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1.
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   

2.
Envelopes of scalar waves are simulated at various distances from an instant point source embedded in a random uniformly scattering medium by means of direct Monte-Carlo modelling of wave-energy transport. Three types of scattering radiation pattern ('indicatrix') are studied, for media specified by (1) a Gaussian autocorrelation function of inhomogeneities, (2) a power-law ('fractal', k -α) inhomogeneity spectrum and (3) the mix of case (1) and the isotropic indicatrix (very small + large inhomogeneities). We look for a model that can qualitatively reproduce the two most characteristic features of real S-wave envelopes of near earthquakes, namely (1) the broadening of the 'direct' wave group with distance and (2) the monotonously decaying shape of the coda envelope that does not deviate strongly from that expected in the isotropic scattering case. Both properties are observed for any band over a wide frequency range (1-40 Hz). The well-studied isotropic scattering model realistically predicts the appearance of codas but fails to predict pulse broadening. The model of large-scale inhomogeneity realistically predicts the mode of pulse broadening but fails to predict codas. We have found that, for a particular frequency band, within each class of inhomogeneity studied, both requirements can be qualitatively satisfied by a certain choice of parameters. In the Gaussian-ACF case, however, this match can be obtained only for a narrow frequency range. In contrast, the fractal case (with a value of exponent a of about 3.5-4) reproduces qualitatively the observed wide-band behaviour, and we consider it a reasonable representation of the gross properties of the earth medium.  相似文献   

3.
Scattering of surface waves modelled by the integral equation method   总被引：1，自引：0，他引：1  
The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.  相似文献   

4.
The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff–Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR . The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR ≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.  相似文献   

5.
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   

6.
Real plane-waves constitute the building blocks for recently developed spectral techniques in synthetic seismology. While providing numerical convenience, real slowness-spectra model certain wave phenomena in a distributed 'unnatural' way, whereas complex spectra model these phenomena in a compact, more 'natural' way. The theory of complex spectra, called by us the 'Spectral Theory of Transients' (STT) and developed elsewhere, is summarized here and contrasted with the real-spectrum approach. Relying strongly on the theory of analytic functions, STT permits the transient responses to be classified and evaluated according to the singularities they introduce in the complex slowness plane. The method is illustrated for a number of 2-D SH -wave model propagation environments, including interface reflection, head waves, multiple encounters with caustics due to concave boundaries or ducting medium inhomogeneities, and diffraction by structures with edges.  相似文献   

7.
Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.  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.
Some remarks on the Gaussian beam summation method   总被引：1，自引：0，他引：1  
Summary. Recently, a method using superposition of Gaussian beams has been proposed for the solution of high-frequency wave problems. The method is a potentially useful approach when the more usual techniques of ray theory fail: it gives answers which are finite at caustics, computes a nonzero field in shadow zones, and exhibits critical angle phenomena, including head waves. Subsequent tests by several authors have been encouraging, although some reported solutions show an unexplained dependence on the 'free' complex parameter ε which specifies the initial widths and phases of the Gaussian beams.
We use methods of uniform asymptotic expansions to explain the behaviour of the Gaussian beam method. We show how it computes correctly the entire caustic boundary layer of a caustic of arbitrary complexity, and computes correctly in a region of critical reflection. However, the beam solution for head waves and in edge-diffracted shadow zones are shown to have the correct asymptotic form, but with governing parameters that are explicitly ε-dependent. We also explain the mechanism by which the beam solution degrades when there are strong lateral inhomogeneities. We compare numerically our predictions for some representative, model problems, with exact solutions obtained by other means.  相似文献   

10.
When interpreting electromagnetic fields observed at the Earth's surface in a realistic geophysical environment it is often necessary to pay special attention to the effects caused by inhomogeneities of the subsurface sedimentary and/or water layer and by inhomogeneities of the Earth's crust. The inhomogeneities of the Earth's crust are expected to be especially important when the electromagnetic field is generated by a source located in a magma chamber of a volcano. The simulation of such effects can be carried out using generalized thin-sheet models, which were independently introduced by Dmitriev (1969 ) and Ranganayaki & Madden (1980 ). In the first part of the paper, a system of integral equations is derived for the horizontal current that flows in the subsurface inhomogeneous conductive layer and for the vertical current crossing the inhomogeneous resistive layer representing the Earth's mantle. The terms relating to the finite thickness of the laterally inhomogeneous part of the model are retained in the equations. This only marginally complicates the equations, whilst allowing for a significant expansion of the approximation limits.
  The system of integral equations is solved using the iterative dissipative method developed by the authors in the period from 1978 to 1988. The method can be applied to the simulation of the electromagnetic field in an arbitrary inhomogeneous medium that dissipates the electromagnetic energy. When considered on a finite numerical grid, the integral equations are reduced to a system of linear equations that possess the same contraction properties as the original equations. As a result, the rate at which the iterative-perturbation sequence converges to the solution remains independent of the numerical grid used for the calculations. In contrast to previous publications on the method, aspects of the algorithm implementation that guarantee its effectiveness and robustness are discussed here.  相似文献   

11.
A Gaussian correlation function characterizes smoothly heterogeneous media, while real heterogeneities in the Earth are often non-Gaussian in nature. Using the Born approximation, mean square amplitudes of the scattered waves have been derived for an elastic media characterized by the Von Karman correlation function. Heterogeneities with different power laws can be defined by the Von Karman correlation function. The sensitivity of fore- and backscattering to heterogeneities with different scales and properties (that is velocity and impedance) is discussed in this paper. The analytical expression for total scattered energy for the incident P waves is also derived for a random medium having the Von Karman correlation function. We find that at high frequencies, the scattered power of converted waves is a function of frequency. In the case of codawave excitation by local earthquakes, which must be handled by the full elastic-wave theory, we can define any type of inhomogeneity by the Von Karman correlation function. It also supports the idea that the lithosphere might have multiple-scale inhomogeneities.  相似文献   

12.
The Born approximation is applied to the modelling of the propagation of deeply turning longperiod body waves through heterogeneities in the lowermost mantle. We use an exact Green's function for a spherically symmetric earth model that also satisfies the appropriate boundary conditions at internal boundaries and the surface of the earth. The scattered displacement field is obtained by a numerical quadrature of the product of the Green's function, the exciting wavefield and structural perturbations. We study three examples: scattering of longperiod P waves from a plume rising from the coremantle boundary (CMB), generation of longperiod precursors to PKIKP by strong, localized scatterers at the CMB, and propagation of corediffracted P waves through largescale heterogeneities in D". The main results are as follows: (1) the signals scattered from a realistic plume are small with relative amplitudes of less than 2 per cent at a period of 20 s, rendering plume detection a fairly difficult task; (2) strong heterogeneities at the CMB of appropriate size may produce observable longperiod precursors to PKIKP in spite of the presence of a diffraction from the PKP B caustic; (3) corediffracted P  waves ( P diff) are sensitive to structure in D" far off the geometrical ray path and also far beyond the entry and exit points of the ray into and out of D"; sensitivity kernels exhibit ringshaped patterns of alternating sign reminiscent of Fresnel zones; (4) P diff also shows a nonnegligible sensitivity to shear wave velocity in D"; (5) down to periods of 40 s, the Born approximation is sufficiently accurate to allow waveform modelling of P diff through largescale heterogeneities in D" of up to 5 per cent.  相似文献   

13.
Today's numerical methods like the Spectral Element Method (SEM) allow accurate simulation of the whole seismic field in complex 3-D geological media. However, the accuracy of such a method requires physical discontinuities to be matched by mesh interfaces. In many realistic earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical cost. In this paper, we address a limited aspect of this problem: an earth model with a thin shallow layer below the free surface in which the elastic and density properties are different from the rest of the medium and in which rapid vertical variations are allowed. We only consider here smooth lateral variations of the thickness and elastic properties of the shallow layer. In the limit of a shallow layer thickness very small compared to the smallest wavelength of the wavefield, by resorting to a second order matching asymptotic approximation, the thin layer can be replaced by a vertically smooth effective medium without discontinuities together with a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows to accurately take into account complex thin shallow structures within the SEM without the classical mesh design and time step constraints. Corrections at receivers and source—when the source is located within the thin shallow layer—have been also derived. Accuracy and efficiency of this formulation are assessed on academic tests. The stability and limitations of this formulation are also discussed.  相似文献   

14.
We present a technique based on the single-scattering approximation that relates time-lapse localized changes in the propagation velocity to changes in the traveltime of singly scattered waves. We describe wave propagation in a random medium with homogeneous statistical properties as a single-scattering process where the fluctuations of the velocity with respect to the background velocity are assumed to be weak. This corresponds to one of two end-member regimes of wave propagation in a random medium, the first being single scattering, and the second multiple scattering. We present a formulation that relates the change in the traveltime of the scattered waves to a localized change in the propagation velocity by means of the Born approximation for the scattered wavefield. We validate the methodology with synthetic seismograms calculated with finite differences for 2-D acoustic waves. Potential applications of this technique include non-destructive evaluation of heterogeneous materials and time-lapse monitoring of heterogeneous reservoirs.  相似文献   

15.
A geomagnetic scattering theory for evaluation of earth structure   总被引：1，自引：0，他引：1  
Summary. Structural features of the Earth's lower crust and upper mantle can be mapped by the analysis of temporal geomagnetic fluctuations using the electromagnetic scattering theory developed in this paper. Decomposing geomagnetic field fluctuations at the Earth's surface into an excitation part and a scattered part forms the basis of a power series development. The vertical field component is interpreted as a scattering of the excitation field. The horizontal gradient and geomagnetic depth sounding methods are special cases of the theory developed. The horizontal gradient sounding method has a tensorial aspect which has not been recognized before; it should be included to obtain correct penetration depth parameter evaluations from field data.  相似文献   

16.
Elastic scattered waves from a continuous and heterogeneous layer   总被引：3，自引：0，他引：3  
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   

17.
We present a regional surface waveform tomography of the Pacific upper mantle, obtained using an automated multimode surface waveform inversion technique on fundamental and higher mode Rayleigh waves, to constrain the   VSV   structure down to ∼400 km depth. We have improved on previous implementations of this technique by robustly accounting for the effects of uncertainties in earthquake source parameters in the tomographic inversion. We have furthermore improved path coverage in the South Pacific region by including Rayleigh wave observations from the French Polynesian Pacific Lithosphere and Upper Mantle Experiment deployment. This improvement has led to imaging of vertical low-velocity structures associated with hotspots within the South Pacific Super-Swell region. We have produced an age-dependent average cross-section for the Pacific Ocean lithosphere and found that the increase in   VSV   with age is broadly compatible with a half-space cooling model of oceanic lithosphere formation. We cannot confirm evidence for a Pacific-wide reheating event. Our synthetic tests show that detailed interpretation of average   VSV   trends across the Pacific Ocean may be misleading unless lateral resolution and amplitude recovery are uniform across the region, a condition that is difficult to achieve in such a large oceanic basin with current seismic stations.  相似文献   

18.
We propose a vertical array analysis method that decomposes complex seismograms into body and surface wave time histories by using a velocity structure at the vertical array site. We assume that the vertical array records are the sum of vertically incident plane P and S waves, and laterally incident Love and Rayleigh waves. Each phase at the surface is related to that at a certain depth by the transfer function in the frequency domain; the transfer function is obtained by Haskell's matrix method, assuming a 1-D velocity structure. Decomposed P , S and surface waves at the surface are estimated from the vertical array records and the transfer functions by using a least-squares method in the frequency domain; their time histories are obtained by the inverse Fourier transform. We carried out numerical tests of this method based on synthetic vertical array records consisting of vertically incident plane P and S waves and laterally incident plane Love and Rayleigh waves. Perfect results of the decomposed P , S , Love and Rayleigh waves were obtained for synthetic records without noise. A test of the synthetic records in which a small amount of white noise was added yielded a reasonable result for the decomposed P , S and surface waves. We applied this method to real vertical array records from the Ashigara valley, a moderate-sized sedimentary valley. The array records from two earthquakes occurring at depths of 123 and 148 km near the array (epicentral distance of about 31 km) exhibited long-duration later phases. The analysis showed that duration of the decomposed S waves was a few seconds and that the decomposed surface waves appeared a few seconds after the direct S -wave arrival and had very long duration. This result indicated that the long-duration later phases were generated not by multireflected S waves, but by basin-induced surface waves.  相似文献   

19.
Although the galvanic distortion due to local, near-surface inhomogeneities is frequency-independent, its effect on the magnetotelluric data becomes, in a 3-D structure, frequency-dependent. Therefore, both the apparent resistivity and the phase responses are disturbed, and a correction should be carried out prior to the 3-D interpretation in order to retrieve the 3-D regional impedance tensor. In many cases, the structure is 2-D for depths corresponding to a first range of periods and 3-D for longer periods (called 2-D/3-D). For these cases, a simple method which allows us to retrieve the 3-D regional impedance tensor (except the static shift) is presented. The method proposed uses the Groom & Bailey decomposition of the distortion matrix for the short periods. Three examples are presented: two using synthetic data and one employing real data. These examples show the effect of the galvanic distortion over a regional 2-D/3-D model and the retrieval of the regional transfer functions from the distorted ones.  相似文献   

20.
Summary. Surface wave behaviour in flat anisotropic structures is first illustrated by performing an exact computation on a simple two-layer model. The variational procedure of Smith & Dahlen is then used to compute the partial derivatives of surface wave phase velocities with respect to the elastic parameters in more realistic earth models. Linear relationships between the partial derivatives for a general anisotropic structure and those for a transversely isotropic structure are derived. When considering waves propagating in a fixed direction, there are only four independent derivatives for Rayleigh waves, and two for Love waves. To avoid the lack of resolution in an inverse method, we propose to use physically constrained models. These results are illustrated by using a model with hexagonal symmetry and a symmetry axis oriented either vertically or horizontally. Quasi-Love- and quasi-Rayleigh-wave partial derivatives are computed for both axis orientations. Modes up to the second overtone and periods ranging between 45 and 130 s have been considered. Finally, anomalies of phase velocity are computed in an oceanic model made of 1/6 oriented olivine crystals with horizontal or vertical preferred orientations of the a -axis.  相似文献   

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