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1.
Summary. The Lanczos method of separating exponentials is applied to the Fourier transform of seismograms in order to separate the various modes that contribute to the given portion of the seismograms. Phase velocities and amplitudes are obtained as functions of the frequency. When applying the method to artificial seismograms, which are built as an exact superposition of a number of modes, the separation is very accurate. The method was also applied to the surface-wave portion of numerical seismograms for a vertical point force in a layered medium. The phase velocity and amplitude of the fundamental mode are obtained. These functions were taken as the first guess in the Backus—Gilbert generalized inverse procedure and the process converged very rapidly. When a perturbation of the phases and amplitudes is taken as the first guess the process converges to the true model when enough data are available. 相似文献
2.
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
3.
Summary. In a spherically symmetric, isotropic earth model the duality between seismic rays and modes can be established completely by application of the principle of stationary phase to the summed normal-mode representation of the time signal. The requirement of stationary phase must be applied not only on the sum over the angular order but also over the radial-order summation.
This approach is illustrated by using asymptotic approximations to the equations for toroidal oscillations. In this way the travel-time formulae for rays from a surface source in a sufficiently smooth earth model are easily derived, and the distribution of modes between the different rays can be found. The result may be used as a selection criterion to reduce the number of modes that must be summed to construct synthetic seismograms at a certain distance and within a certain time window. 相似文献
This approach is illustrated by using asymptotic approximations to the equations for toroidal oscillations. In this way the travel-time formulae for rays from a surface source in a sufficiently smooth earth model are easily derived, and the distribution of modes between the different rays can be found. The result may be used as a selection criterion to reduce the number of modes that must be summed to construct synthetic seismograms at a certain distance and within a certain time window. 相似文献
4.
Danny J. Harvey 《Geophysical Journal International》1981,66(1):37-69
Summary. A normal mode superposition approach is used to synthesize complete seismic codas for flat layered earth models and the P-SV phases. Only modes which have real eigenwavenumbers are used so that the search for eigenvalues in the complex wavenumber plane is confined to the real axis. In order to synthesize early P -wave arrivals by summing a number of'trapped'modes, an anomalously high velocity cap layer is added to the bottom of the structure so that most of the seismic energy is contained in the upper layers as high-order surface waves. Causality arguments are used to define time windows for which the resulting synthetic seismograms are close approximations to the exact solutions without the cap layer. The traditional Thomson—Haskell matrix approach to computing the normal modes is reformulated so that numerical problems encountered at high frequencies are avoided and numerical results of the locked mode approximation are given. 相似文献
5.
Summary. Two methods of computing body wave synthetic seismograms in three-dimensional laterally varying media are discussed. Both these methods are based on the summation of Gaussian beams. In the first, the initial beam parameters are chosen at the source, in the second at the beam endpoints. Both these variants eliminate the ray method singularities. The expansion of the wavefield into plane waves may be considered as the limiting case of the first approach and the Chapman–Maslov method as the limiting case of the second approach. Computer algorithms are briefly described and numerical examples presented. In the first numerical example, the comparisons of the two approaches, based on summing Gaussian beams, with the reflectivity method indicate that the computed synthetic seismograms are satisfactorily accurate even in the caustic region. The next example suggests that the two methods discussed can be simply and effectively applied to 3-D laterally inhomogeneous structures. 相似文献
6.
Enzo Mantovani 《Geophysical Journal International》1978,53(1):55-60
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
7.
Modal synthesis of high-frequency waves in Scotland 总被引：1，自引：0，他引：1
8.
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
9.
As indicated by numerical tests, the computation of synthetic seismograms based on the modal summation technique, for 2-D or 3-D media, requires the computation of the transmission and reflection coupling coefficients, at sharp lateral heterogeneities, with a formulation that takes into account the possible non-Poissonian character of the media. 相似文献
10.
The relation between p-Δcurves for surface and deep focus sources is investigated in order to construct synthetic body wave seismograms for non-zero focal depths by the quantized ray theory algorithm. The transformation of a surface focus p-Δ curve into a deep focus p-Δ curve is denned in terms of that curve which corresponds to surface focus rays reflected from the depth at which the deep focus is located. By analogy with the geometry of the surface focus formulation, paths of integration to obtain absolute travel-time and velocity-depth curves can be denned in the p-Δ plane. Explicit inversion from deep focus data is possible only when the velocity-depth structure above the depth of focus is known. Through a comparison of short period quantized ray theory synthetic seismograms with similar Cagniard-de Hoop computations, it is shown that quantized ray theory can be used for accurate predictions of body wave amplitude behaviour corresponding to a wide range of focal depths. 相似文献
11.
Joseph Ha 《Geophysical Journal International》1984,79(3):863-873
Summary. A set of recurrence relations which are computationally more efficient than those of the reflection matrix method of Kennett & Kerry is presented for P - and SV -wave generation in a ( n + 1) layered medium. The recurrence relations contain no growing terms and thus provide a stable algorithm for computing complete P and SV synthetic seismograms. Our algorithm requires a fewer algebraic operations for computing the reflectivity and transmissivity coefficients, ranging from 15 per cent less for a source in the half-space to 30 per cent less for a source in the top layer, than the reflection matrix method. 相似文献
12.
Summary. High-frequency reflection and refraction seismograms for laterally variable multi-layered elastic media are computed by using the frequency domain elastic Kirchhoff–Helmholtz (KH) theory of Frazer and Sen. Both source and receiver wavefields are expanded in series of generalized rays and then elastic (KH) theory is applied to determine the coupling between each source ray and each receiver ray at each interface. The motion at the receiver is given as a series of integrals, one for each generalized ray. We use geometrical optics and plane wave reflection and transmission coefficients for rapid evaluation of the integrand. When the source or the receiver ray field has caustics on the surface of integration geometrical ray theory breaks down and this gives rise to singularities in the KH integrand. We repair this using methods suggested by Frazer and Sen.
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals. 相似文献
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals. 相似文献
13.
Toshiro Tanimoto 《Geophysical Journal International》1987,89(2):713-740
Summary. The three-dimensional (3-D) shear wave structure of the mantle, down to the depth of about 900 km, is obtained by inverting waveforms of radial component seismograms. Radial component seismograms contain large amplitude overtone signals which circle the Earth as wave packets and are sometimes called X1, X2, X3, … We use data which contain R1, X1 and X2 and filtered between 2 and 10mHz. It is shown that, unless each seismogram is weighted, all seismograms are not fitted uniformly. Only data from large earthquakes are fitted and the final velocity anomalies are biased by the small number of large earthquake data. Resolution is good at shallow depths, becomes worse in the intermediate depth range between about 400 and 500 km and then becomes better at greater depth ranges (600–900km). Even though we use only spheroidal mode data, velocity anomalies in the shallow structure show excellent correlation with the age of the surface rocks of the Earth. In the deeper regions, between about 600 and 900km, South America shows a fast velocity anomaly which may indicate the slab penetration beyond 700 km there. Another region which shows a fast velocity anomaly is the Mariana trench, but other subduction regions do not show such features. 相似文献
14.
Ambient noise Rayleigh wave tomography of New Zealand 总被引：16，自引：0，他引：16
Fan-Chi Lin Michael H. Ritzwoller John Townend Stephen Bannister Martha K. Savage 《Geophysical Journal International》2007,170(2):649-666
We present the first New Zealand-wide study of surface wave dispersion, using ambient noise observed at 42 broad-band stations in the national seismic network (GeoNet) and the Global Seismic Network (GSN). Year-long vertical-component time-series recorded between 2005 April 1 and 2006 March 31 have been correlated with one another to yield estimated fundamental mode Rayleigh wave Green's functions. We filter these Green's functions to compute Rayleigh wave group dispersion curves at periods of 5–50 s, using a phase-matched filter, frequency–time analysis technique. The uncertainties of the measurements are estimated based on the temporal variation of the dispersion curves revealed by 12 overlapping 3-month stacks. After selecting the highest quality dispersion curve measurements, we compute group velocity maps from 7 to 25 s period. These maps, and 1-D shear wave velocity models at four selected locations, exhibit clear correlations with major geological structures, including the Taranaki and Canterbury Basins, the Hikurangi accretionary prism, and previously reported basement terrane boundaries. 相似文献
15.
Summary Reflectivity and ray theories are united to produce a hybrid technique of computing synthetic seismograms for a plane layered medium in subcritical regions. Numerical experiments have indicated that this technique is useful when the depth structure is one composed of thick layers separated by finely layered zones. As the theory for wave propagation in a plane layered medium is well known, the simple SH case is investigated so that the basic idea of the method may be conveyed without an excess of mathematics that would be necessitated if the P-SV problem were considered.
In computing the ray-reflectivity seismogram, the thick layers are treated using asymptotic ray theory while the thin-layered zones are treated as quasiinterfaces where analogues of reflection and transmission coefficients called reflectivities and transmittivities are calculated utilizing a Thomson-Haskell formulation. A stationary phase approximation is employed when evaluating the integral which gives the displacement due to an arbitrary ray propagating in the thick layers of the above-mentioned medium, and the validity of this approximation is discussed.
A comparison of ray, numerical integration (reflectivity) and ray-reflectivity synthetic sections indicates that this method yields quite acceptable results for subcritical reflection work and is suitable for application in seismic interpretation as individual arrivals associated with ray-paths in the thick layers may be identified. Furthermore, the method is quite cost efficient and may be extended to a medium where the thick layers are non-planar using asymptotic ray theory in these layers. 相似文献
In computing the ray-reflectivity seismogram, the thick layers are treated using asymptotic ray theory while the thin-layered zones are treated as quasiinterfaces where analogues of reflection and transmission coefficients called reflectivities and transmittivities are calculated utilizing a Thomson-Haskell formulation. A stationary phase approximation is employed when evaluating the integral which gives the displacement due to an arbitrary ray propagating in the thick layers of the above-mentioned medium, and the validity of this approximation is discussed.
A comparison of ray, numerical integration (reflectivity) and ray-reflectivity synthetic sections indicates that this method yields quite acceptable results for subcritical reflection work and is suitable for application in seismic interpretation as individual arrivals associated with ray-paths in the thick layers may be identified. Furthermore, the method is quite cost efficient and may be extended to a medium where the thick layers are non-planar using asymptotic ray theory in these layers. 相似文献
16.
Summary. Use of paraxially approximated Gaussian beams continues to be actively pursued for construction of synthetic seismograms in complicated environments. How to select the beams in the stack remains a source of difficulty which has primarily been addressed by semi-heuristic considerations. In this paper, the classical example of line-source field reflection from a homogeneous half-space that can sustain a head wave is examined from a plane-wave spectral point of view. The individual beam fields are modelled exactly by the complex source point technique, which emphasizes the complex spectral content of these wave objects. The quality of the paraxial approximation of a typical reflected (Gaussian) beam characterized by different parameters is examined from this perspective, and is compared with uniform and non-uniform asymptotics generated from the exact beam field spectral integral. With this information as background, the reflected field for a real line-source is synthesized by beam superposition. Except for the immediate vicinity of the critical reflection angle, the well-known failure of narrow paraxial beams, no matter how densely stacked, to reproduce the head wave effects is shown to be due to the inadequate spectral content of these beams and not to the failure of beam stacking per se. When the rigorous solutions are used for the narrow-waist beams, even relatively few suffice to yield agreement with the exact solution. This circumstance emphasizes the importance of fully understanding the spectral implications of various beam stacking schemes. 相似文献
17.
Evidence for anisotropy in the upper mantle beneath Eurasia from the polarization of higher mode seismic surface waves 总被引：1，自引：0，他引：1
Summary. Analysis of NORSAR records and a number of Soviet microfilms reveals second-mode surface Caves propagating along paths covering a large part of Eurasia. These second modes in the 6–15-s period band are frequently disturbed by other surface-wave modes and by body-wave arrivals. However, in all cases, where the modes appear to be undisturbed and show normal dispersion, the Second Rayleigh modes have a slowly varying phase difference with the Second Love modes. This coupling has the particle motion of Inclined Rayleigh waves characteristic of surface-wave propagation in anisotropic media, where the anisotropy possesses a horizontal plane of symmetry. Numerical examination of surface wave propagating in Earth models, with an anisotropic layer in the upper mantle, demonstrate that comparatively small thicknesses of material with weak velocity anisotropy can produce large deviations in the polarizations of Inclined Rayleigh Second modes. In many structures, these inclinations are very sensitive to small changes in anisotropic orientation and to small changes in the surrounding isotropic structure. It is suggested that examination of second mode inclination anomalies of second mode surface waves may be a powerful technique for examining the detailed anisotropic structure of the upper mantle. 相似文献
18.
B. R. Cassell 《Geophysical Journal International》1982,69(2):339-354
Summary An effective algorithm for computing synthetic seismograms in laterally inhomogeneous media has been developed. The method, based on zero-order asymptotic ray theory, is primarily intended for use in refraction and reflection studies and provides an economical means of seismic modelling.
A given smoothed velocity-depth-distance model is divided into small squares with constant seismic parameters and first-order interfaces are represented by an arbitrary number of dipping linear segments. The computation of ray propagation and amplitudes through such a model does not involve complicated analytic expressions and therefore minimizes computer time.
Amplitudes are determined by geometrical spreading of spherical wave-fronts and energy partitioning at interfaces. Synthetic seismograms calculated for laterally homogeneous models are in good agreement with those obtained by the Reflectivity Method. 相似文献
A given smoothed velocity-depth-distance model is divided into small squares with constant seismic parameters and first-order interfaces are represented by an arbitrary number of dipping linear segments. The computation of ray propagation and amplitudes through such a model does not involve complicated analytic expressions and therefore minimizes computer time.
Amplitudes are determined by geometrical spreading of spherical wave-fronts and energy partitioning at interfaces. Synthetic seismograms calculated for laterally homogeneous models are in good agreement with those obtained by the Reflectivity Method. 相似文献
19.
Summary. Data from a refraction and a reflection seismic survey in the Black Forest, southwest Germany, are used for extensive one- and two-dimensional modelling. The data are available along approximately the same line, and therefore the same piece of crust is probed by two seismic methods. We utilize this favorable circumstance for detailed model calculations concerning both data sets. Lower crustal properties vary on the scale of a wavelength and thus full solutions of the elastic equations are required: the Reflectivity Method for the evaluation of refraction seismograms and numerical solutions of the acoustic wave equation for the reflection response. Details of the geometry and physical properties of the lamination are derived. Vertical layering on a scale of 100 m is found; horizontal extent of reflecting elements is in the range of a few hundreds of meters; rocks with velocities between 5.6 and 7.2 km/s constitute the lower crust. 相似文献
20.
Summary. A new method is presented for the direct inversion of seismic refraction data in dipping planar structure. Three recording geometries, each consisting of two common-shot profiles, are considered: reversed, split, and roll-along profiles. Inversion is achieved via slant stacking the common-shot wavefield to obtain a delay time–slowness (tau– p ) wavefield. The tau– p curves from two shotpoints describing the critical raypath of refracted and post-critically reflected arrivals are automatically picked using coherency measurements and the two curves are jointly used to calculate velocity and dip of isovelocity lines iteratively, thereby obtaining the final two-dimensional velocity model.
This procedure has been successfully applied to synthetic seismograms calculated for a dipping structure and to field data from central California. The results indicate that direct inversion of closely-spaced refraction/wide-aperture reflection data can practically be achieved in laterally inhomogeneous structures. 相似文献
This procedure has been successfully applied to synthetic seismograms calculated for a dipping structure and to field data from central California. The results indicate that direct inversion of closely-spaced refraction/wide-aperture reflection data can practically be achieved in laterally inhomogeneous structures. 相似文献