Dynamic ray tracing on Lagrangian manifolds

Summary. We show that Maslov's extension of the WKBJ method allows an extension of the dynamic ray tracing to wavefields involving caustics of arbitrary form. If the receiver lies off the caustics, then the synthetic seismogram can be obtained by integrating the DRT system along a single ray joining the receiver to the source which may touch caustics. If the receiver-lies in the vicinity of a caustic then DRT has to be carried out along a bunch of rays covering a neighbourhood of the receiver. Our approach encompasses pre-stressed and/or anisotropic media. Initial boundary conditions for a point source embedded in an anisotropic elastic medium are also presented.     

Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media

Summary. This paper discusses several aspects of the calculation of theoretical seismograms for two-dimensional inhomogeneous media with the method of Gaussian beams. The most important steps of this method, kinematic and dynamic ray tracing, can be performed very efficiently, if the model cross-section is subdivided into triangles with linear velocity laws. Each Gaussian beam is characterized by a complex beam constant ε which determines its width and phase-front curvature. Various possibilities to choose ε are discussed, including cases where beam properties at the beam endpoint (and not at the beginning) are prescribed; for instance, the beam width at the endpoint can be specified. In such cases the beam constant is a function of the radiation angle at the source, and the decomposition of a cylindrical wave into beams has to take this into account by weighting the beams differently, at least in principle. The exact weight function is derived and shown to be reasonably well approximated by the weight function, corresponding to angle-independent ε Theoretical seismograms are presented for a laterally heterogeneous model of the crust–mantle transition which is characterized by complications in the reflection from the transition and in the refraction from below. These complications are modelled by and large with success. The seismograms, however, depend to a certain extent on the choice of the beam constant. Moreover, according to the reciprocity principle calculations with source and receiver interchanged should have the same results as calculations for the original configuration. In practice this is not so, and the difference increases with the strength of lateral heterogeneities. Hence, for a successful application of Gaussian beams the model should not vary too strongly in lateral direction.     

Gaussian beams, complex rays, and the analytic extension of the Green's function in smoothly inhomogeneous media

Summary. Some relations between Gaussian beams, complex rays and the analytic extension of the Green's function in smoothly inhomogeneous media are shown in this paper. It is found that: (1) a single Gaussian beam is a paraxial approximation of the analytical extension of the ray-approximated Green's function in smoothly inhomogeneous media by putting the source point into a complex space. The Gaussian beam approximation of the Green's function has an advantage in computational efficiency and stability and can avoid the singularity problems at caustics, but also introduces a parabolic approximation to the wavefront and an angle-dependent amplitude damping. Therefore the validity of the Gaussian beam approximation should be checked using other methods. (2) Complex-ray tracing, which does not involve the paraxial approximation, can also avoid the singularity problemsm though without the computational efficiency. Therefore, it should be used to verify the Gaussian beam approximation, whenever possible. (3) The decomposition of a plane wave into an ensemble of Gaussian beams is equivalent to approximating the Green's function (the kernel of the ray-Kirchhoff method) with a single Gaussian beam. This introduces a parabolic approximation to the wavefront and a Gaussian windowing for arrival angles which may cause some problems in modelling wave propagation and scattering and has no advantages over other methods. (4) The representation of a point source field by a superposition of Gaussian beams, on the other hand, is equivalent to approximating the Green's function with a bundle of overlapped Gaussian beams. This representation is similar to a Maslov uniform asymptotic representation. It has no caustics and has improved accuracies at the caustics for quasi-plane waves compared to the extended WKBJ method.     

Gaussian beams for surface waves in laterally slowly-varying media

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.     

Some remarks on the Gaussian beam summation method

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.     

Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams

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.     

Range of validity of seismic ray and beam methods in general inhomogeneous media – I. General theory

Summary. The limitations of asymptotic wave theory and its geometrical manifestations are newly formalized and scrutinized. Necessary and sufficient conditions for the existence of acoustic and seismic rays and beams in general inhomogeneous media are expressed in terms of new physical parameters: the threshold frequency ω₀ associated with the P/S decoupling condition, the cut-off frequency ω_c associated with the radiation-zone condition, the total curvature of the wavefront and the Fresnel-zone radius.
The analysis is facilitated with the introduction of a new ancillary functional – the hypereikonal which is capable of representing ordinary as well as evanescent waves. The hypereikonal is the natural extension of the eikonal theory.
With the aid of the above new parameters, simple conditions are obtained for the decoupled far field, the decoupled near field, two point dynamic ray tracing, paraxial wavefields and Gaussian beams.     

Gaussian beams in two-dimensional elastic inhomogeneous media

Summary. Asymptotic high-frequency solutions of elastodynamic equations in two-dimensional laterally inhomogeneous media concentrated close to rays of P - and S -waves are investigated. From a physical point of view, these vectorial solutions correspond to Gaussian beams; the amplitude distribution of their principal components is bell-shaped along the direction perpendicular to the ray. The principal component of the elastodynamic Gaussian beam is controlled by the parabolic equation, which has exactly the same form as the parabolic equation for scalar Gaussian beams. The elastodynamic Gaussian beams are regular everywhere, including caustics.     

Computation of wave fields in inhomogeneous media — Gaussian beam approach

Summary. An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.     

Synthetic body wave seismograms for three-dimensional laterally varying media

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.     

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.     

Gaussian beams in elastic 2-D laterally varying layered structures

Summary. In a paper by Červený & Pšenčik, high-frequency Gaussian beams in elastic 2-D, laterally inhomogeneous, smooth media were investigated as asymptotic high-frequency solutions of elastodynamic equations, concentrated close to rays of P - and S -waves. This paper generalizes the above results for 2-D, laterally inhomogeneous, layered structures. Gaussian beams concentrated close to any multiply-reflected, possibly converted, ray are investigated. Gaussian beams are regular everywhere, including caustic regions. The paraxial ray approximation, which allows the wavefield in the zero-order ray approximation to be evaluated not only directly on the ray, but also in its vicinity, is derived as a limiting case of the Gaussian beams.     

Spectral aspects of the Gaussian beam method: reflection from a homogeneous half-space

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.     

Numerical modelling and inversion of travel times of seismic body waves in inhomogeneous anisotropic media

Summary. Two approaches to travel-time computations in laterally inhomogeneous anisotropic media are presented. The first method is based on ray tracing in an anisotropic inhomogeneous medium, the second on the linearization procedure. The linearization procedure, which can be applied to inhomogeneous, slightly anisotropic media, does not require ray tracing in an anisotropic medium. Applications of linearized equations to the solutions of direct and inverse kinematic problems are discussed. A program package to perform the linearized computations for rather general 2-D laterally inhomogeneous layered structures is described and a numerical example is presented.     

Ray tracing and inhomogeneous dynamic ray tracing for anisotropy specified in curvilinear coordinates

Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.     

中山站高频雷达观测结果初步分析

中山站高频相干散射雷达于2010年4月建成投入观测,并加盟SuperDARN雷达观测网,成为我国探测电离层对流的重要手段。首先简要介绍中山站高频雷达的工作原理及工作模式,然后利用该雷达第一年的观测数据,给出了每个波束观测的电离层回波总数随距离和回波强度、多普勒速度及谱宽的分布,以及电离层回波发生率随频率的变化特性。观测结果和射线追踪的模拟计算表明,由于波束指向不同,电波波矢方向与地磁场形成正交条件的区域有所不同,因此会造成不同频率下,不同波束观测的电离层回波发生率的差别。     

Generalized Born scattering of elastic waves in 3-D media

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.     

Ray-reflectivity method for SH-waves in stacks of thin and thick layers

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.     

Wavefield smoothing and the effect of rough velocity perturbations on arrival times and amplitudes

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.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

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.