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1.
We present a complete ray theory for the calculation of surface-wave observables from anisotropic phase-velocity maps. Starting with the surface-wave dispersion relation in an anisotropic earth model, we derive practical dynamical ray-tracing equations. These equations allow calculation of the observables phase, arrival-angle and amplitude in a ray theoretical framework. Using perturbation theory, we also obtain approximate expressions for these observables. We assess the accuracy of the first-order approximations by using both theories to make predictions on a sample anisotropic phase-velocity map. A comparison of the two methods illustrates the size and type of errors which are introduced by perturbation theory. Perturbation theory phase and arrival-angle predictions agree well with the exact calculation, but amplitude predictions are poor. Many previous studies have modelled surface-wave propagation using only isotropic structure, not allowing for anisotropy. We present hypothetical examples to simulate isotropic modelling of surface waves which pass through anisotropic material. Synthetic data sets of phase and arrival angle are produced by ray tracing with exact ray theory on anisotropic phase-velocity maps. The isotropic models obtained by inverting synthetic anisotropic phase data sets produce deceptively high variance reductions because the effects of anisotropy are mapped into short-wavelength isotropic structure. Inversion of synthetic arrival-angle data sets for isotropic models results in poor variance reductions and poor recovery of the isotropic part of the anisotropic input map. Therefore, successful anisotropic phase-velocity inversions of real data require the inclusion of both phase and arrival-angle measurements.  相似文献   

2.
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   

3.
We present a mathematical framework and a new methodology for the parametrization of surface wave phase-speed models, based on traveltime data. Our method is neither purely local, like block-based approaches, nor is it purely global, like those based on spherical harmonic basis functions. Rather, it combines the well-known theory and practical utility of the spherical harmonics with the spatial localization properties of spline basis functions. We derive the theoretical foundations for the application of harmonic spherical splines to surface wave tomography and summarize the results of numerous numerical tests illustrating the performance of a practical inversion scheme based upon them. Our presentation is based on the notion of reproducing-kernel Hilbert spaces, which lends itself to the parametrization of fully 3-D tomographic earth models that include body waves as well.  相似文献   

4.
We implement the wave equation on a spherical membrane, with a finite-difference algorithm that accounts for finite-frequency effects in the smooth-Earth approximation, and use the resulting 'membrane waves' as an analogue for surface wave propagation in the Earth. In this formulation, we derive fully numerical 2-D sensitivity kernels for phase anomaly measurements, and employ them in a preliminary tomographic application. To speed up the computation of kernels, so that it is practical to formulate the inverse problem also with respect to a laterally heterogeneous starting model, we calculate them via the adjoint method, based on backpropagation, and parallelize our software on a Linux cluster. Our method is a step forward from ray theory, as it surpasses the inherent infinite-frequency approximation. It differs from analytical Born theory in that it does not involve a far-field approximation, and accounts, in principle, for non-linear effects like multiple scattering and wave front healing. It is much cheaper than the more accurate, fully 3-D numerical solution of the Earth's equations of motion, which has not yet been applied to large-scale tomography. Our tomographic results and trade-off analysis are compatible with those found in the ray- and analytical-Born-theory approaches.  相似文献   

5.
A self-adaptive automated parametrization approach is suggested for the sequential inversion of controlled-source seismic tomography and gravity data. The velocities and interfaces are parametrized by their Haar wavelet expansion coefficients. Only those coefficients that are well constrained by the data, as measured by the number of rays that cross the corresponding wavelet function support area and their angular coverage, are inverted for, others are set to zero. This approach results in a reasonable distribution of resolution throughout the model even in cases of irregular ray coverage and does overcome the trade-off between different types of model parameters. A modified sequential inversion approach is suggested to join the traveltimes and gravity anomalies inversion. An algorithm is developed that inverts for smooth velocity and density variations inside the seismic layer, the position of its bottom interface as well as for optimal values of the velocity-to-density regression coefficients. The algorithm makes use of direct (diving), reflected and head (critically refracted) wave traveltimes. The algorithm workflow is demonstrated on a synthetic data example.  相似文献   

6.
Summary. The topological problem underlying remote sensing is analysed by determining the geometric singularities that an unknown surface or structure generically impresses on a sensing wavefield. It is shown that the analytical singularities observed in scattering amplitudes and echograms are produced by the topological singularities of the scattering system. Imposing the principle of structural stability on the inverse scattering problem, the singularities that generically occur in recorded signals, travel-time curves, surface contour maps and Fresnel-zone topographies can, together with the associated high-intensity diffraction patterns, be classified into a few universal standard forms described by catastrophe polynomials. As the source-receiver positions vary, the patterns change their morphologies in terms of specific bifurcation sets. By applying singularity and bifurcation theory to allow the effects of caustics (both in ray and wave theory) to be incorporated into three-dimensional techniques for reconstructing surfaces and subsurface structures from their echoes, the interpretation process is considerably simplified and permits an on-site 3D survey. Universal power laws for singularity-dominated echo amplitude variations with the source frequency are deduced. The shape of a scattering surface is reconstructed using the high-frequency regime alone. Discontinuities in the surface, edges and faults, are discussed in terms of constraint catastrophes and the patterns they produce in echograms are classified.  相似文献   

7.
A general tomographic technique is designed in order (i) to operate in anisotropic media; (ii) to account for the uneven seismic sampling and (iii) to handle massive data sets in a reasonable computing time. One modus operandi to compute a 3-D body wave velocity model relies on surface wave phase velocity measurements. An intermediate step, shared by other approaches, consists in translating, for each period of a given mode branch, the phase velocities integrated along ray paths into local velocity perturbations. To this end, we develop a method, which accounts for the azimuthal anisotropy in its comprehensive form. The weakly non-linear forward problem allows to use a conjugate gradient optimization. The Earth's surface is regularly discretized and the partial derivatives are assigned to the individual grid points. Possible lack of lateral resolution, due to the inescapable uneven ray path coverage, is taken into account through the a priori covariances on parameters with laterally variable correlation lengths. This method allows to efficiently separate the 2ψ and the 4ψ anisotropic effects from the isotropic perturbations. Fundamental mode and overtone phase velocity maps, derived with real Rayleigh wave data sets, are presented and compared with previous maps. The isotropic models concur well with the results of Trampert & Woodhouse. Large 4ψ heterogeneities are located in the tectonically active regions and over the continental lithospheres such as North America, Antarctica or Australia. At various periods, a significant 4ψ signature is correlated with the Hawaii hotspot track. Finally, concurring with the conclusions of Trampert & Woodhouse, our phase velocity maps show that Rayleigh wave data sets do need both 2ψ and 4ψ anisotropic terms.  相似文献   

8.
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   

9.
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   

10.
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   

11.
中国西部干旱区生态景观制图的若干问题   总被引：5，自引：0，他引：5  
制图方法是认识和揭示客观现象与过程的重要途径。生态景观制图作为专题制图学的一个重要方向,通过建立分类体系,制定分类方法,编制专题图件,可以反映自然要素和人为作用的特点,揭示生态景观要素的耦合关系。生态景观制图中专题要素的表现方式,制图过程的数据采集,制图单元的划分,制图的模式选择,以及遥感、GIS与数字制图手段的应用,是十旱Ⅸ景观生态制图的关键问题。生态景观制图巾的地图信息理论、地图模式论、地图分析理论、地图要素综合理论、制冈工艺学等的发展,对于开拓资源环境制图学的新领域具有理论价值,并对生态规划与生态建设具有重要促进作用。  相似文献   

12.
Summary. A method of comparison of exact numerical computations with an asymptotic ray series expansion consisting of the two first terms is proposed. The method makes it unnecessary to derive complicated explicit expressions for the second leading term of the ray series.
As a practical example we consider the anomalous PS arrival generated in the case of a near-vertical incidence of a spherical P wave on a solid/solid boundary. The areas in which the PS wave may be described by two leading terms of the ray series expansion are marked and deviations from the ray theory are analysed.  相似文献   

13.
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   

14.
A method is developed for the reconstruction of a non-uniform distribution of scattering properties in the upper layers of the Earth using data on broadening of an incoherent body-wave group or pulse along a number of rays. The theoretical basis for this reconstruction is a linear integral formula after Bocharov (1985, 1988), which is employed to design a linear inversion procedure. The inversion is performed in terms of a single scalar parameter of effective turbidity. This parameter presents an adequate generalization of the common turbidity parameter used in the isotropic scattering case; it describes, simultaneously, scattering attenuation, pulse broadening and backscattering or coda formation. As a preliminary step, necessary conditions of applicability of the transport equation approach for the analysis of regional high-frequency seismic waves are verified. A new compact derivation of Bocharov's formula is then presented. A linear least-squares inversion procedure for recovering a layered turbidity structure is proposed and tested on synthetic data of onset-to-peak delays of incoherent body-wave pulses. A few practical aspects of the application of the general approach to seismological data are analysed, including the correctness of the low-angle approximation, the use of peak delay observations instead of pulse centroid, the effects of a realistic spatial spectrum of inhomogeneity field, the potential bias produced by intrinsic loss, and the distortions produced by a non-spherical (double dipole) source radiation pattern. The latter point is considered as critically important, as one can expect significant data contamination by nodal arrivals. An efficient robust estimation procedure is designed and tested that is capable of suppressing distortions from nodal and near-nodal data.  相似文献   

15.
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   

16.
Summary . Seismograms recorded at regional distances (2°–12°) are quite complicated due to the waveguide nature of the crust. Generalized ray theory can be used to model the body waves in this distance range but a very large number of rays is required. Here I present a series of approximations to streamline generalized ray theory for the waveguide problem. If a layer over a half-space is used for the structure, then the de Hoop contour for a given ray is most strongly dependent on the fastest velocity of any leg of the ray. This results in analytic approximations to locate the contour. Each ray has two body wave arrivals (a headwave and a reflected arrival) so the displacement response of the ray need only be evaluated at a few points in time about the two arrival times and interpolated in between. A change in structure (increasing crustal thickness or Pn velocity) most strongly affects the relative timing of the headwave and the reflected arrival, so it is possible to 'stretch' or 'squeeze' the waveform of a representative model to simulate a whole suite of models.
Also discussed is the applicability of a single layer over a half-space structure for modelling the observed regional distance waveforms for shallow earthquakes. At periods greater than a few seconds crustal layering can be replaced by a single layer having the appropriate average velocities. Lateral variations in crustal thickness with scale lengths of less than about 100 km can also be modelled with a simple horizontal layer of appropriate average thickness.  相似文献   

17.
Summary. Body wave synthetic siesmograms for laterally varying media are computed by means of a slowness implementation of the extended WKBJ (EWKBJ) theory of Frazer & Phinney. An EWKBJ seismogram is computed by first tracing rays through a particular model to obtain conventional ray information (travel time, ray end point, ray slowness) and then using these data in the finite frequency integral expression for the EWKBJ seismogram. The EWKBJ seismograms compare favourably to geometrical ray theory (GRT) seismograms but are significantly better because of the finite frequency nature of the EWKBJ calculation. More realistic behaviour is obtained with EWKBJ seismograms at normal seismic frequencies near caustics, where the GRT amplitude is infinite, and within geometrical shadow zones where GRT predicts zero amplitudes. In addition the EWKBJ calculation is more sensitive than GRT to focuses and defocuses in the ray field. The major disadvantage of the EWKBJ calculation is the additional computer time over that of GRT, necessary to calculate one seismogram although an EWKBJ seismogram costs much less to compute than a reflectivity seismogram. Another disadvantage of EWKBJ theory is the generation of spurious, non-geometrical phases that are associated with rapidly varying lateral inhomogeneities. Fortunately the amplitudes of these spurious phases are usually much lower than that of neighbouring geometrical phases so that the spurious phases can usually be ignored. When this observation is combined with the moderately increased computational time of the EWKBJ calculation then the gain in finite frequency character significantly outweighs any disadvantages.  相似文献   

18.
High-frequency body waves recorded by a temporary seismic array across the surface rupture trace of the 1992 Landers, California, earthquake were used to determine fault-zone structures down to the seismogenic depth. We first developed a technique to use generalized ray theory to compute synthetic seismograms for arbitrarily oriented tabular low-velocity fault-zone models. We then generated synthetic waveform record sections of a linear array across a vertical fault zone. They show that both arrival times and waveforms of P and S waves vary systematically across the fault due to transmissions and reflections from boundaries of the low-velocity fault zone. The waveform characteristics and arrival-time patterns in the record sections allow us to locate the boundaries of the fault zone and to determine its P - and S -wave velocities independently as well as its depth extent. Therefore, the trade-off between the fault-zone width and velocities can be avoided. Applying the method to the Landers waveform data reveals a low-velocity zone with a width of 270–360 m and a 35–60 per cent reduction in P and S velocities relative to the host rock. The analysis suggests that the low-velocity zone extends to a depth of ∼7 km. The western boundary of the low-velocity zone coincides with the observed main surface rupture trace.  相似文献   

19.
Summary. The Gaussian beam method has recently been introduced into synthetic seismology to overcome shortcomings of the ray method, especially in transition regions due to focusing or diffraction where ray theory fails. One proceeds by discretizing the initial data as a superposition of paraxial Gaussian beams, each of which is then traced through the seismic environment. Since Gaussian beam fields do not diverge in ray transition regions, they are 'uniformly regular' although the quality of this regularity depends on the beam parameters and on the 'numerical distance' which defines the extent of the transitional domain. However, when Gaussian beam patches are used to simulate non-Gaussian initial data, there arise ambiguities due to choice of patch size and location, beam width, etc., which are at the user's disposal. The effects of this arbitrariness have customarily been explored by trial and error numerical experiment but no quantitative recommendations have emerged as yet. As a step toward a priori predictive capability, it is proposed here to perform a systematic study on analytically tractable prototype models of how the parameters and location of a single beam affect the quality of the observed seismic field, especially in ray transition regions. The conversion of ordinary ray fields into beam fields in canonical configurations can be accomplished conveniently by displacing a real source point into a complex coordinate space. Thus, the desired beam solutions can be obtained directly from available ray, and even paraxial ray, fields. Complex ray theory and its implications are reviewed here, with an emphasis on improvements of beam tracking schemes employed at present.  相似文献   

20.
The radiative transfer theory (RTT) describes the energy transport through a random heterogeneous medium, neglecting phase information. It provides an adequate framework for modelling high-frequency seismogram envelopes. For isotropic scattering and sources, the radiative transfer equation (RTE) has been formulated analytically and numerically simulated using Monte Carlo methods for acoustic and elastic media. Here, we derive an exact analytical solution of the RTE in 2-D space for the acoustic case, including anisotropic scattering for a anisotropic point-like impulsive source. For this purpose, we generalize the path integral method, which has been used before in the isotropic case, to take into account the anisotropy of both the source radiation pattern and scattering processes, simultaneously. Then we obtain a general solution, which is written in a closed form in the Fourier space. To illustrate the theoretical results, we compute the full space and time evolution of the specific intensity for an arbitrary case. We also compare the time traces computed from our general solution with cases in which the source and/or the scattering process are isotropic. The importance of taking into account both anisotropies simultaneously becomes obvious in our examples. We also show that at long lapse time, our example approaches the solution of the diffusion equation.  相似文献   

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