Summation of paraxial Gaussian beams and of paraxial ray approximations in inhomogeneous anisotropic structures

Vlastislav Cerveny


Elementary high-frequency approximate solutions of the elastodynamic equation in anisotropic inhomogeneous media, connected with a ray Ω of a zero-order reflected/transmitted wave, are studied. These elementary solutions are called paraxial ray approximations and paraxial Gaussian beams, and are approximately valid not only along the ray Ω, but also in its vicinity. The travel times of paraxial ray approximations are real-valued, but the travel times of paraxial Gaussian beams complex-valued outside the ray Ω. For a fixed travel time, the phase fronts of all elementary solutions are mutually tangent at the relevant point of the ray Ω. With any ray Ω, a three-parameteric system of paraxial ray approximations and a six-parameteric system of paraxial Gaussian beams is connected.

More general solutions of the elastodynamic equation can be obtained by summation of elementary solutions connected with rays passing in the vicinity of the receiver, multiplied by some weighting functions. A general superposition integral is derived which can be used both for the summation of paraxial ray approximations and for summation of paraxial Gaussian beams. The superposition integral is over ray parameters of the near-by rays. It removes certain singularities of standard ray method (caustics, etc.). The integral may be used even for isotropic media, pressure waves in fluids, radio-waves, etc. Individual quantities in the superposition integral can be computed by dynamic ray tracing in wavefront orthonormal coordinates, or by dynamic ray tracing in Cartesian coordinates. The wave under consideration may be generated by an initial surface with the variable initial travel time along it, by a point source with an arbitrary radiation function, and so forth.

The receiver R may be situated arbitrarily in the model, including structural interfaces and the Earth's surface. The superposition is performed over points Rγ, situated on near-by rays specified by ray parameters γ12. Points Rγ on near-by rays may be chosen in different ways. It is customary (but not necessary) to introduce a target surface (possibly curved), passing through the receiver. The points Rγ then represent the end-points or intersections of near-by rays with the target surface. The superposition integrals, however, work even if receivers are not situated on the target surface, but are close to it. For suitably chosen parameters of paraxial ray approximations, the superposition integral yields the Maslov-Chapman integral.

For wave fields generated by a point source with a suitably chosen radiation function, the superposition integrals yield the superposition Green functions, i.e. the paraxial ray approximation Green functions or paraxial Gaussian beam Green function. As a special case, the Maslov-Chapman Green function is obtained.


Anisotropic inhomogeneous layered structures. Paraxial ray approximation. Paraxial Gaussian beams. Summation of paraxial ray approximations. Summation of paraxial Gaussian beams. Superposition Green functions. Maslov-Chapman method.

Whole paper

The paper is available in PostScript (1406 kB !) and GZIPped PostScript (619 kB).

In: Seismic Waves in Complex 3-D Structures, Report 10, pp. 121-159, Dep. Geophys., Charles Univ., Prague, 2000.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .