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1.
Reciprocity theorems for one-way wavefields   总被引：1，自引：0，他引：1  
Acoustic reciprocity theorems have proved their usefulness in the study of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and are thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation into a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theorems" along the same lines as the usual derivation of the 'two-way reciprocity theorems'. However, for the one-way reciprocity theorems it is not directly obvious that the 'contrast term' vanishes when the medium parameters in the two different states are identical. By introducing a modal expansion of the Helraholtz operator, its square root can be derived, which appears to have a symmetric kernel. This symmetry property appears to be sufficient to let the contrast term vanish in the above-mentioned situation.
The one-way reciprocity theorem of the convolution type is exact, whereas the one-way reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators.  相似文献   

2.
Summary . The most complicated part in the computation of ray amplitudes of seismic body waves in laterally inhomogeneous media with curved interfaces lies in the evaluation of the geometrical spreading. Geometrical spreading can be simply expressed in terms of the Jacobian J of the transformation from the Cartesian into ray coordinates. Several systems of ordinary differential equations to compute the function J are suggested. For general three-dimensional media, in which the velocity changes with all the three spatial coordinates, a system of three non-linear ordinary differential equations of the first order is derived. If the velocity does not depend on one coordinate, the system of equations reduces to only one non-linear differential equation. The initial conditions for these differential equations at point (or line) source and at points of intersection of the ray with curved interfaces are presented.  相似文献   

3.
The parameter that defines the ray tracing equations in the direct geometrical approach is the product of the radius of curvature of the wave front by the velocity on the wave front ( RV ). To show this, we derive motion equations for the centre and the radius of curvature of an expanding wave front. The continuity of RV along rays implies Snell's Law. For constant velocities the equation for the radius of curvature reduces to the original Huygens' Principle. The variable RV can be computed during ray tracing and used to determine the local radius of curvature, which in turn can be used in geometrical spreading, amplitude corrections and structure interpretation.  相似文献   

4.
Summary. The transformation of a set of seismograms to the delay time-slowness, τ—p, domain is presented as a sequence of Fourier and Bessel transforms, For a horizontally layered medium, this sequence gives an exact cylindrical wave decomposition of the response to a point source; correctly compensating for the phase shifting and geometrical spreading associated with transmission through the Earth. The resultant τ—p map or 'slant stack' contains true amplitude and phase information. The spatial aliasing properties of the transformation, when applied to a dataset, are greatly improved by the use of only outgoing waves in the Bessel transform. This is equivalent to using Hankel functions rather than Bessel functions, and is justified by the absence of incoming waves from most datasets. The WKBJ approximation to the medium response enables predictions to be made about the shape and amplitude variation with slowness of truncation effects. Theoretically the τ—p transformation is reversible, thus the τ—p domain is a suitable one in which to perform filtering operations before seismogram reconstruction.  相似文献   

5.
Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.  相似文献   

6.
A method for calculating synthetic seismograms in laterally varying media   总被引：2，自引：0，他引：2  
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   

7.
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   

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.
Summary. The ray series solution of the elastodynamic equation of motion for shear waves propagating through a laterally inhomogeneous three-dimensional medium can be simplified by the use of a particular coordinate system that accompanies the wave front along the ray of investigation. The system is entirely determined by parameters that are obtainable from the ray. The transport equations for the principal shear wave components are then no longer coupled, but reduce to the same type of equation which determines the principal compressional wave component.  相似文献   

10.
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   

11.
The generalized ray method in a vertically inhomogeneous model is formulated without any approximation by homogeneous layers. The solution is obtained as an infinite series in multiply 'reflected' waves. Each term can be solved using the exact method or the plane-wave, first-motion or geometrical approximations. It is shown that the first-motion approximation of the series converges rapidly, the ratio of successive terms in the infinite series being-(2 l + 1)(2 l )(6/π)2.
In addition it is shown that the first-motion approximation, which reduces to the geometrical approximation when the latter is valid, is a useful alternative to geometrical ray theory, being more generally valid and being almost as simple to compute.  相似文献   

12.
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   

13.
We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.  相似文献   

14.
This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.  相似文献   

15.
Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.  相似文献   

16.
Geometric ray theory is an extremely efficient tool for modelling wave propagation through heterogeneous media. Its use is, however, only justified when the inhomogeneity satisfies certain smoothness criteria. These criteria are often not satisfied, for example in wave propagation through turbulent media. In this paper, the effect of velocity perturbations on the phase and amplitude of transient wavefields is investigated for the situation that the velocity perturbation is not necessarily smooth enough to justify the use of ray theory. It is shown that the phase and amplitude perturbations of transient arrivals can to first order be written as weighted averages of the velocity perturbation over the first Fresnel zone. The resulting averaging integrals are derived for a homogeneous reference medium as well as for inhomogeneous reference media where the equations of dynamic ray tracing need to be invoked. The use of the averaging integrals is illustrated with a numerical example. This example also shows that the derived averaging integrals form a useful starting point for further approximations. The fact that the delay time due to the velocity perturbation can be expressed as a weighted average over the first Fresnel zone explains the success of tomographic inversions schemes that are based on ray theory in situations where ray theory is strictly not justified; in that situation one merely collapses the true sensitivity function over the first Fresnel zone to a line integral along a geometric ray.  相似文献   

17.
There are three types of surfaces which are used for studying wave propagation in anisotropic media: normal surfaces, slowness surfaces and wave surfaces. Normal surfaces and slowness surfaces have been researched in detail. Wave surfaces are the most complicated and comparatively poorly known compared with the other two. Areas of complicated geometrical structure of the wave surfaces are located in the vicinity of conical acoustic axes. There is an elliptical hole on the quick shear wave surface and complicated folds and cusps on the slow shear wave surface. Decomposition of the slow shear wave surface into smooth sheets is used for the study of its geometrical structure. Complexity of shear wave surfaces can be expressed by the number of waves corresponding to a fixed ray. An original approach to the calculation of wave normals depending on ray direction is presented.  相似文献   

18.
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   

19.
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   

20.
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   

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