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 high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

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.     

The theory of finite frequency body wave synthetic seismograms in inhomogeneous elastic media

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A new method is presented by means of which one can compute finite frequency synthetic seismograms for media whose velocity and density are continuous functions of two or three spatial variables. Basically, the method is a generalization of the familiar phase integral method, to which it reduces in a stratified medium. For a given source location the travel-time and distance functions needed to compute synthetics are obtained by numerically tracing rays through the model. This information is then used to evaluate a double integral over frequency and take-off angle at the source. The solution obtained reduces to the geometrical optics solution wherever that is valid but it also works in shadows and at caustics without knowing explicitly where these may be located. The method can be used as a spectral method, in which the integral over take-off angle is evaluated first, or as a slowness method, in which the frequency integral is evaluated first.     

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.     

Elastic wave propagation in model sediments —I

Summary. A fluid-saturated packing of like elastic spheres is used as a model of an oceanic sediment and a method is presented for calculating the effective velocities of elastic waves in such a medium. In particular the method is applied to low-frequency waves travelling vertically down a cubic packing, saturated with an inviscid fluid and initially at rest under a uniform compressive force. It is found that two waves propagate and moreover, that their velocities are not related through the usual equations of classical elasticity to the effective elastic moduli for static deformation of the packing. For a dry packing, there is found to exist a 'cut-off' frequency above which the wave decays with depth. An extension of the method to slightly viscous fluids is also given.     

Range of validity of seismic ray and beam methods in general inhomogeneous media – II. A canonical problem

Summary. The Green's function, in a constant gradient medium, is derived for an explosive point source, in the frequency and the time domains. The analytical dynamic ray tracing (DRT) solution is rederived with conditions stated in Part I. The Gaussian beam (GB) solution is investigated. New beam parameters and conditions are defined. Comparisons between exact and approximate solutions are undertaken.
For both methods, DRT and GB, conditions of validity are explicit and quantitative. An accuracy criterion is defined in the time domain, and measures a global relative error. The range of validity is expressed in the form of two inequalities for the dynamic ray tracing method and of five inequalities for the Gaussian beam method. Results remain accurate at ray turning points. For the types of medium considered, the breakdown of the dynamic ray tracing method is smoother and better behaved than that of Gaussian beams. As examples, a vertical seismic profiling configuration, and a shallow earthquake are modelled, using Gaussian beams.     

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.     

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.