We study the propagation of rays, paraxial rays, and Gaussian beams in a medium where slowness differs only slightly from that of a reference medium. Ray theory is developed using a Hamiltonian formalism that is independent of the coordinate system under consideration. Let us consider a ray in the unperturbed medium. The perturbation in slowness produces a change of the trajectory of this ray which may be calculated by means of canonical perturbation theory. We define paraxial rays as those rays that propagate in perturbed medium in the vicinity of the perturbed ray. The ray tracing equation for paraxial rays may be obtained by a linearization of the canonical ray equations. The linearized equations are then solved by a propagator method. With the help of the propagator we form beams, i.e. families of paraxial rays that depend on a single beam parameter. The results are very general and may be applied to a number of kinematic and dynamic ray tracing problems, like two-point ray tracing, Gaussian beams, wave front interpolation, etc. The perturbation methods are applied to the study of a few simple problems in which the unperturbed medium is homogeneous. First, we consider a two-dimensional spherical inclusion with a Gaussian slowness perturbation profile. Second, transmission and reflection problems are examined. We compare results for amplitude and travel time computed by exact and perturbed ray theory. The agreement is excellent and may be improved using an iterative procedure by which we change the reference unperturbed ray whenever the perturbation becomes large. Finally, we apply our technique to a three-dimensional problem: we calculate the amplitude perturbation and ray deflection produced by the velocity structure under the Mont Dore volcano (central France). Again a comparison shows excellent agreement between exact and perturbed ray theory.

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Seismic waveform modeling in heterogeneous media by ray perturbation theory

Seismic tomography: a window into deep Earth

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Seismic traveltime tomography of the crust and lithosphere

Paraxial Gaussian beams (GB’s) are collimated wave objects that have found wide application in optical system analysis and design. A GB propagates in physical space according to well-established quasi-geometric-optical rules that can accommodate weakly inhomogeneous media as well as reflection from and transmission through curved interfaces and thin-lens configurations. We examine the GB concept from a broad perspective in the frequency domain (FD) and the short-pulse time domain (TD) and within as well as arbitrarily beyond the paraxial constraint. For the formal analysis, which is followed by physics-matched high-frequency asymptotics, we use a (space–time)–(wavenumber–frequency) phase-space format to discuss the exact complex-source-point method and the associated asymptotic beam tracking by means of complex rays, the TD pulsed-beam (PB) ultrawideband wave-packet counterpart of the FD GB, GB’s and PB’s as basis functions for representing arbitrary fields, GB and PB diffraction, and FD–TD radiation from extended continuous aperture distributions in which the GB and the PB bases, installed through windowed transforms, yield numerically compact physics-matched a priori localization in the plane-wave-based nonwindowed spectral representations.

Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics.

Procedures for accurate ray tracing in complex three-dimensional media with interfaces are proposed. The ray tracing equations and the associated paraxial linear equations are solved either by a numerical solver or by an analytical perturbation approach. Interfaces are described with an explicit representa­ tion or an implicit representation using B-spline inter­ polation. For the implicit representation, we exploit two important properties of B-splines, the convex hull and subdivision properties, in order to determine the intersection of the ray with the interface. At the free surface where the recording system is located, a sampling strategy is proposed: limits of branches at caustics, shadow zones, and medium boundaries are detected for a fixed azimuth while the take-off angle is automatically adjusted in order to have a roughly homogeneous spacing between end points of the rays. The same strategy is also possible for a fixed take-off angle. The assumed continuity of the traveltime surface between two adjacent azimuths enables one to obtain the initial condition of a ray arriving at any station located on the portion of surface delimited by these two azimuths. This procedure al­ lows for the classification of rays arriving at a given station as we show on different synthetic examples.

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Ray tracing in 3-D complex isotropic media: An analysis of the problem

/pdf/gaussian-beam-synthetic-seismograms-1n43ulfy9v.pdf

Gaussian beam synthetic seismograms

/pdf/gaussian-beams-and-paraxial-ray-approximation-in-three-4oikzel8l3.pdf

V. Červeny

Papers

Gaussian beam synthetic seismograms

Gaussian beams and paraxial ray approximation in three-dimensional elastic inhomogeneous media