIMENT OF SHORT WAVE COMMUNICATION AND THE INVESTIGATION OF THE PROPAGATION BEARINGS BETWEEN THE GREAT WALL STATION AND XINXIANG 
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下载免费PDF全文 《极地研究》1990,1(1):67-74
The distance from the Great Wall Station (62.2°S, 58.9°W) to Xinxiang (35.3°N, 113.8°E) is 16981 kin. The path passes through the polar cap absorption region and the auroral absorption zone, and it is across the equator. In this paper firstly the effects of short wave communication and usable time blocks and frequency ranges between the Antarctic Great Wall Station and Xinxiang from December 1985 to March 1986 are introduced. The comparison between the usable frequency ranges with the estimated MUF is made. The upper limit of frequency ranges of communication along the short great circle path basically agrees with the MUF but there is difference between them along the long great circle path. Secondly, the result of the propagation bearings experiment in January to February 1986 is introduced in more detail, The propagation along the great circle path from the Great Wall Station to Xinxiang is the main propagation mode. But the propagation along non great circle paths occurs at times between Great Wa 相似文献
20.
Generalized Born scattering of elastic waves in 3-D media 总被引：1，自引：0，他引：1
It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed. 相似文献